Exponentially fitted finite difference method for singularly perturbed delay differential equations with integral boundary condition

Abstract

In this paper, exponentially fitted finite difference method for solving singularly perturbed delay differential equation with integral boundary condition is considered. To treat the integral boundary condition, Simpson's rule is applied. The stability and parameter 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,  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   where the classical numerical methods fails to give good result and it also improves the results of the methods existing in the literature.

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.…”

Exponentially fitted numerical method for solving singularly perturbed delay reaction-diffusion problem with nonlocal boundary condition

“…The authors in [22] presented a numerical method depends on a FDM on Shishkin mesh to solve the third-order SPDDEs of reaction-diffusion kind with IBC. The authors in [23] used an exponentially fitted numerical scheme to solve SPDDEs of convection-diffusion kind with nonlocal boundary conditions. Debela and Duressa [24] improved the order of accuracy for the method proposed in [23].…”

“…The authors in [23] used an exponentially fitted numerical scheme to solve SPDDEs of convection-diffusion kind with nonlocal boundary conditions. Debela and Duressa [24] improved the order of accuracy for the method proposed in [23]. Kumar and Kumari [25] developed the method based on the idea of Bspline functions and efficient numerical method on a piecewise-uniform mesh was recommended to approximate the solutions of SPDDEs with IBC.…”

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

“…e method has almost firstorder convergence with respect to ε. Debela and Duressa [24] proposed first-order numerical methods using the exponentially fitted finite difference method on a uniform mesh. ey also proposed the accelerated fitted finite difference method for singularly perturbed delay differential equations with nonlocal boundary condition in [25].…”