Solve: Boundary Value Problem
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Boundary Value Problem. See attached file for full problem description.
Consider the boundary value problem:
PDE: u_t = u_xx, (0 < x < 4)
BCs: u_x(0, t) = -2, u_x(4, t) = -2
ICs: u(x, 0) = {0 if 0 <= x <= 2
{2x - 4 if 2 <= x <= 4
(a) Find the steady-state solution. (NOTE: There are Neumann BCs at each end. Generally, this would suggest that there would not be a steady-state solution. In this case, however, the same value of the derivative is given at each end, meaning that heat leaves and enters at the same rate. You will find that the steady-state solution contains an arbitrary constant. So how do you choose this constant? Since the net heat flux into the interval 0 <= x <= 4 is 0, the total heat energy must not depend on time. Choose the constant in the steady-state solution so that the total energy as t -> infinity is the same as the energy at time t = 0.)
(b) Using separation of variables, find the solution u(x, t) to this problem. In order to accomplish this do the following:
Write u(x, t) = w(x) + v(x, t), where w(x) is the particular solution (also the steady-state solution) and v(x, t) is the homogenous solution.
Write down the PDE and the BCs that v(x, t) must satisfy.
Write down the solution for v(x, t) using the separation of variables results.
Choose the arbitrary constants in v(x, t) so that u(x, t) satisfies the initial condition.
(c) Plot u(x, t) as a function of x for several values of t in order to see how the temperature profile evolves from the initial condition towards the steady-state solution. (NOTE: use the subplot command in MATLAB in order to save paper. In each subplot window plot u(x, t) at a given instant in time.)
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Solution Summary
This is a boundary value problem that shows how to find a steady-state solution, find a solution using separation of variables, and plot the function to see the development from initial condition to steady-state solution. This solution is provided as multiple jpeg documents in a .zip file that is attached.
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