mesfin mekuria - Academia.edu (original) (raw)

Papers by mesfin mekuria

Research paper thumbnail of A tension spline fitted numerical scheme for singularly perturbed reaction-diffusion problem with negative shift

Research Square (Research Square), Jan 30, 2023

Objective: A numerical scheme is developed and analyzed for a singularly perturbed reaction-diffu... more Objective: A numerical scheme is developed and analyzed for a singularly perturbed reaction-diffusion problem with negative shift. The influence of the perturbation parameter exhibits boundary layers at the two ends of the domain, and the negative shift causes strong interior layer. The rapidly changing behavior of the solution in the layers brings significant difficulties in solving the problem analytically. The problem is treated by proposing a numerical scheme using the implicit Euler method in the temporal direction and the spline tension method in the spatial direction with uniform meshes. Result: Error estimate is investigated for the developed numerical scheme. The scheme is demonstrated by numerical examples. The theoretical and numerical results show that the method is uniformly convergent.

Research paper thumbnail of Fitted Tension Spline Method for Singularly Perturbed Time Delay Reaction Diffusion Problems

Mathematical Problems in Engineering

A uniformly convergent numerical method is presented for solving singularly perturbed time delay ... more A uniformly convergent numerical method is presented for solving singularly perturbed time delay reaction-diffusion problems. Properties of the continuous solution are discussed. The Crank–Nicolson method is used for discretizing the temporal derivative, and an exponentially fitted tension spline method is applied for the spatial derivative. Using the comparison principle and solution bound, the stability of the method is analyzed. The proposed numerical method is second-order uniformly convergent. The theoretical analysis is supported by numerical test examples for various values of perturbation parameters and mesh size.

Research paper thumbnail of Robust numerical method for singularly perturbed parabolic differential equations with negative shifts

Filomat, 2021

This paper deals with numerical treatment of singularly perturbed parabolic differential equation... more This paper deals with numerical treatment of singularly perturbed parabolic differential equations having delay on the zeroth and first order derivative terms. The solution of the considered problem exhibits boundary layer behaviour as the perturbation parameter tends to zero. The equation is solved using ?-method in temporal discretization and exponentially fitted finite difference method in spatial discretization. The stability of the scheme is proved by using solution bound technique by constructing barrier functions. The parameter uniform convergence analysis of the scheme is carried out and it is shown to be accurate of order O(N-2/N-1+c?+(?t)2) for the case ?= 1/2, where N is the number of mesh points in spatial discretization and ?t is the mesh size in temporal discretization. Numerical examples are considered for validating the theoretical analysis of the scheme.

Research paper thumbnail of An Exponentially Fitted Numerical Scheme via Domain Decomposition for Solving Singularly Perturbed Differential Equations with Large Negative Shift

Journal of Mathematics

In this study, we focus on the formulation and analysis of an exponentially fitted numerical sche... more In this study, we focus on the formulation and analysis of an exponentially fitted numerical scheme by decomposing the domain into subdomains to solve singularly perturbed differential equations with large negative shift. The solution of problem exhibits twin boundary layers due to the presence of the perturbation parameter and strong interior layer due to the large negative shift. The original domain is divided into six subdomains, such as two boundary layer regions, two interior (interfacing) layer regions, and two regular regions. Constructing an exponentially fitted numerical scheme on each boundary and interior layer subdomains and combining with the solutions on the regular subdomains, we obtain a second order ε -uniformly convergent numerical scheme. To demonstrate the theoretical results, numerical examples are provided and analyzed.

Research paper thumbnail of Accurate numerical scheme for singularly perturbed parabolic delay differential equation

BMC Research Notes, 2021

ObjectivesNumerical treatment of singularly perturbed parabolic delay differential equation is co... more ObjectivesNumerical treatment of singularly perturbed parabolic delay differential equation is considered. Solution of the equation exhibits a boundary layer, which makes it difficult for numerical computation. Accurate numerical scheme is proposed using$$\theta$$θ-method in time discretization and non-standard finite difference method in space discretization.ResultStability and uniform convergence of the proposed scheme is investigated. The scheme is uniformly convergent with linear order of convergence before Richardson extrapolation and second order convergent after Richardson extrapolation. Numerical examples are considered to validate the theoretical findings.

