Analytical Techniques for Solving Nonlinear Partial Differential Equations (Synthesis Lectures on Mathematics and Statistics)

Description

This is an introduction to methods for solving nonlinear partial differential equations (NLPDEs). After the introduction of several PDEs drawn from science and engineering, the reader is introduced to techniques used to obtain exact solutions of NPDEs. The chapters include the following topics: Compatibility, Differential Substitutions, Point and Contact Transformations, First Integrals, and Functional Separability. The reader is guided through these chapters and is provided with several detailed examples. Each chapter ends with a series of exercises illustrating the material presented in each chapter. The book can be used as a textbook for a second course in PDEs (typically found in both science and engineering programs) and has been used at the University of Central Arkansas for more than ten years. Preface Acknowledgments Nonlinear PDEs are Everywhere Exercises References Compatibility Charpit's Method Second-Order PDEs Compatibility in (2+1) Dimensions Compatibility for Systems of PDEs Exercises References Differential Substitutions Generalized Burgers' Equation KdV-MKdV Connection Generalized KdV Equation Matrix Hopf–Cole Transformation Darboux Transformations Second-Order Darboux Transformations Darboux Transformations Between Two Diffusion Equations Darboux Transformations Between Two Wave Equations Exercises References Point and Contact Transformations Contact Transformations Hodograph Transformation Legendre Transformation Ampere Transformation Contact Condition Plateau Problem Linearization Well-known Minimal Surfaces Exercises References First Integrals Quasilinear Second-Order Equations Monge–Ampere Equation The Martin Equation First Integrals and Linearization Hyperbolic MA Equations Parabolic MA Equations Elliptic MA Equations Exercises References Functional Separability Exercises References Solutions Author's Biography Blank Page