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 + Bn2 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.
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 + Bn2 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.
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 + Bn2 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.
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 + Bn2
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.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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