Research paper thumbnail of Fitted numerical scheme for singularly perturbed convection-diffusion reaction problems involving delays

Theoretical and Applied Mechanics, 2021

This paper deals with solution methods for singularly perturbed delay differential equations havi... more This paper deals with solution methods for singularly perturbed delay differential equations having delay on the convection and reaction terms. The considered problem exhibits an exponential boundary layer on the left or right side of the domain. The terms with the delay are treated using Taylor?s series approximation and the resulting singularly perturbed boundary value problem is solved using a specially designed exponentially finite difference method. The stability of the scheme is analysed and investigated using a comparison principle and solution bound. The formulated scheme converges uniformly with linear order of convergence. The theoretical findings are validated using three numerical test examples.

Research paper thumbnail of Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation

Advances in Mathematical Physics, 2021

This paper deals with numerical treatment of singularly perturbed parabolic differential equation... more This paper deals with numerical treatment of singularly perturbed parabolic differential equations having large time delay. The highest order derivative term in the equation is multiplied by a perturbation parameter ε , taking arbitrary value in the interval 0 , 1 . For small values of ε , solution of the problem exhibits an exponential boundary layer on the right side of the spatial domain. The properties and bounds of the solution and its derivatives are discussed. The considered singularly perturbed time delay problem is solved using the Crank-Nicolson method in temporal discretization and exponentially fitted operator finite difference method in spatial discretization. The stability of the scheme is investigated and analysed using comparison principle and solution bound. The uniform convergence of the scheme is discussed and proven. The formulated scheme converges uniformly with linear order of convergence. The theoretical analysis of the scheme is validated by considering numer...

Research paper thumbnail of Uniformly Convergent Hybrid Numerical Method for Singularly Perturbed Delay Convection-Diffusion Problems

International Journal of Differential Equations, 2021

This paper deals with numerical treatment of nonstationary singularly perturbed delay convection-... more This paper deals with numerical treatment of nonstationary singularly perturbed delay convection-diffusion problems. The solution of the considered problem exhibits boundary layer on the right side of the spatial domain. To approximate the term with the delay, Taylor’s series approximation is used. The resulting time-dependent singularly perturbed convection-diffusion problems are solved using Crank-Nicolson method for temporal discretization and hybrid method for spatial discretization. The hybrid method is designed using mid-point upwind in regular region with central finite difference in boundary layer region on piecewise uniform Shishkin mesh. Numerical examples are used to validate the theoretical findings and analysis of the proposed scheme. The present method gives accurate and nonoscillatory solutions in regular and boundary layer regions of the solution domain. The stability and the uniform convergence of the scheme are proved. The scheme converges uniformly with almost sec...

Research paper thumbnail of Uniformly Convergent Numerical Method for Singularly Perturbed Delay Parabolic Differential Equations Arising in Computational Neuroscience

Kragujevac Journal of Mathematics, 2022

The motive of this work is to develop ε-uniform numerical method for solving singularly perturbed... more The motive of this work is to develop ε-uniform numerical method for solving singularly perturbed parabolic delay differential equation with small delay. To approximate the term with the delay, Taylor series expansion is used. The resulting singularly perturbed parabolic differential equation is solved by using non-standard finite difference method in spatial direction and implicit Runge-Kutta method for the resulting system of IVPs in temporal direction. Theoretically the developed method is shown to be accurate of order O(N −1 + (∆t) 2 ) by preserving ε-uniform convergence. Two numerical examples are considered to investigate εuniform convergence of the proposed scheme and the result obtained agreed with the theoretical one

Research paper thumbnail of Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation

Advances in Mathematical Physics

This paper deals with numerical treatment of singularly perturbed parabolic differential equation... more This paper deals with numerical treatment of singularly perturbed parabolic differential equations having large time delay. The highest order derivative term in the equation is multiplied by a perturbation parameter ε , taking arbitrary value in the interval 0 , 1 . For small values of ε , solution of the problem exhibits an exponential boundary layer on the right side of the spatial domain. The properties and bounds of the solution and its derivatives are discussed. The considered singularly perturbed time delay problem is solved using the Crank-Nicolson method in temporal discretization and exponentially fitted operator finite difference method in spatial discretization. The stability of the scheme is investigated and analysed using comparison principle and solution bound. The uniform convergence of the scheme is discussed and proven. The formulated scheme converges uniformly with linear order of convergence. The theoretical analysis of the scheme is validated by considering numer...

