A semi-analytical approach to Caputo type time-fractional modified anomalous sub-diffusion equations

This article is devoted to a new semi-analytical algorithm for solving time-fractional modified anomalous sub-diffusion equations (FMASDEs). In this method first, the main problem is reduced to a system of fractional-order ordinary differential equations (FODEs) under known initial value conditions by using the Chebyshev collocation procedure. After that, to solve this system, some auxiliary initial value problems are defined. Next, we find an optimal linear combination of some particular solutions for these problems and finally we use this linear combination to construct a semi-analytical approximate solution for the main problem. To demonstrate the convergence property of the new method, a residual error analysis is performed in details. Some test problems are investigated to show reliability and accuracy of the proposed method. Besides, convergence order's indicators are evaluated for all test problems and are compared with ones of the other methods. Moreover, a comparison between our computed numerical results and the reported results of the other numerical schemes in the literature exhibits that the proposed technique is more precise and reliable. In summary advantages of the proposed method are: high accuracy, easy programming, high experimental convergence order, and solving another types of fractional differential equations.