ISBN:9787302307730
Pricing: 20 yuan
Impression: 1- 1
Binding: paperback
Print date: 20 13- 1-23
Book Introduction: This book closely combines the teaching practice of engineering mathematics, and systematically introduces the establishment of partial differential equation model, several common solutions of three typical equations, special functions, several simple special solutions of linear partial differential equations and some simple special solutions of nonlinear partial differential equations. This book is concise, emphasizing the practical background of mathematical concepts and methods. While paying attention to introducing the necessary theories, we should highlight the methods to solve problems. The content of the book is simple, the methods are diverse, the text is easy to understand, and there are a lot of difficult and difficult examples and exercises. This book can be used as a teaching material for undergraduates and postgraduates majoring in physics, mechanics and engineering, and also as a teaching reference book for undergraduates majoring in information and computational mathematics. In addition, it can also be used as a reference for mathematicians, physicists and engineers.
Derivation and definite solution of typical equations in chapter 1 of the catalogue ............................................................................ 1
1. 1 Derivation of Typical Equation ........................................................................................... 1
1. 1. 1 string vibration equation ........................................................................................ 2
1. 1.2 heat conduction equation ........................................................................................
1. 1.3 transmission line equation ........................................................................................ 6
1. 1.4 electromagnetic field equation ........................................................................................ VII
1.2 definite solution conditions and definite solution problems .................................................................................... 8
1.2. 1 definite solution condition ............................................................................................ 8
1.2.2 definite solution problem ..........................................................................................
1.3 classification of second-order linear partial differential equations ........................................................................ 1 1 exercise 1 ................................................................................................................. 12
Chapter 2 Fourier series method-characteristic expansion method and variable separation method ............................................. 14.
2. 1 Preparatory knowledge ....................................................................................................
2. 1. 1 orthogonal function system ...................................................................................... 15.
2. 1.2 superposition principle of linear equations ........................................................................ 16
2.2 homogenization principle ................................................................................................ 16
2.2. 1 homogenization principle of second-order linear ordinary differential equations with constant coefficients ......................................... 17
2.2.2 Homogenization principle of initial-boundary value problems of string vibration equation and heat conduction equation ........................... 19.
2.3 eigenvalue problem ................................................................................................
2.3. 1 Question ...................................................................................... 20
Stourm-joseph liouville Issue .......................................................................... 2 1
2.3.3 Example ................................................................................................. 22
2.4 Characteristic expansion method ................................................................................................
2.4. 1 Initial-boundary value problem of heat conduction equation ................................................................. 25
2.4.2 Initial-boundary value problem of string vibration equation ................................................................. 27
2.5 Variable Separation ................................................................................................ 29
2.5. 1 Free vibration of bounded string .....................................................................
Four. Table of Contents 2.5.2 ..................................................................... of Heat Conduction on Bounded Rod 33
2.5.3 The definite solution of Laplace equation ................................................................. 34
2.6 Treatment of non-homogeneous boundary conditions ............................................................................... 38
2.7 Physical Meaning, Standing Wave Method and * * * Vibration ............................................................................ 4 1 Exercise 2 ................................................................................................................. 43
The third chapter integral transformation and its application ........................................................................................ 47
3. 1 Fourier transform ................................................................................................ 47
3.2 the application of Fourier transform ...................................................................................... 50
3.2. 1 Initial value problem of heat conduction equation ..................................................................... 50
3.2.2 Initial value problem of string vibration equation ..................................................................... 53
Integral equation .......................................................................................... 56
.3.3 Semi-unbounded Problem: Symmetric Extension Method ............................................................................ 57
3.4 Laplace transform ............................................................................................. 58
3.4. 1 laplace transform concept ........................................................................ 58
3.4.2 Properties of Laplace Transform ........................................................................ 59
Application of Laplace Transform ........................................................................ 6 1 Exercise 3 ................................................................................................................. 65
Chapter IV Initial Value Problems of Hyperbolic Equations-Traveling Wave Method, Spherical Average Method and Dimension Reduction Method ............................ 68
4. 1 traveling wave method for initial value problem of string vibration equation ................................................................. 68
4.2 the physical meaning of D'Alembert formula ........................................................................... 70
4.3 ...................................................... of spherical average method for initial value problem of three-dimensional wave equation 72
4.3. 1 Spherically symmetric solution of three-dimensional wave equation ................................................................. 72
4.3.2 Poisson formula of three-dimensional wave equation ................................................................. 73
4.4 Dimension reduction method for initial value problem of two-dimensional wave equation ............................................................. 75
4.5 Physical Meaning of Poisson Formula and Huygens Principle .............................................................. 77 Exercise 4 ................................................................................................................. 78
Chapter 5 Green's Function Method of Potential Equation ............................................................................. 8 1
5. 1 δ function ........................................................................................................ 8 1
5. The concept of1.1δ-function ................................................................................... 8 1
5. Properties of δ function of1.2 ................................................................................... 82
5.2 Green's Formula and Basic Solution ...................................................................................... 83
Directory V.5.2. 1 green's formula .......................................................................................... 83
5.2.2 Basic understanding of ............................................................................................. 83
5.3 Basic integral formula and some basic properties of harmonic function ................................................... 85
5.4 Green's Function .................................................................................................... 86
5.5 Green's Function in Special Region and the Solution of Dirichlet Boundary Value Problem ........................................ 88
5.5. 1 Green's function and Poisson's formula in the upper half space ........................................................ 88
5.5.2 Green's Function and Poisson's Formula on Sphere ............................................................... 90
5.6 Conformal transformation and its application ...................................................................................... 92
5.6. 1 conformal ............................................................................. analysis function 92
5.6.2 Common conformal mapping ................................................................................ 94
5.6.3 Using conformal mapping to solve two-dimensional stable field problem .................................................... 99 Exercise 5. ............................................................................................................... 10 1
Chapter VI Special Functions and Their Applications ...................................................................................... 104
6. 1 problem derivation .............................................................................................. 104
6.2 Bessel function .............................................................................................. 106
6.2. Series solution of1Bessel equation .................................................................... 106
6.2.2 Properties of Bessel Function ........................................................................... 109
Other types of Bessel functions .................................................................... 1 14.
