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.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
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.
CHN and Swap Heuristic to Solve the Maximum Independent Set ProblemIJECEIAES
We describe a new approach to solve the problem to find the maximum independent set in a given Graph, known also as Max-Stable set problem (MSSP). In this paper, we show how Max-Stable problem can be reformulated into a linear problem under quadratic constraints, and then we resolve the QP result by a hybrid approach based Continuous Hopfeild Neural Network (CHN) and Local Search. In a manner that the solution given by the CHN will be the starting point of the local search. The new approach showed a good performance than the original one which executes a suite of CHN runs, at each execution a new leaner constraint is added into the resolved model. To prove the efficiency of our approach, we present some computational experiments of solving random generated problem and typical MSSP instances of real life problem.
PROBABILISTIC INTERPRETATION OF COMPLEX FUZZY SETIJCSEIT Journal
The innovative concept of Complex Fuzzy Set is introduced. The objective of the this paper to investigate
the concept of Complex Fuzzy set in constraint to a traditional Fuzzy set , where the membership function
ranges from [0, 1], but in the Complex fuzzy set extended to a unit circle in a complex plane, where the
member ship function in the form of complex number. The Compressive study of mathematical operation of
Complex Fuzzy set is presented. The basic operation like addition, subtraction, multiplication and division
are described here. The Novel idea of this paper to measure the similarity between two fuzzy relations by
evaluating δ -equality. Here also we introduce the probabilistic interpretation of the complex fuzzy set
where we attempted to clarify the distinction between Fuzzy logic and probability.
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.
Computer Science
Active and Programmable Networks
Active safety systems
Ad Hoc & Sensor Network
Ad hoc networks for pervasive communications
Adaptive, autonomic and context-aware computing
Advance Computing technology and their application
Advanced Computing Architectures and New Programming Models
Advanced control and measurement
Aeronautical Engineering,
Agent-based middleware
Alert applications
Automotive, marine and aero-space control and all other control applications
Autonomic and self-managing middleware
Autonomous vehicle
Biochemistry
Bioinformatics
BioTechnology(Chemistry, Mathematics, Statistics, Geology)
Broadband and intelligent networks
Broadband wireless technologies
CAD/CAM/CAT/CIM
Call admission and flow/congestion control
Capacity planning and dimensioning
Changing Access to Patient Information
Channel capacity modelling and analysis
Civil Engineering,
Cloud Computing and Applications
Collaborative applications
Communication application
Communication architectures for pervasive computing
Communication systems
Computational intelligence
Computer and microprocessor-based control
Computer Architecture and Embedded Systems
Computer Business
Computer Sciences and Applications
Computer Vision
Computer-based information systems in health care
Computing Ethics
Computing Practices & Applications
Congestion and/or Flow Control
Content Distribution
Context-awareness and middleware
Creativity in Internet management and retailing
Cross-layer design and Physical layer based issue
Cryptography
Data Base Management
Data fusion
Data Mining
Data retrieval
Data Storage Management
Decision analysis methods
Decision making
Digital Economy and Digital Divide
Digital signal processing theory
Distributed Sensor Networks
Drives automation
Drug Design,
Drug Development
DSP implementation
E-Business
E-Commerce
E-Government
Electronic transceiver device for Retail Marketing Industries
Electronics Engineering,
Embeded Computer System
Emerging advances in business and its applications
Emerging signal processing areas
Enabling technologies for pervasive systems
Energy-efficient and green pervasive computing
Environmental Engineering,
Estimation and identification techniques
Evaluation techniques for middleware solutions
Event-based, publish/subscribe, and message-oriented middleware
Evolutionary computing and intelligent systems
Expert approaches
Facilities planning and management
Flexible manufacturing systems
Formal methods and tools for designing
Fuzzy algorithms
Fuzzy logics
GPS and location-based app
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
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.
CHN and Swap Heuristic to Solve the Maximum Independent Set ProblemIJECEIAES
We describe a new approach to solve the problem to find the maximum independent set in a given Graph, known also as Max-Stable set problem (MSSP). In this paper, we show how Max-Stable problem can be reformulated into a linear problem under quadratic constraints, and then we resolve the QP result by a hybrid approach based Continuous Hopfeild Neural Network (CHN) and Local Search. In a manner that the solution given by the CHN will be the starting point of the local search. The new approach showed a good performance than the original one which executes a suite of CHN runs, at each execution a new leaner constraint is added into the resolved model. To prove the efficiency of our approach, we present some computational experiments of solving random generated problem and typical MSSP instances of real life problem.
PROBABILISTIC INTERPRETATION OF COMPLEX FUZZY SETIJCSEIT Journal
The innovative concept of Complex Fuzzy Set is introduced. The objective of the this paper to investigate
the concept of Complex Fuzzy set in constraint to a traditional Fuzzy set , where the membership function
ranges from [0, 1], but in the Complex fuzzy set extended to a unit circle in a complex plane, where the
member ship function in the form of complex number. The Compressive study of mathematical operation of
Complex Fuzzy set is presented. The basic operation like addition, subtraction, multiplication and division
are described here. The Novel idea of this paper to measure the similarity between two fuzzy relations by
evaluating δ -equality. Here also we introduce the probabilistic interpretation of the complex fuzzy set
where we attempted to clarify the distinction between Fuzzy logic and probability.
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.
