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

  1. #1
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    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
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  2. #2
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    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 $a_1 + a_2$.
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  3. #3
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    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)
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  4. #4
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    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.
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  5. #5
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    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)
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