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Collection of PDE animations

Nasser M. Abbasi

April 11, 2018  Compiled on April 11, 2018 at 1:45pm
1 Heat PDE
 1.1 Homogeneous Heat PDE in 1D. bounded domain. (No source term)
  1.1.1 Both ends homogeneous Dirichlet
  1.1.2 Left end Dirichlet nonhomogeneous, right end Dirichlet homogeneous
  1.1.3 Both ends Neumann and homogeneous
  1.1.4 Left end homogeneous Neumann, right end nonhomogeneous Dirichlet
  1.1.5 Both ends homogeneous Neumann (insulated)
  1.1.6 Left end nonhomogeneous time dependent Neumann, right end homogeneous Dirichlet
  1.1.7 Left end nonhomogeneous and time dependent Dirichlet, right end homogeneous Dirichlet
  1.1.8 Both ends Nonhomogeneous Dirichlet (not time dependent)
  1.1.9 Both ends nonhomogeneous and time dependent Dirichlet
  1.1.10 Both ends homogeneous Dirichlet, no source, but added dispersion term
 1.2 Nonhomogeneous heat PDE in 1D bounded domain. (Source term present)
  1.2.1 Both ends nonhomogeneous and time dependent Dirichlet, Source term depends on x only
  1.2.2 Both ends nonhomogeneous and time dependent Dirichlet, Source term depends on x and t
  1.2.3 Both ends homogeneous Dirichlet, Source term depends on x only
  1.2.4 Both ends homogeneous Dirichlet, Source term depends on x and t
  1.2.5 Left end nonhomogeneous Dirichlet, right end homogeneous Dirichlet. source term depends on x and t.
  1.2.6 Both ends nonhomogeneous Dirichlet (not time dependent), Source term depends on x only
  1.2.7 Both ends nonhomogeneous Dirichlet (not time dependent), Source term depends on x and t
 1.3 Homogeneous Heat PDE in 1D, Semi-infinite domain, No source. Laplace Transform method
  1.3.1 Left end nonhomogeneous Dirichlet, No source term, zero initial conditions
  1.3.2 Left end nonhomogeneous and time dependent Dirichlet, No source term, zero initial conditions
2 Wave PDE
 2.1 Homogeneous Wave PDE in 1D on finite domain (plugged string) series solutions
  2.1.1 Both ends of string fixed, initial position zero, Fourier series solution
  2.1.2 Both ends of string fixed, initial velocity zero, Fourier series solution
  2.1.3 Both ends of string fixed, initial velocity zero, d’Alembert solution
  2.1.4 Left end fixed, right end free, initial velocity zero, Fourier series solution
  2.1.5 Left end fixed, right end free. Initial velocity zero. Damping present.
  2.1.6 Both ends fixed. Initial velocity zero. Dispersion term present.
  2.1.7 Both ends fixed. Initial velocity zero. Telegraphy PDE (dispersion and decay)
 2.2 Homogeneous Wave PDE in 1D, semi-infinite domain, no source
  2.2.1 Left end free, initial velocity zero
  2.2.2 Left end fixed, initial velocity zero
 2.3 Wave PDE in 1D, infinite domain, D’Alembert solution
  2.3.1 Initial velocity zero case
 2.4 Homogeneous Wave PDE in 2D. No source
  2.4.1 Rectangular membrane. Fixed on all edges
  2.4.2 circular membrane
 2.5 Solitons wave animation (non-linear wave PDE)
 2.6 Wave PDE with inhomogeneous boundary conditions, 1D (string)
  2.6.1 Analytical solution
  2.6.2 Animation
3 Laplace and Poisson PDE
 3.1 Laplace PDE inside disk
  3.1.1 Analytical solution
 3.2 Laplace PDE outside disk
  3.2.1 Analytical solution
 3.3 Laplace PDE inside square. 3 edges homogeneous boundary conditions.
  3.3.1 Analytical Solution
 3.4 Poisson PDE, Cartesian coordinates, source term depends on y
  3.4.1 Analytical solution
4 Misc. PDE’s
 4.1 FitzHugh-Nagumo in 2D
  4.1.1 Example 1
  4.1.2 Example 2
5 Appendix
 5.1 Summary table
 5.2 Using Mathematica to obtain the eigenvalues and eigenfunctions for heat PDE in 1D
  5.2.1 u(0) = 0,u(L ) = 0
  5.2.2 u′(0) = 0,u′(L ) = 0
  5.2.3 u′(0) = 0,u(L) = 0
  5.2.4 u(0) = 0,u′(L) = 0
  5.2.5                  ′
u(0) = 0,u (L)+ u (L) = 0
  5.2.6         ′        ′
u(0)+ u (0) = 0,u(L ) = 0
  5.2.7 u(− L ) = 0,u (L) = 0