Uniformly convergent numerical method for singularly perturbed convection‐diffusion type problems with nonlocal boundary condition

Abstract

Summary In this article, we consider a class of singularly perturbed differential equations of convection‐diffusion type with nonlocal boundary conditions. A uniformly convergent numerical method is constructed via nonstandard finite difference and numerical integration methods to solve the problem. The nonlocal 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.

Robust Numerical Method for Singularly Perturbed Convection-Diffusion Type Problems With Non-Local Boundary Condition

“…The existence and uniqueness of the solutions of nonlocal boundary value problems have been studied by many authors [3,23]. Some approaches for the numerical solution of singularly perturbed nonlocal boundary value problems have been proposed in [6,7,11,13,14,15,16,18] and [26]. Uniformly convergent numerical methods of order second and high for solving different singularly perturbed problems have been studied in [5,8,9,10,12,13,14,15,16,27] and [36].…”

“…Some approaches for the numerical solution of singularly perturbed nonlocal boundary value problems have been proposed in [6,7,11,13,14,15,16,18] and [26]. Uniformly convergent numerical methods of order second and high for solving different singularly perturbed problems have been studied in [5,8,9,10,12,13,14,15,16,27] and [36]. To the best of our knowledge, the problem under consideration has not been done using nonstandard fitted finite difference method.…”

Parameter-Uniform Numerical Scheme for Singularly Perturbed Delay Parabolic Reaction Diffusion Equations with Integral Boundary Condition

“…ey also proposed the accelerated fitted finite difference method for singularly perturbed delay differential equations with nonlocal boundary condition in [25]. In [26], a uniformly convergent numerical method is constructed and a numerical integration method is used to treat the nonlocal boundary condition. However, up to the best of our knowledge, except the work in [27], not much work has been done to solve the problem under consideration.…”

Exponential Fitted Operator Method for Singularly Perturbed Convection-Diffusion Type Problems with Nonlocal Boundary Condition

“…To the best of our knowledge, the problem under consideration has not been done using the fitted operator method. Motivated by papers [35,36], we develop a uniformly convergent numerical method for solving singularly perturbed problem under consideration.…”