Solving the Telegraph and Oscillatory Differential Equations by a Block Hybrid Trigonometrically Fitted Algorithm

F F Ngwane,S N Jator

doi:10.1155/2015/347864

Abstract

We propose a block hybrid trigonometrically fitted (BHT) method, whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including systems arising from the semidiscretization of hyperbolic Partial Differential Equations (PDEs), such as the Telegraph equation. The BHT is formulated from eight discrete hybrid formulas which are provided by a continuous two-step hybrid trigonometrically fitted method with two off-grid points. The BHT is implemented in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHT is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.

Highlights

In what follows, we consider the numerical solution of the general second order initial value problems (IVPs) of the form y󸀠󸀠 = f (x, y, y󸀠), y (x0) = y0, (1)y󸀠 (x0) = y0󸀠, x ∈ [x0, xN], where f : R × R2m → R2m, N > 0 is an integer, and m is the dimension of the system
Our objective is to present a block hybrid trigonometrically fitted (BHT) that is implemented in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods
The method given in [30] is of an order 2; in this paper, we propose a BHT which is of order 5 and its application is extended to solving Partial Differential Equations (PDEs) such as the Telegraph equation

Summary

Introduction

There are numerous methods for directly solving the special second-order IVPs in which the first derivative does not appear explicitly and it has been shown that these methods have the advantages of requiring less storage space and fewer number of function evaluations (see Hairer [4], Hairer et al [7], Simos [8], Lambert and Watson [9], and Twizell and Khaliq [10]). Vigo-Aguiar [23], Franco and Gomez [24], and Ozawa [25]) Most of these methods are restricted to solving special second-order IVPs in a predictor-corrector mode.

Development of Method

Error Analysis and Stability

Numerical Examples

Hyperbolic PDEs

Conclusion