Application of Chebyshev Polynomials for Solving Abel's Integral Equations of the First and Second Kind

Theory of Approximation and Applications

مقاله 4، دوره 12، شماره 2، اسفند 2018، صفحه 43-60   

نوع مقاله: Research Articles

نویسندگان

Ahmad Shahsavaran^{* 1}؛ M. M. Shamivand²

¹Department of Mathematics

²Department of Mathematics, Islamic Azad University, Borujerd Branch, Borujerd, Iran.

چکیده

In this paper, a numerical implementation of an expansion method is developed for solving Abel's integral equations of the first and second kind. The solution of such equations may demonstrate a singular behaviour in the neighbourhood of the initial point of the interval of
integration. The suggested method is based on the use of Taylor series expansion to overcome the singularity which leads to approximating the unknown function and it's derivatives in terms of Chebyshev polynomials of the first kind. The proposed method, transforms the Abel's integral equations of the first and second kind into a system of linear algebraic equations which can be solved by Gaussian elimination algorithm. Finally, some numerical examples are included to clarify the accuracy and applicability of the presented method which indicate that proposed method is computationally very attractive. In this
paper, all numerical computations were carried out on a PC executing some programs written in maple software.

کلیدواژه‌ها

Abel's integral equations؛ Chebyshev polynomials؛ Taylor series expansion؛ Collocation points