Introduction to Partial Differential Equations

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Presentation transcript:

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

Overview 11/9/2018

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

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

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

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

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

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

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

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

Equilibrium 11/9/2018

Dirichlet Boundary Conditions Steady equation 11/9/2018

Neumann Boundary Conditions Steady equation 11/9/2018