Computer Science
Active and Programmable Networks
Active safety systems
Ad Hoc & Sensor Network
Ad hoc networks for pervasive communications
Adaptive, autonomic and context-aware computing
Advance Computing technology and their application
Advanced Computing Architectures and New Programming Models
Advanced control and measurement
Aeronautical Engineering,
Agent-based middleware
Alert applications
Automotive, marine and aero-space control and all other control applications
Autonomic and self-managing middleware
Autonomous vehicle
Biochemistry
Bioinformatics
BioTechnology(Chemistry, Mathematics, Statistics, Geology)
Broadband and intelligent networks
Broadband wireless technologies
CAD/CAM/CAT/CIM
Call admission and flow/congestion control
Capacity planning and dimensioning
Changing Access to Patient Information
Channel capacity modelling and analysis
Civil Engineering,
Cloud Computing and Applications
Collaborative applications
Communication application
Communication architectures for pervasive computing
Communication systems
Computational intelligence
Computer and microprocessor-based control
Computer Architecture and Embedded Systems
Computer Business
Computer Sciences and Applications
Computer Vision
Computer-based information systems in health care
Computing Ethics
Computing Practices & Applications
Congestion and/or Flow Control
Content Distribution
Context-awareness and middleware
Creativity in Internet management and retailing
Cross-layer design and Physical layer based issue
Cryptography
Data Base Management
Data fusion
Data Mining
Data retrieval
Data Storage Management
Decision analysis methods
Decision making
Digital Economy and Digital Divide
Digital signal processing theory
Distributed Sensor Networks
Drives automation
Drug Design,
Drug Development
DSP implementation
E-Business
E-Commerce
E-Government
Electronic transceiver device for Retail Marketing Industries
Electronics Engineering,
Embeded Computer System
Emerging advances in business and its applications
Emerging signal processing areas
Enabling technologies for pervasive systems
Energy-efficient and green pervasive computing
Environmental Engineering,
Estimation and identification techniques
Evaluation techniques for middleware solutions
Event-based, publish/subscribe, and message-oriented middleware
Evolutionary computing and intelligent systems
Expert approaches
Facilities planning and management
Flexible manufacturing systems
Formal methods and tools for designing
Fuzzy algorithms
Fuzzy logics
GPS and location-based app
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
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.
INDUCTIVE LEARNING OF COMPLEX FUZZY RELATIONijcseit
The objective of this paper to investigate the notion of complex fuzzy set in general view. In constraint to a
traditional fuzzy set, the membership function of the complex fuzzy set, the range from [0.1] extended to a
unit circle in the complex plane. In this article the comprehensive mathematical operations on the complex
fuzzy set are presented. The basic operation of complex fuzzy set such as union, intersection, complement
of complex fuzzy set and complex fuzzy relation are studied. Also vector aggregation and fuzzy relation
over the complex fuzzy set are discussed. Two novel operations of complement and projection for complex
fuzzy relation are introduced.
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
Numerical Solution Of Delay Differential Equations Using The Adomian Decompos...theijes
Adomian Decomposition Method has been applied to obtain approximate solution to a wide class of ordinary and partial differential equation problems arising from Physics, Chemistry, Biology and Engineering. In this paper, a numerical solution of delay differential Equations (DDE) based on the Adomian Decomposition Method (ADM) is presented. The solutions obtained were without discretization nor linearization. Example problems were solved for demonstration. Keywords: Adomian Decomposition, Delay Differential Equations (DDE), Functional Equations , Method of Characteristic.
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.
Second or fourth-order finite difference operators, which one is most effective?Premier Publishers
This paper presents higher-order finite difference (FD) formulas for the spatial approximation of the time-dependent reaction-diffusion problems with a clear justification through examples, “why fourth-order FD formula is preferred to its second-order counterpart” that has been widely used in literature. As a consequence, methods for the solution of initial and boundary value PDEs, such as the method of lines (MOL), is of broad interest in science and engineering. This procedure begins with discretizing the spatial derivatives in the PDE with algebraic approximations. The key idea of MOL is to replace the spatial derivatives in the PDE with the algebraic approximations. Once this procedure is done, the spatial derivatives are no longer stated explicitly in terms of the spatial independent variables. In other words, only one independent variable is remaining, the resulting semi-discrete problem has now become a system of coupled ordinary differential equations (ODEs) in time. Thus, we can apply any integration algorithm for the initial value ODEs to compute an approximate numerical solution to the PDE. Analysis of the basic properties of these schemes such as the order of accuracy, convergence, consistency, stability and symmetry are well examined.
Handling missing data with expectation maximization algorithmLoc Nguyen
Expectation maximization (EM) algorithm is a powerful mathematical tool for estimating parameter of statistical models in case of incomplete data or hidden data. EM assumes that there is a relationship between hidden data and observed data, which can be a joint distribution or a mapping function. Therefore, this implies another implicit relationship between parameter estimation and data imputation. If missing data which contains missing values is considered as hidden data, it is very natural to handle missing data by EM algorithm. Handling missing data is not a new research but this report focuses on the theoretical base with detailed mathematical proofs for fulfilling missing values with EM. Besides, multinormal distribution and multinomial distribution are the two sample statistical models which are concerned to hold missing values.
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.
A DERIVATIVE FREE HIGH ORDERED HYBRID EQUATION SOLVERZac Darcy
Generally a range of equation solvers for estimating the solution of an equation contain the derivative of
first or higher order. Such solvers are difficult to apply in the instances of complicated functional
relationship. The equation solver proposed in this paper meant to solve many of the involved complicated
problems and establishing a process tending towards a higher ordered by alloying the already proved
conventional methods like Newton-Raphson method (N-R), Regula Falsi method (R-F) & Bisection method
(BIS). The present method is good to solve those nonlinear and transcendental equations that cannot be
solved by the basic algebra. Comparative analysis are also made with the other racing formulas of this
group and the result shows that present method is faster than all such methods of the class.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
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.
