Analytical Methods for Solving Nonlinear Partial Differential Equations (Synthesis Lectures on Mathematics & Statistics)

✍ Scribed by Daniel Arrigo

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 181
• Edition: 2nd ed. 2022
• Category: Library

✦ Synopsis

This textbook provides an introduction to methods for solving nonlinear partial differential equations (NLPDEs). After the introduction of several PDEs drawn from science and engineering, readers are introduced to techniques to obtain exact solutions of NLPDEs. The chapters include the following topics: Nonlinear PDEs are Everywhere; Differential Substitutions; Point and Contact Transformations; First Integrals; and Functional Separability. Readers are guided through these chapters and are provided with several detailed examples. Each chapter ends with a series of exercises illustrating the material presented in each chapter. This Second Edition includes a new method of generating contact transformations and focuses on a solution method (parametric Legendre transformations) to solve a particular class of two nonlinear PDEs.

✦ Table of Contents

Preface to Second Edition
Preface to First Edition
Acknowledgements
Contents
1 Nonlinear PDEs are Everywhere
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1.1 Exercises
2 Compatibility
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2.1 Charpit's Method
2.2 Second Order PDEs
2.3 Compatibility in left parenthesis 2 plus 1 right parenthesis(2+1) Dimensions
2.4 Compatibility for Systems of PDEs
2.5 Exercises
3 Differential Substitutions
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3.1 Generalized Burgers' Equation
3.2 KdV-MKdV Connection
3.3 Generalized KdV Equation
3.4 Matrix Hopf-Cole Transformation
3.5 Darboux Transformations
3.5.1 Second Order Darboux Transformations
3.5.2 Darboux Transformations Between Two Diffusion Equations
3.5.3 Darboux Transformations Between Two Wave Equations
3.6 Exercises
4 Point and Contact Transformations
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4.1 Contact Transformations
4.1.1 Hodograph Transformation
4.1.2 Legendre Transformation
4.1.3 Ampere Transformation
4.2 Contact Condition
4.3 Plateau Problem
4.3.1 Linearization
4.3.2 Well Known Minimal Surfaces
4.4 Generating Contact Transformations
4.5 Parametric Legendre Transformations
4.6 Exercises
5 First Integrals
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5.1 Quasilinear Second Order Equations
5.2 Monge-Ampere Equation
5.3 The Martin Equation
5.4 First Integrals and Linearization
5.4.1 Hyperbolic MA Equations
5.4.2 Parabolic MA Equations
5.4.3 Elliptic MA Equations
5.5 Exercises
6 Functional Separability
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6.1 Exercises
A Solutions
Index