Abstract
Many problems in various branches of science, such as physics, chemistry, and engineering have been recently modeled as fractional ODEs and fractional PDEs. Thus, methods to solve such equations, especially in the nonlinear state, have drawn the attention of many researchers. The most important goal of researchers in solving such equations has been set to provide a solution with the possible minimum error. The fractional PDEs can be generally classified into two main types, spatial-fractional, and time-fractional differential equations. This study was designed to provide a numerical solution for the fractional-time diffusion equation using the finite-difference method with Neumann and Robin boundary conditions. The time fraction derivatives in the concept of Caputo were considered, also the stability and convergence of the proposed numerical scheme have been completely proven and a numerical test was also designed and conducted to assess the efficiency and precision of the proposed method. Eventually, it can be said that based on findings, the present technique can provide accurate results.
Highlights
Fractional diffusion equations have significantly drawn the attention of many scholars due to their use in different branches ofscience, including their use in describing some phenomena in physics (Metzler and Klafter, 2000), chemistry and biochemistry (Yuste and Lindenberg, 2002), mechanical engineering (Magin et al, 2009), medicine (Chen et al, 2010), and electronics (Kirane et al, 2013)
The Caputo fractional derivative is a type of fractional derivative in mathematical modeling with experimental data analysis, which is broadly used in different branches of science
We considered a Caputo fractional-time diffusion equationin this article using the finite-difference method (FDM)
Summary
Fractional diffusion equations have significantly drawn the attention of many scholars due to their use in different branches ofscience, including their use in describing some phenomena in physics (Metzler and Klafter, 2000), chemistry and biochemistry (Yuste and Lindenberg, 2002), mechanical engineering (Magin et al, 2009), medicine (Chen et al, 2010), and electronics (Kirane et al, 2013). Considerable numerical methods have been designed for diffusion fractional-time equations. Murio et al devised a stable unconditional implicit numerical method to solve the one-dimensional linear diffusion fractional-time equation on a finite medium (Murio, 2008). In the process of solving fractional-time diffusion equation in this literature, Uim denotes the accurate answer and uim represents the approximate
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