A Derivative Free High Ordered Hybrid Equation Solver Zac Darcy
Generally a range of equation solvers for estimating the solution of an equation contain the derivative of
first or higher order. Such solvers are difficult to apply in the instances of complicated functional
relationship. The equation solver proposed in this paper meant to solve many of the involved complicated
problems and establishing a process tending towards a higher ordered by alloying the already proved
conventional methods like Newton-Raphson method (N-R), Regula Falsi method (R-F) & Bisection method
(BIS). The present method is good to solve those nonlinear and transcendental equations that cannot be
solved by the basic algebra. Comparative analysis are also made with the other racing formulas of this
group and the result shows that present method is faster than all such methods of the class.
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
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.
INDUCTIVE LEARNING OF COMPLEX FUZZY RELATIONijcseit
The objective of this paper to investigate the notion of complex fuzzy set in general view. In constraint to a
traditional fuzzy set, the membership function of the complex fuzzy set, the range from [0.1] extended to a
unit circle in the complex plane. In this article the comprehensive mathematical operations on the complex
fuzzy set are presented. The basic operation of complex fuzzy set such as union, intersection, complement
of complex fuzzy set and complex fuzzy relation are studied. Also vector aggregation and fuzzy relation
over the complex fuzzy set are discussed. Two novel operations of complement and projection for complex
fuzzy relation are introduced.
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
Numerical Solution Of Delay Differential Equations Using The Adomian Decompos...theijes
Adomian Decomposition Method has been applied to obtain approximate solution to a wide class of ordinary and partial differential equation problems arising from Physics, Chemistry, Biology and Engineering. In this paper, a numerical solution of delay differential Equations (DDE) based on the Adomian Decomposition Method (ADM) is presented. The solutions obtained were without discretization nor linearization. Example problems were solved for demonstration. Keywords: Adomian Decomposition, Delay Differential Equations (DDE), Functional Equations , Method of Characteristic.
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.
Second or fourth-order finite difference operators, which one is most effective?Premier Publishers
This paper presents higher-order finite difference (FD) formulas for the spatial approximation of the time-dependent reaction-diffusion problems with a clear justification through examples, “why fourth-order FD formula is preferred to its second-order counterpart” that has been widely used in literature. As a consequence, methods for the solution of initial and boundary value PDEs, such as the method of lines (MOL), is of broad interest in science and engineering. This procedure begins with discretizing the spatial derivatives in the PDE with algebraic approximations. The key idea of MOL is to replace the spatial derivatives in the PDE with the algebraic approximations. Once this procedure is done, the spatial derivatives are no longer stated explicitly in terms of the spatial independent variables. In other words, only one independent variable is remaining, the resulting semi-discrete problem has now become a system of coupled ordinary differential equations (ODEs) in time. Thus, we can apply any integration algorithm for the initial value ODEs to compute an approximate numerical solution to the PDE. Analysis of the basic properties of these schemes such as the order of accuracy, convergence, consistency, stability and symmetry are well examined.
Handling missing data with expectation maximization algorithmLoc Nguyen
Expectation maximization (EM) algorithm is a powerful mathematical tool for estimating parameter of statistical models in case of incomplete data or hidden data. EM assumes that there is a relationship between hidden data and observed data, which can be a joint distribution or a mapping function. Therefore, this implies another implicit relationship between parameter estimation and data imputation. If missing data which contains missing values is considered as hidden data, it is very natural to handle missing data by EM algorithm. Handling missing data is not a new research but this report focuses on the theoretical base with detailed mathematical proofs for fulfilling missing values with EM. Besides, multinormal distribution and multinomial distribution are the two sample statistical models which are concerned to hold missing values.
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.
A DERIVATIVE FREE HIGH ORDERED HYBRID EQUATION SOLVERZac Darcy
Generally a range of equation solvers for estimating the solution of an equation contain the derivative of
first or higher order. Such solvers are difficult to apply in the instances of complicated functional
relationship. The equation solver proposed in this paper meant to solve many of the involved complicated
problems and establishing a process tending towards a higher ordered by alloying the already proved
conventional methods like Newton-Raphson method (N-R), Regula Falsi method (R-F) & Bisection method
(BIS). The present method is good to solve those nonlinear and transcendental equations that cannot be
solved by the basic algebra. Comparative analysis are also made with the other racing formulas of this
group and the result shows that present method is faster than all such methods of the class.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
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.
A Derivative Free High Ordered Hybrid Equation Solver Zac Darcy
Generally a range of equation solvers for estimating the solution of an equation contain the derivative of
first or higher order. Such solvers are difficult to apply in the instances of complicated functional
relationship. The equation solver proposed in this paper meant to solve many of the involved complicated
problems and establishing a process tending towards a higher ordered by alloying the already proved
conventional methods like Newton-Raphson method (N-R), Regula Falsi method (R-F) & Bisection method
(BIS). The present method is good to solve those nonlinear and transcendental equations that cannot be
solved by the basic algebra. Comparative analysis are also made with the other racing formulas of this
group and the result shows that present method is faster than all such methods of the class.
A derivative free high ordered hybrid equation solverZac Darcy
Generally a range of equation solvers for estimating the solution of an equation contain the derivative of
first or higher order. Such solvers are difficult to apply in the instances of complicated functional
relationship. The equation solver proposed in this paper meant to solve many of the involved complicated
problems and establishing a process tending towards a higher ordered by alloying the already proved
conventional methods like Newton-Raphson method (N-R), Regula Falsi method (R-F) & Bisection method
(BIS). The present method is good to solve those nonlinear and transcendental equations that cannot be
solved by the basic algebra. Comparative analysis are also made with the other racing formulas of this
group and the result shows that present method is faster than all such methods of the class.
