Math 435~ PDEs
Miscellaneous
Math 435~ PDEs
Miscellaneous
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Here is
some MATLAB code for solutions to various wave and heat
equation problem: 1. Solution to the heat equation with k = 1 on the real line, with initial temp distribution given by u(x,0) = 1 for |x|<=1, and zero elsewhere: PDE_HeatEqn_RealLine_initStepFn.m 2. Solution to the heat equation with k=1 on the real line, with inital temp distribution given by u(x,0) = e^(-x^2): PDE_HeatEqn_RealLine_initGaussian.m 3. Solution to the wave equation with c = 3 on domain 0 <=x <=2 and initial conditions u(x,0) = 3sin(pi*x) + sin(5*pi*x/2), u_t(x,0) = 0, and boundary conditions homogeneous Dirichlet: PDE_WaveEqn_HomogDirichletBC.m 4. Solution to the wave equation with c = 3 on domain 0 <=x <=2 and initial conditions u_t(x,0) = 3sin(pi*x) + sin(5*pi*x/2), u(x,0) = 0, and boundary conditions homogeneous Dirichlet: PDE_WaveEqn_HomogDirich_initPosZero.m 3. Solution to the wave equation with c = 3 on domain 0 <=x <=2 and initial conditions u(x,0) = 2sin(3*pi*x/4), u_t(x,0) = 0, and boundary conditions homogeneous Dirichlet at x=0, homogeneous Neumann at x = 2: PDE_WaveEqn_MixedNeuDirBCs.m |
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Here is an example of a
solution to Laplace's equation where the domain is an
annulus. Consider this as the steady state solution
for heat flow on the annulus, wherethe outer boundary is
held at a sinusoidaly varying temp and the inner boundary is
held at 0 deg. OR the steady state for an elasic
membrane, whose boundary circles are pinned in the fashion
shown. |
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