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Mathematical and physical equations:

Applicable majors: electronic information science and technology, applied physics.

Prerequisite courses: college physics, advanced mathematics, complex variable function, field theory, vector algebra.

First, the teaching objectives and tasks of the course

Mathematical physical equations are important basic courses and tools in physics, electronic information science and communication science.

Its main feature lies in the close combination of mathematics and physics. Mathematical methods are applied to the analysis of specific problems in practical physics and interdisciplinary science, and mathematical models (partial differential equations) are established through physical processes. By solving and analyzing the model, we can further understand the specific physical process and improve the ability to analyze and solve practical problems.

Mathematical physics method is a pure theoretical course. In teaching, we combine classroom teaching (mainly), after-class exercises and computer exercises, and pay attention to the classroom discussion in the practice class.

The course content includes three parts: the first part is the review of transcendental knowledge such as vector analysis and field theory; The second part is the establishment and routine solution of mathematical and physical equations; Including: definite solution problem, traveling wave method, variable separation method, integral transformation method, Green's function method, variational method and so on. The third part is special function, including legendre polynomials, Bessel function and Sturm-Liu Wei eigenvalue problem.

This course will combine the professional characteristics of applied physics and electronic information, make full use of numerical calculation technology, combine the characteristics of mathematical physics methods, break through the difficulties of the course of mathematical physics methods by optimizing the teaching material system and visual analysis of calculation examples, and improve students' learning interest and ability to analyze and solve problems.

Second, the connection and division of labor between this course and other courses

Before entering this course, students should take courses including: college physics, advanced mathematics, complex variable function, field theory and vector algebra. The study of these courses has laid a good mathematical foundation for this course.

After the completion of this course, you can enter the following courses: four major mechanics, electromagnetic field and microwave technology, modern physical experiments, etc.

Third, the course content and basic requirements

(1) Introduction and review of prerequisite knowledge: (2 class hours)

1, the basic concept of vector, the basis of algebraic operation vector analysis;

2. Basis of field theory (gradient, divergence and curl of vector field);

3. Complex variable function integration;

4. Surplus theory.

2) Establishment and definite solution of mathematical and physical equations: (8 class hours)

1, three basic equations are established: string vibration equation, heat conduction equation and Poisson equation;

2. Definite solution conditions: initial conditions, three boundary conditions, natural boundary conditions and connection conditions.

(3) Traveling Wave Method: (6 class hours)

1, D'Alembert formula, traveling wave solution of one-dimensional problem;

2. Poisson's formula and the average method of transforming three-dimensional problems into one-dimensional problems;

3. Pulse method is used to solve the non-homogeneous problem and delay potential.

(4) Variable separation method: (10 class hour)

1. Free vibration and heat conduction of bound strings;

2. The eigenvalue problem of 2.Sturm-Liouville equation (ordinary differential equation);

3. The definite solution of nonhomogeneous general equation problem:

4. Processing method of non-homogeneous boundary conditions;

5. Separate variables in orthogonal curvilinear coordinates (spherical coordinates and cylindrical coordinates).

(5) Special functions: (12 class hours)

1, the basic properties of legendre polynomials and legendre polynomials;

2. Correlate Legendre function and spherical harmonic function;

3. Separation of variables in spherical coordinate system:

4. Bessel function and its properties, integrating with Bessel function;

5. Calculation and simulation of other column functions and special functions;

6. Separation of variables in column coordinates.

(6) Integral transformation method: (8 class hours)

1, Fourier integral and Fourier transform properties;

2. Solving mathematical equations by Fourier transform;

3. Laplace transform and its properties;

4. Laplace transform method.

(7) Green's function method: (8 class hours)

1, function, Poisson equation boundary value problem, Green's formula;

2. General solution of Green's function;

3. Solving Dirichlet Green's function in some special regions by electric image method;

4. Calculation and simulation of application of Green's function method.

(8) Other common solutions to mathematical and physical equations: (6 class hours)

1, the solution method of nonlinear equation;

2. Integral equation method;

3. Variational method.

1. Basic requirements

This course requires students to understand the methods of establishing mathematical and physical equations, focusing on the establishment and routine solution of three kinds of commonly used partial differential equations; Including: definite solution problem, traveling wave method, variable separation method, integral transformation method, Green's function method, variational method and so on. Master the application of special functions (including legendre polynomials, Bessel function, Sturm-Liuwei eigenvalue problem, etc.). ) in mathematical and physical equations. Learn and improve the ability to analyze and solve practical problems.

2. Key points and difficulties

Emphasis: definite solution problem, traveling wave method, variable separation method, integral transformation method, Green's function method.

