New Nonpolynomial Spline in Compression Method of for the Solution of 1D Wave Equation in Polar Coordinates

We propose a three-level implicit nine point compact finite difference formulation of order two in time and four in space direction, based on nonpolynomial spline in compression approximation in -direction and finite difference approximation in -direction for the numerical solution of one-dimensional wave equation in polar coordinates. We describe the mathematical formulation procedure in detail and also discussed the stability of the method. Numerical results are provided to justify the usefulness of the proposed method. 1. Introduction We consider the one-dimensional wave equation in polar forms: with the following initial conditions: and the following boundary conditions: where and . We assume that the conditions (2) and (3) are given with sufficient smoothness to maintain the order of accuracy in the numerical method under consideration. The study of wave equation in polar form is of keen interest in the fields like acoustics, electromagnetic, fluid dynamics, mathematical physics, and so forth. Efforts are being made to develop efficient and high accuracy finite difference methods for such types of PDEs. During the last three decades, there has been much effort to develop stable numerical methods based on spline approximations for the solution of time-dependent partial differential equations. But so far in the literature, very limited spline methods are there for the wave equation in polar coordinates. In 1968-69, Bickley [1] and Fyfe [2] studied boundary value problems using cubic splines. In 1973, Papamichael and Whiteman [3], and the next year, Fleck [4] and Raggett and Wilson [5] have used a cubic spline technique of lower order accuracy to solve one-dimensional heat conduction equation and wave equation, respectively. Then, Jain et al. [6–9] have derived cubic spline solution for the differential equations including fourth order cubic spline method for solving the nonlinear two point boundary value problems with significant first derivative terms. Recently, Kadalbajoo et al. [10, 11] and Khan et al. [12, 13] have studied parametric cubic spline technique for solving two point boundary value problems. In recent years, Rashidinia et al. [14], Ding and Zhang [15], and Mohanty et al. [16–21] have discussed spline and high order finite difference methods for the solution of hyperbolic equations. In this present paper, we follow the idea of Jain and Aziz [7] by using nonpolynomial spline in compression approximation to develop order four method in space direction for the wave equation in polar co-ordinates. We have shown that our method is in general