This document describes applying He's variational iteration method to solve a subclass of Lane-Emden differential equations. The method constructs a sequence of correction functionals that generate successive approximations to the solution. It introduces a polynomial initial approximation satisfying the boundary conditions. The Lagrange multiplier is determined using variational theory. Estimates are established to prove the iterative sequence converges uniformly to the exact solution. The method is illustrated on some example problems and shown to produce exact polynomial solutions, demonstrating the effectiveness of the variational iteration method.
This research discussed nonoscillatory properties for resolve of nonlinear neutral
differences equation of second order with positive and negative coefficients. The
various new conditions which is ensure that all nonoscillatory solutions tend to zero
or infinity like are given two examples are illustrate the ordinary results.
This research discussed nonoscillatory properties for resolve of nonlinear neutral
differences equation of second order with positive and negative coefficients. The
various new conditions which is ensure that all nonoscillatory solutions tend to zero
or infinity like are given two examples are illustrate the ordinary results.
Dimensionality Reduction Techniques In Response Surface Designsinventionjournals
Dimensionality reduction has enormous applications in various fields in industries. It can be applied in an optimal way with respect to time and cost related to the agricultural sciences, mechanical engineering, chemical technology, pharmaceutical sciences, clinical trials, biological studies, image processing, pattern recognitions etc. Several researchers made attempts on the reduction of the size of the model for different specific problems using some mathematical and statistical techniques identifying and eliminating some insignificant variables.This paper presents a review of the available literature on dimensionality reduction
Lexisearch Approach to Travelling Salesman ProblemIOSR Journals
The aim of this paper is to introduce Lexisearch the structure of the search algorithm does not
require huge dynamic memory during execution. Mathematical programming is concerned with finding optimal
solutions rather than obtaining good solutions. The Lexisearch derives its name from lexicography .This
approach has been used to solve various combinatorial problems efficiently , The Assignment problem, The
Travelling Salesman Problem , The job scheduling problem etc. In all these problems the lexicographic search
was found to be more efficient than the Branch bound algorithms. This algorithm is deterministic and is always
guaranteed to find an optimal solution.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Exact solutions for weakly singular Volterra integral equations by using an e...iosrjce
In this paper, an iterative method proposed by Daftardar-Gejji and Jafari namely (DJM) will be
presented to solve the weakly singular Volterra integral equation (WSVIE) of the second kind. This method is
able to solve large class of linear and nonlinear equations effectively, more accurately and easily. In this
iterative method the solution is obtained in the series form that converges to the exact solution if it exists. The
main contribution of the current paper is to obtain the exact solution rather than numerical solution as done by
some existing techniques. The results demonstrate that the method has many merits such as being derivativefree,
overcome the difficulty arising in calculating Adomian polynomials to handle the nonlinear terms in
Adomian Decomposition Method (ADM). It does not require to calculate Lagrange multiplier as in Variational
Iteration Method (VIM) and no needs to construct a homotopy and solve the corresponding algebraic equations
as in Homotopy Perturbation Method (HPM). The results reveal that the method is accurate and easy to
implement. The software used for the calculations in this study was MATHEMATICA®
10.0.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
In real-life decisions, usually we happen to suffer through different states of uncertainties. In order to counter these uncertainties, in this paper, the author formulated a transportation problem in which costs are triangular intuitionistic fuzzy numbers, supplies and demands are crisp numbers. In this paper, a simple method for solving type-2 intuitionistic fuzzy transportation problem (type-2 IFTP) is proposed and optimal solution is obtained without using intuitionistic fuzzy modified distribution method and intuitionistic fuzzy zero point method. So, the proposed method gives the optimal solution directly. The solution procedure is illustrated with the help of three real life numerical examples. Defect of existing results proposed by Singh and Yadav (2016a) is discussed. Validity of Pandian’s (2014) method is reviewed. Finally, the comparative study, results and discussion are given.
A Mathematical Model to Solve Nonlinear Initial and Boundary Value Problems b...IJERA Editor
In this paper, a novel method called Laplace-differential transform method (LDTM) is used to obtain an
approximate analytical solution for strong nonlinear initial and boundary value problems associated in
engineering phenomena. It is determined that the method works very well for the wide range of parameters and
an excellent agreement is demonstrated and discussed between the approximate solution and the exact one in
three examples. The most significant features of this method are its capability of handling non-linear boundary
value problems.
International Journal of Research in Engineering and Science is an open access peer-reviewed international forum for scientists involved in research to publish quality and refereed papers. Papers reporting original research or experimentally proved review work are welcome. Papers for publication are selected through peer review to ensure originality, relevance, and readability.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Comparision of methods for combination of multiple classifiers that predict b...IJERA Editor
Predictive analysis include techniques fromdata mining that analyze current and historical data and make
predictions about the future. Predictive analytics is used in actuarial science, financial services, retail, travel,
healthcare, insurance, pharmaceuticals, marketing, telecommunications and other fields.Predicting patterns can
be considered as a classification problem and combining the different classifiers gives better results. We will
study and compare three methods used to combine multiple classifiers. Bayesian networks perform
classification based on conditional probability. It is ineffective and easy to interpret as it assumes that the
predictors are independent. Tree augmented naïve Bayes (TAN) constructs a maximum weighted spanning tree
that maximizes the likelihood of the training data, to perform classification.This tree structure eliminates the
independent attribute assumption of naïve Bayesian networks. Behavior-knowledge space method works in two
phases and can provide very good performances if large and representative data sets are available.
AN ALGORITHM FOR SOLVING LINEAR OPTIMIZATION PROBLEMS SUBJECTED TO THE INTERS...ijfcstjournal
Frank t-norms are parametric family of continuous Archimedean t-norms whose members are also strict functions. Very often, this family of t-norms is also called the family of fundamental t-norms because of the
role it plays in several applications. In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated.
Dimensionality Reduction Techniques In Response Surface Designsinventionjournals
Dimensionality reduction has enormous applications in various fields in industries. It can be applied in an optimal way with respect to time and cost related to the agricultural sciences, mechanical engineering, chemical technology, pharmaceutical sciences, clinical trials, biological studies, image processing, pattern recognitions etc. Several researchers made attempts on the reduction of the size of the model for different specific problems using some mathematical and statistical techniques identifying and eliminating some insignificant variables.This paper presents a review of the available literature on dimensionality reduction
Lexisearch Approach to Travelling Salesman ProblemIOSR Journals
The aim of this paper is to introduce Lexisearch the structure of the search algorithm does not
require huge dynamic memory during execution. Mathematical programming is concerned with finding optimal
solutions rather than obtaining good solutions. The Lexisearch derives its name from lexicography .This
approach has been used to solve various combinatorial problems efficiently , The Assignment problem, The
Travelling Salesman Problem , The job scheduling problem etc. In all these problems the lexicographic search
was found to be more efficient than the Branch bound algorithms. This algorithm is deterministic and is always
guaranteed to find an optimal solution.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Exact solutions for weakly singular Volterra integral equations by using an e...iosrjce
In this paper, an iterative method proposed by Daftardar-Gejji and Jafari namely (DJM) will be
presented to solve the weakly singular Volterra integral equation (WSVIE) of the second kind. This method is
able to solve large class of linear and nonlinear equations effectively, more accurately and easily. In this
iterative method the solution is obtained in the series form that converges to the exact solution if it exists. The
main contribution of the current paper is to obtain the exact solution rather than numerical solution as done by
some existing techniques. The results demonstrate that the method has many merits such as being derivativefree,
overcome the difficulty arising in calculating Adomian polynomials to handle the nonlinear terms in
Adomian Decomposition Method (ADM). It does not require to calculate Lagrange multiplier as in Variational
Iteration Method (VIM) and no needs to construct a homotopy and solve the corresponding algebraic equations
as in Homotopy Perturbation Method (HPM). The results reveal that the method is accurate and easy to
implement. The software used for the calculations in this study was MATHEMATICA®
10.0.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
In real-life decisions, usually we happen to suffer through different states of uncertainties. In order to counter these uncertainties, in this paper, the author formulated a transportation problem in which costs are triangular intuitionistic fuzzy numbers, supplies and demands are crisp numbers. In this paper, a simple method for solving type-2 intuitionistic fuzzy transportation problem (type-2 IFTP) is proposed and optimal solution is obtained without using intuitionistic fuzzy modified distribution method and intuitionistic fuzzy zero point method. So, the proposed method gives the optimal solution directly. The solution procedure is illustrated with the help of three real life numerical examples. Defect of existing results proposed by Singh and Yadav (2016a) is discussed. Validity of Pandian’s (2014) method is reviewed. Finally, the comparative study, results and discussion are given.
A Mathematical Model to Solve Nonlinear Initial and Boundary Value Problems b...IJERA Editor
In this paper, a novel method called Laplace-differential transform method (LDTM) is used to obtain an
approximate analytical solution for strong nonlinear initial and boundary value problems associated in
engineering phenomena. It is determined that the method works very well for the wide range of parameters and
an excellent agreement is demonstrated and discussed between the approximate solution and the exact one in
three examples. The most significant features of this method are its capability of handling non-linear boundary
value problems.
