Consider the following boundary value problem for the 2D heat equation: u_t = c^

Question

Consider the following boundary value problem for the 2D heat equation:
u_t = c^2(Nabla)^2u; u(x,0, t) = u(0, y, t) = u(pi, y, t) = 0; u(x,pi , t) = x(pi - x)

(a) What is the steady-state solution, u_ss(x, y)? [Hint: Look at a previous problem on Laplace's equation]. Sketch it.

(b) Define v(x, y, t) = u(x, y, t) - u_ss(x, y), where u_ss(x, y) is your solution to Part (a). Rewrite the PDE including the boundary conditions in terms of v instead of u. The resulting PDE is homogeneous because v(x, y, t) = 0 is a solution.

(c) Write down the general solution to this PDE by adding the steady-state solution to the solution of the related homogeneous problem (which you've already solved!).

Explanation / Answer

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