Difficulties: special function, Green's function method

Preparatory course of numerical calculation methods: mathematical analysis, advanced algebra, ordinary differential equations, functional analysis

I. Basic contents

Absolute error and relative error, influence of error on calculation, stability

I. Basic requirements

1. Understand the concepts of absolute error and relative error.

2. Understand the influence of error on calculation.

3. Understand the concept of stability

Second, the proposed class arrangement:

Chapter ii algebraic interpolation

I. Basic contents

Lagrange interpolation, Newton interpolation, piecewise low-order polynomial interpolation, ENO interpolation, Hermite interpolation, cubic spline interpolation.

Second, the basic requirements

1. Master the construction of Lagrange interpolation polynomial and the estimation of truncation error.

2. Master the construction of Newton interpolation polynomial and the properties of difference quotient.

3. Master the construction and characteristics of piecewise low-order interpolation polynomials.

4. Master the structure and characteristics of ENO interpolation polynomial.

5. Master the construction and characteristics of Hermite interpolation polynomial.

6. Master the construction and characteristics of cubic spline interpolation polynomial.

Three. Suggested curriculum:

1. Lagrange interpolation

2. Newton interpolation

3. Piecewise low-order interpolation

4.Eno interpolation

5. Hermite interpolation

6. Cubic spline interpolation

Chapter III Function Approximation

I. Basic contents

Best uniform approximation polynomial, best square approximation, orthogonal polynomial, least square method, Fourier approximation and fast Fourier transform.

Second, the basic requirements

1. Master the concept of best uniform approximation and understand Chebyshev theorem.

2. Master the concept of optimal square approximation.

3. Master the properties of Legendre orthogonal polynomials and Chebyshev polynomials.

4. Master the least square method of curve fitting.

5. Master Fourier approximation and fast Fourier transform.

Three. Suggested curriculum:

1. Best uniform approximation

2. Best square approximation

3. Orthogonal polynomials

4. Least square method

5. Fourier approximation and fast Fourier transform

Chapter IV Numerical Integration and Numerical Differentiation

I. Basic contents

Interpolation quadrature formula, complex quadrature method and Romberg integral, Gaussian formula, numerical differentiation

Second, the basic requirements

1. Understand the basic idea of numerical quadrature, master the concept of algebraic precision, and master several low-order interpolation quadrature formulas.

2. Master several low-order complex quadrature formulas and understand the idea of Rhomberg algorithm.

3. Understand the idea of Gauss quadrature formula and master the construction of Gauss quadrature formula.

4. Understand the idea of numerical differentiation and master several low-order interpolation derivative formulas.

Three. Suggested curriculum:

1. Interpolation quadrature formula

2. Complex quadrature method and Romberg integral

3. Gauss formula

4. Numerical differentiation

Chapter 5 Numerical Solutions of Ordinary Differential Equations

I. Basic contents

Euler method, Runge-Kutta method, convergence and stability of one-step method, linear multi-step method, equation and higher-order equation.

Second, the basic requirements

1. Master Euler Method

2. Main Runge-Kutta method

3. Understand and master the concepts of convergence and stability of one-step method.

4. Master the idea and construction method of linear multi-step method.

5. Understand the numerical solution of the first-order equation and the idea of transforming the higher-order equation into the first-order equation.

Three. Suggested curriculum:

1. Euler method

2. Runge-Kutta method

3. Convergence and stability of one-step method

4. Linear multi-step method

5. Equations and higher-order equations

Chapter 6 Finding the Root of Equation

I. Basic contents

Finding roots, iterative method, Newton method, chord section method and parabola method, finding algebraic equations of roots

Second, the basic requirements

1. Master the dichotomy

2. Master the construction and convergence conditions of general iterative method.

3. Master the structure and convergence characteristics of Newton method.

4. Master the construction of iterative formulas of chord section method and parabola method.

5. Understand several algorithms for finding roots of algebraic equations.

Three. Suggested curriculum:

1. root search

2. Iterative method

3. Newton method

4. Chord section method and parabola method

5. Find the roots of algebraic equations

Chapter 7: Direct method and iterative method for solving linear equations.

I. Basic contents

Variations of gauss elimination and gauss elimination, Norm and Error Analysis of Vector and Matrix, Jacobian Iteration and Gauss-Cedell Iteration, Convergence of Iteration, Super Relaxation Iteration.

Second, the basic requirements

1. Principal Gaussian elimination method

2. Master several Gaussian elimination methods of deformation.

3. Master the definitions of vector and matrix norm and the concept of matrix condition number.

4. Master Jacobian iteration method and Gauss-Cedell iteration method.

5. Grasp the convergence conditions of iterative method.

6. Understand the idea of super-relaxation iteration method.

Three. Suggested curriculum:

1. Gaussian elimination method

2. The deformation of Gaussian elimination method

3. Vector sum matrix norm and error analysis

4. Jacobian iteration method and Gauss-Cedell iteration method.