Thread: Single second order partial differential equation (heat equation)

1. Single second order partial differential equation (heat equation)

Consider one-dimensional heat conduction equation

ut = kuₓₓ, 0<x<1, 0<t<∞

with boundary conditions: a₁u(0,t)+a₂uₓ(0,t)=0 (a₁,a₂)≠(0,0)
0<t
a₃u(1,t)+a₄uₓ(1,t)=0 (a₃,a₄)≠(0,0)

Initial conditions: u(x,0)= ƒ(x), 0<x<1.

Consider the problems

A. L=1, ƒ(x)=20 (a₁,a₂)=(1,0) and (a₃,a₄)=(1,0)
B. L=1, ƒ(x)=x (a₁,a₂)=(0,1) and (a₃,a₄)=(0,1)
C. L=1, ƒ(x)=20 (a₁,a₂)=(1,0) and (a₃,a₄)=(1,1)

Use k for three different materials: Silver 1.71, Copper 1.14, Cast Iron 0.12 (k cm² sec-¹)

In each case use Maple to plot snapshots of u(x,t) using S5(x,t)
Use Maple to plot u(x,t) at t=0, t=0.5, t=1, t=2 and t=4
Use Maple to plot u(x,t) at x=0, x=0.25, x=0.5, x=0.75 and x=1

For each problem interpret your results physically, comparing and contrasting them with each case

2. Re: Single second order partial differential equation (heat equation)

What is $L$? And can you check those boundary conditions. As given we can't distinguish between $a_1$, $a_2$ and $a_1 + a_2$.

3. Re: Single second order partial differential equation (heat equation)

L is the length and boundary conditions are: a1u(0, t) + a2uₓ(0, t) = 0, (a1, a2) ≠ (0, 0)

0<t

a3u(1, t) + a4uₓ(1, t)= 0 (a3, a4) ≠ (0, 0)

4. Re: Single second order partial differential equation (heat equation)

That looks like $(a_1 + a_2)u(0,t)$ and $(a_3 + a_4)u(1,t)$ which I'm sure isn't correct.

5. Re: Single second order partial differential equation (heat equation)

a1u(0,t) + a2u(index x)(0, t) and a3u(1, t) + a4u(index x)(1, t)