6.3 the application of Bessel function .................................................................................... 1 16
6.4 Legendre function .............................................................................................. 1 19
6.4. Power series solution of1Legendre equation .................................................................... 1 19
The nature of legendre polynomials ....................................................................... 12 1
6.4.3 Joint Legendre Equation .............................................................................. 123
6.5 Application of legendre polynomials ................................................................................ 1 24 exercise 6 ...............................................................................................................125
Chapter VII Special Solution and Special Solution ...................................................................................... 128
7. 1 power series solution of initial value problem of linear evolution equation ........................................................... 128
7.2 migration equation .................................................................................................. 132
7.3 hopf-Cole Transform .......................................................................................... 134
7.3. Hopf-Cole transformation of1............................................................... Berg equation 134.
7.3.2 Generalized Hopf-Cole Transform of KDV Equation ........................................................ 136
7.4 Self-similar solution .................................................................................................. 138
List of intransitive verbs 7.5 Traveling wave solutions ..................................................................................................... 14 1
7.5. 1 direct integration method ..................................................................................... 142
Undetermined derivative method ..................................................................................... 143
Undetermined coefficient method ..................................................................................... 1 45 Exercise 7 ...............................................................................................................1 47 Appendix A Hyperbolic Function ................................................................................................... 1 49 Appendix B Integral Conversion Table .............................................................................................. 1 50 appendix c Bessel function zero table ................................................................................. 1 52 Appendix D Practice Reference Answer ..................................................................................... 1 Cite ................................................................................................................. 16 1
Topic: Mathematical Physics Methods: National Planning Textbook for General Higher Education [15]
Book number: 2 159044
Press: Science
Pricing: 40.0.
ISBN:7030 12 173
Author: Shao Huimin editor
Date of publication:
Version: 1
Format: 16
Introduction:
This book is the research result of the Ministry of Education's "Reform Plan of Teaching Content and Curriculum System of Higher Education Facing 2 1 century", and it is a curriculum textbook facing 2 1 century and a national planning textbook for general higher education during the Tenth Five-Year Plan period.
This book systematically expounds the basic theory of mathematical physics method and its application in physics and engineering technology. The key point is not to blindly pursue the rigor and logic of mathematics, that is, the integrity of pure mathematical theory, but to strive to provide readers with basic concepts, basic theorems and various methods and skills related to mathematical physics methods. Although this book involves some traditional contents, it is stronger in depth and breadth than the previous textbooks; At the same time, the book also adds a lot of contents that reflect the frontier of the discipline, so that students can not only acquire systematic scientific knowledge of related disciplines, but also guide students to enter the frontier of contemporary science. In addition, another feature of this book is that readers can not only obtain simplified and unified basic knowledge of mathematics from the logical structure of this book, but also see unique, concise and practical solutions from the examples in the book.
This book can be used as an undergraduate teaching material for non-mathematics majors of science and engineering in colleges and universities, and can also be used as a reference for graduate students, teachers and scientific and technological personnel of related majors.
Directory:
Chapter 1 Functions of Complex Variables
The concept of 1. 1 complex number
Geometric representation of 1.2 complex number
Operation of 1.3 Complex Numbers
1.4 complex variable function
Limit of 1.5 complex variable function
1.6 Continuity of Complex Variable Function
utilize
Chapter II Analytic Functions
2. 1 derivative of complex variable function
2.2 Cauchy-Riemann condition
2.3 Analysis function
2.4 the relationship between analytic function and harmonic function
2.5 Elementary analytic function
2.6 Application of Analytic Function-Complex Potential of Plane Field
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Chapter III Integration of Complex Variable Functions
3. 1 Basic concepts
3.2 Complex Variable Functions and Integrals
3.3 Cauchy Theorem
3.4 Cauchy integral formula
3.5 Several Inferences of Cauchy Integral Formula
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The fourth chapter is the power series representation of analytic functions.
