Christo Ananth, N. Arabov, D. Nasimov, H. Khuzhayorov, T. AnanthKumar, “Modelling of Commercial Banks Capitals Competition Dynamics”, International Journal of Early Childhood Special Education, Volume 14, Issue 05, 2022,pp. 4124-4132.
Description:
Christo Ananth et al. discussed that according to the observations in this paper, an existing mathematical model of banking capital dynamics should be tweaked. First-order ordinary differential equations with a "predator-pray" structure make up the model, and the indicators are competitive. Numerical realisations of the model are required to account for three distinct sets of initial parameter values. It is demonstrated that a wide range of banking capital dynamics can be produced by altering the starting parameters. One of the three options is selected, and the other two are eliminated. The model is generalized taking into account fractional derivatives of the bank indicators for time, reflecting the rate of their change. Based on numerical calculations, it is established that reduction of the order of derivatives from units leads to a delay of banking capital dynamics. It is shown, that the less the order of derivatives from the unit, the more delay of dynamics of indicators. In all analyzed variants indicators at large times reach their equilibrium values.
Multifactorial Heath-Jarrow-Morton model using principal component analysisIJECEIAES
In this study, we propose an implementation of the multifactor Heath-Jarrow- Morton (HJM) interest rate model using an approach that integrates principal component analysis (PCA) and Monte Carlo simulation (MCS) techniques. By integrating PCA and MCS with the multifactor HJM model, we successfully capture the principal factors driving the evolution of short-term interest rates in the US market. Additionally, we provide a framework for deriving spot interest rates through parameter calibration and forward rate estimation. For this, we use daily data from the US yield curve from June 2017 to December 2019. The integration of PCA, MCS with multifactor HJM model in this study represents a robust and precise approach to characterizing interest rate dynamics and compared to previous approaches, this method provided greater accuracy and improved understanding of the factors influencing US Treasury Yield interest rates.
Applying Convolutional-GRU for Term Deposit Likelihood PredictionVandanaSharma356
Banks are normally offered two kinds of deposit accounts. It consists of deposits like current/saving account and term deposits like fixed or recurring deposits.For enhancing the maximized profit from bank as well as customer perspective, term deposit can accelerate uplifting of finance fields. This paper focuses on likelihood of term deposit subscription taken by the customers. Bank campaign efforts and customer detail analysis caninfluence term deposit subscription chances. An automated system is approached in this paper that works towards prediction of term deposit investment possibilities in advance. This paper proposes deep learning based hybrid model that stacks Convolutional layers and Recurrent Neural Network (RNN) layers as predictive model. For RNN, Gated Recurrent Unit (GRU) is employed. The proposed predictive model is later compared with other benchmark classifiers such as k-Nearest Neighbor (k-NN), Decision tree classifier (DT), and Multi-layer perceptron classifier (MLP). Experimental study concludesthat proposed model attainsan accuracy of 89.59% and MSE of 0.1041 which outperform wellother baseline models.
International Journal of Computational Engineering Research (IJCER) ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology
Multifactorial Heath-Jarrow-Morton model using principal component analysisIJECEIAES
In this study, we propose an implementation of the multifactor Heath-Jarrow- Morton (HJM) interest rate model using an approach that integrates principal component analysis (PCA) and Monte Carlo simulation (MCS) techniques. By integrating PCA and MCS with the multifactor HJM model, we successfully capture the principal factors driving the evolution of short-term interest rates in the US market. Additionally, we provide a framework for deriving spot interest rates through parameter calibration and forward rate estimation. For this, we use daily data from the US yield curve from June 2017 to December 2019. The integration of PCA, MCS with multifactor HJM model in this study represents a robust and precise approach to characterizing interest rate dynamics and compared to previous approaches, this method provided greater accuracy and improved understanding of the factors influencing US Treasury Yield interest rates.
Applying Convolutional-GRU for Term Deposit Likelihood PredictionVandanaSharma356
Banks are normally offered two kinds of deposit accounts. It consists of deposits like current/saving account and term deposits like fixed or recurring deposits.For enhancing the maximized profit from bank as well as customer perspective, term deposit can accelerate uplifting of finance fields. This paper focuses on likelihood of term deposit subscription taken by the customers. Bank campaign efforts and customer detail analysis caninfluence term deposit subscription chances. An automated system is approached in this paper that works towards prediction of term deposit investment possibilities in advance. This paper proposes deep learning based hybrid model that stacks Convolutional layers and Recurrent Neural Network (RNN) layers as predictive model. For RNN, Gated Recurrent Unit (GRU) is employed. The proposed predictive model is later compared with other benchmark classifiers such as k-Nearest Neighbor (k-NN), Decision tree classifier (DT), and Multi-layer perceptron classifier (MLP). Experimental study concludesthat proposed model attainsan accuracy of 89.59% and MSE of 0.1041 which outperform wellother baseline models.
International Journal of Computational Engineering Research (IJCER) ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology
A novel hybrid deep learning model for price prediction IJECEIAES
Price prediction has become a major task due to the explosive increase in the number of investors. The price prediction task has various types such as shares, stocks, foreign exchange instruments, and cryptocurrency. The literature includes several models for price prediction that can be classified based on the utilized methods into three main classes, namely, deep learning, machine learning, and statistical. In this context, we proposed several models’ architectures for price prediction. Among them, we proposed a hybrid one that incorporates long short-term memory (LSTM) and Convolution neural network (CNN) architectures, we called it CNN-LSTM. The proposed CNNLSTM model makes use of the characteristics of the convolution layers for extracting useful features embedded in the time series data and the ability of LSTM architecture to learn long-term dependencies. The proposed architectures are thoroughly evaluated and compared against state-of-the-art methods on three different types of financial product datasets for stocks, foreign exchange instruments, and cryptocurrency. The obtained results show that the proposed CNN-LSTM has the best performance on average for the utilized evaluation metrics. Moreover, the proposed deep learning models were dominant in comparison to the state-of-the-art methods, machine learning models, and statistical models.
Corporate bankruptcy prediction using Deep learning techniquesShantanu Deshpande
Corporate Bankruptcy prediction using Recurrent neural networks – Aim is to build a recurrent neural network-based model to predict whether company will become bankrupt or not using financial ratios of Polish companies.
Methodologies & Tools: CRISP-DM, SMOTE-ENN, GA Algorithm, LSTM network (type of RNN)
A TWO-STAGE HYBRID MODEL BY USING ARTIFICIAL NEURAL NETWORKS AS FEATURE CONST...IJDKP
We propose a two-stage hybrid approach with neural networks as the new feature construction algorithms for bankcard response classifications. The hybrid model uses a very simpleneural network structure as the new feature construction tool in the firststage, thenthe newly created features are used asthe additional input variables in logistic regression in the second stage. The modelis compared with the traditional onestage model in credit customer response classification. It is observed that the proposed two-stage model outperforms the one-stage model in terms of accuracy, the area under ROC curve, andKS statistic. By creating new features with theneural network technique, the underlying nonlinear relationships between variables are identified. Furthermore, by using a verysimple neural network structure, the model could overcome the drawbacks of neural networks interms of its long training time, complex topology, and limited interpretability.
Modelling: What’s next for Financial Services in Europe?GRATeam
This paper outlines a practical roadmap to realising cost savings, delivering a material reduction in the volume and complexity of models by outlining five key principles of model optimisation: develop a comprehensive review of models, harmonise methodologies, re-design model validation/monitoring process, re-think its modelling team’s organisation & governance and build new expertise and recruit talent.
In-spite of large volumes of Contingent Credit Lines (CCL) in all commercial banks, the paucity of Exposure at Default (EAD) models, unsuitability of external data and inconsistent internal data with partial draw-downs has been a major challenge for risk managers as well as regulators in for managing CCL portfolios. This current paper is an attempt to build an easy to implement, pragmatic and parsimonious yet accurate model to determine the exposure distribution of a CCL portfolio. Each of the credit line in a portfolio is modeled as a portfolio of large number of option instruments which can be exercised by the borrower, determining the level of usage. Using an algorithm similar to basic the CreditRisk+ and Fourier Transforms we arrive at a portfolio level probability distribution of usage. We perform a simulation experiment using data from Moody\'s Default Risk Service, historical draw-down rates estimated from the history of defaulted CCLs and a current rated portfolio of such.
COMPARISON OF BANKRUPTCY PREDICTION MODELS WITH PUBLIC RECORDS AND FIRMOGRAPHICScscpconf
Many business operations and strategies rely on bankruptcy prediction. In this paper, we aim to
study the impacts of public records and firmographics and predict the bankruptcy in a 12-
month-ahead period with using different classification models and adding values to traditionally
used financial ratios. Univariate analysis shows the statistical association and significance of
public records and firmographics indicators with the bankruptcy. Further, seven statistical
models and machine learning methods were developed, including Logistic Regression, Decision
Tree, Random Forest, Gradient Boosting, Support Vector Machine, Bayesian Network, and
Neural Network. The performance of models were evaluated and compared based on
classification accuracy, Type I error, Type II error, and ROC curves on the hold-out dataset.
Moreover, an experiment was set up to show the importance of oversampling for rare event
prediction. The result also shows that Bayesian Network is comparatively more robust than
other models without oversampling.
Review Parameters Model Building & Interpretation and Model Tunin.docxcarlstromcurtis
Review Parameters: Model Building & Interpretation and Model Tuning
1. Model Building
a. Assessments and Rationale of Various Models Employed to Predict Loan Defaults
The z-score formula model was employed by Altman (1968) while envisaging bankruptcy. The model was utilized to forecast the likelihood that an organization may fall into bankruptcy in a period of two years. In addition, the Z-score model was instrumental in predicting corporate defaults. The model makes use of various organizational income and balance sheet data to weigh the financial soundness of a firm. The Z-score involves a Linear combination of five general financial ratios which are assessed through coefficients. The author employed the statistical technique of discriminant examination of data set sourced from publically listed manufacturers. A research study by Alexander (2012) made use of symmetric binary alternative models, otherwise referred to as conditional probability models. The study sought to establish the asymmetric binary options models subject to the extreme value theory in better explicating bankruptcy.
