In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
The numerical solution of Huxley equation by the use of two finite difference methods is done. The first one is the explicit scheme and the second one is the Crank-Nicholson scheme. The comparison between the two methods showed that the explicit scheme is easier and has faster convergence while the Crank-Nicholson scheme is more accurate. In addition, the stability analysis using Fourier (von Neumann) method of two schemes is investigated. The resulting analysis showed that the first scheme
is conditionally stable if, r ≤ 2 − aβ∆t , ∆t ≤ 2(∆x)2 and the second
scheme is unconditionally stable.
A numerical solution for Sine-Gordon type system was done by the use of two finite difference schemes, the first is the explicit scheme and the second is the Crank-Nicholson scheme. A comparison between the two schemes showed that the explicit scheme is easier and has faster convergence than the Crank-Nicholson scheme which is more accurate. The MATLAB environment was used for the numerical computations.
Finite element modelling of nonlocal dynamic systems, Modal analysis of nonlocal dynamical systems, Dynamics of damped nonlocal systems, Numerical illustrations
INFLUENCE OF OVERLAYERS ON DEPTH OF IMPLANTED-HETEROJUNCTION RECTIFIERSZac Darcy
In this paper we compare distributions of concentrations of dopants in an implanted-junction rectifiers in a
heterostructures with an overlayer and without the overlayer. Conditions for decreasing of depth of the
considered p-n-junction have been formulated.
The numerical solution of Huxley equation by the use of two finite difference methods is done. The first one is the explicit scheme and the second one is the Crank-Nicholson scheme. The comparison between the two methods showed that the explicit scheme is easier and has faster convergence while the Crank-Nicholson scheme is more accurate. In addition, the stability analysis using Fourier (von Neumann) method of two schemes is investigated. The resulting analysis showed that the first scheme
is conditionally stable if, r ≤ 2 − aβ∆t , ∆t ≤ 2(∆x)2 and the second
scheme is unconditionally stable.
A numerical solution for Sine-Gordon type system was done by the use of two finite difference schemes, the first is the explicit scheme and the second is the Crank-Nicholson scheme. A comparison between the two schemes showed that the explicit scheme is easier and has faster convergence than the Crank-Nicholson scheme which is more accurate. The MATLAB environment was used for the numerical computations.
Finite element modelling of nonlocal dynamic systems, Modal analysis of nonlocal dynamical systems, Dynamics of damped nonlocal systems, Numerical illustrations
INFLUENCE OF OVERLAYERS ON DEPTH OF IMPLANTED-HETEROJUNCTION RECTIFIERSZac Darcy
In this paper we compare distributions of concentrations of dopants in an implanted-junction rectifiers in a
heterostructures with an overlayer and without the overlayer. Conditions for decreasing of depth of the
considered p-n-junction have been formulated.
Stereographic Circular Normal Moment Distributionmathsjournal
Minh et al (2003) and Toshihiro Abe et al (2010) proposed a new method to derive circular distributions from the existing linear models by applying Inverse stereographic projection or equivalently bilinear transformation. In this paper, a new circular model, we call it as stereographic circular normal moment distribution, is derived by inducing modified inverse stereographic projection on normal moment distribution (Akin Olosunde et al (2008)) on real line. This distribution generalizes stereographic circular normal distribution (Toshihiro Abe et al (2010)), the density and distribution functions of proposed model admit closed form. We provide explicit expressions for trigonometric moments.
Numerical Solution of Nth - Order Fuzzy Initial Value Problems by Fourth Orde...IOSR Journals
In this paper, a numerical method for Nth - order fuzzy initial value problems (FIVP) based on
Seikkala derivative of fuzzy process is studied. The fourth order Runge-Kutta method based on Centroidal Mean
(RKCeM4) is used to find the numerical solution and the convergence and stability of the method is proved. This
method is illustrated by solving second and third order FIVPs. The results show that the proposed method suits
well to find the numerical solution of Nth – order FIVPs.
Special Plenary Lecture at the International Conference on VIBRATION ENGINEERING AND TECHNOLOGY OF MACHINERY (VETOMAC), Lisbon, Portugal, September 10 - 13, 2018
http://www.conf.pt/index.php/v-speakers
Propagation of uncertainties in complex engineering dynamical systems is receiving increasing attention. When uncertainties are taken into account, the equations of motion of discretised dynamical systems can be expressed by coupled ordinary differential equations with stochastic coefficients. The computational cost for the solution of such a system mainly depends on the number of degrees of freedom and number of random variables. Among various numerical methods developed for such systems, the polynomial chaos based Galerkin projection approach shows significant promise because it is more accurate compared to the classical perturbation based methods and computationally more efficient compared to the Monte Carlo simulation based methods. However, the computational cost increases significantly with the number of random variables and the results tend to become less accurate for a longer length of time. In this talk novel approaches will be discussed to address these issues. Reduced-order Galerkin projection schemes in the frequency domain will be discussed to address the problem of a large number of random variables. Practical examples will be given to illustrate the application of the proposed Galerkin projection techniques.
Errors in the Discretized Solution of a Differential Equationijtsrd
We study the error in the derivatives of an unknown function. We construct the discretized problem. The local truncation and global errors are discussed. The solution of discretized problem is constructed. The analytical and discretized solutions are compared. The two solution graphs are described by using MATLAB software. Wai Mar Lwin | Khaing Khaing Wai "Errors in the Discretized Solution of a Differential Equation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd27937.pdfPaper URL: https://www.ijtsrd.com/mathemetics/applied-mathamatics/27937/errors-in-the-discretized-solution-of-a-differential-equation/wai-mar-lwin
Dyadics algebra.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Lesson 19: The Mean Value Theorem (Section 021 handout)Matthew Leingang
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
The generalized design principle of TS fuzzy observers for one class of continuous-time
nonlinear MIMO systems is presented in this paper. The problem addressed can be indicated as
a descriptor system approach to TS fuzzy observers design, implying the asymptotic convergence of the state observer error. A new structure of linear matrix inequalities is outlined to possess the observer asymptotic dynamic properties closest to the optimal.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
EXACT SOLUTIONS OF A FAMILY OF HIGHER-DIMENSIONAL SPACE-TIME FRACTIONAL KDV-T...cscpconf
In this paper, based on the definition of conformable fractional derivative, the functional
variable method (FVM) is proposed to seek the exact traveling wave solutions of two higherdimensional
space-time fractional KdV-type equations in mathematical physics, namely the
(3+1)-dimensional space–time fractional Zakharov-Kuznetsov (ZK) equation and the (2+1)-
dimensional space–time fractional Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony
(GZK-BBM) equation. Some new solutions are procured and depicted. These solutions, which
contain kink-shaped, singular kink, bell-shaped soliton, singular soliton and periodic wave
solutions, have many potential applications in mathematical physics and engineering. The
simplicity and reliability of the proposed method is verified.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
A ( )-Stable Order Ten Second Derivative Block Multistep Method for Stiff I...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
Catalan Tau Collocation for Numerical Solution of 2-Dimentional Nonlinear Par...IJERA Editor
Tau method which is an economized polynomial technique for solving ordinary and partial differential
equations with smooth solutions is modified in this paper for easy computation, accuracy and speed. The
modification is based on the systematic use of „Catalan polynomial‟ in collocation tau method and the
linearizing the nonlinear part by the use of Adomian‟s polynomial to approximate the solution of 2-dimentional
Nonlinear Partial differential equation. The method involves the direct use of Catalan Polynomial in the solution
of linearizedPartial differential Equation without first rewriting them in terms of other known functions as
commonly practiced. The linearization process was done through adopting the Adomian Polynomial technique.
