Abstract
We provide numerical solution to the one-dimensional wave equations in polar coordinates, based on the cubic B-spline quasi-interpolation. The numerical scheme is obtained by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a forward difference to approximate the time derivative of the dependent variable. The accuracy of the proposed method is demonstrated by three test problems. The results of numerical experiments are compared with analytical solutions by calculating errors -norm and -norm. The numerical results are found to be in good agreement with the exact solutions. The advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement.
Highlights
The term “spline” in the spline function arises from the prefabricated wood or plastic curve board, which is called spline, and is used by the draftsman to plot smooth curves through connecting the known point
Several numerical schemes for the solution of boundary value problems and partial differential equations based on the spline function have been developed by many researchers
El-Hawary and Mahmoud [6], Mohanty [7], Mohebbi and Dehghan [8], Zhu and Wang [9], Ma et al [10], Dosti and Nazemi [11], Wang et al [12], and other researchers [13,14,15,16] have derived various numerical methods for solution of partial differential equations based on the spline function
Summary
The term “spline” in the spline function arises from the prefabricated wood or plastic curve board, which is called spline, and is used by the draftsman to plot smooth curves through connecting the known point. Several numerical schemes for the solution of boundary value problems and partial differential equations based on the spline function have been developed by many researchers. We provide a numerical scheme to solve singular hyperbolic equation (1) using the derivative of the cubic B-spline quasi-interpolation to approximate the spatial derivative of the differential equations and utilize a forward difference to approximate the time derivative such as [9, 11] shown.
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