Consider the heat equation, Ut = Uzz, u(t,0)=0, u(t, 1) = 0, u(0, 2) = sin(x),...

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Consider the heat equation, Ut = Uzz, u(t,0)=0, u(t, 1) = 0, u(0, 2) = sin(x), t>0, 0<x< 1, (a) Verify that u(t, x) = e-t sin(x) is a solution. (b) Plot, on the same axes, the exact solution as a function of r for the times t = 0.001,0.01,0.1. (c) Plot the surface of the exact solution as a function of 2 € [0, 1] and t€ [0.0.1] using 22 and 11 points respectively. (d) Plot, on the same axes, the numerical solution as a function of r at time T = 0.1 using 20 internal grid points and m = 5, 10, 20 time steps. Use the Crank-Nicolson method. List the script file you actually use. Consider the heat equation, Ut = Uzz, u(t,0)=0, u(t, 1) = 0, u(0, 2) = sin(x), t>0, 0<x< 1, (a) Verify that u(t, x) = e-t sin(x) is a solution. (b) Plot, on the same axes, the exact solution as a function of r for the times t = 0.001,0.01,0.1. (c) Plot the surface of the exact solution as a function of 2 € [0, 1] and t€ [0.0.1] using 22 and 11 points respectively. (d) Plot, on the same axes, the numerical solution as a function of r at time T = 0.1 using 20 internal grid points and m = 5, 10, 20 time steps. Use the Crank-Nicolson method. List the script file you actually use.

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To verify and plot the solutions of the heat equation well need to use numerical methods Specifically well use the CrankNicolson method to solve the e... View the full answer