Search

In this paper, we introduce a new scheme based on rational spline function and nonstandard finite difference technique to solve the time-fractional Swift–Hohenberg equation in the sense of Riemann–Liouville derivative. Via Fourier method, the method is convergent and unconditionally stable. Also, we investigated the existence and uniqueness of the proposed method. Numerical results are demonstrated to validate the applicability and the theoretical results.

We consider singularly perturbed problems of convection-diffusion-reaction type which involve two small parameters. A new discrete cubic spline method is developed for the solution of this problem on a Shishkin mesh. A convergence analysis is given and the method is shown to be almost second-order uniformly convergent with respect to the perturbation parameters 𝜀d and 𝜀c. Numerical results are presented to validate the theoretical results as well as the robustness of the method.

A new discrete spline method is developed to solve the two point fractional Bagley-Torvik equation.The method is based on discrete spline function and a nonstandard Grunwald-Letnikov difference (NSGD) and the weighted and shifted Grunwald-Letnikov difference (WSGD) operators to approximate the fractional derivative. Bounds for Grunwald-Letnikov weights are considered. Convergence analysis is discussed and a class of second order and third order methods are obtained. Illustrative examples are presented to validate the practical usefulness of the methods.

In this paper, we introduce a new technique based on cubic exponential spline functions for computing approximations to the solution of a system of fourth-order boundary value problems associated with obstacle, unilateral and contact problems. It is shown that the present method is of order two and four and gives approximations which are better than those produced by other collocation, finite difference and spline methods. Numerical evidence is presented to illustrate the applicability of the new methods.

The affiliation of the first author W. K. Zahra is Department Engineering Physics and Mathematics, Faculty of Engineering, Tanta University., Tanta, 31521, Egypt.

Sixth order accurate method based on quintic nonpolynomial spline function for the numerical solution of 2μth order two point BVPs (where μ is a positive integer) is presented by transforming the problem into a system of μ second order two point BVPs. The nonpolynomial spline function is used to drive some consistency relations for computing approximation to the solution of this problem. The proposed approach gives better approximations than existing polynomial spline and finite difference methods and has a lower computational cost. Convergence analysis of the proposed method is discussed. Numerical examples are included to illustrate the practical usefulness of our method.

In this paper, quadratic and cubic polynomial and nonpolynomial spline functions based methods are presented to find approximate solutions to second order boundary value problems. Using these spline functions we drive a few consistency relations which to be used for computing approximations to the solution for second order boundary value problems. The present approaches have less computational cost. Convergence analysis of these methods is discussed. Two numerical examples are included to illustrate the practical usefulness of the proposed methods.