The basic concept of 1. 1

1. 1. 1 Basic concepts of partial differential equations

1. 1.2 Three Common Mathematical and Physical Equations

1. 1.3 general problems of mathematical and physical equations

1.2 Derivation of Mathematical and Physical Equations

Derivation of 1.2 1 wave equation

Derivation of 1.2.2 transport equation

1.2.3 derivation of stable field equation

1.3 definite solution conditions and definite solution problems

1.3. 1 initial conditions

1.3.2 boundary conditions

1.3.3 Three kinds of definite solution problems

1.4 Summary of this chapter

Exercise 1

Chapter II Traveling Wave Method

2. 1 D'Alembert formula of one-dimensional wave equation

2. Derivation of1.1D'Alembert formula.

2. 1.2 Physical Meaning of D'Alembert Formula

2. 1.3 Dependency interval and influence area

2.2 Free vibration of semi-infinite strings

2.3 Poisson formula of three-dimensional wave equation

2.3. 1 average method

Poisson formula

2.3.3 Physical Meaning of Poisson Formula

2.4 Forced vibration

2.4. 1 pulse principle

Pure forced vibration

2.4.3 General forced vibration

2.5 General wave problems in three-dimensional unbounded space

2.6 Overview of this chapter

Exercise 2

Chapter III Separation of Variables Method

3. 1 double homogeneous problem

3. 1. 1 Free vibration of bounded strings

3. 1.2 Heat conduction of uniform thin rod

3. 1.3 problem of stable field distribution

3.2 eigenvalue problem

3.2. 1 Sturm-Liuwei equation

3.2.2 eigenvalue problem of Sturm-Liuwei type equation

3.2.3 Properties of Sturm-Liu Wei eigenvalue problem

3.3 Treatment of non-homogeneous equations

3.3. 1 eigenfunction expansion method

Pulse principle method

3.4 Treatment of non-homogeneous boundary conditions

3.4. 1 homogenization principle of boundary conditions

3.4.2 Treatment of other heterogeneous boundary conditions

3.5 Separation of variables in orthogonal curvilinear coordinate system

3.5. 1 definite solution of two-dimensional laplace equation in circular domain

3.5.2 Basic concept of separating variables in orthogonal curvilinear coordinate system

3.5.3 Separation of variables in orthogonal curvilinear coordinate system

3.6 Overview of this chapter

Exercise 3

Chapter IV Special Functions

4. 1 Series Solution of Second Order Linear Ordinary Differential Equation

4. 1. 1 Constants and Singularity of Second Order Linear Ordinary Differential Equations

4. Series solutions in the neighborhood of fixed points of1.2 equation

4. Series Solutions in the Neighborhood of Regular Singularity of1.3 Equation

4.2 legendre polynomials

4.2. 1 legendre polynomials

4.2.2 Differential and Integral Representation of legendre polynomials

4.3 Nature of legendre polynomials

4.3. Generating function of1Legendre function

4.3.2 legendre polynomials's Recursive Formula

Orthogonality of legendre polynomials

4.3.4 Generalized Fourier Series Expansion

4.4 legendre polynomials's application in solving mathematical equations

4.5 Joint Legendre Function

4.5. The eigenvalue problem of1Legendre function

4.5.2 Properties of Joint Legendre Function

4.5.3 Application of Joint Legendre Function in Solving Mathematical Equations

4.6 spherical function

General definition of spherical function

4.6.2 Orthogonality of Spherical Functions

4.6.3 Application of Spherical Function

4.7 Bessel function

4.7. 1 Three Bessel functions (solutions of Bessel equation)

4.7.2 Eigenvalue Problem of Bessel Equation

4.8 Properties of Bessel Function

4.8. Generating Function and Integral Representation of1Bessel Function

4.8.2 Recursive Relation of Bessel Function

4.8.3 Orthogonality of Bessel Function

4.8.4 Generalized Fourier-Bessel Series Expansion

4.9 Other column functions

4.9. 1 spherical Bessel function

4.9.2 Virtual parameter Bessel function

4. The application of10 Bessel function

4. 1 1 Summary of this chapter

Exercise 4

Chapter V Integral Transformation Method

5. 1 Fourier transform

5. 1. 1 Fourier integral

5. 1.2 Fourier transform

5. Physical Meaning of1.3 Fourier Transform

5. Properties of1.4 Fourier Transform

Fourier transform of 5. 1.5 δ function

5.1.6 n-dimensional Fourier transform

5.2 Fourier transform method

5.2. 1 wave problem

5.2.2 Transportation problems

5.2.3 The problem of stable field

5.3 Laplace transform

5.3. 1 laplace transform

5.3.2 Basic Theorem of Laplace Transform

5.3.3 Basic Properties of Laplace Transform

5.4 Application of Laplace Transform

5.4. 1 laplace transform to solve ordinary differential equations

5.4.2 Laplace transform to solve partial differential equations

5.5 Overview of this chapter

Exercise 5

Chapter VI Green's Function Method

6. 1δ function

6. Definition of1.1δ function

6. Properties of1.2 δ function

6. Application of1.3 δ function

6.2 Green's Function Method for Boundary Value Problems of Poisson Equation

6.2. General concept of1Green function

6.2.2 Basic integral formula of Poisson equation

6.3 General solution of Green's function

Green's function in unbounded space

6.3.2 Green's Function for General Boundary Value Problems

Electronic image method

6.3.4 Application of Electric Image Method and Green's Function

6.4 Other Solutions of Green's Function

6.4. 1 eigenfunction expansion method to solve the boundary value problem of Green's function

6.4.2 Solving the variation of Green's function with time by pulse method

6.5 Overview of this chapter

Exercise 6

Chapter VII Other Solutions to Mathematical and Physical Equations

7. 1 continuation method

7. Heat conduction of1.1semi-infinite rod

7. 1.2 Free vibration of bounded strings

7.2 Conformal transformation method

7.2. Definition and Conformal Transformation of1univalent Analytic Function

The Solution of Laplace Equation

7.3 Iterative solution of integral equation

7.3. Several classifications of1integral equation

Iterative solution

7.4 Variational method

7.4. 1 functional and extreme value of functional

Ritz method

Chapter VIII Visual Calculation of Mathematical and Physical Equations

8. 1 visual calculation of separation variable method

The Solution of Poisson Equation in 8. 1. 1 Rectangular Region

Application of separation variable method in electromagnetic field of 8. 1.2 rectangular coordinate system

8.2 Application of Special Functions

8.2. 1 The plane wave is expanded into the superposition of cylindrical waves.

8.2.2 Plane waves are expanded into the superposition of spherical waves.

8.2.3 Application of Special Functions in Wave Problems

8.2.4 Analytical solution of spherical radar cross section

8.3 intuitive calculation of integral transformation method

8.4 Visual Calculation of Green's Function

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