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

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Overview

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What are PDEs?
2 2
2 2
sin cos
u u u u u u
x y u xyt
t x y x x y
     
    
     
An equation containing an unknown function and its
partial derivatives:

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Three Big Classes of Equations
2 2
2 2
u u u
k
t x y
 
  
 
 
  
 
1. Parabolic equations. Two simple example:
• Heat (diffusion) equation describing heat conduction:

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   
2 2
1 2 2 2
, , , ,
u u u u u
v x y t v x y t k
t x y x y
 
    
   
 
    
 
• Convection-diffusion equation describing chemical diffusion
and convection:
Convection Diffusoin

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2 2
2
2 2
u u
c
t x
 

 
2. Hyperbolic equations: Wave equation describing
the string vibration

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2 2 2
2 2 2
0
u u u
x y z
  
  
  
3. Elliptic equations: Laplace’s equation

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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 …

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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 …

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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

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Equilibrium

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Dirichlet Boundary Conditions
2
2
0.1
(0, ) 0, (1, ) 1
( ,0) sin( ).
u u
t x
u t u t
u x x

 
 
 
 

2
2
0
(0) 0, (1) 1
w
x
w w



 
Steady equation

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Neumann Boundary Conditions
2
2
3
0.1
(0, ) 0, (1, ) 0
( ,0) .
u u
t x
u u
t t
x x
u x x
 
 
 
 
 
 

2
2
0
(0) 0, (1) 0
w
x
w w
x x



 
 
 
Steady equation