International Journal of Research in Engineering and Science is an open access peer-reviewed international forum for scientists involved in research to publish quality and refereed papers. Papers reporting original research or experimentally proved review work are welcome. Papers for publication are selected through peer review to ensure originality, relevance, and readability.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Comparision of methods for combination of multiple classifiers that predict b...IJERA Editor
Predictive analysis include techniques fromdata mining that analyze current and historical data and make
predictions about the future. Predictive analytics is used in actuarial science, financial services, retail, travel,
healthcare, insurance, pharmaceuticals, marketing, telecommunications and other fields.Predicting patterns can
be considered as a classification problem and combining the different classifiers gives better results. We will
study and compare three methods used to combine multiple classifiers. Bayesian networks perform
classification based on conditional probability. It is ineffective and easy to interpret as it assumes that the
predictors are independent. Tree augmented naïve Bayes (TAN) constructs a maximum weighted spanning tree
that maximizes the likelihood of the training data, to perform classification.This tree structure eliminates the
independent attribute assumption of naïve Bayesian networks. Behavior-knowledge space method works in two
phases and can provide very good performances if large and representative data sets are available.
AN ALGORITHM FOR SOLVING LINEAR OPTIMIZATION PROBLEMS SUBJECTED TO THE INTERS...ijfcstjournal
Frank t-norms are parametric family of continuous Archimedean t-norms whose members are also strict functions. Very often, this family of t-norms is also called the family of fundamental t-norms because of the
role it plays in several applications. In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated.
Riccati matrix differential equation has long been known to be so difficult to solve analytically and/or numerically. In this connection, most of the recent studies are concerned with the derivation of the necessary conditions that ensure the existence of the solution. Therefore, in this paper, He’s Variational iteration method is used to derive the general form of the iterative approximate sequence of solutions and then proved the convergence of the obtained sequence of approximate solutions to the exact solution. This proof is based on using the mathematical induction to derive a general formula for the upper bound proved to be converging to zero under certain conditions.
Comparative Study of the Effect of Different Collocation Points on Legendre-C...IOSR Journals
We seek to explore the effects of three basic types of Collocation points namely points at zeros of Legendre polynomials, equally-spaced points with boundary points inclusive and equally-spaced points with boundary point non-inclusive. Established in literature is the fact that type of collocation point influences to a large extent the results produced via collocation method (using orthogonal polynomials as basis function). We
analyse the effect of these points on the accuracy of collocation method of solving second order BVP. For equally-spaced points we further consider the effect of including the boundary points as collocation points. Numerical results are presented to depict the effect of these points and the nature of problem that is best handled by each.
Fractional pseudo-Newton method and its use in the solution of a nonlinear sy...mathsjournal
The following document presents a possible solution and a brief stability analysis for a nonlinear system,
which is obtained by studying the possibility of building a hybrid solar receiver; It is necessary to mention that
the solution of the aforementioned system is relatively difficult to obtain through iterative methods since the
system is apparently unstable. To find this possible solution is used a novel numerical method valid for one and
several variables, which using the fractional derivative, allows us to find solutions for some nonlinear systems in
the complex space using real initial conditions, this method is also valid for linear systems. The method described
above has an order of convergence (at least) linear, but it is easy to implement and it is not necessary to invert
some matrix for solving nonlinear systems and linear systems.
Fractional pseudo-Newton method and its use in the solution of a nonlinear sy...mathsjournal
The following document presents a possible solution and a brief stability analysis for a nonlinear system,
which is obtained by studying the possibility of building a hybrid solar receiver; It is necessary to mention that
the solution of the aforementioned system is relatively difficult to obtain through iterative methods since the
system is apparently unstable. To find this possible solution is used a novel numerical method valid for one and
several variables, which using the fractional derivative, allows us to find solutions for some nonlinear systems in
the complex space using real initial conditions, this method is also valid for linear systems. The method described
above has an order of convergence (at least) linear, but it is easy to implement and it is not necessary to invert
some matrix for solving nonlinear systems and linear systems
It is shown that any singular Lagrangian theory: 1) can be formulated without the use of constraints by introducing a Clairaut-type version of the Hamiltonian formalism; 2) leads to a special kind of nonabelian gauge theory which is similar to the Poisson gauge theory; 3) can be treated as the many-time classical dynamics. A generalization of the Legendre transform to the zero Hessian case is done by using the mixed (envelope/general) solution of the multidimensional Clairaut equation. The equations of motion are written in the Hamilton-like form by introducing new antisymmetric brackets. It is shown that any classical degenerate Lagrangian theory is equivalent to the many-time classical dynamics. Finally, the relation between the presented formalism and the Dirac approach to constrained systems is given.