Research paper thumbnail of Higher-Order Uniformly Convergent Numerical Scheme for Singularly Perturbed Differential Difference Equations with Mixed Small Shifts

International Journal of Differential Equations, 2020

This paper deals with numerical treatment of singularly perturbed differential difference equatio... more This paper deals with numerical treatment of singularly perturbed differential difference equations involving mixed small shifts on the reaction terms. The highest-order derivative term in the equation is multiplied by a small perturbation parameter taking arbitrary values in the interval . For small values of , the solution of the problem exhibits exponential boundary layer on the left or right side of the domain and the derivatives of the solution behave boundlessly large. The terms having the shifts are treated using Taylor’s series approximation. The resulting singularly perturbed boundary value problem is solved using exponentially fitted operator FDM. Uniform stability of the scheme is investigated and analysed using comparison principle and solution bound. The formulated scheme converges uniformly with linear order before Richardson extrapolation and quadratic order after Richardson extrapolation. The theoretical analysis of the scheme is validated using numerical test exampl...

Research paper thumbnail of Uniformly convergent numerical scheme for singularly perturbed parabolic delay differential equations

ITM Web of Conferences, 2020

This paper deals with numerical treatment of singularly perturbed parabolic differential difference... more This paper deals with numerical treatment of singularly perturbed parabolic differential difference equations having small shifts on the spatial variable. The considered problem contain small perturbation parameter (ε) multiplied on the diffusion term of the equation. For small values of ε the solution of the problem exhibits a boundary layer on the left or right side of the spatial domain depending on the sign of the convective term. The terms involving the shifts are approximated using Taylor’s series approximation. The resulting singularly perturbed parabolic partial differential equation is solved using implicit Euler method in the temporal discretization with exponentially fitted operator finite difference method in the spatial discretization. The uniform stability of the scheme investigated using comparison principle and discrete solution bound by constructing barrier function. Uniform convergence analysis has been carried out. The scheme gives second order convergence for the case ε...

Research paper thumbnail of Uniformly convergent fitted operator method for singularly perturbed delay differential equations

This paper deals with numerical treatment of singularly perturbed delay differential equations ha... more This paper deals with numerical treatment of singularly perturbed delay differential equations having delay on first derivative term. The solution of the considered problem exhibits boundary layer behaviour on left or right side of the domain depending on the sign of the convective term. The term with the delay is approximated using Taylor series approximation, resulting to asymptotically equivalent singularly perturbed boundary value problem. Uniformly convergent numerical scheme is developed using exponentially fitted finite difference method. The stability of the scheme is investigated using solution bound. The uniform convergence of the scheme is discussed and proved. Numerical examples are considered to validate the theoretical analysis.

Research paper thumbnail of A tension spline fitted numerical scheme for singularly perturbed reaction-diffusion problem with negative shift

Research Square (Research Square), Jan 30, 2023

Objective: A numerical scheme is developed and analyzed for a singularly perturbed reaction-diffu... more Objective: A numerical scheme is developed and analyzed for a singularly perturbed reaction-diffusion problem with negative shift. The influence of the perturbation parameter exhibits boundary layers at the two ends of the domain, and the negative shift causes strong interior layer. The rapidly changing behavior of the solution in the layers brings significant difficulties in solving the problem analytically. The problem is treated by proposing a numerical scheme using the implicit Euler method in the temporal direction and the spline tension method in the spatial direction with uniform meshes. Result: Error estimate is investigated for the developed numerical scheme. The scheme is demonstrated by numerical examples. The theoretical and numerical results show that the method is uniformly convergent.

Research paper thumbnail of Fitted Tension Spline Method for Singularly Perturbed Time Delay Reaction Diffusion Problems

Mathematical Problems in Engineering

A uniformly convergent numerical method is presented for solving singularly perturbed time delay ... more A uniformly convergent numerical method is presented for solving singularly perturbed time delay reaction-diffusion problems. Properties of the continuous solution are discussed. The Crank–Nicolson method is used for discretizing the temporal derivative, and an exponentially fitted tension spline method is applied for the spatial derivative. Using the comparison principle and solution bound, the stability of the method is analyzed. The proposed numerical method is second-order uniformly convergent. The theoretical analysis is supported by numerical test examples for various values of perturbation parameters and mesh size.

Research paper thumbnail of Robust numerical method for singularly perturbed parabolic differential equations with negative shifts

Filomat, 2021

This paper deals with numerical treatment of singularly perturbed parabolic differential equation... more This paper deals with numerical treatment of singularly perturbed parabolic differential equations having delay on the zeroth and first order derivative terms. The solution of the considered problem exhibits boundary layer behaviour as the perturbation parameter tends to zero. The equation is solved using ?-method in temporal discretization and exponentially fitted finite difference method in spatial discretization. The stability of the scheme is proved by using solution bound technique by constructing barrier functions. The parameter uniform convergence analysis of the scheme is carried out and it is shown to be accurate of order O(N-2/N-1+c?+(?t)2) for the case ?= 1/2, where N is the number of mesh points in spatial discretization and ?t is the mesh size in temporal discretization. Numerical examples are considered for validating the theoretical analysis of the scheme.

