Finite difference scheme for singularly perturbed reaction diffusion problem of partial delay differential equation with nonlocal boundary condition

Abstract

This paper investigates singularly perturbed parabolic partial differential equations with delay in space, and the right end plane is an integral boundary condition on a rectangular domain. A small parameter is multiplied in the higher order derivative, which gives boundary layers, and due to the delay term, one more layer occurs on the rectangle domain. A numerical method comprising the standard finite difference scheme on a rectangular piecewise uniform mesh (Shishkin mesh) of $N_{r} \times N_{t}$ N r × N t elements condensing in the boundary layers is suggested, and it is proved to be parameter-uniform. Also, the order of convergence is proved to be almost two in space variable and almost one in time variable. Numerical examples are proposed to validate the theory.

Parameter uniform numerical methods for singularly perturbed delay parabolic differential equations with non-local boundary condition

“…The numerical results are presented for the value of the perturbation parameter { } (for both problems) are stable, and uniformly convergent. From Tables 2 and 4, the performance of the proposed scheme is investigated by comparing with recently published paper in Elango et al (2021). As one sees, the proposed method gives more accurate results.…”

“…(2014), and Kudu, and Amiraliyev, (2015)] and the references therein). However, up to the best of our knowledge, except the work in [Elango et al, (2021)] not much work has been done to solve the problem under consideration. Hence, in this paper, motivated by the works of [Elango et al, (2021)], we construct and analyze a parameter uniform numerical method.…”

“…However, up to the best of our knowledge, except the work in [Elango et al, (2021)] not much work has been done to solve the problem under consideration. Hence, in this paper, motivated by the works of [Elango et al, (2021)], we construct and analyze a parameter uniform numerical method. The proposed method uses fitted operator finite difference method based on the procedures of method of line, which consists of non-standard finite difference operator for the spatial discretization and classical backward Euler method for the time discretization, which treat the problem and first being considered.…”

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

“…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.…”

“…In the present paper, motivated by the studies of [27], we construct and analyze an accurate and uniformly convergent numerical scheme for solving singularly perturbed delay parabolic reaction diffusion equations with integral boundary condition.…”