On New Root Finding Algorithms for Solving Nonlinear Transcendental EquationsAI Publications
In this paper, we present new iterative algorithms to find a root of the given nonlinear transcendental equations. In the proposed algorithms, we use nonlinear Taylor’s polynomial interpolation and a modified error correction term with a fixed-point concept. We also investigated for possible extension of the higher order iterative algorithms in single variable to higher dimension. Several numerical examples are presented to illustrate the proposed algorithms.
The aim of this research is to find accurate solution for the Troesch’s problem by using high performance technique based on parallel processing implementation.
Design/methodology/approach – Feed forward neural network is designed to solve important type of differential equations that arises in many applied sciences and engineering applications. The suitable designed based on choosing suitable learning rate, transfer function, and training algorithm. The authors used back propagation with new implement of Levenberg - Marquardt training algorithm. Also, the authors depend new idea for choosing the weights. The effectiveness of the suggested design for the network is shown by using it for solving Troesch problem in many cases.
Findings – New idea for choosing the weights of the neural network, new implement of Levenberg - Marquardt training algorithm which assist to speeding the convergence and the implementation of the suggested design demonstrates the usefulness in finding exact solutions.
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.
A Tau Approach for Solving Fractional Diffusion Equations using Legendre-Cheb...iosrjce
In this paper, a modified numerical algorithm for solving the fractional diffusion equation is
proposed. Based on Tau idea where the shifted Legendre polynomials in time and the shifted Chebyshev
polynomials in space are utilized respectively.
The problem is reduced to the solution of a system of linear algebraic equations. From the computational point
of view, the solution obtained by this approach is tested and the efficiency of the proposed method is confirmed.
Sinc collocation linked with finite differences for Korteweg-de Vries Fraction...IJECEIAES
A novel numerical method is proposed for Korteweg-de Vries Fractional Equation. The fractional derivatives are described based on the Caputo sense. We construct the solution using different approach, that is based on using collocation techniques. The method combining a finite difference approach in the time-fractional direction, and the Sinc-Collocation in the space direction, where the derivatives are replaced by the necessary matrices, and a system of algebraic equations is obtained to approximate solution of the problem. The numerical results are shown to demonstrate the efficiency of the newly proposed method. Easy and economical implementation is the strength of this method.
Robust Image Denoising in RKHS via Orthogonal Matching PursuitPantelis Bouboulis
We present a robust method for the image denoising task based on kernel ridge regression and sparse modeling. Added noise is assumed to consist of two parts. One part is impulse noise assumed to be sparse (outliers), while the other part is bounded noise. The noisy image is divided into small regions of interest, whose pixels are regarded as points of a two-dimensional surface. A kernel based ridge regression method, whose parameters are selected adaptively, is employed to fit the data, whereas the outliers are detected via the use of the increasingly popular orthogonal matching pursuit (OMP) algorithm. To this end, a new variant of the OMP rationale is employed that has the additional advantage to automatically terminate, when all outliers have been selected.
Numerical solution of oscillatory reaction–diffusion system of $\lambda-omega$ type was done by using two finite difference schemes . The first one is the explicit scheme and the second one is the implicit Crank– Nicholson scheme with averaging of f(u,v) and without averaging of f(u,v). The comparison showed that Crank–Nicholson scheme is better than explicit scheme and Crank–Nicholson scheme with averaging of f(u,v) is more accurate but it needs more time and double storage and double time steps than Crank–Nicholson scheme without averaging of f(u,v).
Stability analysis of steady state solutions of Huxley equation using Fourier mode stability analysis in two cases is investigated. Firstly when the amplitude is constant and secondly when the amplitude is variable and the results were found to be: The solutions u1 = 0 and u1 = 1 are always stable while the solutions u1 = a and u1 = u1(X) are conditionally stable. In the second case, a comparison between the analytical solution and the numerical solution of Galerkin method is done and the results are the same.
Stability study of stationary solutions of the viscous Burgers equation using Fourier mode stability analysis for the stationary solutions u1 = D , where
D is constant and u1 =u1(x),0≤x≤1, in two cases is analyzed. Firstly when the wave amplitude A is constant and secondly when the wave amplitude A is variable. In the case of constant amplitude, the results found to be: The solution u1 = D is always stable while the solution u1 = u1 (x) is conditionally stable. In the case of variable amplitude, it has been found that the solutions u1 = D and u1 = u1 (x) , 0 ≤ x ≤ 1 are conditionally stable.
We study space–time finite element methods for semilinear parabolic problems in (1 + d)–dimensions for d = 2, 3. The discretisation in time is based on the discontinuous Galerkin timestepping method with implicit treatment of the linear terms and either implicit or explicit multistep discretisation of the zeroth order nonlinear reaction terms. Conforming finite element methods are used for the space discretisation. For this implicit-explicit IMEX–dG family of methods, we derive a posteriori and a priori energy-type error bounds and we perform extended numerical experiments. We derive a novel hp–version a posteriori error bounds in the L∞(L2) and L2(H1) norms assuming an only locally Lipschitz growth condition for the nonlinear reactions and no monotonicity of the nonlinear terms. The analysis builds upon the recent work in [60], for the respective linear problem, which is in turn based on combining the elliptic and dG reconstructions in [83, 84] and continuation argument. The a posteriori error bounds appear to be of optimal order and efficient in a series of numerical experiments.
Secondly, we prove a novel hp–version a priori error bounds for the fully–discrete IMEX–dG timestepping schemes in the same setting in L∞(L2) and L2(H1) norms. These error bounds are explicit with respect to both the temporal and spatial meshsizes kn and h, respectively, and, where possible, with respect to the possibly varying temporal polynomial degree r. The a priori error estimates are derived using the elliptic projection technique with an inf-sup argument in time. Standard tools such as Grönwall inequality and discrete stability estimates for fully discrete semilinear parabolic problems with merely locally-Lipschitz continuous nonlinear reaction terms are used. The a priori analysis extends the applicability of the results from [52] to this setting with low regularity. The results are tested by an extensive set of numerical experiments.