Research paper thumbnail of An Exponentially Fitted Numerical Scheme via Domain Decomposition for Solving Singularly Perturbed Differential Equations with Large Negative Shift

Journal of Mathematics

In this study, we focus on the formulation and analysis of an exponentially fitted numerical sche... more In this study, we focus on the formulation and analysis of an exponentially fitted numerical scheme by decomposing the domain into subdomains to solve singularly perturbed differential equations with large negative shift. The solution of problem exhibits twin boundary layers due to the presence of the perturbation parameter and strong interior layer due to the large negative shift. The original domain is divided into six subdomains, such as two boundary layer regions, two interior (interfacing) layer regions, and two regular regions. Constructing an exponentially fitted numerical scheme on each boundary and interior layer subdomains and combining with the solutions on the regular subdomains, we obtain a second order ε -uniformly convergent numerical scheme. To demonstrate the theoretical results, numerical examples are provided and analyzed.

Research paper thumbnail of Accurate numerical scheme for singularly perturbed parabolic delay differential equation

BMC Research Notes, 2021

ObjectivesNumerical treatment of singularly perturbed parabolic delay differential equation is co... more ObjectivesNumerical treatment of singularly perturbed parabolic delay differential equation is considered. Solution of the equation exhibits a boundary layer, which makes it difficult for numerical computation. Accurate numerical scheme is proposed using$$\theta$$θ-method in time discretization and non-standard finite difference method in space discretization.ResultStability and uniform convergence of the proposed scheme is investigated. The scheme is uniformly convergent with linear order of convergence before Richardson extrapolation and second order convergent after Richardson extrapolation. Numerical examples are considered to validate the theoretical findings.

Research paper thumbnail of Fitted numerical scheme for singularly perturbed convection-diffusion reaction problems involving delays

Theoretical and Applied Mechanics, 2021

This paper deals with solution methods for singularly perturbed delay differential equations havi... more This paper deals with solution methods for singularly perturbed delay differential equations having delay on the convection and reaction terms. The considered problem exhibits an exponential boundary layer on the left or right side of the domain. The terms with the delay are treated using Taylor?s series approximation and the resulting singularly perturbed boundary value problem is solved using a specially designed exponentially finite difference method. The stability of the scheme is analysed and investigated using a comparison principle and solution bound. The formulated scheme converges uniformly with linear order of convergence. The theoretical findings are validated using three numerical test examples.

Research paper thumbnail of Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation

Advances in Mathematical Physics, 2021

This paper deals with numerical treatment of singularly perturbed parabolic differential equation... more This paper deals with numerical treatment of singularly perturbed parabolic differential equations having large time delay. The highest order derivative term in the equation is multiplied by a perturbation parameter ε , taking arbitrary value in the interval 0 , 1 . For small values of ε , solution of the problem exhibits an exponential boundary layer on the right side of the spatial domain. The properties and bounds of the solution and its derivatives are discussed. The considered singularly perturbed time delay problem is solved using the Crank-Nicolson method in temporal discretization and exponentially fitted operator finite difference method in spatial discretization. The stability of the scheme is investigated and analysed using comparison principle and solution bound. The uniform convergence of the scheme is discussed and proven. The formulated scheme converges uniformly with linear order of convergence. The theoretical analysis of the scheme is validated by considering numer...

Research paper thumbnail of Uniformly Convergent Hybrid Numerical Method for Singularly Perturbed Delay Convection-Diffusion Problems

International Journal of Differential Equations, 2021

This paper deals with numerical treatment of nonstationary singularly perturbed delay convection-... more This paper deals with numerical treatment of nonstationary singularly perturbed delay convection-diffusion problems. The solution of the considered problem exhibits boundary layer on the right side of the spatial domain. To approximate the term with the delay, Taylor’s series approximation is used. The resulting time-dependent singularly perturbed convection-diffusion problems are solved using Crank-Nicolson method for temporal discretization and hybrid method for spatial discretization. The hybrid method is designed using mid-point upwind in regular region with central finite difference in boundary layer region on piecewise uniform Shishkin mesh. Numerical examples are used to validate the theoretical findings and analysis of the proposed scheme. The present method gives accurate and nonoscillatory solutions in regular and boundary layer regions of the solution domain. The stability and the uniform convergence of the scheme are proved. The scheme converges uniformly with almost sec...

