求解非线性偏微分方程的分析方法

本书是一本获得非线性偏微分方程(NLPDEs)精确解的介绍性书籍。本书包含了非线性PDEs无处不在、相容性、微分替换、点变换与接触变换、第一积分、泛函可分性、解等内容。

Preface
Acknowledgments
1 Nonlinear PDEs are Everywhere
1.1 Exercises
1.2 References
2 Compatibility
2.1 Charpit’s Method
2.2 Second-Order PDEs
2.3 Compatibility in (2 1) Dimensions
2.4 Compatibility for Systems of PDEs
2.5 Exercises
2.6 References
3 Differential Substitutions
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
3.7 References
4 Point and Contact Transformations
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 Exercises
4.5 References
5 First Integrals
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
5.6 References
6 Functional Separability
6.1 Exercises
6.2 References
7 Solutions
Author’s Biography
编辑手记