The results obtained are quite comparable with the standard collocation tau methods for nonlinear partial
differential equations.
SUCCESSIVE LINEARIZATION SOLUTION OF A BOUNDARY LAYER CONVECTIVE HEAT TRANSFE...ijcsa
The purpose of this paper is to discuss the flow of forced convection over a flat plate. The governing partial
differential equations are transformed into ordinary differential equations using suitable transformations.
The resulting equations were solved using a recent semi-numerical scheme known as the successive
linearization method (SLM). A comparison between the obtained results with homotopy perturbation method and numerical method (NM) has been included to test the accuracy and convergence of the method.
Stereographic Circular Normal Moment Distributionmathsjournal
Minh et al (2003) and Toshihiro Abe et al (2010) proposed a new method to derive circular distributions from the existing linear models by applying Inverse stereographic projection or equivalently bilinear transformation. In this paper, a new circular model, we call it as stereographic circular normal moment distribution, is derived by inducing modified inverse stereographic projection on normal moment distribution (Akin Olosunde et al (2008)) on real line. This distribution generalizes stereographic circular normal distribution (Toshihiro Abe et al (2010)), the density and distribution functions of proposed model admit closed form. We provide explicit expressions for trigonometric moments.
Numerical Solution of Nth - Order Fuzzy Initial Value Problems by Fourth Orde...IOSR Journals
In this paper, a numerical method for Nth - order fuzzy initial value problems (FIVP) based on
Seikkala derivative of fuzzy process is studied. The fourth order Runge-Kutta method based on Centroidal Mean
(RKCeM4) is used to find the numerical solution and the convergence and stability of the method is proved. This
method is illustrated by solving second and third order FIVPs. The results show that the proposed method suits
well to find the numerical solution of Nth – order FIVPs.
Special Plenary Lecture at the International Conference on VIBRATION ENGINEERING AND TECHNOLOGY OF MACHINERY (VETOMAC), Lisbon, Portugal, September 10 - 13, 2018
http://www.conf.pt/index.php/v-speakers
Propagation of uncertainties in complex engineering dynamical systems is receiving increasing attention. When uncertainties are taken into account, the equations of motion of discretised dynamical systems can be expressed by coupled ordinary differential equations with stochastic coefficients. The computational cost for the solution of such a system mainly depends on the number of degrees of freedom and number of random variables. Among various numerical methods developed for such systems, the polynomial chaos based Galerkin projection approach shows significant promise because it is more accurate compared to the classical perturbation based methods and computationally more efficient compared to the Monte Carlo simulation based methods. However, the computational cost increases significantly with the number of random variables and the results tend to become less accurate for a longer length of time. In this talk novel approaches will be discussed to address these issues. Reduced-order Galerkin projection schemes in the frequency domain will be discussed to address the problem of a large number of random variables. Practical examples will be given to illustrate the application of the proposed Galerkin projection techniques.
Errors in the Discretized Solution of a Differential Equationijtsrd
We study the error in the derivatives of an unknown function. We construct the discretized problem. The local truncation and global errors are discussed. The solution of discretized problem is constructed. The analytical and discretized solutions are compared. The two solution graphs are described by using MATLAB software. Wai Mar Lwin | Khaing Khaing Wai "Errors in the Discretized Solution of a Differential Equation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd27937.pdfPaper URL: https://www.ijtsrd.com/mathemetics/applied-mathamatics/27937/errors-in-the-discretized-solution-of-a-differential-equation/wai-mar-lwin
Dyadics algebra.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Lesson 19: The Mean Value Theorem (Section 021 handout)Matthew Leingang
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
The generalized design principle of TS fuzzy observers for one class of continuous-time
nonlinear MIMO systems is presented in this paper. The problem addressed can be indicated as
a descriptor system approach to TS fuzzy observers design, implying the asymptotic convergence of the state observer error. A new structure of linear matrix inequalities is outlined to possess the observer asymptotic dynamic properties closest to the optimal.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
EXACT SOLUTIONS OF A FAMILY OF HIGHER-DIMENSIONAL SPACE-TIME FRACTIONAL KDV-T...cscpconf
In this paper, based on the definition of conformable fractional derivative, the functional
variable method (FVM) is proposed to seek the exact traveling wave solutions of two higherdimensional
space-time fractional KdV-type equations in mathematical physics, namely the
(3+1)-dimensional space–time fractional Zakharov-Kuznetsov (ZK) equation and the (2+1)-
dimensional space–time fractional Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony
(GZK-BBM) equation. Some new solutions are procured and depicted. These solutions, which
contain kink-shaped, singular kink, bell-shaped soliton, singular soliton and periodic wave
solutions, have many potential applications in mathematical physics and engineering. The
simplicity and reliability of the proposed method is verified.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
A ( )-Stable Order Ten Second Derivative Block Multistep Method for Stiff I...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
Catalan Tau Collocation for Numerical Solution of 2-Dimentional Nonlinear Par...IJERA Editor
Tau method which is an economized polynomial technique for solving ordinary and partial differential
equations with smooth solutions is modified in this paper for easy computation, accuracy and speed. The
modification is based on the systematic use of „Catalan polynomial‟ in collocation tau method and the
linearizing the nonlinear part by the use of Adomian‟s polynomial to approximate the solution of 2-dimentional
Nonlinear Partial differential equation. The method involves the direct use of Catalan Polynomial in the solution
of linearizedPartial differential Equation without first rewriting them in terms of other known functions as
commonly practiced. The linearization process was done through adopting the Adomian Polynomial technique.
The results obtained are quite comparable with the standard collocation tau methods for nonlinear partial
differential equations.
SUCCESSIVE LINEARIZATION SOLUTION OF A BOUNDARY LAYER CONVECTIVE HEAT TRANSFE...ijcsa
The purpose of this paper is to discuss the flow of forced convection over a flat plate. The governing partial
differential equations are transformed into ordinary differential equations using suitable transformations.
The resulting equations were solved using a recent semi-numerical scheme known as the successive
linearization method (SLM). A comparison between the obtained results with homotopy perturbation method and numerical method (NM) has been included to test the accuracy and convergence of the method.