In their research study on the likelihood of default models examining Russian banks, Anatoly et al. (2014) made use of binary alternative models in predicting the likelihood of default. The study established that preface specialist clustering or mechanical clustering enhances the prediction capacity of the models. Rajan et al. (2010) accentuated the statistical default models as well as inducements. They postulated that purely numerical models disregard the concept that an alteration in the inducements of agents who produce the data may alter the very nature of data. The study attempted to appraise statistical models that unpretentiously pool resources on historical figures devoid of modeling the behavior of driving forces that generates these data. Goodhart (2011) sought to assess the likelihood of small businesses to default on loans. Making use of data on business loan assortment, the study established the particular lender, loan, and borrower characteristics as well as modifications in the economic environments that lead to a rise in the probability of default. The results of the study form the basis for the scoring model. Focusing on modeling default possibility, Singhee & Rutenbar (2010) found the risk as the uncertainty revolving around an enterprise’s capacity to service its obligations and debts.
Using the logistic model to forecast the probability of bank loan defaults, Adam et al. (2012) employed a data set with demographic information on borrowers. The authors attempted to establish the risk factors linked to borrowers are attributable to default. The identified risk factors included marital status, gender, occupation, age, and loan duration. Cababrese (2012) employed three accepted data mining algorithms, naïve Bayesian classifiers, artificial neural network decision trees coupled with a logical regression model to formulate a prediction m ...
International journal of engineering and mathematical modelling vol1 no1_2015_2IJEMM
Default risk has always been a matter of importance for financial managers and scholars. In this paper we apply an intensity-based approach for default estimation with a software simulation of the Cox-Ingersoll-Ross model. We analyze the possibilities and effects of a non-linear dependence between economic and financial state variables and the default density, as specified by the theoretical model. Then we perform a test for verifying how simulation techniques can improve the analysis of such complex relations when closed-form solutions are either not available or hard to come by.
Environmental Pollution RecommendationThere is a concern in yo.docxSALU18
Environmental Pollution Recommendation
There is a concern in your community regarding the environment. You've been tasked to research and present the concerns to your local or state government (California)
Perform an internet search to identify an instance of environmental pollution in your state.
Create a 5-to 8-slide PowerPoint® presentation or a 350-to 525-word proposal.
· Identify the effects of this pollution on human health and the environment.
· Explain the causes of this pollution.
· Recommend ways to prevent/clean up this type of environmental pollution.
· Include appropriate images.
Use at least 2 outside references.
Format your presentation and references consistent with APA guidelines.
· For Online and Directed Study students, these are Microsoft® PowerPoint® presentations with notes similar to what you would present orally.
Learning Objectives
After completing this chapter, you should be able to:
• Define a model and describe how models can be used to analyze operating
problems.
• Discuss the nature of forecasting.
• Explain how forecasting can be applied to problems.
• Describe methods of forecasting, including judgment and experience, time-series
analysis, and regression and correlation.
• Construct forecasting models.
• Estimate forecasting errors.
6 .Thinkstock
Models and Forecasting
von70154_06_c06_139-178.indd 139 3/6/13 3:18 PM
CHAPTER 6Section 6.1 Introduction to Models and Decision Making
6.1 Introduction to Models and Decision Making
In order for an organization to design, build, and operate a production facility that is capable of meeting customer demand for services (such as health care) or goods (such as ceiling fans), it is necessary for management to obtain an estimate or forecast of demand
for its products. A forecast is a prediction of the future. It often examines historical data to
determine relationships among key variables in a problem and uses those relationships to
make statements about the future value of one or more of the variables. Once an organiza-
tion has a forecast of demand, it can make decisions regarding the volume of product that
needs to be produced, the number of workers to hire, and other key operating variables.
A model is an abstraction from the real problem of the key variables and relationships in
order to simplify the problem. The purpose of modeling is to provide the user with a bet-
ter understanding of the problem and with a means of manipulating the results for what-
if analyses. Forecasting uses models to help organizations predict important parameters.
Demand is one of those parameters, but cost, revenue, profits, and other variables can also
be forecasted. The purpose of this chapter is to discuss models and describe how they can
be applied to business problems, and to explain forecasting and its role in operations.
Stages in Decision Making
Organizational performance is a result of the decisions that management makes over a
period of time: ...
Multi-dimensional time series based approach for Banking Regulatory Stress Te...Genpact Ltd
Under regulatory paradigm of banking risk management, banks are required to perform stress testing of internally computed risk parameters to ensure holding of adequate amount of capital to offset the effects of downturn events. For this purpose, most of the contemporary stress-testing practices are limited to one dimensionality of the calculation, where endogenous risk parameters are predicted by modeling and scenario based values of exogenous parameters (macroeconomic variables).
Call for Papers - Journal of Electrical Systems (JES), E-ISSN: 1112-5209, ind...Christo Ananth
At the forefront of technological innovation and scholarly discourse, the Journal of Electrical Systems (JES) is a peer-reviewed publication dedicated to advancing the understanding and application of electrical systems, communication systems and information science. With a commitment to excellence, we provide a platform for researchers, academics, and professionals to contribute to the ever-evolving field of electrical engineering, communication technology and Information Systems.
The mission of JES is to foster the exchange of knowledge and ideas in electrical and communication systems, promoting cutting-edge research and facilitating discussions that drive progress in the field. We aim to be a beacon for those seeking to explore, challenge, and revolutionize the way we harness, distribute, and utilize electrical energy and information systems..
Call for Papers - Utilitas Mathematica, E-ISSN: 0315-3681, indexed in ScopusChristo Ananth
Utilitas Mathematica Journal is a broad scope journal that publishes original research and review articles on all aspects of both pure and applied mathematics. This journal is the official publication of the Utilitas Mathematica Academy, Canada. It enjoys good reputation and popularity at international level in terms of research papers and distribution worldwide. Offers selected original research in Pure and Applied Mathematics and Statistics. UMJ coverage extends to Operations Research, Mathematical Economics, Mathematics Biology and Computer Science. Published in association with the Utilitas Mathematica Academy. The leadership of the Utilitas Mathematica Journal commits to strengthening our professional community by making it more just, equitable, diverse, and inclusive. We affirm that our mission, Promote the Practice and Profession of Statistics, can be realized only by fully embracing justice, equity, diversity, and inclusivity in all of our operations. Individuals embody many traits, so the leadership will work with the members of UMJ to create and sustain responsive, flourishing, and safe environments that support individual needs, stimulate intellectual growth, and promote professional advancement for all
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Price prediction has become a major task due to the explosive increase in the number of investors. The price prediction task has various types such as shares, stocks, foreign exchange instruments, and cryptocurrency. The literature includes several models for price prediction that can be classified based on the utilized methods into three main classes, namely, deep learning, machine learning, and statistical. In this context, we proposed several models’ architectures for price prediction. Among them, we proposed a hybrid one that incorporates long short-term memory (LSTM) and Convolution neural network (CNN) architectures, we called it CNN-LSTM. The proposed CNNLSTM model makes use of the characteristics of the convolution layers for extracting useful features embedded in the time series data and the ability of LSTM architecture to learn long-term dependencies. The proposed architectures are thoroughly evaluated and compared against state-of-the-art methods on three different types of financial product datasets for stocks, foreign exchange instruments, and cryptocurrency. The obtained results show that the proposed CNN-LSTM has the best performance on average for the utilized evaluation metrics. Moreover, the proposed deep learning models were dominant in comparison to the state-of-the-art methods, machine learning models, and statistical models.
Corporate bankruptcy prediction using Deep learning techniquesShantanu Deshpande
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We propose a two-stage hybrid approach with neural networks as the new feature construction algorithms for bankcard response classifications. The hybrid model uses a very simpleneural network structure as the new feature construction tool in the firststage, thenthe newly created features are used asthe additional input variables in logistic regression in the second stage. The modelis compared with the traditional onestage model in credit customer response classification. It is observed that the proposed two-stage model outperforms the one-stage model in terms of accuracy, the area under ROC curve, andKS statistic. By creating new features with theneural network technique, the underlying nonlinear relationships between variables are identified. Furthermore, by using a verysimple neural network structure, the model could overcome the drawbacks of neural networks interms of its long training time, complex topology, and limited interpretability.
Modelling: What’s next for Financial Services in Europe?GRATeam
This paper outlines a practical roadmap to realising cost savings, delivering a material reduction in the volume and complexity of models by outlining five key principles of model optimisation: develop a comprehensive review of models, harmonise methodologies, re-design model validation/monitoring process, re-think its modelling team’s organisation & governance and build new expertise and recruit talent.
In-spite of large volumes of Contingent Credit Lines (CCL) in all commercial banks, the paucity of Exposure at Default (EAD) models, unsuitability of external data and inconsistent internal data with partial draw-downs has been a major challenge for risk managers as well as regulators in for managing CCL portfolios. This current paper is an attempt to build an easy to implement, pragmatic and parsimonious yet accurate model to determine the exposure distribution of a CCL portfolio. Each of the credit line in a portfolio is modeled as a portfolio of large number of option instruments which can be exercised by the borrower, determining the level of usage. Using an algorithm similar to basic the CreditRisk+ and Fourier Transforms we arrive at a portfolio level probability distribution of usage. We perform a simulation experiment using data from Moody\'s Default Risk Service, historical draw-down rates estimated from the history of defaulted CCLs and a current rated portfolio of such.
COMPARISON OF BANKRUPTCY PREDICTION MODELS WITH PUBLIC RECORDS AND FIRMOGRAPHICScscpconf
Many business operations and strategies rely on bankruptcy prediction. In this paper, we aim to
study the impacts of public records and firmographics and predict the bankruptcy in a 12-
month-ahead period with using different classification models and adding values to traditionally
used financial ratios. Univariate analysis shows the statistical association and significance of
public records and firmographics indicators with the bankruptcy. Further, seven statistical
models and machine learning methods were developed, including Logistic Regression, Decision
Tree, Random Forest, Gradient Boosting, Support Vector Machine, Bayesian Network, and
Neural Network. The performance of models were evaluated and compared based on
classification accuracy, Type I error, Type II error, and ROC curves on the hold-out dataset.
