Wavelet Based Numerical Methods for Fractional Differential Equations

Abstract

This dissertation discusses wavelet numerical schemes for solving linear and nonlinear fractional di erential equations. A numerical method based on the Boubaker wavelet for the solution of linear and nonlinear fractional di erential equations and fractional delay di erential equations is presented. Method of steps is used to deal with frac tional delay di erential equations. Furthermore, we have utilized Quasilinearization technique in conjunction with Boubaker wavelet method to reduce nonlinear fractional di erential equations to system of linear fractional di erential equations. We have de rived and constructed the Boubaker wavelet and Boubaker wavelet operational matrix of fractional integration for the solution of Caputo-Hadamard fractional di erential equation. We have discussed the error analysis of the Boubaker wavelet. We have also provided numerical simulation of the proposed wavelet method to check the reliability and applicability of the proposed method. In the second part of the thesis, we have developed a technique using shifted Gegenbauer wavelets. We have constructed operational matrix of shifted Gegenbauer wavelets, fractional order integration and fractional order derivative to solve the ini tial value problem. Furthermore, solution of nonlinear fractional di erential equation is obtained by the proposed method in conjunction with the Adomian decomposition method. We have investigated the convergence of the proposed method. We have pre sented numerical examples and graphical results to check the reliability and e ciency of the proposed method. Lastly, we have developed the Katugampola Gegenbauer wavelet (KGW) on the interval [a,(ρ + a ρ ) 1/ρ), ρ > 0, a ∈ R, for the numerical solutions of Katugampola fractional di erential equations. We have proved that these wavelets are orthonormal within the interval [a,(ρ+a ρ ) 1/ρ). We have constructed the matrix for KGW, the KGW integration matrix, and the KGW operational matrix of derivative. We have combined the KGW method with Adomian decomposition method for the solution of nonlinear