MODELING OF REDISTRIBUTION OF INFUSED DOPANT IN A MULTILAYER STRUCTURE DOPANT...mathsjournal
In this paper we used an analytical approach to model nonlinear diffusion of dopant in a multilayer structure with account nonstationary annealing of the dopant. The approach do without crosslinking solutions at
the interface between layers of the multilayer structure. In this paper we analyzed influence of pressure of
vapor of infusing dopant during doping of multilayer structure on values of optimal parameters of technological process to manufacture p-n-junctions. It has been shown, that doping of multilayer structures by
diffusion and optimization of annealing of dopant gives us possibility to increase sharpness of p-n-junctions
(single p-n-junctions and p-n-junctions within transistors) and to increase homogeneity of dopant distribution in doped area.
MODELING OF REDISTRIBUTION OF INFUSED DOPANT IN A MULTILAYER STRUCTURE DOPANT...mathsjournal
In this paper we used an analytical approach to model nonlinear diffusion of dopant in a multilayer structure with account nonstationary annealing of the dopant. The approach do without crosslinking solutions at
the interface between layers of the multilayer structure. In this paper we analyzed influence of pressure of
vapor of infusing dopant during doping of multilayer structure on values of optimal parameters of technological process to manufacture p-n-junctions. It has been shown, that doping of multilayer structures by
diffusion and optimization of annealing of dopant gives us possibility to increase sharpness of p-n-junctions
(single p-n-junctions and p-n-junctions within transistors) and to increase homogeneity of dopant distribution in doped area.
In this paper, an accurate numerical method is presented to find the numerical solution of the singularinitial value problems. The second-order singular initial value problem under consideration is transferredinto a first-order system of initial value problems, and then it can be solved by using the fifth-order RungeKutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methodsis confirmed by solving three model examples, and the obtained approximate solutions are compared withthe existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numericalmethod for solving the singular initial value problems.
ACCURATE NUMERICAL METHOD FOR SINGULAR INITIAL-VALUE PROBLEMSaciijournal
In this paper, an accurate numerical method is presented to find the numerical solution of the singular
initial value problems. The second-order singular initial value problem under consideration is transferred
into a first-order system of initial value problems, and then it can be solved by using the fifth-order Runge
Kutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methods
is confirmed by solving three model examples, and the obtained approximate solutions are compared with
the existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numerical
method for solving the singular initial value problems.
ACCURATE NUMERICAL METHOD FOR SINGULAR INITIAL-VALUE PROBLEMSaciijournal
In this paper, an accurate numerical method is presented to find the numerical solution of the singular initial value problems. The second-order singular initial value problem under consideration is transferred into a first-order system of initial value problems, and then it can be solved by using the fifth-order Runge Kutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methods is confirmed by solving three model examples, and the obtained approximate solutions are compared with the existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numerical method for solving the singular initial value problems.
Accurate Numerical Method for Singular Initial-Value Problemsaciijournal
In this paper, an accurate numerical method is presented to find the numerical solution of the singular initial value problems. The second-order singular initial value problem under consideration is transferred into a first-order system of initial value problems, and then it can be solved by using the fifth-order Runge Kutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methods is confirmed by solving three model examples, and the obtained approximate solutions are compared with the existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numerical method for solving the singular initial value problems.
Accurate Numerical Method for Singular Initial-Value Problemsaciijournal
In this paper, an accurate numerical method is presented to find the numerical solution of the singular
initial value problems. The second-order singular initial value problem under consideration is transferred
into a first-order system of initial value problems, and then it can be solved by using the fifth-order Runge
Kutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methods
is confirmed by solving three model examples, and the obtained approximate solutions are compared with
the existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numerical
method for solving the singular initial value problems.
The numerical solution of helmholtz equation via multivariate padé approximationeSAT Journals
Abstract
In this study, we consider the numerical solution of the Helmholtz equation, arising from numerous physical phenomena and
engineering applications including energy systems. We make use of a Padé Approximation method for the solution of the
Helmholtz equation. Firstly, Helmholtz partial differential equation had been converted to power series by two-dimensional
differential transformation, then the numerical solution of the equation was put into Padé series form for accelerating
convergence and decreasing computational time. Hereby, we obtained numerical solution of Helmholtz type partial differential
equation.
Key Words: Helmholtz equation, Two-dimensional differential transformation, Multivariate Padé approximation,
power series
Sparse data formats and efficient numerical methods for uncertainties in nume...Alexander Litvinenko
Description of methodologies and overview of numerical methods, which we used for modeling and quantification of uncertainties in numerical aerodynamics
MODELING OF REDISTRIBUTION OF INFUSED DOPANT IN A MULTILAYER STRUCTURE DOPANT...mathsjournal
In this paper we used an analytical approach to model nonlinear diffusion of dopant in a multilayer structure with account nonstationary annealing of the dopant. The approach do without crosslinking solutions at
the interface between layers of the multilayer structure. In this paper we analyzed influence of pressure of
vapor of infusing dopant during doping of multilayer structure on values of optimal parameters of technological process to manufacture p-n-junctions. It has been shown, that doping of multilayer structures by
diffusion and optimization of annealing of dopant gives us possibility to increase sharpness of p-n-junctions
(single p-n-junctions and p-n-junctions within transistors) and to increase homogeneity of dopant distribution in doped area.
Modeling of Redistribution of Infused Dopant in a Multilayer Structure Dopant...mathsjournal
In this paper we used an analytical approach to model nonlinear diffusion of dopant in a multilayer structure with account nonstationary annealing of the dopant. The approach do without crosslinking solutions at the interface between layers of the multilayer structure. In this paper we analyzed influence of pressure of vapor of infusing dopant during doping of multilayer structure on values of optimal parameters of technological process to manufacture p-n-junctions. It has been shown, that doping of multilayer structures by diffusion and optimization of annealing of dopant gives us possibility to increase sharpness of p-n-junctions (single p-n-junctions and p-n-junctions within transistors) and to increase homogeneity of dopant distribution in doped area.
Anomalous Diffusion Through Homopolar Membrane: One-Dimensional Model by Guilherme Garcia Gimenez and Adélcio C Oliveira* in Evolutions in Mechanical Engineering
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
ON INCREASING OF DENSITY OF ELEMENTS IN A MULTIVIBRATOR ON BIPOLAR TRANSISTORSijcsitcejournal
In this paper we consider an approach to increase density of elements of a multivibrator on bipolar transistors.
The considered approach based on manufacturing a heterostructure with necessity configuration,
doping by diffusion or ion implantation of required areas to manufacture the required type of conductivity
(p or n) in the areas and optimization of annealing of dopant and/or radiation defects to manufacture more
compact distributions of concentrations of dopants. We also introduce an analytical approach to prognosis
technological process.
Optimization of technological process to decrease dimensions of circuits xor ...ijfcstjournal
The paper describes an approach of increasing of integration rate of elements of integrated circuits. The
approach has been illustrated by example of manufacturing of a circuit XOR. Framework the approach one
should manufacture a heterostructure with specific configuration. After that several special areas of the
heterostructure should be doped by diffusion and/or ion implantation and optimization of annealing of dopant
and/or radiation defects. We analyzed redistribution of dopant with account redistribution of radiation
defects to formulate recommendations to decrease dimensions of integrated circuits by using analytical
approaches of modeling of technological process.
Influence of Overlayers on Depth of Implanted-Heterojunction RectifiersZac Darcy
In this paper we compare distributions of concentrations of dopants in an implanted-junction rectifiers in a
heterostructures with an overlayer and without the overlayer. Conditions for decreasing of depth of the
considered p-n-junction have been formulated.
Similar to FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC CONVECTION-DIFFUSION TYPE (20)
THE PRESSURE SIGNAL CALIBRATION TECHNOLOGY OF THE COMPREHENSIVE TEST SYSTEMieijjournal
The pressure signal calibration technology of the comprehensive test system which involved pressure
sensors was studied in this paper. The melioration of pressure signal calibration methods was elaborated.
Compared with the calibration methods in the lab and after analyzing the relevant problems,the
calibration technology online was achieved. The test datum and reasons of measuring error analyzed,the
uncertainty evaluation was given and then this calibration method was proved to be feasible and accurate.
8 th International Conference on Education (EDU 2023)ieijjournal
8th International Conference on Education (EDU 2023) will provide an excellent
international forum for sharing knowledge and results in theory, methodology and
applications impacts and challenges of education. The conference documents practical and
theoretical results which make a fundamental contribution for the development of
Educational research. The goal of this conference is to bring together researchers and
practitioners from academia and industry to focus on Educational advancements and
establishing new collaborations in these areas. Original research papers, state-of-the-art
reviews are invited for publication in all areas of Education
Informatics Engineering, an International Journal (IEIJ)ieijjournal
Informatics is a rapidly developing the field. The study of informatics involves human-computer
interaction and how an interface can be built to maximize user efficiency. Due to the growth in IT,
individuals and organizations increasingly process information digitally. This has led to the study of
informatics to solve privacy, security, healthcare, education, poverty, and challenges in our environment.
The Informatics Engineering, an International Journal (IEIJ) is an open access peer-reviewed journal that
publishes articles which contribute new results in all areas of Informatics. The goal of this journal is to
bring together researchers and practitioners from academia and industry to focus on the human use of
computing fields such as communication, mathematics, multimedia, and human-computer interaction
design and establishing new collaborations in these areas.
Big data is a prominent term which characterizes the improvement and availability of data in all three
formats like structure, unstructured and semi formats. Structure data is located in a fixed field of a record
or file and it is present in the relational data bases and spreadsheets whereas an unstructured data file
includes text and multimedia contents. The primary objective of this big data concept is to describe the
extreme volume of data sets i.e. both structured and unstructured. It is further defined with three “V”
dimensions namely Volume, Velocity and Variety, and two more “V” also added i.e. Value and Veracity.
Volume denotes the size of data, Velocity depends upon the speed of the data processing, Variety is
described with the types of the data, Value which derives the business value and Veracity describes about
the quality of the data and data understandability. Nowadays, big data has become unique and preferred
research areas in the field of computer science. Many open research problems are available in big data
and good solutions also been proposed by the researchers even though there is a need for development of
many new techniques and algorithms for big data analysis in order to get optimal solutions. In this paper,
a detailed study about big data, its basic concepts, history, applications, technique, research issues and
tools are discussed.
LOW POWER SI CLASS E POWER AMPLIFIER AND RF SWITCH FOR HEALTH CAREieijjournal
This research was to design a 2.4 GHz class E Power Amplifier (PA) for health care, with 0.18um
Semiconductor Manufacturing International Corporation CMOS technology by using Cadence software.
And also RF switch was designed at cadence software with power Jazz 180nm SOI process. The ultimate
goal for such application is to reach high performance and low cost, and between high performance and
low power consumption design. This paper introduces the design of a 2.4GHz class E power amplifier and
RF switch design. PA consists of cascade stage with negative capacitance. This power amplifier can
transmit 16dBm output power to a 50Ω load. The performance of the power amplifier and switch meet the
specification requirements of the desired
PRACTICE OF CALIBRATION TECHNOLOGY FOR THE SPECIAL TEST EQUIPMENT ieijjournal
For the issues encountered in the special test equipment calibration work, based on the characteristics of
special test equipment, the calibration point selection,classification of calibration parameters and
calibration method of special test equipment are briefly introduced in this paper, at the same time, the
preparation and management requirements of calibration specification are described.
8th International Conference on Signal, Image Processing and Embedded Systems...ieijjournal
8th International Conference on Signal, Image Processing and Embedded Systems (SIGEM 2022) is a forum for presenting new advances and research results in the fields of Digital Image Processing and Embedded Systems.
Informatics Engineering, an International Journal (IEIJ)ieijjournal
Informatics is rapidly developing field. The study of informatics involves human-computer interaction and how an interface can be built to maximize user-efficiency. Due to the growth in IT, individuals and organizations increasingly process information digitally. This has led to the study of informatics to solve privacy, security, healthcare, education, poverty, and challenges in our environment.
International Conference on Artificial Intelligence Advances (AIAD 2022) ieijjournal
International Conference on Artificial Intelligence Advances (AIAD 2022) will act as a major forum for the presentation of innovative ideas, approaches, developments, and research projects in the area advanced Artificial Intelligence. It will also serve to facilitate the exchange of information between researchers and industry professionals to discuss the latest issues and advancement in the research area. Core areas of AI and advanced multi-disciplinary and its applications will be covered during the conferences.
Informatics Engineering, an International Journal (IEIJ)ieijjournal
Call for papers...!!!
Informatics Engineering, an International Journal (IEIJ)
https://airccse.org/journal/ieij/index.html
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8th International Conference on Artificial Intelligence and Soft Computing (A...ieijjournal
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8th International Conference on Artificial Intelligence and Soft Computing (AIS 2022)
August 20 ~ 21, 2022, Chennai, India
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Informatics Engineering, an International Journal (IEIJ)
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International Conference on Artificial Intelligence Advances (AIAD 2022)ieijjournal
International Conference on Artificial Intelligence Advances (AIAD 2022)
August 27 ~ 28, 2022, Virtual Conference
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LOW POWER SI CLASS E POWER AMPLIFIER AND RF SWITCH FOR HEALTH CAREieijjournal
This research was to design a 2.4 GHz class E Power Amplifier (PA) for health care, with 0.18um
Semiconductor Manufacturing International Corporation CMOS technology by using Cadence software.
And also RF switch was designed at cadence software with power Jazz 180nm SOI process. The ultimate goal for such application is to reach high performance and low cost, and between high performance and low power consumption design. This paper introduces the design of a 2.4GHz class E power amplifier and
RF switch design. PA consists of cascade stage with negative capacitance. This power amplifier can
transmit 16dBm output power to a 50Ω load. The performance of the power amplifier and switch meet the specification requirements of the desired.
