Beyond the tanh method-looking for explicit travelling wave solutions to partial differential equations

Dissertation (MSc)--University of Pretoria, 2022.In this work, we focus on a general procedure for finding exact travelling wave solutions for evolution equations with polynomial nonlinearites. Mathematically, looking for travelling wave solutions is asking the question whether a given PDE has solutions invariant under a Galilean transformation; in such a case, it can be reduced to an ODE. We discuss the existence of travelling wave solutions by using phase plane analysis. We show that popular methods such as the tanh-method, G0/G-method and many more are special cases of the presented approach. Analytical solutions to several examples of nonlinear equations are illustrated. In the application, we use the Maple program to compute solutions to nonlinear systems of equations.Mathematics and Applied MathematicsMScUnrestricte

A GOOD INITIAL GUESS FOR APPROXIMATING NONLINEAR OSCILLATORS BY THE HOMOTOPY PERTURBATION METHOD

A good initial guess and an appropriate homotopy equation are two main factors in applications of the homotopy perturbation method. For a nonlinear oscillator, a cosine function is used in an initial guess. This article recommends a general approach to construction of the initial guess and the homotopy equation. Duffing oscillator is adopted as an example to elucidate the effectiveness of the method

Exploring Advanced Analysis Technique for Shallow Water Flow Models with Diverse Applications

The present article focuses on the analytical approach using fractional orders and its application in the dynamics of physical processes. Fractional order models align better with experimental data compared to non-fractional ones. This study primarily focuses on employing the new approximate analytical method to solve shallow water models with fractional orders. Numerical examples within the Caputo fractional derivative showcase the method’s application. Results for both integer and fractional orders are graphically depicted, demonstrating the fractional solutions’ closeness to actual data. Analysis of 3D and 2D fractional order graphs reveals convergence toward integer order graphs as fractional derivatives approach non-fractional ones. This method shows promise for direct application in solving targeted problems and can be easily adapted for other fractional nature problems

Study on the nonlinear vibration of embedded carbon nanotube via the Hamiltonian-based method:

This article mainly studies the vibration of the carbon nanotubes embedded in elastic medium. A new novel method called the Hamiltonian-based method is applied to determine the frequency property of the nonlinear vibration. Finally, the effectiveness and reliability of the proposed method is verified through the numerical results. The obtained results in this work are expected to be helpful for the study of the nonlinear vibration

Analytical and Numerical Methods for Differential Equations and Applications

The book is a printed version of the Special issue Analytical and Numerical Methods for Differential Equations and Applications, published in Frontiers in Applied Mathematics and Statistic

Traveling Wave Solutions for the Generalized Zakharov Equations

We use the bifurcation method of dynamical systems to study the traveling wave solutions for the generalized Zakharov equations. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow-up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended