This paper presents numerical treatments for a class of singularly perturbed parabolic partial differential equations with nonlocal boundary conditions. The problem has strong boundary layers at x = 0 and x = 1. The nonstandard finite difference method was developed to solve the considered problem in the spatial direction, and the implicit Euler method was proposed to solve the resulting system of IVPs in the temporal direction. The nonlocal boundary condition is approximated by Simpsons 13 rule. The stability and uniform convergence analysis of the scheme are studied. The developed scheme is second-order uniformly convergent in the spatial direction and first-order in the temporal direction. Two test examples are carried out to validate the applicability of the developed numerical scheme. The obtained numerical results reflect the theoretical estimate.

Objectives In this article, a singularly perturbed delay reaction-diffusion problem with nonlocal boundary conditions is considered. The exponential fitting factor is introduced to treat the solutions inside the boundary layer which occur due to perturbation parameter. The considered problem has interior layer at $$s = 1$$ s = 1 and strong boundary layers at $$s = 0$$ s = 0 and $$s= 2$$ s = 2 . We proposed an exponentially fitted finite difference method to solve the considered problem. The nonlocal boundary condition is treated using Composite Simpson’s $$\frac{1}{3}$$ 1 3 rule. Result The stability and uniform convergence analysis of the proposed approach are established. The error estimation of the developed method is shown to be second-order uniform convergent. Two test examples were carried out to validate the applicability of the developed numerical method. The numerical results reflect the theoretical estimations.

Objectives: In this article, singularly perturbed delay reaction-diffusion problem with nonlocal boundary condition is considered. The exponential fitting factor is introduced to treat the solutions inside the boundary layer which occur due to perturbation parameter. The considered problem has interior layer at x = 1 and strong boundary layers at x = 0 and x = 2. We proposed an exponentially fitted finite difference method to solve the considered problem. The nonlocal boundary condition is treated using Composite Simpson’s 1/3 rule. Result: The stability and uniform convergence analysis of the proposed approach are established. The error estimation of the developed method is shown to be second-order uniform convergent. Two test examples are carried out to validate the applicability of the developed numerical method. The numerical results reflect the theoretical estimations. AMS Subject Classification: Primary 65L11, 65L12, 65L20, 65L70.

In this paper, we study the numerical method for solving singularly perturbed partial delay differential equations with integral boundary conditions. Due to a small perturbation parameter acted on the higher derivative, solution of the problem exhibit a boundary layers at left and right end plane of the domain. An interior layer is also formed because of the presence of large delay on the space variable. A hybrid numerical method is proposed in the spatial direction, and an implicit Euler method is used in temporal direction. The proposed hybrid scheme constitute of cubic spline method in the boundary layer region and a classical finite difference method in the outer layer region. The integral boundary condition is treated using Simpson’s 1/3 rule. The uniform stability and convergence analysis for the proposed scheme is studied. The developed method is uniformly convergent with second-order in space and first-order in time. Two numerical test examples are considered to validate a theoretical result. The numerical results are in agreement with the theoretical estimations. MSC Classification: 65M06 , 65M12 , 65M22 , 65M25