Alaofi, Zaki Mrzog (2023) Numerical Treatment of Some Types of Nonlinear Partial Differential Equations. PhD thesis, Victoria University.

Abstract

Over the last decade, many quality research papers and monographs have been published focusing on numerical approximations of nonlinear partial differential equations (NPDEs). These equations are very important in mathematics and relevant to the study of various real-life phenomena from nature, physics, engineering and sciences. In this thesis, the cubic B-spline (CBS), non-polynomial spline, fractional calculus and Adomian decomposing methods are used to approximate solutions to the dissipative wave, the dispersive partial differential, coupled nonlinear nonhomogeneous Klein–Gordon, linear space-fractional telegraph partial differential and generalised Burgers–Huxley equations. These approximate solutions have been proven to be stable and convergent in various studies. The numerical examples considered in this paper illustrate the efficiency of the method compared with those used in recent works published in this field. This thesis investigates the treatment of some PDEs using numerical methods. One of the objectives of this thesis is to obtain accurate and constant numerical solutions to nonlinear integer and fractional order NPDEs. The first chapter presents a general introduction, motivation for the study, research questions, contributions and objectives of the research and the research methodology, and outlines the thesis organisation. Chapter 2 covers four main topics: PDEs, the B-spline method, the fractional calculus method and the Adomian method. Chapter 3 focuses on numerical approximations to solve the dissipative wave equation based on the CBS method. The steps followed involve the governing equation and derivation of the proposed method; the initial state; stability analysis; and numerical examples. Chapter 4 applies the non-polynomial spline method to identify an approximation solution for the third-order dispersive PDE. The steps followed involve analysis of the method; error analysis; stability analysis; and numerical examples. Chapter 5 provides an approximate analysis for coupled nonlinear non-homogeneous Klein–Gordon equations using the CBS method. The steps followed involve the numerical method; stability analysis; and numerical examples. Chapter 6 discusses the fractional calculus method for solving the linear spacefractional telegraph PDE. The steps followed involve derivation of the method; the spline relations; stability analysis; and numerical examples. Chapter 7 investigates solution of the generalised Burgers–Huxley equation with high-order nonlinearity terms using the Adomian decomposition method. The steps followed involve global exponential stability; construction of the adaptive boundary control; the initial boundary value problem; and numerical examples. The last chapter provides a summary of the thesis and makes some suggestions for further research.

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