Moreover, an experiment was set up to show the importance of oversampling for rare event
prediction. The result also shows that Bayesian Network is comparatively more robust than
other models without oversampling.
Review Parameters Model Building & Interpretation and Model Tunin.docxcarlstromcurtis
Review Parameters: Model Building & Interpretation and Model Tuning
1. Model Building
a. Assessments and Rationale of Various Models Employed to Predict Loan Defaults
The z-score formula model was employed by Altman (1968) while envisaging bankruptcy. The model was utilized to forecast the likelihood that an organization may fall into bankruptcy in a period of two years. In addition, the Z-score model was instrumental in predicting corporate defaults. The model makes use of various organizational income and balance sheet data to weigh the financial soundness of a firm. The Z-score involves a Linear combination of five general financial ratios which are assessed through coefficients. The author employed the statistical technique of discriminant examination of data set sourced from publically listed manufacturers. A research study by Alexander (2012) made use of symmetric binary alternative models, otherwise referred to as conditional probability models. The study sought to establish the asymmetric binary options models subject to the extreme value theory in better explicating bankruptcy.
In their research study on the likelihood of default models examining Russian banks, Anatoly et al. (2014) made use of binary alternative models in predicting the likelihood of default. The study established that preface specialist clustering or mechanical clustering enhances the prediction capacity of the models. Rajan et al. (2010) accentuated the statistical default models as well as inducements. They postulated that purely numerical models disregard the concept that an alteration in the inducements of agents who produce the data may alter the very nature of data. The study attempted to appraise statistical models that unpretentiously pool resources on historical figures devoid of modeling the behavior of driving forces that generates these data. Goodhart (2011) sought to assess the likelihood of small businesses to default on loans. Making use of data on business loan assortment, the study established the particular lender, loan, and borrower characteristics as well as modifications in the economic environments that lead to a rise in the probability of default. The results of the study form the basis for the scoring model. Focusing on modeling default possibility, Singhee & Rutenbar (2010) found the risk as the uncertainty revolving around an enterprise’s capacity to service its obligations and debts.
Using the logistic model to forecast the probability of bank loan defaults, Adam et al. (2012) employed a data set with demographic information on borrowers. The authors attempted to establish the risk factors linked to borrowers are attributable to default. The identified risk factors included marital status, gender, occupation, age, and loan duration. Cababrese (2012) employed three accepted data mining algorithms, naïve Bayesian classifiers, artificial neural network decision trees coupled with a logical regression model to formulate a prediction m ...
International journal of engineering and mathematical modelling vol1 no1_2015_2IJEMM
Default risk has always been a matter of importance for financial managers and scholars. In this paper we apply an intensity-based approach for default estimation with a software simulation of the Cox-Ingersoll-Ross model. We analyze the possibilities and effects of a non-linear dependence between economic and financial state variables and the default density, as specified by the theoretical model. Then we perform a test for verifying how simulation techniques can improve the analysis of such complex relations when closed-form solutions are either not available or hard to come by.
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Environmental Pollution Recommendation
There is a concern in your community regarding the environment. You've been tasked to research and present the concerns to your local or state government (California)
Perform an internet search to identify an instance of environmental pollution in your state.
Create a 5-to 8-slide PowerPoint® presentation or a 350-to 525-word proposal.
· Identify the effects of this pollution on human health and the environment.
· Explain the causes of this pollution.
· Recommend ways to prevent/clean up this type of environmental pollution.
· Include appropriate images.
Use at least 2 outside references.
Format your presentation and references consistent with APA guidelines.
· For Online and Directed Study students, these are Microsoft® PowerPoint® presentations with notes similar to what you would present orally.
Learning Objectives
After completing this chapter, you should be able to:
• Define a model and describe how models can be used to analyze operating
problems.
• Discuss the nature of forecasting.
• Explain how forecasting can be applied to problems.
• Describe methods of forecasting, including judgment and experience, time-series
analysis, and regression and correlation.
• Construct forecasting models.
• Estimate forecasting errors.
6 .Thinkstock
Models and Forecasting
von70154_06_c06_139-178.indd 139 3/6/13 3:18 PM
CHAPTER 6Section 6.1 Introduction to Models and Decision Making
6.1 Introduction to Models and Decision Making
In order for an organization to design, build, and operate a production facility that is capable of meeting customer demand for services (such as health care) or goods (such as ceiling fans), it is necessary for management to obtain an estimate or forecast of demand
for its products. A forecast is a prediction of the future. It often examines historical data to
determine relationships among key variables in a problem and uses those relationships to
make statements about the future value of one or more of the variables. Once an organiza-
tion has a forecast of demand, it can make decisions regarding the volume of product that
needs to be produced, the number of workers to hire, and other key operating variables.
A model is an abstraction from the real problem of the key variables and relationships in
order to simplify the problem. The purpose of modeling is to provide the user with a bet-
ter understanding of the problem and with a means of manipulating the results for what-
if analyses. Forecasting uses models to help organizations predict important parameters.
Demand is one of those parameters, but cost, revenue, profits, and other variables can also
be forecasted. The purpose of this chapter is to discuss models and describe how they can
be applied to business problems, and to explain forecasting and its role in operations.
Stages in Decision Making
Organizational performance is a result of the decisions that management makes over a
period of time: ...
Multi-dimensional time series based approach for Banking Regulatory Stress Te...Genpact Ltd
Under regulatory paradigm of banking risk management, banks are required to perform stress testing of internally computed risk parameters to ensure holding of adequate amount of capital to offset the effects of downturn events. For this purpose, most of the contemporary stress-testing practices are limited to one dimensionality of the calculation, where endogenous risk parameters are predicted by modeling and scenario based values of exogenous parameters (macroeconomic variables).
Call for Papers - Journal of Electrical Systems (JES), E-ISSN: 1112-5209, ind...Christo Ananth
At the forefront of technological innovation and scholarly discourse, the Journal of Electrical Systems (JES) is a peer-reviewed publication dedicated to advancing the understanding and application of electrical systems, communication systems and information science. With a commitment to excellence, we provide a platform for researchers, academics, and professionals to contribute to the ever-evolving field of electrical engineering, communication technology and Information Systems.
The mission of JES is to foster the exchange of knowledge and ideas in electrical and communication systems, promoting cutting-edge research and facilitating discussions that drive progress in the field. We aim to be a beacon for those seeking to explore, challenge, and revolutionize the way we harness, distribute, and utilize electrical energy and information systems..
Call for Papers - Utilitas Mathematica, E-ISSN: 0315-3681, indexed in ScopusChristo Ananth
Utilitas Mathematica Journal is a broad scope journal that publishes original research and review articles on all aspects of both pure and applied mathematics. This journal is the official publication of the Utilitas Mathematica Academy, Canada. It enjoys good reputation and popularity at international level in terms of research papers and distribution worldwide. Offers selected original research in Pure and Applied Mathematics and Statistics. UMJ coverage extends to Operations Research, Mathematical Economics, Mathematics Biology and Computer Science. Published in association with the Utilitas Mathematica Academy. The leadership of the Utilitas Mathematica Journal commits to strengthening our professional community by making it more just, equitable, diverse, and inclusive. We affirm that our mission, Promote the Practice and Profession of Statistics, can be realized only by fully embracing justice, equity, diversity, and inclusivity in all of our operations. Individuals embody many traits, so the leadership will work with the members of UMJ to create and sustain responsive, flourishing, and safe environments that support individual needs, stimulate intellectual growth, and promote professional advancement for all
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Published since 2004, Periódico Tchê Química (PQT) is a is a triannual (published every four months), international, fully peer-reviewed, and open-access Journal that welcomes high-quality theoretically informed publications in the multi and interdisciplinary fields of Chemistry, Biology, Physics, Mathematics, Pharmacy, Medicine, Engineering, Agriculture and Education in Science.
Researchers from all countries are invited to publish on its pages. The Journal is committed to achieving a broad international appeal, attracting contributions, and addressing issues from a range of disciplines. The Periódico Tchê Química is a double-blind peer-review journal dedicated to express views on the covered topics, thereby generating a cross current of ideas on emerging matters
Call for Papers - PROCEEDINGS ON ENGINEERING SCIENCES, P-ISSN-2620-2832, E-IS...Christo Ananth
Proceedings on Engineering Sciences examines new research and development at the engineering. It provides a common forum for both front line engineering as well as pioneering academic research. The journal's multidisciplinary approach draws from such fields as Automation, Automotive engineering, Business, Chemical engineering, Civil engineering, Control and system engineering, Electrical and electronic engineering, Electronics, Environmental engineering, Industrial and manufacturing engineering, Industrial management, Information and communication technology, Management and Accounting, Management and quality studies, Management Science and Operations Research, Materials engineering, Mechanical engineering, Mechanics of Materials, Mining and energy, Safety, Risk, Reliability, and Quality, Software engineering, Surveying and transport, Architecture and urban engineering.