4. 1 complex term series
4.2 Series of complex variable functions
4.3 power series
4.4 Power Series Expansion of Analytic Functions
4.5 Isolated Singularities of Analytic Functions
4.6 Properties of Analytic Function at Infinite Point
4.7 Analytical continuation
4.8 Application
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Chapter 5 residue theory and its application
5. 1 Basic theory of residue
5.2 Using residue theorem to calculate real integral
5.3 Logarithmic residual and radial angle principle
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Chapter VI Generalized Functions
6. 1 δ function
6.2 Introduction to Generalized Functions
6.3 Basic operations of generalized functions
6.4 Fourier Transform of Generalized Functions
6.5 General solution
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Chapter 7 expansion method of completely orthogonal function system
7. 1 orthogonality
7.2 Zero function
7.3 Integrity
7.4 Promotion
Chapter VIII Sturm-Liu Wei eigenvalue problem.
8. Formulation of1eigenvalue problem
8.2 Main conclusions of eigenvalue problem
8.3 Other types of eigenvalue problems
Chapter 9 Fourier series and Fourier transform
9. 1 Periodic Function and Fourier Series
9.2 Complete orthogonal function system
9.3 Properties of Fourier Series
9.4 Application of Fourier Series
9.5 Fourier function series on finite interval
9.6 Fourier series in complex exponential form
9.7 the connection between Fourier expansion and Laurent expansion
9.8 Fourier integration and transformation
9.9 Properties of Fourier Transform
9. Brief Introduction of10 Wavelet Transform
9. Three definitions of11
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Chapter 10 Laplace Transform
The concept of Laplace transform 10. 1
Laplace transform of 10.2 basic function
Properties of Laplace Transform of 10.3
10.4 Laplace inverse transform
10.5 application
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Chapter 11 Series Solutions of Second Order Linear Ordinary Differential Equations
The series solution of the neighborhood of 1 1. 1 invariant point
The Series Solution of the Neighborhood of 1 1.2 Regular Singularity
1 1.3 the second solution
Asymptotic solution of 1 1.4 irregular singularity
1 1.5 asymptotic expansion and steepest descent method
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Chapter 12 Mathematical Model-Definite Solution Problem
12. 1 Introduction
Establishment of 12.2 mathematical model
12.3 definite solution conditions
12.4 definite solution problem
12.5 solution method
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Chapter XIII Classification of Second Order Linear Partial Differential Equations
The basic concept of 13. 1
13.2 classification and standardization of second-order linear partial differential equations
13.3 Further Simplification of Second Order Linear Constant Coefficient Partial Differential Equation
Physical Connotation of 13.4 Three Equations
Characteristics of 13.5 second-order linear partial differential equation
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Chapter 14 Traveling Wave Method
General solution of 14. 1
14.2 traveling wave solution
14.3 D'Alembert formula
Free vibration of 14.4 semi-infinite string
Free vibration of a string fixed at both ends
14.6 homogenization principle (Duhamel principle)
14.7 nonlinear partial differential equation
utilize
Chapter 15 Separation of Variables Method
15. 1 separation variable
Separation of variables in 15.2 rectangular coordinate system
15.3 method of separating variables in cylindrical coordinate system
15.4 separation variables in spherical coordinate system
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Chapter 16 Legendre Function
The definition and expression of 16. 1 legendre polynomials
16.2 properties of legendre polynomials
Legendre function of the second kind Q 1(x)
The eigenvalue problem of Legendre equation 16.4
16.5 joint Legendre equation and its solution
16.6 spherical harmonic function
16.7 application
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Chapter 17 Bessel function
17. 1 Bessel equation and its solution
17.2 integer order (first kind) Bessel function
17.3 modified Bessel equation and its solution
17.4 spherical Bessel equation and spherical Bessel function
17.5 generalized Bessel function
17.6 application
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Chapter 18 Integral Transformation Method
18. 1 Fourier transform
18.2 Laplace transform
18.3 Fourier sine transform
18.4 Fourier cosine transform
18.5 hankel transform
18.6 is applicable to bounded areas.
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Chapter 19 Variational Method
The basic concept of 19. 1
Extreme value of 19.2 functional
19.3 the relationship between functional extremum and mathematical physics problems
19.4 A Direct Method for Finding Functional Extremum —— Ritz Method
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Chapter 20 Green's Function Method
20. 1 Green formula
20.2 Green's function method for steady-state boundary value problems
20.3 Green's function method for heat conduction problems
20.4 Green's Function Method for Wave Problems
20.5 Determination of Green's Function
20.6 application
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Chapter 21 Conformal Transformation Method
Conformal transformation of 2 1. 1 and its basic problems
Several conformal transformations commonly used in 2 1.2
Transformation of 2 1.3 Polygon
2 1.4 application
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Main bibliography