Fitted non-polynomial spline method for singularly perturbed differential difference equations with integral boundary condition

The aim of this paper is to present fitted non-polynomial spline method for singularly perturbed differential-difference equations with integral boundary condition. The stability and uniform convergence of the proposed method are proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter, ε and mesh size, h. The numerical results are tabulated in terms of maximum absolute errors and rate of convergence and it is observed that the present method is more accurate and uniformly convergent for h ≥ ε where the classical numerical methods fails to give good result and it also improves the results of the methods existing in the literature

Parameter-uniformly convergent numerical scheme for singularly perturbed delay parabolic differential equation via extended B-spline collocation

This paper presents a parameter-uniform numerical method to solve the time dependent singularly perturbed delay parabolic convection-diffusion problems. The solution to these problems displays a parabolic boundary layer if the perturbation parameter approaches zero. The retarded argument of the delay term made to coincide with a mesh point and the resulting singularly perturbed delay parabolic convection-diffusion problem is approximated using the implicit Euler method in temporal direction and extended cubic B-spline collocation in spatial orientation by introducing artificial viscosity both on uniform mesh. The proposed method is shown to be parameter uniform convergent, unconditionally stable, and linear order of accuracy. Furthermore, the obtained numerical results agreed with the theoretical results

Boundary Layer Resolving Exact Difference Scheme for Solving Singularly Perturbed Convection-Diffusion-Reaction Equation

This paper considers the numerical treatment of singularly perturbed time-dependent convection-diffusion-reaction equation. The diffusion term of the equation is multiplied by a small perturbation parameter (ε), which takes an arbitrary value in the interval (0, 1]. For small values of ε, the solution of the equation exhibits an exponential boundary layer which makes it difficult to solve it analytically or using classical numerical methods. We proposed numerical schemes using the Crank–Nicolson method in time derivative discretization and the nonstandard finite difference method (exact finite difference method) in space derivative discretization on a uniform and piecewise uniform Shishkin mesh. The existence of unique discrete solutions and the stability of the schemes are discussed and proved. Uniform convergence of the schemes is proved. The formulated schemes converge uniformly with linear order of convergence. The method on Shishkin mesh possesses boundary layer resolving property. We validated the methods by considering two numerical examples for different values of ε and mesh length

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

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 ε taking arbitrary values in the interval 0,1. For small values of ε, 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 examples for different values of ε and mesh number N

Accelerated Exponentially Fitted Operator Method for Singularly Perturbed Problems with Integral Boundary Condition

In this paper, we consider a class of singularly perturbed differential equations of convection diffusion type with integral boundary condition. An accelerated uniformly convergent numerical method is constructed via exponentially fitted operator method using Richardson extrapolation techniques and numerical integration methods to solve the problem. The integral boundary condition is treated using numerical integration techniques. Maximum absolute errors and rates of convergence for different values of perturbation parameter and mesh sizes are tabulated for the numerical example considered. The method is shown to be ε-uniformly convergent

A Numerical Approach for Diffusion-Dominant Two-Parameter Singularly Perturbed Delay Parabolic Differential Equations

A numerical scheme is developed to solve a large time delay two-parameter singularly perturbed one-dimensional parabolic problem in a rectangular domain. Two small parameters multiply the convective and diffusive terms, which determine the width of the left and right lateral surface boundary layers. Uniform mesh and piece-wise uniform Shishkin mesh discretization are applied in time and spatial dimensions, respectively. The numerical scheme is formulated by using the Crank–Nicolson method on two consecutive time steps and the average central finite difference approximates in spatial derivatives. It is proved that the method is uniformly convergent, independent of the perturbation parameters, where the convection term is dominated by the diffusion term. The resulting scheme is almost second-order convergent in the spatial direction and second-order convergent in the temporal direction. Numerical experiments illustrate theoretical findings, and the method provides more accurate numerical solutions than prior literature

A Uniformly Convergent Collocation Method for Singularly Perturbed Delay Parabolic Reaction-Diffusion Problem

In this article, a numerical solution is proposed for singularly perturbed delay parabolic reaction-diffusion problem with mixed-type boundary conditions. The problem is discretized by the implicit Euler method on uniform mesh in time and extended cubic B-spline collocation method on a Shishkin mesh in space. The parameter-uniform convergence of the method is given, and it is shown to be ε-uniformly convergent of OΔt+N−2ln2N, where Δt and N denote the step size in time and number of mesh intervals in space, respectively. The proposed method gives accurate results by choosing suitable value of the free parameter λ. Some numerical results are carried out to support the theory

Uniformly convergent numerical scheme for singularly perturbed parabolic delay differential equations

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 > N−1 and first order convergence for the case ε « N−1, where N is number of mesh interval. Test examples and numerical results are considered for validating the theoretical analysis of the scheme

Graded mesh B-spline collocation method for two parameters singularly perturbed boundary value problems

The solutions of two parameters singularly perturbed boundary value problems typically exhibit two boundary layers. Because of the presence of these layers standard numerical methods fail to give accurate approximations. This paper introduces a numerical treatment of a class of two parameters singularly perturbed boundary value problems whose solution exhibits boundary layer phenomena. A graded mesh is considered to resolve the boundary layers and collocation method with cubic B-splines on the graded mesh is proposed and analyzed. The proposed method leads to a tri-diagonal linear system of equations. The stability and parameters uniform convergence of the present method are examined. To verify the theoretical estimates and efficiency of the method several known test problems in the literature are considered. Comparisons to some existing results are made to show the better efficiency of the proposed method. Summing up: • The present method is found to be stable and parameters uniform convergent and the numerical results support the theoretical findings. • Experimental results show that the present method approximates the solution very well and has a rate of convergence of order two in the maximum norm. • Experimental results show that cubic B-spline collocation method on graded mesh is more efficient than cubic B-spline collocation method on Shishkin mesh and some other existing methods in the literature