Comparative Analysis of Different Numerical Methods of Solving First Order Di...ijtsrd
A mathematical equation which relates various function with its derivatives is known as a differential equation.. It is a well known and popular field of mathematics because of its vast application area to the real world problems such as mechanical vibrations, theory of heat transfer, electric circuits etc. In this paper, we compare some methods of solving differential equations in numerical analysis with classical method and see the accuracy level of the same. Which will helpful to the readers to understand the importance and usefulness of these methods. Chandrajeet Singh Yadav"Comparative Analysis of Different Numerical Methods of Solving First Order Differential Equation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-4 , June 2018, URL: http://www.ijtsrd.com/papers/ijtsrd13045.pdf http://www.ijtsrd.com/mathemetics/applied-mathematics/13045/comparative-analysis-of-different-numerical-methods-of-solving-first-order-differential-equation/chandrajeet-singh-yadav
Welcome to the first live UiPath Community Day Dubai! Join us for this unique occasion to meet our local and global UiPath Community and leaders. You will get a full view of the MEA region's automation landscape and the AI Powered automation technology capabilities of UiPath. Also, hosted by our local partners Marc Ellis, you will enjoy a half-day packed with industry insights and automation peers networking.
📕 Curious on our agenda? Wait no more!
10:00 Welcome note - UiPath Community in Dubai
Lovely Sinha, UiPath Community Chapter Leader, UiPath MVPx3, Hyper-automation Consultant, First Abu Dhabi Bank
10:20 A UiPath cross-region MEA overview
Ashraf El Zarka, VP and Managing Director MEA, UiPath
10:35: Customer Success Journey
Deepthi Deepak, Head of Intelligent Automation CoE, First Abu Dhabi Bank
11:15 The UiPath approach to GenAI with our three principles: improve accuracy, supercharge productivity, and automate more
Boris Krumrey, Global VP, Automation Innovation, UiPath
12:15 To discover how Marc Ellis leverages tech-driven solutions in recruitment and managed services.
Brendan Lingam, Director of Sales and Business Development, Marc Ellis
Pushing the limits of ePRTC: 100ns holdover for 100 daysAdtran
At WSTS 2024, Alon Stern explored the topic of parametric holdover and explained how recent research findings can be implemented in real-world PNT networks to achieve 100 nanoseconds of accuracy for up to 100 days.
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
The Metaverse and AI: how can decision-makers harness the Metaverse for their...Jen Stirrup
The Metaverse is popularized in science fiction, and now it is becoming closer to being a part of our daily lives through the use of social media and shopping companies. How can businesses survive in a world where Artificial Intelligence is becoming the present as well as the future of technology, and how does the Metaverse fit into business strategy when futurist ideas are developing into reality at accelerated rates? How do we do this when our data isn't up to scratch? How can we move towards success with our data so we are set up for the Metaverse when it arrives?
How can you help your company evolve, adapt, and succeed using Artificial Intelligence and the Metaverse to stay ahead of the competition? What are the potential issues, complications, and benefits that these technologies could bring to us and our organizations? In this session, Jen Stirrup will explain how to start thinking about these technologies as an organisation.
PHP Frameworks: I want to break free (IPC Berlin 2024)Ralf Eggert
In this presentation, we examine the challenges and limitations of relying too heavily on PHP frameworks in web development. We discuss the history of PHP and its frameworks to understand how this dependence has evolved. The focus will be on providing concrete tips and strategies to reduce reliance on these frameworks, based on real-world examples and practical considerations. The goal is to equip developers with the skills and knowledge to create more flexible and future-proof web applications. We'll explore the importance of maintaining autonomy in a rapidly changing tech landscape and how to make informed decisions in PHP development.
This talk is aimed at encouraging a more independent approach to using PHP frameworks, moving towards a more flexible and future-proof approach to PHP development.
Climate Impact of Software Testing at Nordic Testing DaysKari Kakkonen
My slides at Nordic Testing Days 6.6.2024
Climate impact / sustainability of software testing discussed on the talk. ICT and testing must carry their part of global responsibility to help with the climat warming. We can minimize the carbon footprint but we can also have a carbon handprint, a positive impact on the climate. Quality characteristics can be added with sustainability, and then measured continuously. Test environments can be used less, and in smaller scale and on demand. Test techniques can be used in optimizing or minimizing number of tests. Test automation can be used to speed up testing.
