Wavelet Quasilinearization Methods for Fractional Differential Equations

Abstract

The main objective of this thesis is to develop numerical methods for solving nonlinear fractional ordinary differential equations, nonlinear fractional partial differential equations, and linear and nonlinear fractional delay differential equation. Some methods are proposed by utilizing wavelets operational matrix methods and quasilinearization technique, these methods are used for the solution of nonlinear fractional differential equations, we call these methods as wavelet quasilinearization techniques. According to the wavelet quasilinearization techniques, we convert the fractional nonlinear differential equation to fractional discretize differential equation by using quasilinearization technique and apply wavelet methods at each iteration of quasilinearization technique to get the solution. We established a technique by utilizing both the Haar wavelet and Picard technique for solving the fractional nonlinear differential equation. While some methods based on the wavelets methods and method of steps, used for the solution of linear and nonlinear fractional delay differential equation. These techniques converts the fractional linear or nonlinear delay differential equation on a given interval to an fractional linear or nonlinear differential equation without delay over that interval, by using the function defined on previous interval, and then apply the wavelet method on the obtained fractional differential equation to find the solution on a given interval. The same procedure provides the solution on next intervals. We also developed a method, Gegenbauer wavelet operational matrix method, by using Gegenbauer polynomials. The Gegenbauer wavelet matrix, Gegenbauer wavelet operational matrix of fractional integration and Gegenbauer wavelet operational matrix of fractional integration for boundary value problems are derived, constructed and utilized for the solution of fractional differential equations. The convergence and supporting analysis of our methods are also investigated. The comparison analysis of methods with other existing numerical methods is also performed.