A. A. Salama

Fourth-order finite difference method for solving Burgers’ equation

Abstract In this paper, we present fourth-order finite difference method for solving nonlinear one-dimensional Burgers’ equation. This method is unconditionally stable. The convergence analysis of the present method is studied and an upper bound for the error is derived. Numerical comparisons are made with most of the existing numerical methods for solving this equation.

FOURTH-ORDER FINITE-DIFFERENCE METHOD FOR THIRD-ORDER BOUNDARY-VALUE PROBLEMS

ABSTRACT In this article, we introduce a finite-difference method to solve linear and nonlinear third-order boundary-value problems. We use only four grid points in this method of solution. This method is convergent to fourth-order accuracy. In addition, we show that this method is unconditionally stable. The Falkner-Skan equation and the Blasius equation are considered as special cases of nonlinear problems. Numerical examples are given to illustrate the method and its convergence.

Numerical methods based on extended one-step methods for solving optimal control problems

In this article, we develop extended one-step methods for solving the dynamic system of optimal control problems governed by ordinary differential equations. The control variables are approximated by polynomial functions. The proposed problem is reduced to either a constrained or unconstrained minimization problem according to the nature of the dynamic system and the given conditions. Numerical results and comparisons with other methods are presented.

A class of exponential methods for stiff initial-value problems

In this paper, we derive a class of exponential methods for solving the stiff initial-value problems. The present methods are unconditionally stable and satisfy a discrete maximum principle. These include an even order of accuracy when the perturbation parameter, e, is fixed and have the property that if e is of order h they reduce to first order accuracy. Also, these methods are optimal when e-0. Finally, good results and comparison with the uniform second-order scheme are considered.

Interval Schemes for Singularly Perturbed Initial Value Problems

The modified exponential interval schemes are introduced for the solution of singularly perturbed initial value problems. We give the outline of constructing the schemes of k-th order, then we construct four schemes for k = 1 and k = 2. These schemes are uniformly convergent of second and third order accuracy. Also, we introduce the idea of optimal convergence. Numerical results and comparisons with other schemes are presented.

An exponential method for a singular perturbation problem

A fourth-order tridiagonal method of exponential type for singular perturbation problem is derived. This method is unconditionally stable and have the property that if ∊ is of order h it reduces to second-order accuracy. Numerical results and comparison to other methods are considered.

A higher order uniformly convergent method for singularly perturbed delay parabolic partial differential equations

ABSTRACT In this article, we aim to introduce a high-order uniformly convergent method to solve singularly perturbed delay parabolic convection diffusion problems exhibiting a regular boundary layer. The domain is discretized by a uniform mesh in the time direction and a piecewise-uniform Shishkin mesh for the spatial direction. We use the Crank–Nicolson method for the time derivative and we develop a fourth-order compact difference method to solve the set of ordinary differential equations at each time level. The stability analysis and the truncation error are discussed. Parameter-uniform error estimates are derived and it is shown that the method is -uniformly convergent of second-order accurate in time, and in the spatial direction it is of second-order outside region of boundary layer, and of almost fourth-order inside the layer region. Numerical examples are presented to verify the theoretical results and to confirm the efficiency and high accuracy of the proposed method.

Optimal extended one-step schemes of exponential type for stiff initial-value problems

Extended one-step schemes of exponential type are introduced for the numerical solution of stiff initial-value problems. These schemes are uniformly convergent of third and fourth orders of accuracy. In addition, we show that these schemes are optimal when ϵ → 0. Numerical results and comparisons with other schemes are presented.

Modified oci scheme for two-point boundary-value problems

A sixth-order compact implicit tridiagonal scheme for solving a two-point boundary-value problem (TPBVP) is presented. This scheme is unconditionally stable. Numerical example and comparisons with other schemes are given.