Preface

Chapter 8 **General Functions Method in Transport Boundary Value**

Chapter 9 **Solution of Nonlinear Partial Differential Equations by New Laplace Variational Iteration Method 153**

**Problems of Elasticity Theory 129**

Lyudmila Alexeyeva

Tarig M. Elzaki

**VI** Contents

Differential equations are mathematical equations that relate some functions with their de‐ rivatives. The functions usually represent some physical quantities and their derivatives represent their rates of change and the equation relates the two together. For example, in fluid dynamics, the Navier-Stokes equations are a system of mathematical equations that relate the velocities of the fluid to partial derivatives of velocity and pressure. The editor of the present book has worked on solving the Navier-Stokes equations in cylindrical coordi‐ nates for multiphase flows where the equations are coupled to the continuity and level set distance function equations. Such work now in press has revealed an analytical procedure to solve this system of equations by defining a composite velocity formulation for the sum of three principal directions of flow and connecting this to the level set function and its deriva‐ tives. It has been shown that it is possible to solve analytically multiphase flow using level set methods for vertical and horizontal tubes. It has been shown that in this pursuit the structure of the governing equations for multiphase flow has some interesting symmetries, which reduce the composite formulation above ordinary differential equations. Further analysis using pseudo-exact differential equations results in Abel-type equations emerging in the analysis. It is a worthy exercise to correctly reduce a system of partial differential equations to ordinary differential equations and hence prove the existence and uniqueness of solutions to such mathematical problems. For this reason, the editor of this book has been motivated to introduce various topics welcomed from an international audience of mathe‐ maticians and researchers to contribute various aspects of the theory and application of dif‐ ferential equations to the current project.

The editor has incorporated contributions from a diverse group of leading researchers in the field of differential equations. This book aims to provide an overview of the current knowl‐ edge in the field of differential equations. The main subject areas are divided into general theory and applications. These include fixed point approach to solution existence of differ‐ ential equations, existence theory of differential equations of arbitrary order, topological methods in the theory of ordinary differential equations, impulsive fractional differential equations with finite delay and integral boundary conditions, an extension of Massera's the‐ orem for n-dimensional stochastic differential equations, phase portraits of cubic dynamic systems in a Poincare circle, differential equations arising from the three-variable Hermite polynomials and computation of their zeros and reproducing kernel method for differential equations. Applications include local discontinuous Galerkin method for nonlinear Ginz‐ burg-Landau equation, general function method in transport boundary value problems of theory of elasticity and solution of nonlinear partial differential equations by new Laplace variational iteration method.

Existence/uniqueness theory of differential equations is presented in this book with applica‐ tions that will be of benefit to mathematicians, applied mathematicians and researchers in the field. The book is written primarily for those who have some knowledge of differential equations and mathematical analysis. The authors of each section bring a strong emphasis on theoretical foundations to the book.

> **Terry E. Moschandreou** University of Western Ontario Canada

**Section 1**

**Theory of Differential Equations**

**Theory of Differential Equations**

Existence/uniqueness theory of differential equations is presented in this book with applica‐ tions that will be of benefit to mathematicians, applied mathematicians and researchers in the field. The book is written primarily for those who have some knowledge of differential equations and mathematical analysis. The authors of each section bring a strong emphasis

> **Terry E. Moschandreou** University of Western Ontario

> > Canada

on theoretical foundations to the book.

VIII Preface

**Chapter 1**

Provisional chapter

**Fixed Point Theory Approach to Existence of Solutions**

DOI: 10.5772/intechopen.74560

In this chapter, we introduce a generalized contractions and prove some fixed point theorems in generalized metric spaces by using the generalized contractions. Moreover, we will apply the fixed point theorems to show the existence and uniqueness of solution to the ordinary difference equation (ODE), Partial difference equation (PDEs) and frac-

Keywords: fixed point, contraction, generalized contraction, differential equation, partial

The study of differential equations is a wide field in pure and applied mathematics, chemistry, physics, engineering and biological science. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics investigated the existence and uniqueness of solutions, but applied mathematics focuses on the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

solutions. Instead, solutions can be approximated using numerical methods.

Fixed Point Theory Approach to Existence of Solutions

**with Differential Equations**

with Differential Equations

Poom Kumam

Poom Kumam

Abstract

1. Introduction

Piyachat Borisut, Konrawut Khammahawong and

Piyachat Borisut, Konrawut Khammahawong and

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74560

tional boundary value problem.

differential equation, fractional difference equation

#### **Fixed Point Theory Approach to Existence of Solutions with Differential Equations** Fixed Point Theory Approach to Existence of Solutions with Differential Equations

DOI: 10.5772/intechopen.74560

Piyachat Borisut, Konrawut Khammahawong and Poom Kumam Piyachat Borisut, Konrawut Khammahawong and Poom Kumam

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74560

### Abstract

In this chapter, we introduce a generalized contractions and prove some fixed point theorems in generalized metric spaces by using the generalized contractions. Moreover, we will apply the fixed point theorems to show the existence and uniqueness of solution to the ordinary difference equation (ODE), Partial difference equation (PDEs) and fractional boundary value problem.

Keywords: fixed point, contraction, generalized contraction, differential equation, partial differential equation, fractional difference equation