FROM FPGA TO ASIC IMPLEMENTATION OF AN OPENRISC BASED SOC FOR VOIP APPLICATIONieijjournal
ASIC (Application Specific Integrated Circuit) design verification takes as long as the designers take to describe, synthesis and implement the design. The hybrid approach, where the design is first prototyped on an FPGA (Field-Programmable Gate Array) platform for functional validation and then implemented as
an ASIC allows earlier defect detection in the design process and thus allows a significant time saving.
This paper deals with a CMOS standard-cell ASIC implementation of a SoC (System on Chip) based on the
OpenRISC processor for Voice over IP (VoIP) application; where a hybrid approach is adopted. The
architecture of the design is mainly based on the reuse of IPs cores described at the RTL level. This RTL code is technology-independent; hence the design can be ported easily from FPGA to ASIC.
Informatics Engineering, an International Journal (IEIJ)ieijjournal
Informatics is a rapidly developing the field. The study of informatics involves human-computer interaction and how an interface can be built to maximize user efficiency. Due to the growth in IT, individuals and organizations increasingly process information digitally. This has led to the study of informatics to solve privacy, security, healthcare, education, poverty, and challenges in our environment.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC CONVECTION-DIFFUSION TYPE
1. Informatics Engineering, an International Journal (IEIJ), Vol.5, No.1, March 2021
DOI : 10.5121/ieij.2021.5101 1
FITTED OPERATOR FINITE DIFFERENCE METHOD
FOR SINGULARLY PERTURBED PARABOLIC
CONVECTION-DIFFUSION TYPE
Tesfaye Aga Bullo and Gemechis File Duressa
Department of Mathematics, College of Natural Science, Jimma University, Jimma, P.O.
Box 378, Ethiopia
ABSTRACT
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
KEYWORDS
Singularly perturbation; convection-diffusion; boundary layer; Richardson extrapolation; fitted operator.
1. INTRODUCTION
Singularly perturbed parabolic partial differential equations are types of the differential equations
whose highest order derivative multiplied by small positive parameter. Basic application of these
equations are in the Navier stoke’s equations, in modeling and analysis of heat and mass transfer
process when the thermal conductivity and diffusion coefficients are small and the rate of reaction
is large [6, 11]. The presence of a small parameter in the given differential equation leads to
difficulty to obtain satisfactory numerical solutions. Thus, numerical treatment of the singularly
perturbed parabolic initial-boundary value problem is problematic because of the presence of
boundary layers in its solution. Hence, time-dependent convection-diffusion problems have been
largely discovered by researchers. Wherever they either use time discretization followed by the
spatial discretization or discretizes all the variables at once.
Researchers like, Clavero et al. [2] used the classical implicit Euler to discretize the time variable
and the upwind scheme on a non-uniform mesh for the spatial variable. The scheme was shown to
be first-order accurate. To get a higher-order method, Kadalbajoo and Awasthi [9] used the Crank
Nicolson finite difference method to discretize the time variable along with the classical finite
difference method on a piecewise uniform mesh for the space variable. Gowrisankar and Natesan
[13] working the backward Euler to discretize the time variable and the classical upwind finite
difference method on a layer adapted mesh for the space variable. To hypothesis the mesh, the
authors used equidistribution of a positive monitor function which is a linear combination of a
constant and the second-order spatial derivative of the singular component of the solution at each
time level. Their analysis provided a first-order accuracy in time and almost first-order accuracy in
2. Informatics Engineering, an International Journal (IEIJ), Vol.9, No.5, March 2021
2
space. Again, Growsiankor and Natesan [12] again constructed a layer adapted mesh together with
the classical upwind finite difference method to discretize the spatial variable.
In [9], Kadalbajoo et al. used the B-spline collocation method on a piecewise uniform mesh to
discretize the spatial variable and the classical implicit Euler for time variable to obtain a higher-
order accuracy in space. Natesan and Mukherjee [8] working using the Richardson extrapolation
method to post-process a uniformly convergent method to a second-order accuracy in both
variables. But, these methods mostly treated the stated problem for small order of convergence.
Thus, it is necessary to improve the accuracy with higher order of convergence for solving
singularly perturbed parabolic convection-diffusion types.
2. FORMULATION OF NUMERICAL METHOD
Consider the singularly perturbed parabolic initial-boundary value problems of the form
2
2
( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )
u u u
x t x t a x t x t b x t u x t f x t
t x x
(1)
( , ) : (0,1) (0, ]
x t T
subject to the initial and boundary conditions:
0 0
1 1
( ,0) ( ) on : ( ,0): 0 1
(0, ) ( ) on : (0, ): 0
(1, ) ( ), on : (1, ): 0
x
u x s x S x x
u t q t S t t T
u t q t S t t T
(2)
where is a perturbation parameter that satisfy 0 1
. Functions ( , )
a x t , ( , )
b x t and
( , )
f x t are assumed to be sufficiently smooth functions on the given domain such that for the
constants and .
( , ) 0,
( , ) 0.
a x t
b x t
(3)
Under sufficient smoothness and compatibility conditions imposed on the functions
0 1
( ), ( ), ( )
s x q t q t and ( , )
f x t , the initial-boundary value problem admits a unique solution
( , )
u x t to the assumed condition in Eq. (3), ( , ) 0
a x t which exhibits a boundary layer of width
( )
O near the boundary 1
x of the domain , [8].
Consider Eq. (1) on a particular domain ( , ) : (0,1) (0,1]
x t with the conditions in Eq. (2)
and let and
M N be positive integers. When working on , we use a rectangular grid
k
h
whose
nodes are
,
m n
x t
for 0,1, . . .
m M
and 0,1, . . .
n N
. Here, 0 1
0 . . . 1
M
x x x
and
0 1
0 . . . 1
N
t t t
such grids are called tensor-product grids.
3. Informatics Engineering, an International Journal (IEIJ), Vol.9, No.5, March 2021
3
1 1
, , 0,1,2, . . . , and , , 0,1,2, . . . ,
n m
t nk k n N x mh h m M
N M
(4)
Let denote the approximate solution ( , )
n
m m n
u u x t
at an arbitrary point( , )
m n
x t . Then,
considering the uniform discretization formula given in Eq. (4) in the t direction only, we have
the finite difference approximation for the derivatives with respect to t as:
1
1 1
2
2
( ) ( )
( ) ( ) ...
2
n
n n n
u u u u
x x k
x x
t k t
(5)
Substituting Eq. (5) into Eq. (1) transforms the singularly perturbed partial differential equation to
ordinary differential equation at each ( 1)th
n level of the form:
1 1
2
1
1 1 1
1
2
1 ( )
( ) ( ) ( ) ( ) ( ) ( )
n n n
n
n n n
u u u
f
a b u
d d x
x x x x x x
dx dx k k
(6)
where
1
1
2
2
( )
2
n
u
k
x
t
. With the conditions: 0 1
(0, ) ( ) and (1, ) ( ), 0 1
u t q t u t q t t
Let denoting:
1
1
1 1 1
1
1
1 ( )
( ) ( ) ( ) ( ) ( ) ( )
n n
n
n n n
n u u
f
a b u
B
d x
x x x x x x
dx k k
(7)
Then, Eq. (6) re-written as the second order singularly perturbed ordinary boundary value problem
1
2
1
2
( ) ( )
n
n
u
B
d
x x
dx
(8)
Subject to the boundary conditions:
1 1
0 1
0
(0, ) ( ) and (1, ) ( ) , 0 1
n n
M
u t q t u u t q t u t
.
A general j step method for the solution of Eq. (8) can be written as:
1
2
2
1 1
1 1
2
1 0 1
n
j j
n n
i i
m m i
i i m i
h
d u
u e u g
dx
(9)
Where '
i
e s and '
i
g s are arbitrary constant.
To determine the coefficients '
i
e s and '
i
g s , write the local truncation error (
1
1
n
m
LT
) for 2
j ,
in expanding form as:
4. Informatics Engineering, an International Journal (IEIJ), Vol.9, No.5, March 2021
4
1 1 1
1
1 1 2 1
1
1 1 1
2 2 2
2
0 1 1 2 1
2 2 2
( ) ( ) ( )
( ) ( ) ( )
n n n
n
m m m
m
n n n
m m m
u u u
h
LT x e x e x
d u d u d u
g x g x g x
dx dx dx
(10)
Assume that the function
1( )
n
u x
has continuous derivatives of sufficiently higher order.
Expanding the terms
1
1
( )
n
m
u x
and
1
2
1
2
( )
n
m
d u
x
dx
by Taylor’s series expansion about the point
m
x then substituting into Eq. (11) and grouping like terms gives:
1
1
1
1 2 2
1
1
2
2
2 0 1 2
2
1 1
3 4
2 2 0 2
3 4
0 2
3 4
5
2 0 2
5
5
1 ( ) (1 ) ( )
( ) ( )
2
( ) ( ) ( ) ( )
6 6 24 24 2 2
( ) (
120 120 6 6
n
n
n
m m
m
n
m
n n
m m
u
h
h h
h
du
LT e e x h e x
dx
d u
e g g g x
dx
e d u e g g d u
g g x x
dx dx
e g g d u
x
dx
1 1
6
2 0 2
6
6
) ( ) ( ) ...
720 720 24 24
n n
m m
h
e g g d u
x
dx
(11)
Method given in Eq. (9), for 2
j is of order four if all the coefficients given in Eq. (11) are equal
to zero except after the coefficient of
6
h , which gives:
1 2 0 2 1
1 5
2, 1, and
12 6
e e g g g
Because of 2
j and the values of this system of equations, Eq. (9) becomes:
2 2 2 2
1 1 1 1 1 1
2
1 1 1 1
2 2 2
2 ( ) 10( ) ( )
12
n n n n n n
m m
m m m m
h d u d u d u
u u u
dx dx dx
(12)
Where
4 6
1
2
6
( )
240
n
m
h d u
dx
Using the denotation in Eq. (7) into Eq. (13), we have:
1 1 1 1 1 1
2
1 1 1 1
2
12
2 10
n n n n n n
m m
m m m m
h
u u u B B B
(13)
Consider the following Taylor’s series expansion about the point m
x as:
5. Informatics Engineering, an International Journal (IEIJ), Vol.9, No.5, March 2021
5
1 1 1
2
1 2
2
( )
n n n
m m
m
h
du du d u
h O
dx dx dx
and
1 1 1
2
1 2
2
( )
n n n
m m
m
h
du du d u
h O
dx dx dx
(14)
With the central difference approximation:
1 1 1 1 1
1 1
2
1 1 1 1
3 4
2
2
2
and
2
n n n n n
n n
m
m m
m m m m
h
u u u u u
du d u
dx h dx
(15)
where
2 3
1
3
3
( )
6
n
m
h d u
dx
and
2 4
1
4
4
( )
12
n
m
h d u
dx
Substituting Eq. (15) into Eq. (14), also gives the central difference approximation:
1 1 1 1 1 1 1 1
1 1 1 1 1 1
5 6
3 4 4 3
and
2 2
n n n n n n n n
m m
m m m m m m
du u u u du u u u
dx h dx h
(16)
Where
2
5 3 4 ( )
h
h O
and
2
6 3 4 ( )
h
h O
Now, considering Eqs. (15) and (16) into Eq. (7) at m
x x
and 1
m
x x
gives:
1
1
1 1 1 1 1 1 1 1
3 1
1 1 1 1 1 1 1 1
1
1 1 1 1 1 1 1
3 1
1 1
1
1
1 1
1 1
1 1
( 4 3 )
2
1 1
( )
2
(3 4
2
n
m
n n n n n n n n n
m
m m m m m m m m
n
m
n n n n n n n n
m m m m m m
m m
n
m
n n n
m
m m
a
B u u u a b u f u
h k k
a
B u u a b u f u
h k k
a
B u u
h
1 1 1 1 1 1
3 1
1 1 1 1 1 1
1 1
)
n n n n n n
m m m m m m
u a b u f u
k k
(17)
Substituting Eq. (17) into Eq. (13) and grouping like terms, we get:
1 1
1
1 1
1 1
1 1
2
1 1
1 1
1 1
2
1 1
1
1 1
1 1
1 1 1 1
2
1
1
3
12 1 5
2 2
2 2
24 1
10
3
12 1 5 1
10
2 2
n n
n
m
m m
n n
m m
n n
m m
n n
m m
n n
n
m
m m
n n n n n
m
m m m m
n
m
h
h
h
a a
a
b u
h k h h
a a
b u
h k h
a a
a
b u u u u
h k h h k
f
1 1 1
1
10 n n n
m m
m
f f T
(18)
Where the truncation error
1
n
m
T
is given by:
1 1 1 1
1 2 3 5 6
1 1
12
n n n n
m m m m
T a a a
6. Informatics Engineering, an International Journal (IEIJ), Vol.9, No.5, March 2021
6
Introducing fitting parameter into Eq. (18), let denote
h
and then evaluate the limit of Eq.
(18) as 0
h we get:
1 1 1
1 1
(1)
0
1 1 1
1 1
0
lim
2lim 2
n n n
m m
h
n n n
m
m m
h
a u u
u u u
(19)
On the other hand, as the full prove given by Roos et. al., [6], the asymptotic expansion for the
solution Eq. (1) is given:
1
(1)
1
1 ( )
1
0
( ) ( )
n x
n a
n
u e
x u x A
(20)
where
1
0 ( )
n
u x
is the solution of the reduced problem, and A is an arbitrary constant. From Eq.
