论文标题
通过具有分数基函数的光谱方法的分数弗雷德·局部分化方程的数值解
Numerical solution of fractional Fredholm integro-differential equations by spectral method with fractional basis functions
论文作者
论文摘要
本文提出了一种有效的光谱方法，用于求解分数弗雷姆·局部差异方程。解决此类问题的解决方案的不平滑度导致基于经典多项式（例如Chebyshev，Legendre，Laguerre等）的光谱方法的性能，并具有低融合顺序。因此，解决此类问题的经典数值方法的发展成为一个具有挑战性的问题。由于非平滑溶液具有与分数功率多项式相同的渐近行为，因此，分数基函数是克服光谱方法准确性的缺点的最佳候选者。另一方面，分数多项式函数的分数整合在分数多项式类别中，这是使用分数基函数的主要优点之一。在本文中，引入了基于分数Chelyshkov基础函数的隐式光谱搭配方法。该方法的框架是将问题与分数操作集成矩阵一起利用光谱搭配方法的非线性方程式减少。使用牛顿的迭代方法解决了获得的代数系统。研究了该方法的收敛分析。与其他现有方法相比，数值示例显示了该方法对流畅和非平滑溶液的问题的效率。
This paper presents an efficient spectral method for solving the fractional Fredholm integro-differential equations. The non-smoothness of the solutions to such problems leads to the performance of spectral methods based on the classical polynomials such as Chebyshev, Legendre, Laguerre, etc, with a low order of convergence. For this reason, the development of classic numerical methods to solve such problems becomes a challenging issue. Since the non-smooth solutions have the same asymptotic behavior with polynomials of fractional powers, therefore, fractional basis functions are the best candidate to overcome the drawbacks of the accuracy of the spectral methods. On the other hand, the fractional integration of the fractional polynomials functions is in the class of fractional polynomials and this is one of the main advantages of using the fractional basis functions. In this paper, an implicit spectral collocation method based on the fractional Chelyshkov basis functions is introduced. The framework of the method is to reduce the problem into a nonlinear system of equations utilizing the spectral collocation method along with the fractional operational integration matrix. The obtained algebraic system is solved using Newton's iterative method. Convergence analysis of the method is studied. The numerical examples show the efficiency of the method on the problems with smooth and non-smooth solutions in comparison with other existing methods.
