Generalization of Numerical Methods for Fractional Differential Equations

Abstract

In the last decades, fractional calculus has gained much importance over ordinary calculus due to its diverse applications in many areas of medical science, engineering, and physics. Therefore, there exists a strong motivation to develop reliable and proficient numerical methods. The main focus of interest of this study is to develop some reliable, more efficient, and accurate numerical techniques for obtaining the approximate solution of various classes of fractional differential equations (FDEs) which includes Caputo fractional differential equations (CFDEs) and Caputo-Hadamard fractional differential equations (CHFDEs). In this work, wavelets are the primary part of developing numerical schemes for obtaining approximate solutions of linear and nonlinear FDEs, and linear and nonlinear fractional delay differential equations (FDDEs). We have developed a numerical method by generalizing the sine-cosine wavelets known as fractional-order generalized sine-cosine wavelets (FGSCW) for the solution of linear and nonlinear CHFDEs. We have also addressed a novel Vieta-Lucas wavelets method (VLW) for solving linear and nonlinear FDDEs. We have proposed a new wavelet method known as the Krawtchouk wavelets method (KW) by considering a discrete class of Krawtchouk orthogonal polynomials for the solution of initial value problem (IVP), boundary value problem (BVP), and delay problem in the sense of both CFDEs and CHFDEs. We have proposed a numerical method known as Hahn wavelets (HW) by considering the discrete Hahn orthogonal polynomials for the solution of nonlinear CFDEs and nonlinear CHFDEs. We have constructed operational matrices for the FGSCWs, VLWs, KWs, and HWs. We have obtained an exact formula for the Riemann-Liouville fractional integration (RLFI) of VLWs and HWs. We have constructed operational matrices of RLFI by the VLWs, KWs, and HWs. We have constructed operational matrices of Hadamard fractional integral (HFI) by the FGSCWs, KWs, and HWs. The quasilinearization technique (QT) is used to transform the nonlinear FDEs into a discrete linear form. The proposed techniques have been analyzed for convergence. Construction of operational matrices, orthogonality of wavelets, and procedure of implementation are also documented. We have used MATLAB2014a for the numerical simulations of our proposed methods and comparison analysis is also performed with other existing numerical methods to show the effectiveness of the proposed methods.