A tale of scale & speed: How the US Navy is enabling software delivery from l...sonjaschweigert1
Rapid and secure feature delivery is a goal across every application team and every branch of the DoD. The Navy’s DevSecOps platform, Party Barge, has achieved:
- Reduction in onboarding time from 5 weeks to 1 day
- Improved developer experience and productivity through actionable findings and reduction of false positives
- Maintenance of superior security standards and inherent policy enforcement with Authorization to Operate (ATO)
Development teams can ship efficiently and ensure applications are cyber ready for Navy Authorizing Officials (AOs). In this webinar, Sigma Defense and Anchore will give attendees a look behind the scenes and demo secure pipeline automation and security artifacts that speed up application ATO and time to production.
We will cover:
- How to remove silos in DevSecOps
- How to build efficient development pipeline roles and component templates
- How to deliver security artifacts that matter for ATO’s (SBOMs, vulnerability reports, and policy evidence)
- How to streamline operations with automated policy checks on container images
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdfPaige Cruz
Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack.
I, a former op, would like to extend an invitation to all application developers to join the observability party will share these foundational concepts to build on:
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
Essentials of Automations: The Art of Triggers and Actions in FMESafe Software
In this second installment of our Essentials of Automations webinar series, we’ll explore the landscape of triggers and actions, guiding you through the nuances of authoring and adapting workspaces for seamless automations. Gain an understanding of the full spectrum of triggers and actions available in FME, empowering you to enhance your workspaces for efficient automation.
We’ll kick things off by showcasing the most commonly used event-based triggers, introducing you to various automation workflows like manual triggers, schedules, directory watchers, and more. Plus, see how these elements play out in real scenarios.
Whether you’re tweaking your current setup or building from the ground up, this session will arm you with the tools and insights needed to transform your FME usage into a powerhouse of productivity. Join us to discover effective strategies that simplify complex processes, enhancing your productivity and transforming your data management practices with FME. Let’s turn complexity into clarity and make your workspaces work wonders!
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
Free Complete Python - A step towards Data Science
11.[36 49]solution of a subclass of lane emden differential equation by variational iteration method
1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.4, 2012
36
Solution of a Subclass of Lane-Emden Differential Equation by
Variational Iteration Method
Bhupesh K. Tripathi
Department of Mathematics, C.M.P. College, University of Allahabad, Allahabad 211002, India
E-mail: bupeshkt@gmail.com
Abstract
In this paper we apply He’s variational iteration method to find out an appropriate solution to a class of
singular differential equation under imposed conditions by introducing and inducting in a polynomial pro
satisfying the given subject to conditions at the outset as selective function to the solution extracting
process. As for as application part is concerned, Illustrative examples from the available literature when
treated all over reveal and out show that the solution deduced by proposed method is exact and again
polynomial. Overall, a successful produce of exact solutions by proposed process itself justify the
effectiveness and efficiency of the method so very much.
Keywords: He’s variational iteration method, Lane-Emden differential equation, exact solution,
polynomial, Lagrange multiplier.
1. Introduction
The universe is filled with numerous scientific advances and full off due observations that had tempted and
motivated to ponder on with outmost concern and curiosity all about. So happened realised seriously,
considered accordingly, analysed with all effort either implicitly or explicitly to the field of
multi-disciplinary sciences through modelling into suitable mathematical preposition like in the form of a
singular second order differential equation endowed with known boundary or other subject to conditions.
Further onwards for the sake of convenience and ascertaining definiteness to not only systematic
characteristics but also to the corresponding proper solution out of these prolific existing thought provoking
spectral problems of science thoroughly, the area of interest and investigation into such variety of subject
matter is constricted and limited to only some of the phenomenon occurring in mathematical
physics ,astrophysics, biological science of human physiology and chemical kinetics inter alia the theory of
stellar structure , the thermal behaviour of a spherical cloud of a gas , the isothermal gas spheres, the
thermionic emission of currents , the degeneration of white-dwarf of a star, the thermal distribution profile
in a human head , the radial stress within a circular plane , the elastic pressure under normal pressure the
oxygen tension in a spherical cell with Michaelis–Menton oxygen uptake kinetics, the reactants
concentration in a chemical reactor, the radial stress on a rotationally symmetric shallow membrane cap, the
temperature present in an anti-symmetric circular plate and many more like
problems[3,4,5,9,11,12,13,18,25,40] .Thereupon the considered range of mathematically modelled
problems may be affined to the a special class of Lane-Emden differential equation for apropos
interpretation and comprehensive investigation. Let the Lane-Emden differential equation considered with
composite imposed condition be
(1.1)
Subject to conditions
or
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Where f(x, y) is a real valued continuous function, g(x) C [0 1] and A, B, C, D are real constant.
