Weijiu Liu

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Introduction to Partial Differential Equations

• Weijiu Liu
• Department of Mathematics
• University of Central Arkansas

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Overview
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What are PDEs?
An equation containing an unknown function and
its partial derivatives
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Three Big Classes of Equations

• Parabolic equations. Two simple example
• Heat (diffusion) equation describing heat
  conduction

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• Convection-diffusion equation describing chemical
  diffusion
• and convection

Convection
Diffusoin
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• Hyperbolic equations Wave equation describing
• the string vibration

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1. Elliptic equations Laplaces 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
• DAlemberts 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 Laplaces 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
Steady equation
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Neumann Boundary Conditions
Steady equation