Question

Consider the following initial-boundary-value problem $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} - \alpha xu$, $0 < x < L$, $t > 0$, $\alpha > 0$ (const.) $\frac{\partial u}{\partial x}(0, t) = 0$, $\frac{\partial u}{\partial x}(L, t) = 0$, $t > 0$, u(x, 0) = f(x)$, $0 < x < L$ (a) Solve the steady-state problem Ans. $u_E(x) = 0$ (b) Use the method of separation of variables to solve the full initial-boundary value problem. Use this result to determine if a steady state is attained and if it agrees with the solution from part (a). Ans. $u(x, t) = A_0 \exp(-\alpha t) + \sum_{n=1}^{\infty} A_n \exp[-\left(\frac{n^2 \pi^2}{L^2} + \alpha\right)t] \cos \frac{n\pi}{L} x$ where $A_0 = \frac{1}{L} \int_0^L f(x)dx$, $A_n = \frac{2}{L} \int_0^L f(x) \cos \frac{n\pi}{L} x dx$ $\lim_{t \to \infty} u(x, t) = 0 \implies u_E(x) = 0$

          Consider the following initial-boundary-value problem
$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} - \alpha xu$, $0 < x < L$, $t > 0$, $\alpha > 0$ (const.)
$\frac{\partial u}{\partial x}(0, t) = 0$, $\frac{\partial u}{\partial x}(L, t) = 0$, $t > 0$,
u(x, 0) = f(x)$, $0 < x < L$
(a) Solve the steady-state problem Ans. $u_E(x) = 0$
(b) Use the method of separation of variables to solve the full initial-boundary value
problem. Use this result to determine if a steady state is attained and if it agrees with the
solution from part (a).
Ans. $u(x, t) = A_0 \exp(-\alpha t) + \sum_{n=1}^{\infty} A_n \exp[-\left(\frac{n^2 \pi^2}{L^2} + \alpha\right)t] \cos \frac{n\pi}{L} x$
where $A_0 = \frac{1}{L} \int_0^L f(x)dx$, $A_n = \frac{2}{L} \int_0^L f(x) \cos \frac{n\pi}{L} x dx$
$\lim_{t \to \infty} u(x, t) = 0 \implies u_E(x) = 0$

Consider the following initial-boundary-value problem
(∂ u)/(∂ t) = (∂^2 u)/(∂ x^2) - α xu, 0 < x < L, t > 0, α > 0 (const.)
(∂ u)/(∂ x)(0, t) = 0, (∂ u)/(∂ x)(L, t) = 0, t > 0,
u(x, 0) = f(x),0 < x < L(a) Solve the steady-state problem Ans.uE(x) = 0(b) Use the method of separation of variables to solve the full initial-boundary value
problem. Use this result to determine if a steady state is attained and if it agrees with the
solution from part (a).
Ans.u(x, t) = A0 (-αt) + ∑n=1^∞ An [-((n^2 π^2)/(L^2) + α)t] cos(nπ)/(L) xwhereA0 = (1)/(L) ∫0^L f(x)dx,An = (2)/(L) ∫0^L f(x) cos(nπ)/(L) x dxlimt →∞ u(x, t) = 0 uE(x) = 0

Added by Stephen B.

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