Consider the following nonhomogeneous heat equation problem:
∂u/∂t - ∂^2u/∂x^2 = sin(3πx), 0<t
u(x,0) = 5sin(3πx) + 4sin(9πx) + 3x + 2
u(0,t) = 2
u(1,t) = 5
With a suitable shifting function s(x), we can express the solution by
u(x,t) = w(x,t) = s(x)
where w(x,t) is the solution to some related heat equation with homogeneous BCs.
(a) Find a suitable shifting function that satisfies the nonhomogeneous BCs.
(b) Derive the heat equation problem that w(x,t) satisfies given your choice of shifting function in part (a). Make sure to include the BCs and IC satisfied by w(x,t).
(c) Solve for w(x,t) by the method of eigenfunction expansion. Make sure to use the appropriate eigenfunctions, which you do not need to re-derive.
(d) Write out the solution, u(x,t), to the original problem.
Consider the following nonhomogeneous heat equation problem:
u_x,0 = 5sin(3Tx) + 4sin(9Tx) + 3x + 2
u(0,t) = 2
u(1,t) = 5
With a suitable shifting function s(x), we can express the solution by
u_x,t = w_x,t = s_x
where w_x,t is the solution to some related heat equation with homogeneous BCs.
(a) Find a suitable shifting function that satisfies the nonhomogeneous BCs.
(b) Derive the heat equation problem that w_x,t satisfies given your choice of shifting function in part (a). Make sure to include the BCs and IC satisfied by w,t.
(c) Solve for w_x,t by the method of eigenfunction expansion. Make sure to use the appropriate eigenfunctions, which you do not need to re-derive.
(d) Write out the solution, u_x,t, to the original problem.