Call for Papers - Onkologia i Radioterapia, P-ISSN-1896-8961, E-ISSN 2449-916...Christo Ananth
Onkologia I Radioterapia is an international peer reviewed journal which publishes on both clinical and pre-clinical research related to cancer. Journal also provide latest information in field of oncology and radiotherapy to both clinical practitioner as well as basic researchers. Submission for publication can be submitted through online submission, Editorial manager system, or through email as attachment to journal office. For any issue, journal office can be contacted through email or phone for instatnt resolution of issue. Onkologia I Radioterapia is a peer-reviewed scopus indexed medical journal publishing original scientific (experimental, clinical, laboratory), review and case studies (case report) in the field of oncology and radiotherapy. In addition, publishes letters to the Editorial Board, reports on scientific conferences, book reviews, as well as announcements about planned congresses and scientific congresses. Oncology and Radiotherapy appear four times a year. All articles published with www.itmedical.pl and www.medicalproject.com.pl is now available on our new website
Call for Papers - Journal of Indian School of Political Economy, E-ISSN 0971-...Christo Ananth
The journal is published every quarter and contains 200 pages in each issue. It is devoted to the study of Indian economy, polity and society. Research papers, review articles, book reviews are published in the journal. All research papers published in the journal are subject to an intensive refereeing process. Each issue of the journal also includes a section on documentation, which reproduces extensive excerpts of relevant reports of committees, working groups, task forces, etc., which may not be readily accessible, official documents compiled from scattered electronic and/or other sources and statistical supplement for ready reference of the readers. It is now in its nineteenth year of publication. So far, five special issues have been brought out, namely: (i) The Scheduled Castes: An Inter-Regional Perspective, (ii) Political Parties and Elections in Indian States : 1990-2003, (iii) Child Labour, (iv) World Trade Organisation Agreements, and (v) Basel-II and Indian Banks
Call for Papers - Journal of Ecohumanism (JoE), ISSN (Print): 2752-6798, ISSN...Christo Ananth
Journal of Ecohumanism is an Open Access international peer-reviewed journal for scholars, researchers, and students who are interested in the fields of Environmental Humanities, Ecohumanism, Ecology, Literary Theory and Cultural Criticism, Economic and Business Studies, Law and Legal Studies in a broad interdisciplinary field of Social Sciences and Humanities. Journal of Ecohumanism is an Open Access peer reviewed journal, allowing users to freely access, download, copy, distribute, print, search, or link to full-text articles for any lawful purpose without requiring permission from the publisher or author. JoE follows a strict double, blind review policy for all the submissions which is embedded in our general publication ethics and supported by rigorous academic scrutiny of papers published. Materials published in the journal do not necessarily represent the views of its editorial board and reviewers
Call for Papers- Journal of Wireless Mobile Networks, Ubiquitous Computing, a...Christo Ananth
JoWUA is an online peer-reviewed journal and aims to provide an international forum for researchers, professionals, and industrial practitioners on all topics related to wireless mobile networks, ubiquitous computing, and their dependable applications. JoWUA consists of high-quality technical manuscripts on advances in the state-of-the-art of wireless mobile networks, ubiquitous computing, and their dependable applications; both theoretical approaches and practical approaches are encouraged to submit. All published articles in JoWUA are freely accessible in this website because it is an open access journal. JoWUA has four issues (March, June, September, December) per year with special issues covering specific research areas by guest editors. The editorial board of JoWUA makes an effort for the increase in the quality of accepted articles compared to other competing journals
Call for Papers - International Journal of Intelligent Systems and Applicatio...Christo Ananth
International Journal of Intelligent Systems and Applications in Engineering (IJISAE) is an international and interdisciplinary journal for both invited and contributed peer reviewed articles that intelligent systems and applications in engineering at all levels. The journal publishes a broad range of papers covering theory and practice in order to facilitate future efforts of individuals and groups involved in the field. IJISAE, a peer-reviewed double-blind refereed journal, publishes original papers featuring innovative and practical technologies related to the design and development of intelligent systems in engineering. Its coverage also includes papers on intelligent systems applications in areas such as nanotechnology, renewable energy, medicine engineering, Aeronautics and Astronautics, mechatronics, industrial manufacturing, bioengineering, agriculture, services, intelligence based automation and appliances, medical robots and robotic rehabilitations, space exploration and etc.
Call for Papers- Special Issue: Recent Trends, Innovations and Sustainable So...Christo Ananth
Energy Systems Modelling is growing in relevance on providing insights and strategies to plan a carbon-neutral future. The implementation of an effective energy transition plan faces multiple challenges, spanning from the integration of the operations of different energy carriers and sectors to the consideration of multiple spatial and temporal resolutions. Demand-side management has to be applied to multi-carrier energy system models lacks; prosumers is explored only in a limited manner; In General, multi-scale modelling frameworks should be established and considered both in the dimensions of time, space, technology and energy carrier; long term energy system models tend to address uncertainty scarcely; there is a lack of studies modelling uncertainties related to emerging technologies and; modelling of energy consumer behaviour is one of the major aspect of future research. The increased pressure in decarbonizing the energy system has renewed the interest in energy system modelling, with several reviews trying to convey a comprehensive description of the utilized methodologies as well as providing new insights on how they can be used to answer new questions
Call for Papers - Educational Administration: Theory and Practice, E-ISSN: 21...Christo Ananth
The Educational Administration: Theory and Practice publishes prominent empirical and conceptual articles focused on timely and critical leadership and policy issues of educational organizations. The journal embraces traditional and emergent research paradigms, methods, and issues. The journal particularly promotes the publication of rigorous and relevant scholarly work that enhances linkages among and utility for educational policy, practice, and research arenas.
The goal of the editorial team and the journal’s editorial board is to promote sound scholarship and a clear and continuing dialogue among scholars and practitioners from a broad spectrum of education. Educational Administration: Theory and Practice presents prominent empirical and conceptual articles focused on timely and critical leadership and policy issues facing educational organizations. As an editorial team, we embrace traditional and emergent theoretical frameworks, research methods, and topics. We particularly promote the publication of rigorous and relevant scholarly work with utility for educational policy, practice, and research.
The journal’s primary focus is on studies of educational leadership, organizations, leadership development, and policy as they relate to elementary and secondary levels of education. Examinations of leadership and policy that fall outside K-12 are considered insofar as there are meaningful connections to the K-12 arena (e.g., college pipeline). International comparative investigations are welcome to the extent they have implications for a broad audience.s.
Bharatiya Shiksha Shodh Patrika is a half yearly refereed UGC care listed journal of Social Sciences Journal of Education. It is a bilingual (Hindi and English) Journal being published regularly since 1982 by Bharatiya Shiksha Shodh Sansthan, Saraswatikunj, Nirala Nagar, Lucknow, Uttar Pradesh, India. Bharatiya Shiksha Shodh Sansthan is an apex Research Institute of Vidya Bharti. The objective of this Journal is to provide an academic forum for Teachers, Teacher Educators, Research Scholars, Policy Makers. Administrators, other Research Workers to encourage original and critical thinking in the field of Education and allied disciplines through presentation of novel ideas, critical appraisal of contemporary educational problems and views and experiences on improved educational practices. However, articles from other disciplines related to contemporary educational issues of relevance may be accepted for publication subject to the approval of the Review Committee.
Call for Papers - African Journal of Biological Sciences, E-ISSN: 2663-2187, ...Christo Ananth
African Journal of Biological Sciences is an International peer-reviewed, Open Access journal that publishes original research articles as well as review articles in all areas of Biological Sciences. It operates a fully open access publishing model which allows open global access to its published content. This model is supported through Article Processing Charges. For more information on Article Processing charges click here. Its scope embraces Animal Sciences, Biochemistry, Bioinformatics, Biotechnology, Botany, Cell Biology, Developmental Biology, Ecology, Environmental Sciences, Ethno Medicine, Food Science, Freshwater Biology, Genetics, Immunology, Marine Biology, Microbiology, Molecular Biology, Physiology, Plant Sciences, Structural Biology,Toxicology,Zoology etc.
It is essential that authors prepare their manuscripts according to established specifications. Failure to follow them may result in papers being delayed or rejected. Therefore, contributors are strongly encouraged to read the author guidelines carefully before preparing a manuscript for submission. The manuscripts should be checked carefully for grammatical, punctuation errors. All papers are subjected to peer review. All articles published in this journal represent the opinion of the authors and not reflect the official policy of the Journal of African Journal of Biological Sciences
Wind Energy Harvesting: Technological Advances and Environmental ImpactsChristo Ananth
Christo Ananth, Rajini K R Karduri, "Wind Energy Harvesting: Technological Advances
and Environmental Impacts", International Journal of Advanced Research in Basic Engineering Sciences and Technology (IJARBEST), Volume 6,Issue 2,February 2020,pp:77-84
Hydrogen Economy: Opportunities and Challenges for a Sustainable FutureChristo Ananth
Christo Ananth, Rajini K R Karduri, "Hydrogen Economy: Opportunities and Challenges
for a Sustainable Future", International Journal of Advanced Research in Basic Engineering Sciences and Technology (IJARBEST), Volume 6,Issue 2,February 2020,pp:69-76
The Economics of Transitioning to Renewable Energy SourcesChristo Ananth
Christo Ananth, Rajini K R Karduri, "The Economics of Transitioning to Renewable Energy Sources", International Journal of Advanced Research in Basic Engineering Sciences and Technology (IJARBEST), Volume 6,Issue 2,February 2020,pp:61-68
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
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Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
HEAP SORT ILLUSTRATED WITH HEAPIFY, BUILD HEAP FOR DYNAMIC ARRAYS.
Heap sort is a comparison-based sorting technique based on Binary Heap data structure. It is similar to the selection sort where we first find the minimum element and place the minimum element at the beginning. Repeat the same process for the remaining elements.
Online aptitude test management system project report.pdfKamal Acharya
The purpose of on-line aptitude test system is to take online test in an efficient manner and no time wasting for checking the paper. The main objective of on-line aptitude test system is to efficiently evaluate the candidate thoroughly through a fully automated system that not only saves lot of time but also gives fast results. For students they give papers according to their convenience and time and there is no need of using extra thing like paper, pen etc. This can be used in educational institutions as well as in corporate world. Can be used anywhere any time as it is a web based application (user Location doesn’t matter). No restriction that examiner has to be present when the candidate takes the test.
Every time when lecturers/professors need to conduct examinations they have to sit down think about the questions and then create a whole new set of questions for each and every exam. In some cases the professor may want to give an open book online exam that is the student can take the exam any time anywhere, but the student might have to answer the questions in a limited time period. The professor may want to change the sequence of questions for every student. The problem that a student has is whenever a date for the exam is declared the student has to take it and there is no way he can take it at some other time. This project will create an interface for the examiner to create and store questions in a repository. It will also create an interface for the student to take examinations at his convenience and the questions and/or exams may be timed. Thereby creating an application which can be used by examiners and examinee’s simultaneously.
