On the application of a numerical method to the resolution of fractional order differential equations

Abstract

Fractional differential equations (FDEs) are becoming increasingly popular as a modeling tool to describe a wide range of natural phenomena in physics, chemistry, biology, and so on. These FDEs help scientists to understand, analyze, and make predictions about the modeled system in one case– when their solutions are available. But most FDEs do not have exact solutions and even if there are exact solutions, they can not be evaluated exactly. Thus, one has to rely on numerical methods to obtain their approximate solutions. The purpose of this thesis is to present an efficient computational method for finding numerical solutions of some important fractional differential equations that do not have exact solutions: the fractional biological-population model, fractional cancer tumor models, time-fractional advection-diffusion equation, and time-fractional Swift-Hohenberg equation. Those models are solved by using the reproducing kernel Hilbert space method (RKHSM). The main advantages of this method that encouraged us to use it are its flexibility and simplicity. The convergence analysis and error estimations associated with the RKHSM are discussed to confirm the applicability theoretically. The impact of the fractional derivative on each model is discussed. We also illustrate the profiles of several representative numerical solutions of these models. By testing some examples for each model, we demonstrated the potentiality, validity, and effectiveness of the RKHSM. The computational results are compared with other available results in which these comparisons indicate the superiority and accuracy of the RKHSM in solving complex problems