Optimal order a posteriori error bounds for semilinear parabolic equations are derived by using discontinuous Galerkin method in time and continuous (conforming) finite element in space. Our main tools in this analysis the reconstruction in time and elliptic reconstruction in space with Gronwall’s lemma and continuation argument.
Optimal order a posteriori error estimates for fully discrete semilinear parabolic problems in $푳_\infty (푳_ퟐ )$ − norm are presented. Standard conforming Galerkin (continuous) finite element method is employed in space discretisation and backward Euler method is used in time discretisation. The main tools in deriving these error estimates are the elliptic reconstruction technique, energy arguments, with Grönwall's lemma and continuation argument. These error bounds can be used in designing efficient adaptive finite element schemes which lead to significant savings in the computational costs such as implementation time, memory and data size.
Optimal order a posteriori error bounds in L∞(L2) norm are derived for semidiscrete semilinear parabolic problems. Standard continuous Galerkin (conforming) finite element method is employed. Our main tools in deriving these error estimates are the elliptic reconstruction technique which is first introduced by Makridakis and Nochetto [5], with the aid of Gronwall’s lemma and continuation argument.
More from Mohammad Sabawi Lecturer at Mathematics Department/College of Educations for Women/Tikrit University (7)
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Numerical Solution and Stability Analysis of Huxley Equation
1. Raf. J. of Comp. & Math’s. , Vol. 2, No. 1, 2005
85
Numerical Solution and Stability Analysis of Huxley Equation
Saad Manaa Mohammad Sabawi
College of Computers Sciences and Mathematics
Mosul University
Received on: 25/01/2005 Accepted on: 05/04/2005
ﺍﻟﻤﻠﺨﺹ
ﺘﻤﻌﺎﺩﻟﺔ ﺤل ﻡHuxleyﺍﻟﻤﻨﺘﻬﻴﺔ ﻗﺎﺕ ﺍﻟﻔﺭﻭ ﻁﺭﺍﺌﻕ ﻤﻥ ﻁﺭﻴﻘﺘﻴﻥ ﺒﺎﺴﺘﺨﺩﺍﻡ:ﻫﻲ ﺍﻷﻭﻟﻰ
ﻁﺭﻴﻘﺔ ﻫﻲ ﻭﺍﻟﺜﺎﻨﻴﺔ ﺍﻟﺼﺭﻴﺤﺔ ﺍﻟﻁﺭﻴﻘﺔCrank-Nicholsonﻜﻠﺘـﺎ ﻨﺘـﺎﺌﺞ ﺒﻴﻥ ﻤﻘﺎﺭﻨﺔ ﻋﻤل ﺘﻡ ﺇﺫ
ﺍﻟﺜﺎﻨﻴﺔ ﺍﻟﻁﺭﻴﻘﺔ ﻜﺎﻨﺕ ﺤﻴﻥ ﻓﻲ ﹰﺎﺘﻘﺎﺭﺒ ﻭﺍﻷﺴﺭﻉ ﺍﻷﺴﻬل ﻫﻲ ﺍﻷﻭﻟﻰ ﺍﻟﻁﺭﻴﻘﺔ ﺇﻥ ﺘﺒﻴﻥ ﻭﻗﺩ ﺍﻟﻁﺭﻴﻘﺘﻴﻥ
ﺍﻷﺩﻕ ﻫﻲ.ﻁﺭﻴﻘﺔ ﺒﺎﺴﺘﺨﺩﺍﻡ ﺍﻟﻁﺭﻴﻘﺘﻴﻥ ﻜﻠﺘﺎ ﺍﺴﺘﻘﺭﺍﺭﻴﺔ ﺩﺭﺍﺴﺔ ﻜﺫﻟﻙ ﺘﻤﺕ ﻭﻟﻘﺩFourier
( vonn Neumann)ـﺎﻥﻜـ ﺇﺫﺍ ـﺸﺭﻭﻁﻤـ ـﻭﻨﺤـ ـﻰﻋﻠـ ـﺴﺘﻘﺭﺓﻤـ ـﻰﺍﻷﻭﻟـ ـﺔﺍﻟﻁﺭﻴﻘـ ﺇﻥ ـﻴﻥﺘﺒـ ﺇﺫ
( )
( )2
2
4
2
,
4
2
xa
x
t
ta
r
∆−
∆
≤∆
∆−
≤
β
β
ﻏﻴﺭ ﻨﺤﻭ ﻋﻠﻰ ﻤﺴﺘﻘﺭﺓ ﺍﻟﺜﺎﻨﻴﺔ ﺍﻟﻁﺭﻴﻘﺔ ﻜﺎﻨﺕ ﺤﻴﻥ ﻓﻲ
ﻤﺸﺭﻭﻁ.
ABSTRACT
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, ( )
( )2
2
4
2
,
4
2
xa
x
t
ta
r
∆−
∆
≤∆
∆−
≤
β
β and the second
scheme is unconditionally stable.
2. SaadManaa &Mohammad Sabawi
86
1. Introduction
It is probably not an overstatement to say that almost all partial
differential equations (PDEs) that arise in a practical setting are solved
numerically on a computer. Since the development of high-speed computing
devices the numerical solution of PDEs has been in active state with the
invention of new algorithms and the examination of the underlying theory.
