Numerical treatment for singularly perturbed nonlinear differential difference equations with negative shift

Abstract

This paper is devoted to the numerical study of the boundary value problems for singularly perturbed nonlinear differential difference equations with negative shift. The highest derivative of such types of boundary value problems is multiplied by a small parameter which is called the singular perturbation parameter. Due to the presence of the singular perturbation parameter the solution of such problems exhibits layer behavior and the classical numerical methods to approximate the solution do not converge uniformly with respect to the singular perturbation parameter. Thus, in general, to tackle the boundary value problems for singularly perturbed nonlinear differential difference equations with negative shift, one encounters three difficulties: (i) due to the presence of nonlinearity, (ii) due to the presence of terms containing shifts and (iii) due to the presence of singular perturbation parameter. To resolve the first difficulty, we use quasilinearization process to linearize the nonlinear differential equation. After applying the quasilinearization process to the nonlinear problem, a sequence of linearized problems is obtained. We show that the solution of the sequence of the linearized problems converge quadratically to the solution of the original nonlinear problem. To resolve the second difficulty, Taylor's series is used to tackle the terms containing shift provided the shifts is of small order of singular perturbation parameter and when shift is of capital order of singular perturbation, a special type of mesh is used. Finally, to resolve the third difficulty, we use a piecewise uniform mesh which is dense in the boundary layer region and coarse in the outer region and standard finite difference operators are used to approximate the derivatives. The difference scheme so obtained is shown to be parameter uniform by establishing the parameter-uniform error estimates. To demonstrate the efficiency of the method, some numerical experiments are carried out.