Search

In this paper, we study the numerical method for a class of nonlinear singularly perturbed delay differential equations using parametric cubic spline. Quasilinearization process is applied to convert the nonlinear singularly perturbed delay differential equations into a sequence of linear singularly perturbed delay differential equations. When the delay is not sufficiently smaller order of the singular perturbation parameter, the approach of expanding the delay term in Taylor’s series may lead to bad approximation. To handle the delay term, we construct a special type of mesh in such a way that the term containing delay lies on nodal points after discretization. The parametric cubic spline is presented for solving sequence of linear singularly perturbed delay differential equations. The error analysis of the method is presented and shows second-order convergence. The effect of delay parameter on the boundary layer behavior of the solution is discussed with two test examples.

In this paper, we consider two-parameter singularly perturbed boundary value problems via fitted splines. We propose an exponentially fitting factor to the highest order derivative of the problem and discretize by parametric cubic spline functions on a uniform mesh. An error analysis of the method is presented. The efficiency of the method is demonstrated by test examples which support the theoretical results.

In this paper, non-polynomial spline method for solving one dimensional nonlinear Benjamin-Bona-Mahony-Burgers equation is presented. Stability analysis of the present method is analyzed by means of Von Neumann process and the method is proven to be unconditionally stable. Truncation error of the proposed method is also discussed. Few numerical evidences are given to prove the validation of the proposed method.

AbstractThis paper applies differential transform – Pade technique for treating non-linear singular boundary value problems arising in various physical problems of science and engineering. Comparisons are made between the results of the proposed method, and the exact solutions. The results show that the proposed method is an attractive method for solving non-linear singular boundary value problems.

This article contributes a numerical technique for a class of singularly perturbed time delayed parabolic partial differential equation. A priori results of maximum principle, stability and bounds are discussed. The continuous problem is semi-discretized by the Crank–Nicolson based scheme in the temporal direction and then discretized by the tension spline scheme on non-uniform Shishkin mesh. Error estimation for the discretized problem is derived. To validate the theoretical findings, the numerical outcomes for linear and nonlinear problems are tested.