The parameter ‘k’ is a real number greater than or equal to one. If k =1 or k=2 the problem (1.1) reduces to
cylindrical or spherical type by virtue of corroborating perspective symmetries properly. To begin with the
mode of analytical realisation, as of now, we again opt to restrict the domain of class of differential
equation (1.1) in the larger interest of finding any suitable, simple and effective methodology enabling
some better and appropriate much needed solution to the corresponding accustomed and coherent subclass
of problems. Subsequently as to the sufficient interest towards such class of problems a sincere attempt is
made ahead via the technique of variational iteration method successfully.
Now consider a specific subclass of problems as follow
(1.2)
Subject to conditions
Or
where A, B, C and D are real constants and the parameter k≥1.
However, p(x) and q(x) are polynomials of suitable degree and ‘a’ is any real number. Solution to the class
of problems (1.1) exists and is unique as well [15, 33, 38]. The point x=0 is a singular point of the problem
may offer a peculiar behavior to the solution in the neighborhood of that point like out showing a rapid
change, partly skeptical and chaotic towards some solution procuring process making one unable to
understand about the behavior of the solution over there at ‘x’(=0)equals to zero. However, due to
singularity to the extreme of the solution domain any of the numerical scheme may again face convergence
problem. However, the singular behavior could not impede the keen interest of researchers related to the
field of study of such kind of thought provoking problems any more.
In recent past, with regard to finding the solution to the Lane-Emden equation so far several other methods
like B-Spline method, Homotopy method, Finite element method, Lie group analysis, Modified Variational,
iteration method, Adomian method, Modified Adomian Decomposition method, Multi-integral method,
Differential method, Projection method, Legendre wavelets method, Taylors series method, Rational
Chebyshev collocation method ,Pseudo spectral methods have had been discussed and applied gracefully
[2,7,8,10,14,16,17,23,27,28,34,36,37,42,43].
The method under consideration that is to be put forward and proposed to be applied upon, is a method
none other than the He’s variational iteration method often ascribed to and eulogised for solving famous
subtle and meticulous problems like Autonomous ordinary differential system, Nonlinear oscillations,
Nonlinear relaxation phenomena in polycrystalline solids, Nonlinear thermo elasticity, Cubic nonlinear
Schrodinger equation, Ion acoustic plasma wave, Nonlinear oscillators with discontinuities, non-Newtonian
flows, Burger’s and coupled Burger’s equation, General Riccati equation, Multispecies Lotaka –Volterra
equations, Rational solution of Toda lattice equation, Helmholtz equation ,Generalized KdV equation and
Nonlinear differential equations of fractional order[1,6,20,21,29,30,31,32,35,39,40].
2. He’s Variational Iteration Method (V I M)
Variational iteration method may be understood like simultaneous toning up of Lagrange multiplier and
variational theory complimenting each other in unison. In as much as the type of such consequential mutual
indiscrete coexistence happen to be deduced out of two different mathematical concepts altogether
sometimes also referred to as modified Lagrange multiplier method previously put forward by Inokuti
Sakine and Mura[19] and later on envisioned and improvised by Chinese mathematician J.H. He have
outreached and surpassed a milestone for known to have solved plenty of challenging problems with
perfection, accuracy and great efficiency.Wich is what that itself speaks the volumes of its ability to elicit
solution out of a diversifying class of problems. In order to incorporate and treat on by this very method
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further on consider a general differential equation in operator form as
D where D is the usual differential operator (2.11)
y(x) is twice continuously differentiable function on a domain and g(x) is real valued inhomogeneous term.
The relation (2.1) may be decomposed as follows
L (2.12)
where L and N are linear and nonlinear differential operators respectively.
The variation iteration method acquires high efficiency and real potential to the required process of finding
solution systematically by successive generation of recursive relations of correctional functional with
respect to (2.12) via variational theory. Observing the success and usefulness of the proposed variant on so
many other intrigue solution desired problems it is expedient to introduce and treat the given class of
problems similarly. It is important to note that the variation iteration method accumulates its inner
efficiency and enough potential needed for the solution exhibiting procedure with regard to (2.12) is by
virtue of successive generation of recursive correctional functional systematically with the help of well
thought exotic concept of variational theory. Eventually, therefore for finding a just and acceptable solution
to the class of the problems (1.1) we adhere to construct a sequence of integral equations also called as
correctional functional to the problems (2.12) as follows
( ) = ( ) + ( ) ((L ( ( )) +N ( ) g ( )) ds , n 0 (2.13)
Where is Lagrange multiplier determined optimally satisfying all stationary conditions after
variational method is applied to (2.3).However, there exists one more important feature responsible for ease
and utility of the proposed method realized so all over is the assumption and choice of considering the
inconvenient highly nonlinear and complicated dependent variables as restricted variables so as to
minimizing the magnitude of the undesired error creeping into the susceptible solution finding process
of the general problem(1.1).The emblem aforementioned ‘ is the restricted variation, which means
=0.Eventually, after ‘ ’ is determined , a proper and suitable selective function may it be a linear one
or appropriately nonlinear with respect to (2.2) is assumed as an initial approximation for finding next
successive iterative function by recursive sequence of correction functionals anticipating to satisfy the
given boundary conditions. On few occasions it is witnessed that finally or preferably the limiting value
(as of sequential approximations incurred after due process of iteration leads to exact solution.
