1. Introduction to Partial Differential Equations
Weijiu Liu
Department of Mathematics
University of Central Arkansas

2. Overview
11/9/2018

3. What are PDEs? An equation containing an unknown function and its
partial derivatives:
11/9/2018

4. Three Big Classes of Equations
Parabolic equations. Two simple example:
Heat (diffusion) equation describing heat conduction:
11/9/2018

5. Convection-diffusion equation describing chemical diffusion
and convection:
Convection
Diffusoin
11/9/2018

6. Hyperbolic equations: Wave equation describing the string vibration
11/9/2018

7. Elliptic equations: Laplace’s equation
11/9/2018

8. What we want to do about the equations
Find an exact solution
Find a numerical solution
Study their well-posedness
Study their stability
Design a control law to force their solution to your desired one
More …
11/9/2018

9. What are methods to find a solution
Separation of variables
Fourier series
Fourier transformation
Laplace transformation
Method of characteristics
Green functions
D’Alembert’s formulas
Symmetry analysis
More …
11/9/2018

10. Tentative schedule Derivation of the heat equation, 1 week
Separation of variables for solving the heat equation and Laplace’s equation, 2 weeks
Fourier series, 3 weeks
Derivation of the wave equation, 1 week
Separation of variables for solving the wave equation, 1week
High dimensional equations, 3 weeks
Non-homogeneous problems, 2 weeks
Equations on infinite domains, 1 week
11/9/2018

11. Equilibrium
11/9/2018

12. Dirichlet Boundary Conditions
Steady equation
11/9/2018

13. Neumann Boundary Conditions
Steady equation
11/9/2018