# Numerical solution for a class of variable order fractional integral-differential equation with Atangana-Baleanu-Caputo fractional derivative

Document Type : Original Paper

Authors

Department of Mathematics, Faculty of Science, University of Birjand, Birjand, Iran.

Abstract

In this paper we consider fractional integral-differential equations of variable order containing Atangana-Baleanu-Caputo fractional derivatives as follows:
\begin{align*}
\mathfrak{D}_{\alpha(t)}^{ABC}&\Big[u(x,t).g(x,t)\Big]+\frac{\partial u(x,t)}{\partial t}+\int_{0}^{t}u(x,Y)dY\nonumber\\
&+\int_{0}^{t}u(x,Y).k(x,Y)dY=f(x,t),
\end{align*}
We try to solve this equation using a numerical method based on matrix operators including Chebyshev polynomials. By using these operational matrixes the fractional order integral-differential equation is transformed into an algebraic system which by solving them, we will obtain the numerical answer of the above fractional integral-differential equation. To show the accuracy and efficiency of this method, we have calculated some numerical examples by MATLAB software.

Keywords

Main Subjects

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### History

• Receive Date: 14 October 2020
• Revise Date: 28 December 2020
• Accept Date: 19 February 2021