However, to our class of problems we consider a polynomial pre satisfying either boundary condition or
initial condition corresponding to the problem as selective function that is likely to produce well desired
exact solution.
2.1Variational Method and generalized Lagrange Multiplier
In order to avert the inconvenience caused by the presence of singularity the model (1.2) is required to be
treated by modifying the problem without changing the status of referred physical phenomenon.
Accordingly, the modified imposed value problem is
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for all ‘ x’ belonging to [0 1] (2.11)
Thus the sequential correctional funtionals corresponding to (2.11) may be defined as follows
( ) = ds n 0 (2.12)
where (x) is the initial selective function and is iterate of the correctional functional. Now
optimal value of (s) is identified naturally by taking variation with respect to (x) and subject to
restricted variation of unpleasing terms of i.e. (x) =0. Consequently to embark on the relation
(x) = (x) + (s)( + )ds , n 0 (2.13)
Further on by virtue of integrating ( ) by parts and subject to the restricted variation of =0) we
have then for n 0
(x) = (1 + -1) ) (x) + ( (s) (s)) |s=x + (s) ds,
This mathematical variational equation asserts that the stationary conditions are
= 0 (2.14)
=0 (2.15)
s (2.16)
The relations (2.14), (2.15) and (2.16) altogether implies that
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(s
(2.17)
Henceforth, the sequence of correction functionals are now given by
(x) = (x) + ) ( + n 0 (2.18)
Moreover, the relation (2.18) can be rewritten as
(x) = (x) + ) ( + n 0 (2.19)
In view of (2.11), (2.18) and (2.19) it is plausible to observe that by variational theory the process of
optimization self endeavors to multiply the relation (1.2) by ‘ ’power of the variable ‘x’ to come to
succor and modify the considered model problem in the neighborhood of singular point so that the
proposed method becomes expedient and can be expedited elegantly to conclude about innate and cohesive
grid scientific behavior of hitherto discussed well defined class of applied nature of problems.Again,we
also observe that the execution process carried out this way facilitate to express all the derivative terms as a
total differential of some function henceforth manifesting the differential equation like semi-exact. Which is
why, we can visualize the Lagrange multiplier as semi integrating factor for our model problems.
Also clearly would it be deduced from (2.19) that the limit of the convergent iterative sequence
satisfying given conditions is the desired exact solution to (1.1).
2.2 Convergence of Iterative Sequence
Essentially our prime motive in this section is to establish the convergence of the considered sequence of
correctional functionals generated out after VIM is executed onto the class (1.1) with regard to establish
(2.19) observe that
= + (x)) is the ‘ ’ partial sum of the infinite series
+ (x) (2.21)
Then necessarily the convergence of infinite series (2.21) implies the convergence of intermediary iterative
sequence of partial sums of the auxiliary series (2.21) as well. Suppose be the
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initial selective function of polynomial function consuming the given conditions in the problem.
Then the first successive variational iterate is given by
( ) = (s) a )) ds (2.22)
On integration by parts and erstwhile appliance of the proper stationary conditions we have
| (x) (x)|=| (s) + ds| (2.23)
This implies that
| ( ≤ (s)|+ | (s) | (| a ||p(s) | + ds (2.24)
Similarly, the relation (2.19) on carrying out similar simplifications and using stationary conditions, imply
or, | (x)| | p(s)|)( ds
≤ ( ds
≤ ds (2.25)
And, above all
| (x)|=| ) ds|
or,| (x)| | p(s)| ds , 2
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≤ ds, , s
(2.26)
Now, choose
M =sup (| (s) | , ) (2.27)
0≤s≤ n
Then, again observe and proceed to establish the inequality
(s) (s)| (2.28)
Obviously, relations (2.24), (2.25), (2.26) and (2.27) together imply that
| (x)- (x) | ds = M (2.29)
As well as, | (x)| ds (2.30)
s
Using (2.29) in (2.30) we find that
| (x)| M ds =
s
Thus, the statement (2.28) is true for natural number n=1
As usual, suppose that
(s) (s)| holds for some, (2.31)
Then, again relations (2.24) and (2.27) altogether imply that
| – (x) | ( (s) (s) |ds
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s
or, | (x) (x)| ds
s
That implies by (2.27) and (2.31)
| (x) (x)| M ds=
Therefore, by Principle of Induction
| | holds (2.32)
Now we claim that the series (2.21) converges both absolutely and uniformly for all [0 1] using (2.32)
Since,| (x)|+ (x) (x)| | (x)|+ =| (x)|+ ( 1) [01] (2.33)
Therefore the series + (x) converge uniformly [01] and by virtue of
(2.33) sequence of its partial sums converges to solution function of the given class of
problems.
