Algebraic Approaches to Partial Differential Equations pdf epub mobi txt 电子书 下载 2025

想要找书就要到 小美书屋

立刻按 ctrl+D收藏本页

你会得到大惊喜!!

This book presents the various algebraic techniques for solving partial differential equations to yield exact solutions, techniques developed by the author in recent years and with emphasis on physical equations such as: the Maxwell equations, the Dirac equations, the KdV equation, the KP equation, the nonlinear Schrodinger equation, the Davey and Stewartson equations, the Boussinesq equations in geophysics, the Navier-Stokes equations and the boundary layer problems. In order to solve them, I have employed the grading technique, matrix differential operators, stable-range of nonlinear terms, moving frames, asymmetric assumptions, symmetry transformations, linearization techniques and special functions. The book is self-contained and requires only a minimal understanding of calculus and linear algebra, making it accessible to a broad audience in the fields of mathematics, the sciences and engineering. Readers may find the exact solutions and mathematical skills needed in their own research.

The author received his Ph.D. from Rutgers University, USA in 1992. He is currently a research professor at the Chinese Academy of Sciences’ Institute of Mathematics, and has been working on representation theory and applied partial differential equations for twenty years, during which he has published over fifty substantial research papers and two monographs on mathematics.

Part I Ordinary Differential Equations
1 First-Order Ordinary Differential Equations . . . . . . . . . . . . . 1
1.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Special Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Higher Order Ordinary Differential Equations . . . . . . . . . . . . 17
2.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Method of Undetermined Coefficients . . . . . . . . . . . . . . . . 21
2.3 MethodofVariationofParameters . . . . . . . . . . . . . . . . . 25
2.4 Series Method and Bessel Functions . . . . . . . . . . . . . . . . 29
3 Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1 Gamma and Beta Functions . . . . . . . . . . . . . . . . . . . . . 37
3.2 Gauss Hypergeometric Functions . . . . . . . . . . . . . . . . . . 43
3.3 Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 Weierstrass’s Elliptic Functions . . . . . . . . . . . . . . . . . . . 54
3.5 Jacobian Elliptic Functions . . . . . . . . . . . . . . . . . . . . . 61
Part II Partial Differential Equations
4 First-Order or Linear Equations . . . . . . . . . . . . . . . . . . . . 67
4.1 MethodofCharacteristics . . . . . . . . . . . . . . . . . . . . . . 68
4.2 Characteristic Strip and Exact Equations . . . . . . . . . . . . . . 71
4.3 Polynomial Solutions of Flag Equations . . . . . . . . . . . . . . 74
4.4 Use of Fourier Expansion I . . . . . . . . . . . . . . . . . . . . . 93
4.5 Use of Fourier Expansion II . . . . . . . . . . . . . . . . . . . . . 100
4.6 Calogero–Sutherland Model . . . . . . . . . . . . . . . . . . . . . 117
4.7 MaxwellEquations . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.8 Dirac Equation and Acoustic System . . . . . . . . . . . . . . . . 134
5 Nonlinear Scalar Equations . . . . . . . . . . . . . . . . . . . . . . . 141
5.1 KorteweganddeVriesEquation . . . . . . . . . . . . . . . . . . 142
xxiii
xxiv Contents
5.2 Kadomtsev and Petviashvili Equation . . . . . . . . . . . . . . . . 149
5.3 EquationofTransonicGasFlows . . . . . . . . . . . . . . . . . . 155
5.4 Short-Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . 161
5.5 Khokhlov and Zabolotskaya Equation . . . . . . . . . . . . . . . . 168
5.6 Equation of Geopotential Forecast . . . . . . . . . . . . . . . . . . 172
6 Nonlinear Schrödinger and Davey–Stewartson Equations . . . . . . 179
6.1 Nonlinear Schrödinger Equation . . . . . . . . . . . . . . . . . . . 179
6.2 Coupled Schrödinger Equations . . . . . . . . . . . . . . . . . . . 187
6.3 DaveyandStewartsonEquations . . . . . . . . . . . . . . . . . . 201
7 Dynamic Convection in a Sea . . . . . . . . . . . . . . . . . . . . . . 213
7.1 Equations andSymmetries . . . . . . . . . . . . . . . . . . . . . . 213
7.2 Moving-Line Approach . . . . . . . . . . . . . . . . . . . . . . . 216
7.3 Cylindrical Product Approach . . . . . . . . . . . . . . . . . . . . 219
7.4 Dimensional Reduction . . . . . . . . . . . . . . . . . . . . . . . 223
8 Boussinesq Equations in Geophysics . . . . . . . . . . . . . . . . . . 231
8.1 Two-Dimensional Equations . . . . . . . . . . . . . . . . . . . . . 231
8.2 Three-Dimensional Equations and Symmetry . . . . . . . . . . . . 247
8.3 Asymmetric Approach I . . . . . . . . . . . . . . . . . . . . . . . 249
8.4 Asymmetric Approach II . . . . . . . . . . . . . . . . . . . . . . 255
8.5 Asymmetric Approach III . . . . . . . . . . . . . . . . . . . . . . 261
9 Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . 269
9.1 Background and Symmetry . . . . . . . . . . . . . . . . . . . . . 269
9.2 Asymmetric Approaches . . . . . . . . . . . . . . . . . . . . . . . 273
9.3 Moving-Frame Approach I . . . . . . . . . . . . . . . . . . . . . 285
9.4 Moving-Frame Approach II . . . . . . . . . . . . . . . . . . . . . 296
10 Classical Boundary Layer Problems . . . . . . . . . . . . . . . . . . 317
10.1 Two-Dimensional Problem . . . . . . . . . . . . . . . . . . . . . 317
10.2 Three-Dimensional Problem: General . . . . . . . . . . . . . . . . 326
10.3 Uniform Exponential Approaches . . . . . . . . . . . . . . . . . . 332
10.4 Distinct Exponential Approaches . . . . . . . . . . . . . . . . . . 344
10.5 Trigonometric and Hyperbolic Approaches . . . . . . . . . . . . . 350
10.6 Rational Approaches . . . . . . . . . . . . . . . . . . . . . . . . . 362
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
· · · · · · (收起)