(e) Show that v(x, t) = u(x, t) — uɛ(x, t) is solution of a homogeneous problem (with a different initial condition) and solve it using the method of separation of variables and the superposition principle. The solution has the form v(x,t) = Go(t) + Σ Gn(t) cos(nлx) n=1 Enter the functions Go (t) and Gn(t) (in that order) into the answer box below, separated with a comma. (f) The solution u(x, t) of the original problem, written as a cosine Fourier series with coefficients that are functions of t, has the form 8 u(x,t) = H₁(t) + Σ H₂(t) cos(nлx) n=1 Enter the functions Ho(t) and H₂(t) (in that order) into the answer box below, separated with a comma.
I wrote out the question as requested and tried to re-upload the 2 questions I'm interested in. I need the solution to e and f after we find the value of Cn for that question based on your solution before. I hope this clears things up.
Consider the following heat equation on a rod (which includes heat loss through the lateral side and fixed heat flow on the boundary):
∂u/∂t = ∂2u/∂x2 − 9u, 0 < x < 1, t > 0,
subject to the boundary conditions
ux(0, t) = 3, ux(1, t) = 18, t > 0,
and the initial condition
u(x, 0) = 0, 0 < x < 1.
(a) Find the equilibrium solution uE(x) for this problem by solving a certain boundary value problem on (0, 1).
(b) Use integration by parts and the given boundary conditions that uE(x) satisfies to compute
A0 = 1∫0 u′′E (x) dx
Enter A0. (Do not use the explicit formula for uE(x) obtained in part (a).)
(c) Use integration by parts and the given boundary conditions that uE(x) satisfies to show that
1∫0 u′′E (x) cos(nπx) dx = An + B*n^2 1∫0 uE(x) cos(nπx) dx, n ≥ 1, (1)
for some constants B and An. (Do not use the explicit formula for uE(x) obtained in part (a).) Enter the values of B and An (in that order) into the answer box below, separated with a comma.
(d) Use the differential equation that uE(x) satisfies together with the results obtained in (b) and (c) to deduce the Fourier cosine series representation for uE(x) on (0, 1) which is of the form
a0/2 + ∞Σn =1 an cos(nπx).
Enter the values of a0 and an (in that order) into the answer box below, separated with a comma.
(e) Show that v(x, t) = u(x, t) − uE(x, t) is solution of a homogeneous problem (with a different initial condition) and solve it using the method of separation of variables and the superposition principle. The solution has the form
v(x, t) = G0(t) + ∞Σn =1 Gn(t) cos(nπx)
Enter the functions G0(t) and Gn(t) (in that order) into the answer box below, separated with a comma.
(f) The solution u(x, t) of the original problem, written as a cosine Fourier series with coefficients that are functions of t, has the form
u(x, t) = H0(t) + ∞Σn =1 Hn(t) cos(nπx)
Enter the functions H0(t) and Hn(t) (in that order) into the answer box below, separated with a comma.
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