Cubic Spline Chebyshev Polynomial Approximation for Solving Boundary Value Problems

A. Ji̇moh, M. H. Sulaiman, A. S. Mohammed
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Abstract

In this work, a Chebyshev polynomial spline function is derived and used to approximate the solution of the second order two-point boundary value problems of variable coefficients with the associated boundary conditions. In deriving the method, the cubic spline Chebyshev polynomial approximation, $S(x)$ is made to satisfy certain conditions for continuity and smoothness of functions. Numerical examples are presented to illustrate the applications of this method. The solution, $y(x)$ of these examples are obtained at some nodal points in the interval of consideration. The absolute errors in each example are estimated, and the comparison of exact values, and approximate values by the present method and other methods in literature at the nodal points are presented graphically. The comparison shows that the proposed method produces better results than Approaching Spline Techniques and collocation method.
解决边界值问题的立方样条切比雪夫多项式逼近法
在这项工作中,推导出了切比雪夫多项式样条函数,并将其用于近似求解可变系数的二阶两点边界值问题及相关边界条件。在推导该方法时,立方样条切比雪夫多项式近似值 $S(x)$ 满足函数连续性和平滑性的某些条件。本文通过数值示例来说明该方法的应用。这些示例的解$y(x)$ 是在考虑区间的一些节点上得到的。对每个例子的绝对误差进行了估算,并以图形方式展示了精确值与本方法和文献中其他方法在节点上的近似值的比较。比较结果表明,所提出的方法比逼近样条线技术和搭配法取得了更好的结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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