Elegant scheme for solving Caputo-time-fractional integro-differential equations

Abstract

The aim of this letter is to construct an analytical series solution in the sense of fractional calculus to a class of Caputo-fractional integro-differential equations of the form
\begin{equation*}
\begin{aligned}
D^{\alpha}_t w(t)&= p(t)w(t)+g(t)+ \int_0^t k(t,\tau) w^m(\tau) d\tau \\
w(0) &= w_0,
\end{aligned}
\end{equation*}
where $t\in [0,1]$, $m\in \mathbb{N}$, $\alpha\in (0,1]$ is the Caputo fractional derivative order, and $k(t,\tau)$ is an analytic separable function by means of fractional derivative. A modified version of the Taylor power series method will be utilized in this work which stands on finding recursively the series coefficients by fractionally differentiating the equation under consideration. The method is tested on different examples and closed-form solutions are successfully obtained.