This is one of the most active areas in applied mathematics and it has
a great impact on science and engineering because of the ease and efficiency
it has shown in solving even the most complicated problems. The basic idea
of the method of finite differences is to cast the continuous problem
described by the PDE and auxiliary conditions into a discrete problem that
can be solved by a computer in finitely many steps. The discretization is
accomplished by restricting the problem to a set of discrete points. By
systematic procedure, we then calculate the unknown function at those
discrete points. Consequently, a finite difference technique yields a solution
only at discrete points in the domain of interest rather than, as we expect for
an analytical calculation, a formula or closed-form solution valid at all
points of the domain [11]. Manoranjan et al [12] obtained estimates for the
critical lengths of the domain at which bifurcation occurs in the cases
,2/10,,0 ≤<= aab and 1.
Manoranjan [13] studied in detail the solutions bifurcating from the
equilibrium state au = . Eilbeck and Manoranjan [3] considered different
types of basis functions for the pseudo-spectral method applied to the
nonlinear reaction-diffusion equation in 1- and 2- space dimensions. Eilbeck
[4] extended the pseudo-spectral method to follow steady state solutions as a
function of the problem parameter, using path-following techniques. Fath
and Domanski [6] studied the cellular differentiation in a developing
organism via a discrete bistable reaction-diffusion model and they used the
numerical simulation to support their expectations of the qualitative
behavior of the system. Lewis and Keener [10] studied the propagation
failure using the one –dimensional scalar bistable equation by a passive gap
and they used the numerical simulation in their study. Binczak et al [1]
compared the numerical predictions of the simple myelinated nerve fibers
with the theoretical results in the continuum and discrete limits. Broadbridge
et al [2] re-examined the derivation of the gene- transport equations and
used the Gaussian clump of alleles by the use of a numerical method-of-
lines by using PDETWO program. Lefantzi et al [9] presented their findings
for various orders of spatial discretizations as applied to SAMR (Structured
Adaptive Mesh Refinement) simulations.
3. Numerical Solution and Stability…
87
In this paper, the numerical solution of Huxley equation by using two
finite difference methods and stability analysis of these two methods are
analyzed.
2. The Mathematical Model
One of the famous non-linear reaction-diffusion equations is the
generalized Burgers-Huxley (gBH) equation:
( )( ) ( )
( )0,1aand,0,0,0
112
2
∈>≥≥
−−=
∂
∂
−
∂
∂
+
∂
∂
δβα
βα δδδ
auuu
x
u
x
u
u
t
u
If we take 0,0,1 ≠≠= βαδ and , equation (1) becomes the following
Burgers-Huxley (BH) equation:
( )( ) ( )212
2
auuu
x
u
x
u
u
t
u
−−=
∂
∂
−
∂
∂
+
∂
∂
βα
Equation (2) shows a prototype model for describing the interaction
mechanism, convection transport. When 1,0 == δβ and , equation (1) is
reduced to Burgers equation which describes the far field of wave
propagation in nonlinear dissipative systems
( )302
2
=
∂
∂
−
∂
∂
+
∂
∂
x
u
x
u
u
t
u
α
When 1,0 == δα and , equation (1) is reduced to the Huxley equation
which describes nerve pulse propagation in nerve fibers and wall motion in
liquid crystal
( )( ) ( )412
2
auuu
x
u
t
u
−−=
∂
∂
−
∂
∂
β
It is known that nonlinear diffusion equations (3) and (4) play
important roles in nonlinear physics. They are of special significance for
studying nonlinear phenomena [19]. Zeldovich and Frank- Kamenetsky
formulated the equation (4) in 1938 as a model for flame front propagation
and for this reason this equation sometimes named Zeldovich-Frank-
Kamenetsky (ZF) equation, which has been extensively studied as a simple
nerve model [1]. In 1952 Hodgkin and Huxley proposed their famous
Hodgkin-Huxley model for nerve propagation. Because of the mathematical
complexity of this model, it led to the introduction of the simpler Fitzhugh-
Nagumo system. The simplified model of the Fitzhugh-Nagumo system is
Huxley equation [18]. Because Huxley equation is a special case of
Fitzhugh-Nagumo system, it is sometimes named Fitzhugh-Nagumo (FN)
equation [5] or the reduced Nagumo equation or Nagumo equation [15]. In
sixties, Fitzhugh used equation (4) as an approximate equation for the
4. SaadManaa &Mohammad Sabawi
88
description of dynamics of the giant axon. This equation was among the first
models of excited media [8].
In this paper, we shall take the Huxley equation as a model problem [12]:
( )( )
[ ]
( ) ( ) ( )
( ) ( ) ( )6,,
50,0,0,
0,LL,-x
1
2
2
2
btLutLu
HbHxHbxu
t
auuu
x
u
t
u
==−
>≥+−=
≥∈
−−=
∂
∂
−
∂
∂
β
For the purpose of numerical calculations, we shall take:
( ) .30,10,10,1,1,0,1 ≤≤≤<≤≤=∈= tHandbLaβ
3. Derivation of the Explicit Scheme Formula of Huxley Equation
[14] is ( ){ }ctLxLtxR ≤≤≤≤−= 0,:, Assume that the rectangle
subdivided into ( )1−n by( )1−m rectangles with sides kthx =∆=∆ , .
