Next Article in Journal
An Algebraic Inequality with Applications to Certain Chen Inequalities
Previous Article in Journal
Using the Supremum Form of Auxiliary Functions to Study the Common Coupled Coincidence Points in Fuzzy Semi-Metric Spaces
Previous Article in Special Issue
On the Bernstein Affine Fractal Interpolation Curved Lines and Surfaces
Article

Fractional Bernstein Series Solution of Fractional Diffusion Equations with Error Estimate

1
Department of Applied Mathematics, Abu Dhabi University, Abu Dhabi 59911, UAE
2
Elemantary Mathematics Education Program, Faculty of Education, Mugla Sitki Kocman University, Mugla 48000, Turkey
3
Department of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, UKM Bangi 43600, Malaysia
*
Author to whom correspondence should be addressed.
Received: 15 December 2020 / Revised: 1 January 2021 / Accepted: 4 January 2021 / Published: 7 January 2021
(This article belongs to the Special Issue Fractional Calculus, Wavelets and Fractals)
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. View Full-Text
Keywords: bernstein series; fractional calculus; diffusion equations; error estimate bernstein series; fractional calculus; diffusion equations; error estimate
Show Figures

Figure 1

MDPI and ACS Style

Alshbool, M.H.; Isik, O.; Hashim, I. Fractional Bernstein Series Solution of Fractional Diffusion Equations with Error Estimate. Axioms 2021, 10, 6. https://doi.org/10.3390/axioms10010006

AMA Style

Alshbool MH, Isik O, Hashim I. Fractional Bernstein Series Solution of Fractional Diffusion Equations with Error Estimate. Axioms. 2021; 10(1):6. https://doi.org/10.3390/axioms10010006

Chicago/Turabian Style

Alshbool, Mohammed H., Osman Isik, and Ishak Hashim. 2021. "Fractional Bernstein Series Solution of Fractional Diffusion Equations with Error Estimate" Axioms 10, no. 1: 6. https://doi.org/10.3390/axioms10010006

Find Other Styles
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Back to TopTop