Fractional Bernstein Series Solution of Fractional Diffusion Equations with Error Estimate

Axioms

MohammedHamed Alshbool¹  Ishak Hashim²  Osman Isik³ 

[1] Department of Applied Mathematics, Abu Dhabi University, Abu Dhabi 59911, UAE;Department of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, UKM Bangi 43600, Malaysia;Elemantary Mathematics Education Program, Faculty of Education, Mugla Sitki Kocman University, Mugla 48000, Turkey;

关键词: bernstein series;    fractional calculus;    diffusion equations;    error estimate;   

DOI  :  10.3390/axioms10010006

来源: DOAJ

【 摘 要 】

In the present paper, we introduce the fractional Bernstein series solution (FBSS) to solve the fractional diffusion equation, which is a generalization of the classical diffusion equation. The Bernstein polynomial method is a promising one and can be generalized to more complicated problems in fractional partial differential equations. To get the FBSS, we first convert all terms in the problem to matrix forms. Then, the fundamental matrix equation is obtained and thus, the solution is obtained. Two error estimation methods based on a residual correction procedure and the consecutive approximations are incorporated to find the estimate and bound of the absolute error. The perturbation and stability analysis of the method is given. We apply the method to some illustrative examples. The numerical results are compared with the exact solutions and known second-order methods. The outcomes of the numerical examples are very encouraging and show that the FBSS is highly useful in solving fractional partial problems. The results show the accuracy and effectiveness of the method.

【 授权许可】

Unknown