Start at the bottom row, where ,01 == tt and the initial condition is [12]:
( ) ( ) ( ) .1,...,3,2,, 2
1 −=+−== niHxHbxftxu iii
A method for computing the approximations to ( )txu , at grid points in
successive rows will be developed
( ){ } ( )7,...,4,3,2,1,...,4,3,2:, mjnitxu ji =−=
The difference formulas used for ( )txut , and ( )txuxx , are:
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )9
,,2,
,
8
,,
,
2
2
hO
h
thxutxuthxu
txu
kO
k
txuktxu
txu
xx
t
+
−+−+
=
+
−+
=
Where the grid points are:
kttktthxxhxx jjjjiiii −=+=−=+= −+−+ 1111 ,,,
Neglecting the terms ( )kΟ and ( )2
hΟ , and use approximation jiu ,
for ( )ji txu , in equations (8) and (9), which are in turn substituted in
equation (4), we get
( )( ) ( )101
2u
,,,2
,1,,1,1ji,
auuu
h
uuu
k
u
jijiji
jijijiji
−−=
+−
−
− −++
β
From equation (10), we have
( ) ( )( ) ( )1112 ,,,,1,,1,1, auuukuuuruu jijijijijijijiji −−++−=− −++ β
Where 2
/ hkr =
5. Numerical Solution and Stability…
89
After some mathematical manipulation, we obtain
( ) ( ) ( ) ( ) ( )12121 ,
2
,,,1,11, jijijijijiji uaukuakruuru −++−−++= +−+ ββ
Equation (12) represents the explicit finite difference formula for
equation (4). Equation (12) is employed to create ( )thj 1+ row across the
grid, assuming that approximations in the jth row are known.
Notice that this formula explicitly gives the value 1, +jiu in terms of
jiji uu ,,1 ,− , and jiu ,1+ .
4. Stability Analysis of the Explicit Scheme Using Fourier
(von Neumann) Method
The basic idea of this method is to replace the solution of the finite
difference method mnu , at time t by ( ) xi
et γ
ψ , where 0,1 >−= γi [16].
To apply von Neumann method to equation (4), we resort to the linearized
stability analysis [7], we have
( )132
2
ua
x
u
t
u
β−
∂
∂
=
∂
∂
The finite difference explicit formula for (13) is:
( )
( )14
2
,2
,1,,1,1,
mn
mnmnmnmnmn
ua
x
uuu
t
uu
β−
∆
+−
=
∆
− −++
Substituting ( ) xi
mn etu γ
ψ=, in (14), we have
( ) ( ) ( ) ( ) ( )
( )
( )
( ) ( ) ( )
( )
[ ] ( )
( ) ( ) ( )[ ] ( )ttaeetrttt
etaeee
x
t
e
t
ttt
eta
x
etetet
t
etett
xixi
xixixixixi
xi
xxixixxixixi
ψβψψψ
βψ
ψψψ
βψ
ψψψψψ
γγ
γγγγγ
γ
γγγγγ
∆−−+=−∆+
⇒−−+
∆
=
∆
−∆+
⇒−
∆
+−
=
∆
−∆+
∆−∆
∆−∆
∆−∆+
2
2
2
2
2
)()(
Where 2
/ hkr =
6. SaadManaa &Mohammad Sabawi
90
( ) ( ) ( )[ ] ( )
( )[ ] ( )
( )[ ] ( )
( )[ ] ( )
( ) ( )[ ] ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( )[ ] ( )
( ) ( ) ( ) ξβγψψ
ψβγ
ψβγψψψ
ψβγψ
ψβγψ
ψβγψ
ψβγψ
ψβγψ
ψβγγγγψψψ
=∆−∆−=∆+
⇒∆−∆−=
∆−∆−=∆+
⇒∆−∆−=
∆−∆−−−=
∆−∆−−=
∆−−∆=
∆−−∆=
∆−−∆−∆+∆+∆=−∆+
taxrttt
ttaxr
ttaxtrttt
ttaxtr
ttaxtr
ttaxtr
ttaxtr
ttaxtr
ttaxixxixtrttt
2/sin41/
2/sin41
2/sin4
2/sin4
)2/sin21(12
cos12
1cos2
2cos2
2sincossincos
2
2
2
2
2
Where ξ can be visualized as the amplification factor and we get
( ) ( ) ( )15/ ξψψ =∆+ ttt
As we advance the solution from a particular plane ( )tψ to the next
plane ( )tt ∆+ψ , ( ) ( )ttt ψψ −∆+ must start decreasing or alternatively ( )tψ
must be bounded function, i.e. ( )tψ should not tend infinity for large t .
From equation (15), for boundedness of (15), we need
( ) ( )
( ) ( )1612/sin41
1
1/1
2
≤∆−∆−
⇒≤
⇒≤+
taxr
tt
βγ
ξ
ψψ
In the above inequality, the right-side inequality is:
( ) 12/sin41 2
≤∆−∆− taxr βγ
Implies 0>r and this is always true.
Hence, in order that (16) is to be satisfied, we need
( )
( )
( )
( )2/sin
42
1
2/sin42
2/sin42
2/sin411
2
2
2
2
xr
ta
taxr
taxr
taxr
∆≥
∆
−
⇒∆+∆≥
⇒∆−∆−≤−
⇒∆−∆−≤−
γ
β
βγ
βγ
βγ
For some β , ( )2/sin2
x∆γ is unity and hence the above condition
reduces to
7. Numerical Solution and Stability…
91
( )
( )
( )
( )
( )18
4
2
,(17),xt /r
17
4
2
2
2
2
haveweinequalityfromSince
xa
x
t
ta
r
∆+
∆
≤∆
∆∆=
∆−
≤
β
β
This precisely the conditions imposed on the explicit scheme to be
stable.
