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

✍ Scribed by Steven G. Krantz (editor), Daniel J. Arrigo

• Publisher: MORGAN & CLAYPOOL
• Year: 2019
• Tongue: English
• Leaves: 167
• Series: Synthesis Lectures on Mathematics and Statistics
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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
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