Numerical solutions of nonlinear partial differential equations with the help of haar wavelets

This thesis consists of six chapters. In the first chapter after giving some general knowledge about wavelets, Haar wavelets, function approximation by Haar wavelets and Haar integrals that used in the thesis are introduced. The parts in the second to sixth chapters are the original parts of the thesis where considered problems are solved by Haar wavelet method. In the second chapter, regularized long wave (RLW) equation is considered. Haar wavelet method is applied to three model problems: single soliter wave, interaction of two solitary waves and wave undulation problems. The obtained numerical results are compared with exact solution and with existing results in the literature. The error norms and the invariants are given in tables. Also the numerical results are plotted. In the third chapter, Korteweg-de Vries (KdV) equation is regarded. Single soliton wave problem and interaction of two soliton waves problem are solved by Haar wavelet method. The obtained numerical results are given in tables and are plotted. Also they are compared with the analytical solution and with those already exist in the literature. In the fourth chapter, coupled nonlinear Schrödinger-KdV (NLS-KdV) equation is taken into account. Numerical solutions and invariants are computed with Haar wavelet method. After tabulating and plotting the obtained numerical results they are compared with the exact solution and with available results in the literature. In the fifth chapter, fourth order parabolic partial differential equations are considered. With the help of Haar wavelet method, variable and constant coefficient, homogeneous and non-homogeneous model problems are solved numerically. The obtained numerical results are compared with the analytical solution and with previous results that exist in the literature. Also numerical results are depicted graphically. In the sixth chapter, Fractional Burgers' and Fractional Coupled Burgers' equations are taken into account. The fractional derivatives are discretized by a formula which is known as L1 formula in the literature and then the considered model problems are solved numerically by the aid of Haar wavelet method. The results are given in tables and are plotted. After that they are compared with the exact solution and with those already exist in the literature. The numerical computations in this thesis are done with the codes written in free software GNU Octave and Python programming language.