5. Derivation of the Crank-Nicholson Scheme Formula of Huxley
Equation
The is diffusion term xxu in this method is represented by central
differences, with their values at the current and previous time steps averaged
[17]:
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )20
,,
u
(19)
,,2,,,2,
2
1
t
2
k
txuktxu
h
thxutxuthxukthxuktxukthxu
uxx
−+
=
++−−+++++−+−
=
By using the approximation jiu , for ( )ji txu , in equations (19) and
(20), which are in turn substituted into equation (4), we have
( )( ) ( )211
2
2
2
2
,,,2
,1,,1
2
1,11,1,1,1,
auuu
h
uuu
h
uuu
k
uu
jijiji
jijijijijijijiji
−−=
+−
−
+−
−
− −++−++++
β
From (21), we get
( )
( ) ( )( )( )auuukuuur
uuuruu
jijijijijiji
jijijijiji
−−++−=
+−−−
−+
+−++++
,
2
,,,1,,1
1,11,1,1,1,
22
222
β
2
/ hkr = Where
After some mathematical manipulation, we get
( ) ( ) ( ) ( )
( ) ( ) ( )221,...,4,3,2,12
22222
,
2
,
,,1,11,1,11,1
−=−++
−−++=+++− +−++++−
niuauk
uakruururuur
jiji
jijijijijiji
β
β
Equation (22) represents the Crank-Nicholson formula for equation (4).
The terms on the right hand side of equation (22) are all known.
Hence, the equations in (22) form a tridiagonal linear algebraic system
BAX = .
The boundary conditions are used in the first and last equations only
i.e. .,, 1,,1,1,1 jbuuandbuu jnjnjj ∀==== ++
Equations in (22) are especially pleasing to view in their tridiagonal
matrix form BAX = , where A is the coefficient matrix, X is the unknown
vector and B is the known vector as shown below:
8. SaadManaa &Mohammad Sabawi
92
=
+−
−+−
−+−
−+−
−+
+
+
+
+
+
1j1,-n
1j2,-n
1,
1,3
1,2
u
u
:
:
22
22
22
22
22
jp
j
j
u
u
u
rr
rrr
rrr
rrr
rr
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( )
+−−+
−++−−++
−++−−++
−++−−++
+−++−−+
−−
−−−−−
+−
rbukarru
uaukukaruu
uaukukaruu
uaukukaruur
ruuaukukarrb
jnjn
jnjnjnjnjn
jpjpjpjpjp
jjjjj
jjjj
2222
12222r
:
12222r
:
12222
122222
,1,2
,2
2
,2,2,1,3
,
2
,,,1,1
,3
2
,3,3,4,2
,3,2
2
,2,2
β
ββ
ββ
ββ
ββ
When the Crank-Nicholson scheme is implemented with a computer,
the linear system BAX = can be solved by either direct means or by
iteration.In this paper, the Gaussian elimination method (direct method) has
been used to solve the algebraic system AX = B.
6. Stability Analysis of the Crank- Nicholson Scheme Using Fourier
(von Neumann) Method
The finite difference Crank-Nicholson formula for (13) is:
( ) ( )
( )23u-
2
2
2
2
mn,2
,1,,1
2
1,11,1,1,1,
βa
x
uuu
x
uuu
t
uu mnmnmnmnmnmnmnmn
∆
+−
+
∆
+−
=
∆
− −++−++++
Substituting ( ) xi
mn etu γ
ψ=, in equation (23), we have
10. SaadManaa &Mohammad Sabawi
94
Nicholson scheme which is more accurate than the explicit scheme and the
results of this paper are affirming the analytical results which obtained by
Manoranjan et al [12] as shown below:
.( ) ( )aLiftastxu −<∞→→ 1/0, π(1) If0=bthen
.( ) π<∞→→ Liftasatxu ,(2) Ifab =then
.( ) aLiftastxu /1, π<∞→→(3) If1=bthen
as shown in figure (1) and table (1). In addition, from stability analysis, we
concluded that the explicit scheme is conditionally stable if
( )
( )2
2
4
2
,
4
2
xa
x
t
ta
r
∆−
∆
≤∆
∆−
≤
β
β
while the Crank-Nicholson scheme is
unconditionally stable.
Figure (1)
Figure(1)Explains the solution of the Huxley equation by the use of
Crank-Nicholson scheme for various values of H at a=b=0.8.
The figure shows that the solution of the problem converges to the
steady state solution u = a = 0.8 as t gets large at specific boundary
condition b = 0.8.
11. Numerical Solution and Stability…
Table (1)
Explicit
b=0.25, a=0.25,
H=0.1
Crank-Nicholson
b=0.25, a=0.25,
H=0.1
Explicit
b=0.25,
a=0.25, H=0.3
Crank-Nicholson
b=0.25, a=0.25,
H=0.3
0.1000 0.1000 0.3000 0.3000
0.1287 0.1274 0.2910 0.2914
0.1573 0.1513 0.2817 0.2836
0.1771 0.1706 0.2751 0.2772
0.1933 0.1862 0.2696 0.2720
0.2056 0.1987 0.2654 0.2678
0.2152 0.2087 0.2621 0.2644
0.2228 0.2167 0.2595 0.2616
0.2287 0.2232 0.2575 0.2594
0.2333 0.2284 0.2559 0.2576
0.2369 0.2326 0.2546 0.2561
0.2397 0.2359 0.2536 0.2549
0.2420 0.2387 0.2528 0.2540
0.2437 0.2409 0.2522 0.2532
0.2451 0.2426 0.2517 0.2526
0.2461 0.2440 0.2514 0.2521
0.2470 0.2452 0.2511 0.2517
0.2476 0.2461 0.2508 0.2514
Table (1) shows the solution of Huxley equation by the use of
Crank-Nicholson scheme and explicit scheme for some values of a, b, and H.
The table above explains that the solution of the two schemes
converges to the steady state solution u = a = 0.25 and the number of steps
which are needed to reach the solution u = a = 0.25 in the explicit scheme is
less than the number of steps in the Crank-Nicholson scheme at specific
boundary condition b = 0.25 and H = 0.1, 0.3.
Acknowledgement
The authors would like to express their gratitude and indebtedness to
every person helped in this paper.
95
12. SaadManaa &Mohammad Sabawi
96
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