Chapter 8 **General Functions Method in Transport Boundary Value Problems of Elasticity Theory 129** Lyudmila Alexeyeva

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

Preface

ferential equations to the current project.

variational iteration method.

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‐

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