Examination System is very useful for Teachers/Professors. As in the teaching profession, you are responsible for writing question papers. In the conventional method, you write the question paper on paper, keep question papers separate from answers and all this information you have to keep in a locker to avoid unauthorized access. Using the Examination System you can create a question paper and everything will be written to a single exam file in encrypted format. You can set the General and Administrator password to avoid unauthorized access to your question paper. Every time you start the examination, the program shuffles all the questions and selects them randomly from the database, which reduces the chances of memorizing the questions.
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
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NUMERICAL SIMULATIONS OF HEAT AND MASS TRANSFER IN CONDENSING HEAT EXCHANGERS...ssuser7dcef0
Power plants release a large amount of water vapor into the
atmosphere through the stack. The flue gas can be a potential
source for obtaining much needed cooling water for a power
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moisture, it could reduce its total cooling water intake
requirement. One of the most practical way to recover water
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reduced in a condensing heat exchanger by acid condensation. reduced in a condensing heat exchanger by acid condensation.
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condensation in a flue gas condensing heat exchanger was
developed using MATLAB. Governing equations based on
mass and energy balances for the system were derived to
predict variables such as flue gas exit temperature, cooling
water outlet temperature, mole fraction and condensation rates
of water and sulfuric acid vapors. The equations were solved
using an iterative solution technique with calculations of heat
and mass transfer coefficients and physical properties.
Tutorial for 16S rRNA Gene Analysis with QIIME2.pdf
Modelling of Commercial Banks Capitals Competition Dynamics
1. International Journal of Early Childhood Special Education (INT-JECSE)
DOI:10.9756/INTJECSE/V14I5.476 ISSN: 1308-5581 Vol 14, Issue 05 2022
4124
Modelling of Commercial Banks Capitals Competition Dynamics
N.Arabov,
Samarkand State University, Uzbekistan.
D.Nasimov,
Academy of Public Administration, Uzbekistan.
H.Khuzhayorov,
Samarkand State University, Uzbekistan.
Christo Ananth*,
Samarkand State University, Uzbekistan.
T. AnanthKumar,
IFET College of Engineering, India.
Abstract---According to the observations in this paper, an existing mathematical model of banking capital dynamics
should be tweaked. First-order ordinary differential equations with a "predator-pray" structure make up the model,
and the indicators are competitive. Numerical realisations of the model are required to account for three distinct sets
of initial parameter values. It is demonstrated that a wide range of banking capital dynamics can be produced by
altering the starting parameters. One of the three options is selected, and the other two are eliminated. The model is
generalized taking into account fractional derivatives of the bank indicators for time, reflecting the rate of their
change. Based on numerical calculations, it is established that reduction of the order of derivatives from units leads
to a delay of banking capital dynamics. It is shown, that the less the order of derivatives from the unit, the more
delay of dynamics of indicators. In all analyzed variants indicators at large times reach their equilibrium values.
Keywords---Credit Resources, Deposits, Dynamics of the Banking Capital, Fractional Derivatives, Loan,
Mathematical Model, Own Capital of the Bank.
I. Introduction
Business entities, enterprises, and entrepreneurs need capital to establish new high-tech manufacturing facilities,
especially early in the project life cycle. Some businesses deposit any excess cash with the bank during the same
period. Bank capital is "dynamic," meaning it can change over time as a result of this constant change. The
modeling of fund demand and supply, risks and interbank competition, borrower-client late payments, changes in
market conditions, and unfavorable economic processes at the local and global levels is difficult to accomplish.
Loan officers consider the borrower's ability to repay the loan in full and on time when determining whether or not
to extend a loan to him or her. Many changes occur during a project's implementation that cannot be considered
when determining the feasibility of debtors. The mathematical model of commercial banks' credit and deposit
operations should be dynamic, allowing for a time-series analysis of key financial and economic indicators. It has
been said (Capinski and Zastawniak 2003; Bensoussan& Zhang 2009; Greene 2012).Everyone knows that bank
lending carries risk. The credit rating of a debtor is an essential factor in determining credit risk. The volatility of
credit ratings can impact a bank's ability to conduct normal business. Most mathematical models, credit risk
management methods, and other applications assume that debtors' credit ratings transitional probabilities are
constant and similar. It is also well-known that debtors' credit ratings change over time due to various factors. The
variability of income over time is a popular indicator for assessing long-term financial risk (Lando 2004; Bielecki,
Rutkowski 2007; Kwok 2008). When all else is equal, portfolio diversification can help reduce market volatility. So
the total variance is the sum of each component of the portfolio. Diversification reduces overall income volatility
while lowering risk. Banks should be cautious when engaging in risky activities. Banks today are much more than
just deposit takers. It is necessary to recognise their full commercial status. The mathematical modelling of
commercial bank financial transactions is becoming increasingly difficult due to increased risk. Banks want to make
as much money as possible by giving customers loans (both individuals and legal entities). The client is concerned
about reducing the amount of additional interest charged. As a result, the bank's and borrower's interests clash.
Samarskii and Mikhailov's (2002; Brauer and Castillo-Chavez 2012; Hastings 2013) principles have been
successfully applied in modeling biological and ecological systems, wage and employment changes, and other areas.
Theoretically, such models can be built between any two competitors. These models can be simplified using Lotka-
Volterra equations. The following authors were particularly interested in models like these when it came to market
competition dynamics (Tseng et al. 2014; Cooper, Nakanishi 1988; Lakka et al. 2013; Michalakelis et al. 2012;
2. International Journal of Early Childhood Special Education (INT-JECSE)
DOI:10.9756/INTJECSE/V14I5.476 ISSN: 1308-5581 Vol 14, Issue 05 2022
4125
Marasco et al. 2016). Also, competition models can assess bank health and economic growth (Jayakumar et al.
2018). Competition models are used to study American bank deposit and loan dynamics (Sumarti et al. 2014; Ansori
et al. 2019). These models have many uses, but they also have some flaws. (Marasco et al. 2016) expresses concern:
For example, economic factors that affect market share dynamics are constant over time; b) the models are
frequently not adequately connected to economic theory; and c) the effectiveness of proposed models is determined
by estimations of model parameters that determine competition roles. Unfortunately, the data used to derive these
estimates is scarce, and numerical methods such as genetic algorithms and other approaches are used. These models
can predict a wide range of competitive economic processes. As stated by (Comes 2012), a bidirectional capital
transfer from the Mother Bank to the Subsidiary Bank and vice versa is considered in a three-level Lotka-Volterra
(TLVR) model. In the TLVR model, the banking sector's equilibrium is analysed using the Fokker-Planck-
Kolmogorov stochastic equation solution. Lotka-Volterra models for n-level banking can be created from scratch
(Marasco et al. 2016).
Many researchers are now interested in fractional differential equations because they can accurately describe
various phenomena. These systems are studied using fractional-order differential equations. Fractional-order
systems are preferred because they allow more model freedom. These evolution equations produce fractional
Brownian motion. Non-Markovian despite the motions being Gaussian (Das et al. 2011). The authors solved the
fractional-order Lotka–Volterra equations using a nonlinear analytical method called homotopy perturbation. To
derive fractional Lotka–Volterra equations from classical ones, fractional derivatives are used instead of first-order
time derivatives. The fractional predator-prey and rabies models are numerically solved in (Ahmed et al. 2007). The
scientists claim the system is as stable as an integer-order system for fractional-order systems. The authors used
powerful analytical methods like the homotopy perturbation method to approximate an analytical solution for the
fractional-order Lotka–Volterra model. The Caputo-Fabrizio fractional derivative and fractional calculus are used to
study a three-dimensional Lotka-Volterra differential equation (Khalighi et al., 2021). The nonlinear population
model's fractional operator derivatives produce a variety of dynamical behaviors. [Traditional] Many generalized
fractional Lotka–Volterra models are proposed (Elettreby et al. 2017; Owolabi 2021; Amirian et al. 2020).
Fractional competitive models for commercial banks appeared in the 1990s. By Fatmawati et al., the Lotka-Volterra
competition model's parameter values are estimated using the genetic algorithm method. Atangana-Baleanu and
Caputo derivatives are used to study commercial and rural bank competition in 2019. To summarise, both model
operators provide useful data at the fractional-order parameter's points of interest. Wang et al. (2019) also use a
competition model to study bank data dynamics, with encouraging results. These papers look at fractional operators
(Caputo and its variants) and integers in general (including the Atangana–Baleanu case). Among Baleanu's works,
Atangana stands head and shoulders above the rest. Because the Caputo-Fabrizio operator is a fractional operator of
the fractal-fractional operator, this issue affects both operators (Atangana et al., 2020). This study compares integer-
order relevant results for accurate data to fractal and fractional order parameter values. A study found that the
Caputo-Fabrizio derivative fractal-fractional outperforms the integer-order definition. This short analysis shows that
fractional calculus began to be applied widely in the analysis of bank activity. Mathematical models with fractional
derivatives describe the bank data more adequately. A mathematical model for banking data through with fractal-
fractional operator in the sense of Caputo derivative is also presented by Li et al (2020). It was shown that with
varying fractal and fractional orders of derivatives one can obtain the best fitting to the data.
In this paper, we consider a mathematical model describing the dynamics of commercial bank capital where
various bank indicators have a competitive character. The model by its structure is similar to the three-level Lotka-
Volterra (TLV) model. However, here process and its indicators are absolutely others. First, we give general
information on mathematical models. Then a modified mathematical model for bank capital dynamics is developed
and numerically analyzed. Further, the model is generalized using fractional derivatives. Based on the numerical
analysis the influence of the order of fractional derivative on the dynamics of the banking capital is established. In
conclusion, we summarize the results.
II. Mathematical Models
In the classical form, the Lotka-Volterra “pray-predator” model can be written in the following form [7, 9]
𝑑𝑥1
𝑑𝑡
= (𝑎 − 𝑏𝑥2)𝑥1,
𝑑𝑥2
𝑑𝑡
= (−𝑐 + 𝑑 ⋅ 𝑥1)𝑥2, − 1
3. International Journal of Early Childhood Special Education (INT-JECSE)
DOI:10.9756/INTJECSE/V14I5.476 ISSN: 1308-5581 Vol 14, Issue 05 2022
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where𝑥1(𝑡), 𝑥2(𝑡) − prays and predators numbers, respectively, 𝑎, 𝑏, 𝑐, 𝑑 constant coefficients. A mathematical
model to describe the behaviour of deposit and loan volumes between two commercial banks can be written as
(Ansori et al. 2019).
𝑑𝑥1
𝑑𝑡
= 𝑎𝑥1 (1 −
𝑥1
𝐾𝐷
) − 𝑏𝑥1,
𝑑𝑥2
𝑑𝑡
= 𝑐𝑥2 (1 −
𝑥2
𝐾𝐿
𝑥1
𝐾𝐷
) − 𝑑𝑥2, (2)
where𝑥1(𝑡), 𝑥2(𝑡) are amounts of deposit and loan, respectively, 𝑎, 𝑏, 𝑐, 𝑑, 𝐾𝐿,𝐾𝐷 −constant parameters.
It is common to refer to banking micro-variables as a complex dynamic system in the banking industry because
of their high variability. Deposits and loans (based on the transport equation) are two types of financial instruments:
reserves and equity. Reserves and equity are two financial instruments (based on Selyutin and Rudenko, 2013).
Stochastic differential equations (in the form of a probability distribution) [30] develop two systems of differential
equations related to one another for the elements of a bank's balance sheet, which are then applied to the elements of
the bank's balance sheet. In both systems, two variables are used: the deposit and the loan. Ansori et al. (2019) use a
logistic model to compare the deposit and loan volumes of two different banks in their study. Savings and debt
transfer are two aspects of four different models that are discussed further below. Depositors in bank 1 transfer their
funds to bank 2, whereas borrowers transfer debts owed to bank 2 to bank 1. This is the model that is produced as a
result.
𝑑𝐷1
𝑑𝑡
= 𝑔𝐷1𝐷1 (1 −
𝐷1
𝐾𝐷1
) − 𝑤1𝐷1 − 𝑎2𝐷1𝐷2,
𝑑𝐿1
𝑑𝑡
= 𝑔𝐿1𝐿1 (1 −
𝐿1
𝐾𝐿1
𝐷1
𝐾𝐷1
) − 𝑏1𝐿1 − 𝑐2𝐿2𝐿1,
𝑑𝐷2
𝑑𝑡
= 𝑔𝐷2𝐷2 (1 −
𝐷2
𝐾𝐷2
) − 𝑤2𝐷21
+ 𝑎1𝐷1𝐷2,
𝑑𝐿2
𝑑𝑡
= 𝑔𝐿2𝐿2 (1 −
𝐿2
𝐾𝐿2
𝐷2
𝐾𝐷2
) − 𝑏2𝐿2 + 𝑐2𝐿2𝐿1,
where 𝐷𝑖 and 𝐿𝑖 denote the volume of deposit and loan for the bank 𝑖 , 𝑔𝐷𝑖, 𝐾𝐷𝑖, 𝑤𝑖, 𝑎𝑖, 𝑔𝐿𝑖, 𝐾𝐿𝑖, 𝑏𝑖, 𝑐𝑖 are
constant positive parameters, 𝑖 = 1; 2.
Dynamics of maximum profit by commercial banks (𝑥1(𝑡)) and rural banks (𝑥2(𝑡)) in Indonesia in [27] is
written in the form
𝑑𝑥1
𝑑𝑡
= 𝑎1𝑥1 (1 −
𝑥1
𝑐1
) − 𝑏1𝑥1𝑥2,
𝑑𝑥2
𝑑𝑡
= 𝑎2𝑥2 (1 −
𝑥2
𝑐2
) − 𝑏2𝑥1𝑥2, (3)
where𝑎1, 𝑎2, 𝑏1, 𝑏2,𝑐1, 𝑐2 are constants.
Model (3) depicts the competition between commercial and rural banks in an environment where fractal-
fractional operations are not used. By Atangana et al. (2020) the model (3) is generalized, where instead of
𝑑𝑥1
𝑑𝑡
and
𝑑𝑥2
𝑑𝑡
the fractal-fractional Caputo-Fabrisio operators 𝐷
𝐶𝐹
𝑜,𝑡
𝑢,𝑣
(𝑥1(𝑡)) and 𝐷
𝐶𝐹
𝑜,𝑡
𝑢,𝑣
(𝑥2(𝑡)) are used.
The equilibrium state of the models (1), (2), (3) can be obtained by setting
𝑑𝑥1
𝑑𝑡
= 0,
𝑑𝑥2
𝑑𝑡
= 0, 𝐷
𝐶𝐹
𝑜,𝑡
𝑢,𝑣
(𝑥1(𝑡)) = 0, 𝐷
𝐶𝐹
𝑜,𝑡
𝑢,𝑣
(𝑥2.
(𝑡)) = 0.
The TLV model that represents a tri-trophic capital chain between Mother bank, Subsidiary bank and individuals
has the form [18]
𝑑𝑥1
𝑑𝑡
= 𝑥1(𝑡)(𝑎1 − 𝑏1𝑥2(𝑡) + 𝑐1𝑥3(𝑡)), (4)
𝑑𝑥2
𝑑𝑡
= 𝑥2(𝑡)(−𝑎2 + 𝑏2𝑥1(𝑡)),(5)
𝑑𝑥3
𝑑𝑡
= 𝑥3(𝑡)(𝑎3 − 𝑏3𝑥1(𝑡)),(6)
where𝑥1(𝑡), 𝑥2(𝑡), 𝑥3(𝑡) is the number of Mother bank, Subsidiary bank and Individuals, respectively,
𝑎𝑖, 𝑏𝑖,𝑐1, 𝑖 = 1,2,3 are positive constants.
Some other approach to modelling of dynamics of the banking capital is given in [32]. The model is presented as
𝜕𝐷
𝜕𝑡
= (𝑟𝐷 − 𝑎11)𝐷 + 𝑎12𝐶 + 𝑎13𝑟𝐷𝑌(𝑡), (7)
𝑑𝐾
𝑑𝑡
= −𝑎21𝐾 + 𝑎22𝐶 +
𝑎23
𝑟𝐾(𝑡)
𝑌′(𝑡)(8)
𝑑𝐶
𝑑𝑡
= 𝑎31(𝑟𝐾𝐾 − 𝑟𝐷𝐷) − 𝑎32𝐶,(9)
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where 𝐷(𝑡) - the volume of deposits of clients at the moment 𝑡, 𝐾(𝑡) - size of a credit portfolio of the bank at the
moment 𝑡, 𝐶(𝑡) - own capital of the bank which has been saved up by some moment 𝑡, 𝑌(𝑡) - the aggregate income
of clients of the bank at the moment 𝑡 in annual expression, 𝑟𝐷, 𝑟𝐾 - interest rates under deposits and bank credits,
respectively, 𝑎11, 𝑎12, 𝑎13, 𝑎21, 𝑎22, 𝑎31, 𝑎32 - positive constants.
In the models (4) - (6) and (7) - (9) three spices are used in conjunction with one another. We can use these
models to solve ordinary differential equations, but first, we must ensure that the solution is stable. Changing the
model parameters can achieve an equilibrium state between three different spices in the presence of a set of model
constants. This article also discusses the model parameters that affect the solution's stability. The stability of the
bank's differential equation model solution systems is inversely proportional. At the equilibrium points of their
respective models of equations, (4) to (6) and (7) to (9) respectively, are the steady solutions to the system of
equations.
When the models (7) - (9) are examined in detail, it is discovered that a stable solution to the system of equations
can only be found within a very narrow range of the model parameters. A variant of this model, which incorporates
some additional features, is presented here due to this research. This assumption was made when formulating
equation (8), which states that the rate at which the credit portfolio dynamics changes is influenced by the credit
policy of the financial institution. Possessing large amounts of own capital on a broad basis allows financial
institutions to broaden the scope of their credit portfolios while also increasing the rate at which their credit
portfolios grow (Vlasenko 2013). Addition of the member a22 C to the right-hand side of equation (8) was used to
take into consideration the given circumstance. Another assumption is being used in this situation. We believe that
the volume of involved capital from investors, i.e. deposits, as a component of the bank's financial resources has an
impact on the increase in the dynamics of a credit portfolio's growth. Because of this assumption, we replace 𝑎22𝐶
on 𝑎22𝐷 in (8). Thus, equation (8) takes the form
𝑑𝐾
𝑑𝑡
= 𝑎22𝐷 − 𝑎21𝐾 +
𝑎23
𝑟𝐾(𝑡)
𝑌′(𝑡). (10)
The equations (7) and (9) do not change. Then the system of equations (7) - (9) with taking into account (10) we
write in the following form
𝜕𝐷
𝜕𝑡
= (𝑟𝐷 − 𝑎11)𝐷 + 𝑎12𝐶 + 𝑎13𝑟𝐷𝑌(𝑡), (11)
𝑑𝐾
𝑑𝑡
= 𝑎22𝐷 − 𝑎21𝐾 +
𝑎23
𝑟𝐾(𝑡)
𝑌′(𝑡), (12)
𝑑𝐶
𝑑𝑡
= 𝑎31(𝑟𝐾𝐾 − 𝑟𝐷𝐷) − 𝑎32𝐶. (13)
(Atangana et al. 2020; Li et al. 2020) have recently begun to use fractional derivatives in analyzing bank activity
models. They are considered more general and realistic than models with integer-order derivatives because they can
describe memory and heredity processes. Based on these works, we propose the generalization of models (4)-(6) and
(11), (12) and (13).
a) Model (4) – (6)
𝑑𝛼𝑥1
𝑑𝑡𝛼 = 𝑥1(𝑡)(𝑎1 − 𝑏1𝑥2(𝑡) + 𝑐1𝑥3(𝑡)), (14)
𝑑𝛽𝑥2
𝑑𝑡𝛽 = 𝑥2(𝑡)(−𝑎2 + 𝑏2𝑥1(𝑡)), (15)
𝑑𝛾𝑥3
𝑑𝑡𝛾 = 𝑥3(𝑡)(𝑎3 − 𝑏3𝑥1(𝑡)), (16)
b) Model (11) – (13)
𝑑𝛼𝐷
𝑑𝑡𝛼 = (𝑟𝐷 − 𝑎11)𝐷 + 𝑎12𝐶 + 𝑎13𝑟𝐷𝑌,(17)
𝑑𝛽𝐾
𝑑𝑡𝛽 = 𝑎22𝐷 − 𝑎21𝐾 +
𝑎23
𝑟𝐾(𝑡)
𝑌′(𝑡). (18)
𝑑𝛾𝐶
𝑑𝑡𝛾 = 𝑎31(𝑟𝐾𝐾 − 𝑟𝐷𝐷) − 𝑎32𝐶, (19)
where𝛼, 𝛽, 𝛾is the order of derivatives.
