The Numerical Solution of Fractional Partial Differential Equations Via Haar Wavelet

Abstract

In this dissertation, Haar wavelet collocation method is implemented to one and higher dimensional fractional partial differential equations (FPDEs) sub ject to the initial and Dirichlet boundary conditions. A new technique de pending on Haar wavelet is constructed to compute the numerical solutions of different types of time- and space-time-FPDEs. The method is implemented to linear as well as nonlinear FPDEs. The technique is also developed for systems of FDEs and FPDEs. The fractional derivative involved is evaluated using the Caputo definition. The method is semi-analytic as it involves the exact integration of the Caputo fractional derivative. The suggested tech nique is applied to the nonlinear fractional Burger’s equations, KdV-Burger’s equations, fractional Fisher’s equation and fractional advection equation. For linear FPDEs, the system of equations obtained after discretization is linear and then Guass elimination method is applied. In case of nonlinear FPDEs, the system of equations obtained is nonlinear and so Newton’s or Broyden’s method is employed. New algorithms are designed for the newly developed techniques. The techniques developed for all types of FPDEs are imple mented and tested through MATLAB software. Several numerical experiments are performed to test the accuracy, conver gence and efficiency of the newly constructed technique. The validity of the newly developed technique is tested upon various linear and nonlinear prob lems from literature. Some of the bench-mark problems are solved through the proposed technique and the results acquired are compared with the ex isting methods in literature. The numerical results obtained for a number of test problems are also compared with the exact solutions. The performance of the proposed technique are shown by computing the errors, computational cost and rates of convergence. The numerical results demonstrate that the proposed technique is simply as well as widely applicable, efficient and accu rate.