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.

\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**

- Atangana-Baleanu-Caputo fractional derivative
- Chebyshev polynomials
- Operational matrixes
- Fractional integral-differential equations

**Main Subjects**

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June 2021

Pages 253-270

**Receive Date:**14 October 2020**Revise Date:**28 December 2020**Accept Date:**19 February 2021**First Publish Date:**07 May 2021