Partial Differential Equations

This course aims to introduce you to some basic concepts and properties of first-order partial differential equations, wave equation, heat equation, and Laplace's equation as well as the numerical finite-difference methods for solving partial differential equations. This course will also develop your skills in solving some important partial differential equations with various auxiliary conditions. What you learned in this course will have a wide application in real life applications as well as for numerous further graduate courses.

Course Content:

• Definition of partial differential equation (PDE); linearity, order, solution, heterogeneity of a partial differential equation; Uniqueness of the solution to a partial differential equation; Types of partial differential equation.
• Geometric method and Cartesian method for solving the first-order partial differential equation with constant or variable coefficients; Operator factorization method for solving the second-order constant coefficient PDEs.
• Derivations of the D'Alembert solution formula for the wave equation and the solution formula for the heat equation on the whole axis; Introduction to the reflection method for the solution formulas of the wave equations and heat equations on the half line with Dirichlet boundary condition and Neumann boundary condition;
• Introduction to the method of separation of variables and solve wave equations, heat equations, and Laplace equations on finite intervals.
• Fourier series expansion including Fourier cosine series, Fourier sine series, and full Fourier series.
• Finite-difference method for solving PDEs.