Find the steady state solution of the following modified...

Name: find the steady state solution of the following modified heat equation an advection term has been included u uxt on x in 0l ut uxx beta ux 0 beta is a constant l is a constant with bcs u0 44687
Uploaded: 2023-11-01T11:39:12-08:00
Duration: 2 min 47 s
Channel: Adi S
Description: find the steady state solution of the following modified heat equation an advection term has been included u uxt on x in 0l ut uxx beta ux 0 beta is a constant l is a constant with bcs u0 44687

Find the steady state solution of the following modified heat equation. (An advection term has been included.) $u = u(x,t)$ on $x \in [0,L]$ $u_t - u_{xx} + \beta u_x = 0$ $\beta$ is a constant $L$ is a constant with BC's: $u(0,t) - 5u_x(0,t) = 3$ $u(L,t) = 10$ with IC: $u(x,0) = 3x$

We will solve the heat equation u_t = 3 u_xx, 0 < x < 2, t ≥ 0 with boundary/initial conditions: u(0,t) = 0, u(2,t) = 0, and u(x, 0) = { 3, 0 < x ≤ 1; 0, 1 < x < 2 This models temperature in a thin rod of length L = 2 with thermal diffusivity ̑ = 3 where the temperature at the ends is fixed at 0 and the initial temperature distribution is u(x, 0). For extra practice we will solve this problem from scratch. Separate Variables. Find Eigenfunctions for X(x). The problem splits into cases based on the sign of ̑. (Notation: For the cases below, use constants a and b) • Case 1: ̑ = 0 X(x) = Plugging the boundary values into this formula gives 0 = X(0) = 0 = X(2) = So X(x) = which means u(x, t) = We can ignore this case. • Case 2: ̑ = -̳^2 < 0 (In your answers below use gamma instead of lambda) X(x) = Plugging the boundary values into this formula gives 0 = X(0) = 0 = X(2) = So X(x) = which means u(x, t) = We can ignore this case.

Adi S.