3. Illustrative Problems
The proposed method is justified by successful implementation of VIM on some of the specific problems of
linear and nonlinear type often referred, discussed and had been attempted to solve by other different
methods in literature available so far.
3.1Example1:
Consider the following boundary value problem [21,34]
(x) + (x) + = (3.11)
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Subject to ,
Solution: To solve (5.1) we construct correction functional as follows
( ) = ( ) + ( (s) s ( )) ds n 0
Where ‘ (s)’ is optimally identified Lagrange multiplier similar to (2.27).Then the first iterative solution is
( ) = ( ) + ( (s) (s) s ( )) ds
Let (x) be the selective polynomial function satisfying the given boundary conditions. We may simply
choose selective function as
(x) = a a
Then the first iterate is as follows
(x) = a a + ( (s) (s) s ( )) ds
Now on performing simplifications, we get
(x)= a a (a + (a
Further onwards imposition of boundary condition on (x) asserts that ‘a=1’enabling, (x) =
as the produced exact solution to the problem.
3.2Example2:
Consider the boundary value problem [7, 34,35]
) (3.21)
y (0) =1 , y (1) =0
Solution: The correction functional for the problem (3.21) is
(x) = (x) + ds (3.22)
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‘ (s) ’is usual optimally identified Lagrange multiplier
Let (x) = a (1 as selective initial approximation function to induce successive first iterate as
(x) = a (1
Since solution to (1.2) type of boundary value problems are unique, therefore upon matching the boundary
condition we get ‘a=1’ rendering (x) = (1 as the exact solution to (3.21)
3.3Example3:
Let the nonlinear boundary value problem [42]
(3.31)
y(0) =2, (0) = 0
Solution: The correctional functional with respect to (3.31) is given by
(x) = (x) + ds for n=0, 1… (3.32)
Let, (x) = = 2+a b be the selective initial approximation function .Then by VIM,
First iterative approximate solution to (3.32) simplifies to
(x)=2+ (3.32)
Then on matching the given initial condition and applying unique feature of solution again implies that a=0
and b=1, exhibiting (x)= 2+ , the exact solution to the problem.
However if we consider differential equation (3.31) along boundary conditions =2 and (1) =3 then
(3.33) similarly provides exact solution to the boundary value problem as well.
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3.4 Example4:
Consider the boundary value problem [17, 42]
+ + = 6 (3.41)
Subject to =0 , (1) = 1
Solution: The correctional functional for boundary value problem (3.41) is as follows
(x) = (x) + ds for n=0, 1, 2…… (3.42)
Let ( ) = a + (1-a) be the selective function satisfying the given boundary condition then
Then the first iterate by variation iteration method from (3.42) is given by
Now matching the conditions at the end points of the solution domain and using the fact that the solution to
such boundary value problem are unique we get, . Hence the method producing the exact solution
.
3.5Example5
Consider the problem[10,33]
(3.51)
Subject to
Solution: If ‘λ ’is the Lagrange multiplier then the first correctional functional is given by
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(3.52)
Let us consider as selective function
Then on inserting the value of in (3.52) we get
(3.53)
Hence upon imposing the given initial condition in (3.53) and using the criterion of uniqueness of the
solution we have a=-1 and b=1 felicitating y(x) = .
Moreover, if we again consider the problem (3.51) along with condition = = then on the basis
of similar logic on (3.53) we get an exact solution to the considered boundary value problem.
4. Conclusions
This is pertinent to note that He’s variation iteration method applies successfully to a linear as well as to a
nonlinear class of boundary or initial value problems of type (1.2). Frontier examples of relevance that have
had occurred time and again and had been dealt by some other method of solution are taken and solved to
focus and assert that a proper selection of selective function and henceforth imposition of boundary or
initial condition as we please on iterative correctional function may lead to an exact solution. However
sometimes necessity of uniqueness of solution is also assumed during solution maneuvering process.
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