Research paper thumbnail of Uniformly Convergent Numerical Method for Singularly Perturbed Delay Parabolic Differential Equations Arising in Computational Neuroscience

Kragujevac Journal of Mathematics, 2022

The motive of this work is to develop ε-uniform numerical method for solving singularly perturbed... more The motive of this work is to develop ε-uniform numerical method for solving singularly perturbed parabolic delay differential equation with small delay. To approximate the term with the delay, Taylor series expansion is used. The resulting singularly perturbed parabolic differential equation is solved by using non-standard finite difference method in spatial direction and implicit Runge-Kutta method for the resulting system of IVPs in temporal direction. Theoretically the developed method is shown to be accurate of order O(N −1 + (∆t) 2 ) by preserving ε-uniform convergence. Two numerical examples are considered to investigate εuniform convergence of the proposed scheme and the result obtained agreed with the theoretical one

Research paper thumbnail of Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation

Advances in Mathematical Physics

This paper deals with numerical treatment of singularly perturbed parabolic differential equation... more This paper deals with numerical treatment of singularly perturbed parabolic differential equations having large time delay. The highest order derivative term in the equation is multiplied by a perturbation parameter ε , taking arbitrary value in the interval 0 , 1 . For small values of ε , solution of the problem exhibits an exponential boundary layer on the right side of the spatial domain. The properties and bounds of the solution and its derivatives are discussed. The considered singularly perturbed time delay problem is solved using the Crank-Nicolson method in temporal discretization and exponentially fitted operator finite difference method in spatial discretization. The stability of the scheme is investigated and analysed using comparison principle and solution bound. The uniform convergence of the scheme is discussed and proven. The formulated scheme converges uniformly with linear order of convergence. The theoretical analysis of the scheme is validated by considering numer...

Research paper thumbnail of Higher-Order Uniformly Convergent Numerical Scheme for Singularly Perturbed Differential Difference Equations with Mixed Small Shifts

International Journal of Differential Equations, 2020

This paper deals with numerical treatment of singularly perturbed differential difference equatio... more This paper deals with numerical treatment of singularly perturbed differential difference equations involving mixed small shifts on the reaction terms. The highest-order derivative term in the equation is multiplied by a small perturbation parameter taking arbitrary values in the interval . For small values of , the solution of the problem exhibits exponential boundary layer on the left or right side of the domain and the derivatives of the solution behave boundlessly large. The terms having the shifts are treated using Taylor’s series approximation. The resulting singularly perturbed boundary value problem is solved using exponentially fitted operator FDM. Uniform stability of the scheme is investigated and analysed using comparison principle and solution bound. The formulated scheme converges uniformly with linear order before Richardson extrapolation and quadratic order after Richardson extrapolation. The theoretical analysis of the scheme is validated using numerical test exampl...

Research paper thumbnail of Uniformly convergent numerical scheme for singularly perturbed parabolic delay differential equations

ITM Web of Conferences, 2020

This paper deals with numerical treatment of singularly perturbed parabolic differential difference... more This paper deals with numerical treatment of singularly perturbed parabolic differential difference equations having small shifts on the spatial variable. The considered problem contain small perturbation parameter (ε) multiplied on the diffusion term of the equation. For small values of ε the solution of the problem exhibits a boundary layer on the left or right side of the spatial domain depending on the sign of the convective term. The terms involving the shifts are approximated using Taylor’s series approximation. The resulting singularly perturbed parabolic partial differential equation is solved using implicit Euler method in the temporal discretization with exponentially fitted operator finite difference method in the spatial discretization. The uniform stability of the scheme investigated using comparison principle and discrete solution bound by constructing barrier function. Uniform convergence analysis has been carried out. The scheme gives second order convergence for the case ε...

Research paper thumbnail of Uniformly convergent fitted operator method for singularly perturbed delay differential equations

This paper deals with numerical treatment of singularly perturbed delay differential equations ha... more This paper deals with numerical treatment of singularly perturbed delay differential equations having delay on first derivative term. The solution of the considered problem exhibits boundary layer behaviour on left or right side of the domain depending on the sign of the convective term. The term with the delay is approximated using Taylor series approximation, resulting to asymptotically equivalent singularly perturbed boundary value problem. Uniformly convergent numerical scheme is developed using exponentially fitted finite difference method. The stability of the scheme is investigated using solution bound. The uniform convergence of the scheme is discussed and proved. Numerical examples are considered to validate the theoretical analysis.