# Math 497A Summer Schedule

This special topics course is part of the international undergraduate program organized by Penn State and Peking University.

Summer Program Schedule:

Main topics for the summer program: (37 lectures)

Introduction (1.1, 1.2, 1.3, 1.4)

Elliptic equations in 1D: Two point boundary value problems

General solutions and the Green's function (2.1.1, 2.1.2)

Smoothness and the maximum principle (2.1.3, 6.1)

A finite difference approximation (2.2.1, 2.2.2)

Gaussian elimination of a tridiagonal system (2.2.3)

The finite difference equation (2.3.1)

The maximum principle and convergence (2.3.4, 2.3.5)

Elliptic equations in 2D: Poisson's equation

General solutions in a rectangular domain (7.1)

General solutions in a disc (7.2.1)

Maximum principle for Harmonic functions (6.4.1)

A finite difference approximation (7.5)

An error estimate (7.6)

Finite element approximation* (2 lectures)

Solution of the linear system* (2 lectures)

Parabolic equations in 1D: The heat equation

General solutions: separation of variables (3.2)

Fourier coefficients (3.3, 3.4)

Neumann problem (3.6)

Fundamental solutions and the heat kernel*

Energy estimate (3.7)

Maximum principle (6.2.1)

A finite difference scheme (4.1)

Von Neumann stability analysis (4.3.1) (2 lectures)

Numerical stability by energy method (4.5)

Implicit methods (4.4)

Maximum principle for the difference equation (6.2.3, 6.2.4)

Truncation error (10.3)

Convergence analysis (10.3)

The heat equation in 2D

Finite difference approximation*

Stability analysis*

Finite element method*

Hyperbolic equations in 1D: the wave equation

General solutions: separation of variables (5.1)

Energy estimate (5.2)

Finite difference scheme (5.3)

Stability analysis (5.3.1)

Summer Program Schedule:

Main topics for the summer program: (37 lectures)

Introduction (1.1, 1.2, 1.3, 1.4)

Elliptic equations in 1D: Two point boundary value problems

General solutions and the Green's function (2.1.1, 2.1.2)

Smoothness and the maximum principle (2.1.3, 6.1)

A finite difference approximation (2.2.1, 2.2.2)

Gaussian elimination of a tridiagonal system (2.2.3)

The finite difference equation (2.3.1)

The maximum principle and convergence (2.3.4, 2.3.5)

Elliptic equations in 2D: Poisson's equation

General solutions in a rectangular domain (7.1)

General solutions in a disc (7.2.1)

Maximum principle for Harmonic functions (6.4.1)

A finite difference approximation (7.5)

An error estimate (7.6)

Finite element approximation* (2 lectures)

Solution of the linear system* (2 lectures)

Parabolic equations in 1D: The heat equation

General solutions: separation of variables (3.2)

Fourier coefficients (3.3, 3.4)

Neumann problem (3.6)

Fundamental solutions and the heat kernel*

Energy estimate (3.7)

Maximum principle (6.2.1)

A finite difference scheme (4.1)

Von Neumann stability analysis (4.3.1) (2 lectures)

Numerical stability by energy method (4.5)

Implicit methods (4.4)

Maximum principle for the difference equation (6.2.3, 6.2.4)

Truncation error (10.3)

Convergence analysis (10.3)

The heat equation in 2D

Finite difference approximation*

Stability analysis*

Finite element method*

Hyperbolic equations in 1D: the wave equation

General solutions: separation of variables (5.1)

Energy estimate (5.2)

Finite difference scheme (5.3)

Stability analysis (5.3.1)

Good.