The system of the ordinary differential equations (11) - (13) in the matrix form we write as
𝑑𝑋
𝑑𝑡
= 𝐴𝑋 + 𝐵, (20)
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where𝑋 = (
𝐷
𝐾
𝐶
) , 𝐴 = (
𝑟𝐷 − 𝑎110𝑎12
𝑎22 − 𝑎210
−𝑟𝐷𝑎31𝑎31𝑟𝐾 − 𝑎32
) , 𝐵 = (
𝑎13𝑟𝐷𝑌(𝑡)
𝑎23
𝑟𝐾(𝑡)
𝑌′(𝑡)
0
).
To solve (20) it is necessary to set the initial conditions 𝑋(0) = 𝑋0, where𝑋0 = (𝐷0𝐾0𝐶0)𝑇
, 𝐷0, 𝐾0,𝐶0are initial
values of 𝐷, 𝐾 ⥂, 𝐶, i.e.𝐷(0) = 𝐷0, 𝐾(0) = 𝐾0𝐶(0) = 𝐶0, T is a transposition sign.
The equilibrium state of the system (20) is defined as a stationary solution
𝑑𝑋
𝑑𝑡
= 0, that gives a system of linear
algebraic equations 𝐴𝑋 = −𝐵with respect to 𝐷, 𝐾 ⥂, 𝐶. The solution of the last system of equations at 𝑌(𝑡) = 𝑌0 =
𝑐𝑜𝑛𝑠𝑡is
𝐷0
= 𝑎13𝑎32𝑎21𝑟𝐷𝑌0/𝑑, 𝐾0
= 𝑎13𝑎32𝑎22𝑟𝐷𝑌0/𝑑, 𝐶0
= 𝑎13𝑎31𝑟𝐷(𝑎22𝑟𝐾 − 𝑎21𝑟𝐷)𝑌0/𝑑, (21)
where 𝑑 = 𝑎13𝑎12𝑎21𝑟𝐷 + 𝑎11𝑎32𝑎21 − 𝑎31𝑎12𝑎22𝑟𝐾 − 𝑎32𝑎21𝑟𝐷.
The system (12) solution must be stable for the equilibrium state to exist. The eigenvalues of matrix A will be
investigated in more detail later on. At least one of the three eigenvalues should be complex-conjugated with
negative real parts, and the other two should be positive real parts. It satisfies the Raus-Gurvits requirements (Walter
1998). Fractional derivatives of the Riemann-Liouville, Grünwald-Letnikov, Caputo, Caputo-Fabrizio, Atangana-
Baleanu, and other functions can be used.
III.Numerical Analysis of Models
Here we give the numerical analysis of the system of ordinary differential equations describing the dynamics of
the banking capital. At first, the system of equations with integer derivatives is analyzed. Then we give a short
information on fractional-order derivatives. At last, we numerically analyze the system of ordinary differential
equations of fractional order (17) - (19). In order to begin, we must first solve the system of equations (11) - (13).
This system is solved by using a standard library of the programming language Matlab, which is available online.
Figure 1 depicts some of the outcomes. Table 1 shows the calculated values of parameters at their initial and
intermediate stages.
3.1. Numerical Analysis of the System of the Equation of Integer Order
Table 1: The Initial and Intermediate Calculated Values of Parameters for the Problem
Parameters, units Variant 1, Fig.1а Variant 2, Fig.1b Variant 3, Fig.1c
𝑟𝐷, 1/𝑦𝑒𝑎𝑟 0,125 0,103 0,139
𝑟𝐾, 1/𝑦𝑒𝑎𝑟 0,234 0,225 0,236
𝑎11, 1/𝑦𝑒𝑎𝑟 0,139 0,247 0,225
𝑎12, 1/𝑦𝑒𝑎𝑟 0,00001 0,00001 0,01
𝑎13, 1/𝑦𝑒𝑎𝑟 0,219 0,263 0,104
𝑎21, 1/𝑦𝑒𝑎𝑟 0,212 0,209 0,104
𝑎22, 1/𝑦𝑒𝑎𝑟 0,195 0,192 0,149
𝑎23, 1/𝑦𝑒𝑎𝑟 0 0 0
𝑎31, 1/𝑦𝑒𝑎𝑟 0,474 0,493 0,351
𝑎32, 1/𝑦𝑒𝑎𝑟 0,101 0,102 0,108
𝜆1, 1/𝑦𝑒𝑎𝑟 -0,014 -0,209 0,123+0,045i
𝜆2, 1/𝑦𝑒𝑎𝑟 -0,212 -0,143 -0,123- 0,045i
𝜆3, 1/𝑦𝑒𝑎𝑟 -0,101 -0,102 -0,052
𝑌0,bln$ 150 150 150
𝐷0,bln$ 350 20 50
𝐾0,bln$ 200 10 40
𝐶0,bln$ 150 30 30
𝐷0
,bln$ 297,146 28,487 27,271
𝐾0
,bln$ 273,315 26,195 39,173
𝐶0
,bln$ 124,777 14,326 17,845
Figure 1 depicts the relationship between various financial indicators and the initial data. Some indicators are
both decreasing and increasing in a monotonous fashion. The local minimum and maximum values break up the
data's monotony. The values of the initial parameters impact the nature of the dependencies as well (see Figure 1).
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Figure 1b illustrates this by showing that, while D(t) in Figures 1a and 1c is monotonically decreasing, it is
increasing in Figure 1. In Figures 1a and 1c, non-monotonic K(t) dynamics are shown, whereas in Figure 1,
monotonically increasing K(t) dynamics are shown. In Figure 1c, the eigenvalues of matrix A are represented by one
real and two complex-conjugated eigenvalues. Figure 1c shows the real Eigenvalues of Matrix A, as received.
Expect the oscillatory dynamics of financial indicators in this environment to continue. As shown in Table 1, the
fluctuations of the indicators are not well expressed and are only apparent for indicator C due to the small values of
the imaginary parts of eigenvalues (t). Matrix A has eigenvalues that can be described as either all negative real or as
two complex-conjugated with negative real parts and one real negative (as shown in Figures 1a and b), depending on
the initial parameter variations (Fig 1 c). K0/D0, the bank's credit portfolio ratio to client investors' deposits, is an
essential financial indicator to watch when the bank is in equilibrium. The liquidity of a commercial bank's balance
sheet can be gauged using this coefficient (Vlasenko 2013). The higher the value, the more difficult it is for the bank
to serve its customers due to the lack of liquidity. This indicates the bank's inefficiency in financial and economic
activity, reflected in its low profitability. As we investigate each parameter, we will calculate this factor (Tab. 1).
For the variant 1,2 with Fig. 1а,b we have K^0/D^0=0,92, and for the variant 3 with Fig.1c: K^0/D^0=1,44. The
above critical values cannot be compared to these. Financially, the bank prefers Variant 3 with Fig.1c.
In the model, C(t) represents only the capitalized profit, which is the factor a31 considers. Growing crediting
dynamics K(t) does not always mean growing crediting dynamics C(t). This analysis excludes time-revalued basic
capital and additional investor investments but not credit-depositary commercial activity (e.g., a bank's loan
portfolio). All of this clearly reflected Dynamics C. (t). Inflation-driven revaluation of basic capitals requires
consideration of financial and economic indicators.
3.2. Fractional Derivatives
There are many definitions of fractional-order derivatives. Detailed information on this topic can be found in
(Podlubny 1999; Samko et al. 1993; Yang 2019). Here we give some definitions of the most used fractional
derivatives. The Grünwald-Letnikov derivative of the function 𝑓(𝑡)is defined as,
𝐷𝛼
𝑓(𝑡) = 𝑙𝑖𝑚
ℎ→∞
1
ℎ𝛼
∑ (−1)𝑟
0≤𝑟<𝑚 (
𝛼
𝑟
) 𝑓(𝑡 + (𝛼 − 𝑟)ℎ), (22)
where𝛼 is the order of derivative, 𝑚 is the smallest natural number such that
𝑝 ≤ 𝑚, ,
N
r (
𝛼
𝑟
) =
𝑝(𝑝−1)⋅⋅⋅(𝑝−𝑟+1)
𝑟(𝑟−1)⋅⋅⋅1
. (23)
The Riemann-Liouville fractional derivative of order 𝛼 is given by
𝐷
𝑎 𝑡
𝛼
𝑓(𝑡) =
1
𝛤(𝑛 − 𝛼)
𝑑𝑛
𝑑𝑡𝑛
∫ (𝑡 − 𝜏)
𝑡
𝑎
𝑛−𝛼−1
𝑓(𝜏)𝑑𝜏, 𝑛 − 1 < 𝛼 < 𝑛
where𝛤(⋅) is the gamma function.
