A New Parameter-Uniform Discretization of Semilinear Singularly Perturbed Problems

Abstract

In this paper, we present a numerical approach to solving singularly perturbed semilinear convection-diffusion problems. The nonlinear part of the problem is linearized via the quasilinearization technique. We then design and implement a fitted operator finite difference method to solve the sequence of linear singularly perturbed problems that emerges from the quasilinearization process. We carry out a rigorous analysis to attest to the convergence of the proposed procedure and notice that the method is first-order uniformly convergent. Some numerical evaluations are implemented on model examples to confirm the proposed theoretical results and to show the efficiency of the method.

A Robust Numerical Method for a Singularly Perturbed Semilinear Problem with Integral Boundary Conditions

“…Kumar et al [20] considered highorder convergent methods for singularly perturbed quasilinear problems with integral boundary conditions. Munyakazi and Kehinde [21] suggest a new parameter-uniform discretization of semilinear singularly perturbed problems. The concepts of existence and uniqueness can be examined to find solutions for such problems [22,23].…”

Fourth-order fitted mesh scheme for semilinear singularly perturbed reaction–diffusion problems

“…This type of problem shares the basic behavior of singularly perturbed reaction–diffusion boundary value problem. Authors in [ 23 ] presented a numerical approach to solving singularly perturbed semilinear convection–diffusion problems. The nonlinear part of the problem is linearized via the quasilinearization technique.…”

Editorial for the Special Issue “Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms”

“…In the paper [22], the authors present a numerical approach to solving singularly perturbed semilinear convection-diffusion problems. The nonlinear part of the problem is linearized via the quasilinearization technique.…”