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