Exponentially fitted difference scheme for singularly perturbed mixed integro-differential equations

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Musa Cakir ◽  
Baransel Gunes
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Difference Scheme ◽  
Singularly Perturbed ◽  
Stability And Convergence ◽  
Uniform Mesh ◽  
Numerical Examples ◽  
Exponentially Fitted ◽  
The Stability ◽  
Exponentially Fitted Difference Scheme

Abstract In this study, singularly perturbed mixed integro-differential equations (SPMIDEs) are taken into account. First, the asymptotic behavior of the solution is investigated. Then, by using interpolating quadrature rules and an exponential basis function, the finite difference scheme is constructed on a uniform mesh. The stability and convergence of the proposed scheme are analyzed in the discrete maximum norm. Some numerical examples are solved, and numerical outcomes are obtained.

scholarly journals Exponentially Fitted Numerical Method for Singularly Perturbed Differential-Difference Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Habtamu Garoma Debela ◽  
Solomon Bati Kejela ◽  
Ayana Deressa Negassa
Keyword(s):  
Numerical Method ◽  
Difference Equations ◽  
Oscillatory Behavior ◽  
Singularly Perturbed ◽  
Stability And Convergence ◽  
Uniform Mesh ◽  
Small Shift ◽  
Exponentially Fitted ◽  
Maximum Absolute Errors ◽  
The Stability

This paper presents a numerical method to solve singularly perturbed differential-difference equations. The solution of this problem exhibits layer or oscillatory behavior depending on the sign of the sum of the coefficients in reaction terms. A fourth-order exponentially fitted numerical scheme on uniform mesh is developed. The stability and convergence of the proposed method have been established. The effect of delay parameter (small shift) on the boundary layer(s) has also been analyzed and depicted in graphs. The applicability of the proposed scheme is validated by implementing it on four model examples. Maximum absolute errors in comparison with the other numerical experiments are tabulated to illustrate the proposed method.

scholarly journals A Numerical Technique for Solving Nonlinear Singularly Perturbed Delay Differential Equations

2018 ◽  
Vol 23 (1) ◽  
pp. 64-78 ◽  
Author(s):  
A.S.V. Ravi Kanth ◽  
P. Murali Mohan Kumar
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Technique ◽  
Singularly Perturbed ◽  
Spline Method ◽  
Numerical Examples ◽  
Exponentially Fitted ◽  
Delay Differential ◽  
Quasilinearization Technique

This paper presents a numerical technique for solving nonlinear singu- larly perturbed delay differential equations. Quasilinearization technique is applied to convert the nonlinear singularly perturbed delay differential equation into a se- quence of linear singularly perturbed delay differential equations. An exponentially fitted spline method is presented for solving sequence of linear singularly perturbed delay differential equations. Error estimates of the method is discussed. Numerical examples are solved to show the applicability and efficiency of the proposed scheme.

scholarly journals Robust numerical method for singularly perturbed differential equations having both large and small delay

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Habtamu Garoma Debela
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Large Delay ◽  
Content Type ◽  
Small Delay ◽  
Numerical Experimentation ◽  
The Stability

PurposeThe purpose of this study is to develop stable, convergent and accurate numerical method for solving singularly perturbed differential equations having both small and large delay.Design/methodology/approachThis study introduces a fitted nonpolynomial spline method for singularly perturbed differential equations having both small and large delay. The numerical scheme is developed on uniform mesh using fitted operator in the given differential equation.FindingsThe stability of the developed numerical method is established and its uniform convergence is proved. To validate the applicability of the method, one model problem is considered for numerical experimentation for different values of the perturbation parameter and mesh points.Originality/valueIn this paper, the authors consider a new governing problem having both small delay on convection term and large delay. As far as the researchers' knowledge is considered numerical solution of singularly perturbed boundary value problem containing both small delay and large delay is first being considered.

A parameter-uniform finite difference scheme for singularly perturbed parabolic problem with two small parameters

2021 ◽  
Author(s):  
Tesfaye Aga Bullo ◽  
Guy Aymard Degla ◽  
Gemechis File Duressa
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Variable Parameter ◽  
Accurate Solution ◽  
Singularly Perturbed ◽  
Finite Difference Approximation ◽  
Parabolic Problems ◽  
Difference Approximation ◽  
Uniform Mesh

A parameter-uniform finite difference scheme is constructed and analyzed for solving singularly perturbed parabolic problems with two parameters. The solution involves boundary layers at both the left and right ends of the solution domain. A numerical algorithm is formulated based on uniform mesh finite difference approximation for time variable and appropriate piecewise uniform mesh for the spatial variable. Parameter-uniform error bounds are established for both theoretical and experimental results and observed that the scheme is second-order convergent. Furthermore, the present method produces a more accurate solution than some methods existing in the literature.   

scholarly journals Finite Difference Solution to a Nonlinear Time-Fractional Logistic Model

2021 ◽  
Vol 13 (2) ◽  
pp. 60
Author(s):  
Yuanyuan Yang ◽  
Gongsheng Li
Keyword(s):  
Finite Difference ◽  
Theoretical Analysis ◽  
Difference Scheme ◽  
Logistic Model ◽  
Stability And Convergence ◽  
Numerical Examples ◽  
Implicit Finite Difference Scheme ◽  
Finite Difference Solution ◽  
Implicit Finite Difference ◽  
Convergence Of The Scheme

We set forth a time-fractional logistic model and give an implicit finite difference scheme for solving of the model. The L^2 stability and convergence of the scheme are proved with the aids of discrete Gronwall inequality, and numerical examples are presented to support the theoretical analysis.

scholarly journals Some Second-Order σ Schemes Combined with an H1-Galerkin MFE Method for a Nonlinear Distributed-Order Sub-Diffusion Equation

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 187
Author(s):  
Yaxin Hou ◽  
Cao Wen ◽  
Hong Li ◽  
Yang Liu ◽  
Zhichao Fang ◽  
...  
Keyword(s):  
Finite Element ◽  
Diffusion Equation ◽  
Mixed Finite Element ◽  
Second Order ◽  
Stability And Convergence ◽  
Numerical Examples ◽  
The Stability ◽  
High Order Time ◽  
Distributed Order ◽  
Convergence Of The Scheme

In this article, some high-order time discrete schemes with an H 1 -Galerkin mixed finite element (MFE) method are studied to numerically solve a nonlinear distributed-order sub-diffusion model. Among the considered techniques, the interpolation approximation combined with second-order σ schemes in time is used to approximate the distributed order derivative. The stability and convergence of the scheme are discussed. Some numerical examples are provided to indicate the feasibility and efficiency of our schemes.

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