APPROXIMATE SOLUTIONS FOR SEVERAL CLASSES OF FRACTIONAL DIFFERENTIAL EQUATIONS BASED ON RESIDUAL POWER SERIES TECHNIQUE

Abstract

Fractional differential equations (FDEs) have become a significant issue in applied and mathematical analysis. In addition, accurate modelling of many natural phenomena using fractional differential equations is essential to understanding their structure, behaviour and construction. This thesis investigates analytical approximation methods based on the residual power series (RPS) approach for solving a class of FDEs and fractional integro-differential equations arising in engineering and applied science. The methodology optimises the approximate solutions by minimising the residual error functions to generate a fractional power series with a highly convergent rate. Convergence analysis of the proposed method is provided together with some numerical applications to confirm the theoretical aspect of the method. The fractional derivative is considered in a Caputo sense. However, this thesis covers many topics of fractional models. First, a class of fractional integro-differential equations of the Volterra type are used to obtain an accurate analytic-numeric solution based on fractional power series expansion. Second, the RPS expansion principle is directly used to express the approximate solutions for mixed integro-differential equations of fractional-order in convergent series formula with computable components. In addition, mathematical preliminaries, properties and analysis of the RPS algorithm are investigated and some illustrative numerical examples are included to demonstrate efficiency, accuracy, and applicability of the RPS method. In each example, the approximate solutions are compared with identified analytical solutions which found in good agreement with each other. Third, a reliable treatment based on the concept of residual error is employed to address the series solution of the differential logistic system in the fractional sense. The proposed technique is a combination of the generalised Taylor series and minimising the residual error functions. Some realistic applications of population growth models are given with a numerical comparison between the RPS method and the optimal homotopy asymptotic method to verify the theoretical statement. The numerical results highlight the universality of the proposed algorithm in obtaining series solutions consistently and show that approximate values are acceptable in terms of stability and accuracy. Finally, a novel approach for obtaining the numerical solution for a class of fractional Bagley-Torvik problems (FBTP) is presented. Meanwhile, the RPS description is given in detail to approximate the solution of FBTPs by highlighting all the steps necessary to implement the algorithm to address some tested problems. The results indicate that the RPS algorithm is reliable and suitable for solving a wide range of fractional differential equations in physics and engineering. The main feature of the proposed method is that it can be directly applied to solve nonlinear fractional problems without the need for unphysical restrictive assumptions, linearization, perturbation, or guessing the initial data. For numerical implementation, all symbolic and digital calculations are performed using the Wolfram Mathematica 10 software package.