Partial Differential Equations

Partial Differential Equations

1. Intro to PDEs

   • Superposition

2. 1st Order PDEs

   • Constant Coefficients
     • Chain rule
     • Method of characteristics
       • dy/dx=b/a
     • aUx+bUy=0
   • Variable Coefficients
     • aUx+bUy=cU+d
   • Initial Conditions, t=0. Boundary Conditions, x=0
   • Picard-Lindelof Theorem

3. 2nd Order PDEs

   • aUxx+2bUxy+cUyy+dUx+eUy+fU=h(x,y)
     • b^2-ac>0 hyper
     • b^2-ac=0 para
     • b^2-ac<0 ellip
   • Change of variables

4. Wave equation

   • Utt-c^2Uxx=0
     • Separation of Variables
       • Eigenvalue Problem
   • D'Alembert's Solution
   • Dirichlet/Neumann BCs
   • Odd/even extensions
   • Fourier Series

5. Elliptic Equation

   • Laplace Equation Uxx+Uyy=0
     • Dirichlet BP
     • Harmonic
       • Cauchy-Riemann uy=ux, ux=-uy
     • Cartesian/Polar
   • Poisson Equation Uxx+Uyy=f(x,y)
     • Mean Value Property
   • Max/min principle

6. Heat equation

   • Heat Kernel
   • Fourier-Poisson
   • Dirac's delta

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