Exponential Cubic B-spline Collocation Method for Solving the One Dimensional Wave Equation Subject to an Integral Conservation Condition

In this paper, a collocation method is presented based on the exponential B-spline functions for the numerical solution of the one-dimensional wave equation subject to an integral conservation condition. The wave equation is fully-discretized by using the exponential cubic B-spline collocation for spatial discretization and the finite difference method for the time discretization. The difference scheme is analyzed to be unconditionally stable. The results of a numerical experiment show the stability and efficiency of the proposed method.


Introduction
Consider the one-dimensional nonlinear wave equation with the following conditions , , ) and ) ( 2 t l are known functions. It plays a very important role in studying plasma physics, thermoelasticity, chemical heterogeneity and etc. The existence and uniqueness of the solution of this problem was discussed in [1]. Many numerical methods of this problem, such as finite difference schemes [2], the method of lines [3], a second kind Chebyshev polynomial approach [4], a meshless method using radial basis functions [5], have been presented.
The exponential B-splines are picewise functions, which have a free parameter p. By means of this collocation method, some partial differential equations have been solved numerically [6][7][8][9]. In this paper, we apply the exponential cubic B-spline collocation method to solve the one-dimensional wave equation subject to an integral conservation condition.

Collocation Method
The domain [a,b] [0,T] is divided into an M N mesh, with h=(b-a)/M, and =T/N, respectively. x j =a+jh, for j=0,1,2, ,M is the jth node. t k =k, k=0,1,,N, is the time level for the kth step.
An approximate solution u(x,t) to the analytical solution U(x, t) of equation (1) is given by are time-dependent unknowns to be determined. The exponential cubic B-spline where p is a free parameter, and Then, we have At k=0, the term u -1 in (12) can be determined by (3), that is If θ=0, the difference scheme will be explicit. If θ=1/2, it will be a Crank-Nicolson scheme. If θ=1, it will be implicit.
To obtain a unique solution, two additional equations are needed. Thus, by terms of (4) and (5), we have ) ( 1 To start iterations of this system (12), the initial parameters can be obtained from the following conditions:

Stability Analysis
The stability of the presented method is investigated by Von Neumann method, which is applicable to linear scheme. Hence, following [10], the homogeneous part of the nonlinear equation (1) is linearized by assuming all nonlinear terms equal zero. Substituting (10) into the linear scheme of (12), we obtain the following equation The trial solutions (one Fourier mode out of the full solution) at a given point Thus the proposed numerical scheme is unconditionally stable.

Conclusion
For solving the one-dimensional wave equation subject to an integral conservation condition, the exponential cubic B-spline collocation method is presented. The numerical experiment is proposed to illustrate the effectiveness of this method. The accuracy of calculation can be improved by choosing appropriate parameters θ and p.