Embedded Files
The following is a collection of various Desmos presentations that I made for Math 462 - Partial Differential Equations. Many of these consist of a plot of a solution u(x,t) of a PDE. The graph shown is the graph of u(x,t) for a fixed value of time t. The value of t can then be changed to dynamically see how the solution changes in time. The best way to do this is to click the "play" arrow next to the variable t, and it will show an animation of the solution.
Wave Equation on the Real Line: d'Alembert's Solution - You can enter whatever initial position/velocity you want.
Heat Equation on Half-Line: Example with Dirichlet Condition u(0, t) = 0 - This is one example whose initial temperature is 1 across the entire rod. If you click the button to make the other graph appear, you will see the odd reflection.
Heat Equation on Half-Line: Examples with Neumann Condition u_x(0, t) = 0 - There are two examples here. You can also make the even reflections appear.
Wave Equation on Half-Line: Dirichlet Condition u(0, t) = 0 - You can enter whatever initial position/velocity you want.
Wave Equation on Half Line: Neumann Condition u_x(0, t) = 0 - You can enter whatever initial position/velocity you want.
Heat Equation on Half-Line: Example with Inhomogeneous Dirichlet Condition u(0, t) = 1 - This is one example whose initial temperature is 0 across the entire rod. But then heat enters the rod from the left endpoint.
Wave Equation on Half Line: Arbitrary Inhomogeneous Dirichlet Condition u(0, t) = h(t) - You can change h(t) to be whatever you want, but the the initial position/velocity are both 0 here. Imagine someone grabbed the left endpoint of the string and started moving it up and down according to the function h(t). We didn't discuss this one in class, but it is cool!
Heat Equation on Finite Interval: Dirichlet Boundary Conditions - This only has the first four terms of the series solution, but you can change the coefficients to whatever you want.
Heat Equation on Finite Interval: Neumann Boundary Conditions - This only has the first five terms of the series solution, but you can change the coefficients to whatever you want.
Wave Equation on Finite Interval: Dirichlet Boundary Conditions - This only has the first four A_n terms of the series solution, but you can change the coefficients to whatever you want. All B_n = 0 (this corresponds to zero initial velocity).
Wave Equation on Finite Interval: Neumann Boundary Conditions - This only has the first five A_n terms of the series solution, but you can change the coefficients to whatever you want. All B_n = 0 (this corresponds to zero initial velocity).
Heat Equation on Finite Interval: Robin Boundary Conditions - This only has the first five terms of the series solution, but you can change the coefficients to whatever you want. The length of the rod is L = 3 and the boundary conditions are u_x(0, t) - u(0, t) = 0 and u_x(L, t) + u(L, t) = 0.
Wave Equation on Finite Interval: Robin Boundary Conditions - This only has the first five terms of the series solution, but you can change the coefficients to whatever you want to change the initial position. The initial velocity is 0. The length of the string is L = 3 and the boundary conditions are u_x(0, t) - u(0, t) = 0 and u_x(L, t) + u(L, t) = 0.
Dirichlet Heat Equation on Interval - for arbitrary initial condition.
Dirichlet Wave Equation on Interval - for arbitrary initial conditions.
Neumann Heat Equation on Interval - for arbitrary initial condition.
Neumann Wave Equation on Interval - for arbitrary initial conditions.
Dirichlet Damped Wave Equation on Interval - for arbitrary initial position. Initial velocity is always assumed to be 0. You can adjust the damping parameter (called a) to be anywhere between 0 and 2*pi*c/l.
Dirichlet Overdamped Wave Equation on Interval - for arbitrary initial position. Initial velocity is always assumed to be 0. You can adjust the damping parameter (called a) to be anywhere between 2*pi*c/l and 4*pi*c/l.
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