(20), we have:
1
1
1
1
(1)
(1)( )
1 1
0
0
(1)
(1)( 1 )
1 1
1 0
0
lim (0)
lim (0)
n
n
n
n
m
n n
m
h
m
n n
m
h
a a
e e
a a
e e
u u A
u u A
(21)
Putting Eq. (21) into Eq. (19), gives the value of fitting parameter as:
1 1
(1) (1)
coth
2 2
n n
a a
(22)
The obtained scheme, after introducing the fitted parameter which can be written as the three-
term recurrence relation:
1 1 1 1 1 1 1 1
1 1
n n n n n n n n
m m m m m m
m m
E u F u G u H TE
(23)
For
1,2, . . . , ; 0,1, . . ., and
k
m M n N
h
Where
1 1 1 1 1
1 1 1
12 1
(3 ) 5 1
2
n n n n n
m m
m m m
E a a a kb
h
1 1 1 1
1 1
12
2 10 10
n n n n
m m
m m
F a a kb
h
1 1 1 1 1
1 1 1
12 1
( 3 ) 5 1
2
n n n n n
m m
m m m
G a a a kb
h
1 1 1 1
1 1 1 1
10 10
n n n n n n n
m m m
m m m m
H u u u k f f f
7. Informatics Engineering, an International Journal (IEIJ), Vol.9, No.5, March 2021
7
With the truncation error
1 1
n n
m m
TE kT
Since the initial and boundary conditions are given in Eq. (2), we have
0
( ,0) ( ) ,
m
u x s x u
0,1, . . .
m M
,
1
0 0
(0, ) ( ) n
u t q t u
and
1
1
(1, ) ( ) n
M
u t q t u
0,1, . . .
n N
3. RICHARDSON EXTRAPOLATION
Richardson extrapolation technique is a convergence acceleration technique that consists of two
computed approximations of a solution (on two nested meshes) [16]. The linear combination turns
out to be a better approximation. The truncation error of the schemes in Eq. (18) is given by:
2 4 6
4
1 1 1 1 1
1 2 3 5 6
1 1
2 6
12 6 ( ) ( )
240
n n n n n
m m m
m m h
u h d u
T a a a k x O k
t dx
Hence, we have
4
( , ) n
m n m h
u x t U C k
(24)
Where ( , ) and n
m n m
u x t U are exact and approximate solutions respectively, C is a constant
independent of mesh sizes h and k.
Let
2N
M
be the mesh obtained by bisecting each mesh interval in
N
M
and denote the
approximation of the solution on
2N
M
by
n
m
U . Consider Eq. (24) works for any , 0
h k , which
implies:
4
( , ) , ( , )
n N N
m n m m n
M M
h
u x t U C k R x t
(25)
So that, it works for any
, 0
2
k
h
yields:
4 2 2
( , ) , ( , )
2
n N N
m n m m n
M M
h
k
u x t U C R x t
(26)
Where the remainders,
N
M
R and
2N
M
R are
2 4
( )
k h
O . A combination of inequalities in Eqs. (25)
and (26) leads to
2 4
( , ) 2 ( )
n n
m n m m k h
u x t U U O
which suggests that
2
ext
n n n
m m m
U U U
(27)
is also an approximation of ( , )
m n
u x t .
Using this approximation to evaluate the truncation error, we obtain
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8
2 4
( , )
ext
n
m n m k h
u x t U C
(28)
4. STABILITY AND CONVERGENCE ANALYSIS
A partial differential equation is well-posed if its solution exists, and depends continuously on the
initial and boundary conditions as given in [5 - 6]. The Von Neumann stability technique is applied
to investigate the stability of the developed scheme in Eq. (23), by assuming that the solution of
Eq. (23) at the grid point
,
m n
x t
is given by:
m
n i
n
m e
u
(29)
where 1,
i and is the real number and is the amplitude factor. Now, putting Eq. (29) into
the homogeneous part of Eq. (23) gives:
1 1 1
10
i i
i i
n n n
m m m
e e
e e
E F G
Since, the value of 1
i and using the relations: cos sin
i
e i
, the above equation
becomes:
1 1 1 1 1
10 2cos
cos sin
n n n n n
m m m m m
F E G i E G
(30)
From the trigonometric definition, the value
maxcos 1
which simplifies the relation and from
Eq. (3), we have: ( , ) 0 and ( , ) 0
a x t b x t
. Thus, Eq. (30), show that the
generalization for stability:
2
2
1 1
124 124
1
12 12 144
144 4 24 2 4 6
n m
m m
k a a
h h
Therefore, 1
Hence, the developed scheme in Eq. (23) is stable. Thus, the developed scheme in Eq. (23) is
unconditionally stable by Lax Richtmyer's definition, [5].
To investigate the consistency of the method, we have the truncation error from Eq. (23) given as:-
1 1,
n n
m m
TE kT
where
1 1 1 1
1 2 3 5 6
1 1
12
n n n n
m m m m
T a a a
, which re-arranged as:
2
4
1
2
6 ( ) ( )
n
m h
u
TE k x O
t
(31)
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9
Thus, Eq. (31) vanishes as 0 and 0
k h
. Hence, the scheme is consistent with the order of
2
h
O k
. Therefore, the scheme in Eq. (23), is convergent by Lax’s equivalence theorem [5].
5. NUMERICAL EXAMPLES, RESULTS, AND DISCUSSION
Example 1: Consider the singularly perturbed parabolic initial-boundary value problem:
2
2
(1 (1 )) ( , ), ( , ) (0,1) (0,1]
u u u
x x f x t x t
t x x
( ,0) ( ), 0 1
(0, ) (1, ) 0, 0 1
u x s x x
u t u t t
We choose the initial data ( )
s x and the source function ( , )
f x t to fit with the exact solution:
1 1 1
( , ) ( (1 ) )
x
t
e e e e
u x t x
As the exact solution for this example is known, for each , we calculate absolute maximum error
by:
,
,
( , )
max ( , )
M N
i j
ext
M N n
m n m
x t Q
E u x t u
, where
( , ) and
ext
n
m n m
u x t u
respectively, denote
the exact and numerical solution. Besides, we determined the corresponding order of convergence
by:
, 2 ,2
, log( ) log( )
log(2)
M N M N
M N E E
P
Table 1: Comparison of Maximum absolute errors of the solution for Example 1 at the
number of intervals /
M N
32/10 64/ 20 128/ 40 256/80 512/160 1024/ 320
Present method
0
10 2.2301e-05 7.0395e-06 2.0147e-06 5.4221e-07 1.4096e-07 3.5958e-08
2
10 6.3211e-03 1.5103e-03 2.9093e-04 6.7722e-05 1.6736e-05 4.1635e-06
4
10 8.9601e-03 4.7439e-03 2.4923e-03 1.2798e-03 6.4877e-04 3.2620e-04
Meth0d in [3]
0
10 6.8921e-04 3.7085e-04 1.9290e-04 9.8440e-05 4.9739e-05 ---
2
10 7.1532e-02 4.5000e-02 2.6393e-02 1.4579e-02 7.1423e-03 ---
4
10 9.3382e-02 5.5430e-02 3.9185e-02 2.1997e-02 1.1787e-02 ---
Method in [12]
0
10 6.8921e-04 3.7085e-04 1.9290e-04 9.8440e-05 4.9739e-05 2.5002e-05
2
10 3.6778e-02 2.2324e-02 1.2953e-02 7.2482e-03 3.9080e-03 2.0439e-03
4
10 6.6889e-02 3.7859e-02 2.0136e-02 1.0334e-02 5.2851e-03 2.6686e-03
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Table 2: Comparisons of the corresponding order of convergence for Example 1 at the
number of intervals /
M N
32/10 64/ 20 128/ 40 256/80 512/160
Present method
0
10 1.6636 1.8049 1.8936 1.9436 1.9709
2
10 2.0653 2.3761 2.1030 2.0167 2.0071
4
10 0.9174 0.9286 0.9616 0.9801 0.9920
Meth0d in [3]
0
10 0.8941 0.9430 0.9705 0.9849 ---
2
10 0.6687 0.7698 0.8563 1.0294 ---
4
10 0.7525 0.5004 0.8330 0.9001 ---
Meth0d in [12]
0
10 0.8941 0.9430 0.9705 0.9849 0.9923
2
10 0.7203 0.7853 0.8376 0.8912 0.9351
4
10 0.8211 0.9108 0.9624 0.9674 0.9858
Table 3: Maximum absolute errors and order of convergence before and after extrapolation for Example 1
at number of intervals M N
Extrapolation 32
M 64
M 128
M 256
M 512
M
6
2
Before 6.1530e-03 2.1777e-03 1.0045e-03 4.8793e-04 2.4106e-04
1.4985 1.1163 1.0417 1.0173 ---
After 3.8402e-03 7.1539e-04 1.6307e-04 3.9538e-05 9.8539e-06
2.4244 2.1332 2.0442 2.0045 ---
8
2
Before 1.0587e-02 4.7844e-03 1.6105e-03 5.7997e-04 2.7077e-04
1.1459 1.5708 1.4735 1.0989 ----
After 8.8146e-03 3.9827e-03 1.0913e-03 2.0090e-04 4.6361e-05
1.1461 1.8677 2.4415 2.1155 ---
10
2
Before 1.0639e-02 5.4361e-03 2.7310e-03 1.2154e-03 4.0731e-04
0.9687 0.9931 1.1680 1.5772 ---
After 8.8727e-03 4.7569e-03 2.4582e-03 1.0617e-03 2.8578e-04
0.8994 0.9524 1.2112 1.8934 --
Table 4: Maximum absolute errors with and without fitting factor for Example1 at
number of intervals M N
32 64 128 256 512
With fitting factor
6
2 3.8402e-03 7.1539e-04 1.6307e-04 3.9538e-05 9.8539e-06
10
2 8.8727e-03 4.7569e-03 2.4582e-03 1.0617e-03 2.8578e-04
14
2 8.8727e-03 4.7570e-03 2.4744e-03 1.2659e-03 6.5539e-04
Without fitting factor
6
2 1.1678e-01 3.0846e-02 7.2058e-03 1.8335e-03 4.9938e-04
10
2 7.7390e-01 7.6766e-01 5.7619e-01 3.4266e-01 1.3274e-01
14
2 1.4193e+00 9.7878e-01 9.7093e-01 9.6259e-01 8.9013e-01
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Figure 1: The physical behavior of the solutions of Example 1 for
2
10
64 and
M N
Figure 2: Pointwise absolute errors of Example 1 for
2
10
32 and
M N
Example 2: Consider the following time-dependent convection-diffusion problem:
2 3
2 2 3
2
1 1
(1 sin( )) (1 sin( )) 1 (1 )sin( )
2 2 2
( , ) (0,1) (0,1]
( ,0) 0, 0 1
(0, ) (1, ) 0, 0 1
u u u t
x x x u x x t t t
t x x
x t
u x x
u t u t t
As the exact solution ( , )
u x t is unknown, we use the double mesh principle
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Table 5: Comparison of maximum absolute errors obtained by present method for Example 2 at the
number of intervals /
M N
32/16 64/ 32 128/ 64 256/128
With fitting factor
0
10 3.8982e-05 1.0566e-05 2.7520e-06 7.0238e-07
1
10 3.1182e-04 8.6811e-05 2.2884e-05 5.8797e-06
2
10 1.6457e-03 6.4773e-04 1.9137e-04 5.0168e-05
3
10 1.7926e-03 9.8907e-04 5.1915e-04 2.5586e-04
4
10 1.7926e-03 9.8907e-04 5.1957e-04 2.6634e-04
Without fitting factor
0
10 4.0628e-05 1.0975e-05 2.8544e-06 7.2796e-07
1
10 5.1449e-04 1.3026e-04 3.2566e-05 8.1845e-06
2
10 3.6028e-02 1.5329e-02 4.5671e-03 9.9486e-04
3
10 1.1039e-01 9.7292e-02 7.4558e-02 4.6373e-02
4
10 1.2702e-01 1.2773e-01 1.2467e-01 1.1760e-01
Table 6: Comparisons of the corresponding order of convergence for Example 2 at /
M N
32/16 64/ 32 128/ 64
With fitting factor
0
10 1.8834 1.9409 1.9702
1
10 1.8448 1.9235 1.9605
2
10 1.3452 1.7590 1.9315
3
10 0.8579 0.9299 1.0208
4
10 0.8579 0.9288 0.9640
Without fitting factor
0
10 1.8883 1.9430 1.9713
1
10 1.9817 2.0000 1.9924
2
10 1.2329 1.7469 2.1987
3
10 0.1822 0.3840 0.6851
4
10 -0.0080 0.0350 0.0842
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Figure 3: Pointwise absolute errors of Example 2 for
4
10
64 and
M N
It can be seen from the results obtained and presented in Tables 1, 3, 4, and 5 show that the
numerical method presented in this paper produces more accurate numerical solutions for
singularly perturbed parabolic convection-diffusion problems with the right boundary layer.
Results presented in Tables 2, 3, and 6, shows that the maximum absolute errors and the
corresponding rate of convergence calculated using the present method is more accurate with a
higher rate of convergence than some existing methods. For each , and
M N
in Tables 2 - 6
shows the effectiveness of applying the fitted operator finite difference method with Richardson
extrapolation to obtain a more accurate numerical solution. The three figures (Figure 1 -3), show
that the stated problem has the right boundary layer so that maximum absolute errors existing in
the layer region.
6. CONCLUSION
In this paper, we have discussed a fitted operator finite difference method with Richardson
extrapolation for solving singularly perturbed parabolic convection-diffusion problem. First, the
Backward-Euler method is applied concerning the time derivative which gives a second-order
singularly perturbed ordinary differential equation with two-point boundary value points. Second,
applying the fitted operator finite difference method on the obtained ordinary differential equation
results in the two-level in the time direction and three-term recurrence relations in spatial
derivatives that can be solved by the Thomas algorithm. In this work, the two main contributions
to obtain a more accurate numerical solution with the higher order of convergence are using second-
order Richardson extrapolation in the time direction and introduce the fitting parameter on the
spatial direction.
Numerical simulations are provided to support the theoretical descriptions and to demonstrate the
effectiveness and betterment of using the proposed method. We point out that the fitting operator
method used in this work can be extended to other types of singularly perturbed time-dependent
problems.
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