An alternative differentiation operator to the Grünwald-Letnikov and Riemann-Liouville operators was proposed
by Caputo,
𝐷𝛼
𝑓(𝑡) =
1
𝛤(𝑛−𝛼)
∫ (𝑡 − 𝜏)
𝑡
𝑎
𝑛−𝛼−1 𝑑𝑛𝑓(𝜏)
𝑑𝑡𝑛 𝑑𝜏, 𝑛 − 1 < 𝛼 < 𝑛, 𝑛 = [𝛼] + 1. (24)
Caputo's operator has some advantages over the Riemann-Liouville operator: For example, the Riemann-
Liouville fractional derivative's Laplace transform yields boundary conditions containing the lower and upper limits
of the derivative. These situations are difficult to describe physically. The Caputo derivative's Laplace transform
yields integer order boundary conditions. As an extra, the Caputo-Riemann-Liouville derivative of a constant is zero
while it is not. We also use the Caputo derivative for its advantages. Because, we will be solving the equations using
the finite-difference method, we will briefly discuss fractional derivative approximations.
3.3. Numerical Analysis of the System of the Equation of Fractional Order
To solve (17) – (19) we also use the finite difference method, where for the discretization of the fractional
derivatives a numerical integration of the Caputo representation will be used. So, we have the following
approximation in the case α≤1,n=1,a=0 (Liu et al. 2006; Sweilam et al. 2012; Xia et al. 2009).
𝑑𝛼
𝑓(𝑡𝑖)
𝑑𝑡𝛼
=
1
𝛤(1 − 𝛼)
∫ (𝑡𝑖 − 𝑠)
𝑡𝑖
𝑎
𝑛−𝛼−1
𝑑𝑓(𝑠)
𝑑𝑡
𝑑𝑠 =
1
𝛤(1 − 𝛼)
∑ ∫
𝑗ℎ
(𝑗−1)ℎ
𝑖
𝑗=1
[
𝑓𝑗 − 𝑓𝑗−1
ℎ
+ 𝑜(ℎ)] (𝑖ℎ − 𝑠)−𝛼
𝑑𝑠 =
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4130
=
1
𝛤(2 − 𝛼)ℎ𝛼
∑ ∫
𝑗ℎ
(𝑗−1)ℎ
𝑖
𝑗=1
(𝑓𝑗 − 𝑓𝑗−1)[(𝑖 − 𝑗 + 1)1−𝛼
− (𝑖 − 𝑗1−𝛼)] +
+
1
𝛤(2−𝛼)
∑ ∫
𝑗ℎ
(𝑗−1)ℎ
𝑖
𝑗=1 [(𝑖 − 𝑗 + 1)1−𝛼
− (𝑖 − 𝑗1−𝛼)]𝑜(ℎ2−𝛼) = (22)
= 𝜎𝛼,ℎ ∑ 𝜔𝑗
𝛼
(𝑓𝑖−𝑗+1 − 𝑓𝑖−𝑗)
𝑖
𝑗=1
+ 𝑜(ℎ),
whereℎ is the mesh step,
,
2
1
,
h
h 𝜔𝑗
𝛼
= 𝑗1−𝛼
− (𝑗 − 1)1−𝛼
.
The equations (17) - (19) with use (22) were approximate in the following form
𝜏1−𝛼
Г(2 − 𝛼)
[∑
𝐷𝑘+1 − 𝐷𝑘
𝜏
((𝑖 − 𝑘 + 1)1−𝛼
− (𝑖 − 𝑘)1−𝛼)
𝑖−1
𝑘=0
+
𝐷𝑖+1 − 𝐷𝑖
𝜏
] =
= (𝑟𝐷 − 𝑎11)𝐷𝑖 + 𝑎12𝐶𝑖 + 𝑎13𝑟𝐷𝑌𝑖, (23)
𝜏1−𝛽
Г(2 − 𝛽)
[∑
𝐾𝑘+1 − 𝐾𝑘
𝜏
((𝑖 − 𝑘 + 1)1−𝛽
− (𝑖 − 𝑘)1−𝛽)
𝑖−1
𝑘=0
+
𝐾𝑖+1 − 𝐾𝑖
𝜏
] =
= 𝑎22𝐷𝑖 − 𝑎21𝐾𝑖 +
𝑎23
𝑟𝐾
𝑌𝑖
′
, (24)
𝜏1−𝛾
Г(2 − 𝛾)
[∑
𝐶𝑘+1 − 𝐶𝑘
𝜏
((𝑖 − 𝑘 + 1)1−𝛾
− (𝑖 − 𝑘)1−𝛾)
𝑖−1
𝑘=0
+
𝐶𝑖+1 − 𝐶𝑖
𝜏
] =
= 𝑎31(𝑟𝐾𝐾𝑖 − 𝑟𝐷𝐷𝑖) − 𝑎32𝐶𝑖, (25)
where i
D
𝐾𝑖,𝐶𝑖are grid functions in a time point 𝑡𝑖, corresponding to 𝐷(𝑡),𝐾(𝑡), 𝐶(𝑡), respectively, 𝜏 is the time
step.
From the explicit finite difference schemes (23) - (25) we define 𝐷𝑖+1,𝐾𝑖+1,𝐶𝑖+1
𝐷𝑖+1 = ((𝑟𝐷 − 𝑎11)Г(2 − 𝛼)𝜏𝛼
+ 1)𝐷𝑖 + (𝑎12𝐶𝑖 + 𝑎13𝑟𝐷𝑌)Г(2 − 𝛼)𝜏𝛼
−
− [∑
𝐷𝑘+1−𝐷𝑘
𝜏
((𝑖 − 𝑘 + 1)1−𝛼
− (𝑖 − 𝑘)1−𝛼)
𝑖−1
𝑘=0 +
𝐷𝑖+1−𝐷𝑖
𝜏
], (26)
𝐾𝑖+1 = (1 − 𝑎21Г(2 − 𝛽)𝜏𝛽)𝐾𝑖 + (𝑎22𝐷𝑖 + 𝑎23𝑌𝑖
′
/𝑟𝐾)Г(2 − 𝛽)𝜏𝛽
−
− [∑
𝐾𝑘+1−𝐾𝑘
𝜏
((𝑖 − 𝑘 + 1)1−𝛽
− (𝑖 − 𝑘)1−𝛼)
𝑖−1
𝑘=0 +
𝐾𝑖+1−𝐾𝑖
𝜏
], (27)
𝐶𝑖+1 = (1 − 𝑎32Г(2 − 𝛾)𝜏𝛾)𝐶𝑖 + 𝑎31(𝑟𝐾𝐾𝑖 − 𝑟𝐷𝐷𝑖)Г(2 − 𝛾)𝜏𝛾
−
− [∑
𝐶𝑘+1−𝐶𝑘
𝜏
((𝑖 − 𝑘 + 1)1−𝛾
− (𝑖 − 𝑘)1−𝛾)
𝑖−1
𝑘=0 +
𝐶𝑖+1−𝐶𝑖
𝜏
]. (28)
The relations (26) - (28) define solutions on the time layer 𝑡𝑗+1 based on solutions on the previous layers.
According to (26)-(28), numerical calculations were carried out for those three variants, the parameters of which
are given in Table 1. Some results are shown in Fig. 2-4. The orders of fractional derivatives𝛼, 𝛽, 𝛾 were chosen in
different ways with a sequential decrease from 1. In Fig. 2-4 for comparison with the classical case, the graphs from
Fig. 1 are also shown. In Fig. 1 a,b,c the parameters 𝛼, 𝛽, 𝛾 alternately take the value 0.9, while the rest are equal to
one. There is slight lagging dynamics on the graphs. In fig. 2 d,e,f parameters, alternately take values of 0.8, while
the rest have values of 0.9. The decrease in the parameter values from unity, in this case, is more significant than in
Fig. 2 a,b,c. In this case, a more significant delay in dynamics of 𝐷,𝐾,𝐶 is observed. Consequently, a decrease in the
order of fractional derivatives in the model (17) - (19) leads to lagging dynamics of all indicators 𝐷,𝐾,𝐶. In this
case, the smaller the values 𝛼, 𝛽, 𝛾 from unity, the more pronounced the phenomenon of delay. With increasing
time, the retarded dynamics of 𝐷,𝐾,𝐶weakens and asymptotically go to the stationary values determined for the
model with integer derivatives. Consequently, the indicators𝐷,𝐾,𝐶 reach the same equilibrium values, only with a
certain delay, the magnitude and duration of which is determined by the values 𝛼, 𝛽, 𝛾. The calculation results for
options 2 and 3 are shown in Fig. 3,4 for the same sets of parameters 𝛼, 𝛽, 𝛾. For these variants, similar results were
obtained as in the first variant. Here, a decrease in the values of parameters 𝛼, 𝛽, 𝛾 from unity leads to the same
lagging dynamics, although the dynamics of indicators 𝐷,𝐾,𝐶 is completely different here.
8. International Journal of Early Childhood Special Education (INT-JECSE)
DOI:10.9756/INTJECSE/V14I5.476 ISSN: 1308-5581 Vol 14, Issue 05 2022
4131
IV.Conclusion
In this paper, a modified mathematical model of the dynamics of the capital of a commercial bank is developed.
A numerical calculation was carried out for three variants of initial parameters, for which the dynamics of various
indicators of the bank's activity is shown, such as D(t) - the volume of deposits of clients at the moment t, K(t) - the
size of a credit portfolio of the bank at the moment t, C(t) - own capital of the bank which has been saved up by
some moment t. It is shown, that depending on the values of the initial parameters, it is possible to obtain different
dynamics of these indicators. Further, the model is generalized using fractional derivatives of the above indicators,
reflecting their rate of change over time. Based on the numerical analyses of the model, the role of the order of
fractional derivatives on the dynamics of bank capital is estimated. It is shown, that a decrease in the order of the
derivatives from unity leads to lagging dynamics of indicators. In this case, the more the orders of the derivatives
decrease from unity, the more pronounced the delay effect is. In all the variants considered, over time, the indicators
of bank capital reach their equilibrium values, which are the same for both the model with integer derivatives and
the model with fractional derivatives.
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