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Math Help - PDE

  1. #1
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    PDE

    I'm in an elementary PDE's class and I'm trying to do problems in my textbook. I'm having trouble starting. I was wondering if someone could show me how to do this example, so I can model similar solutions after it.

    Find the solution and give a physical interpretation of the problem uxx = 4ut (i.e. second partial derivative of u with respect to x = 4 times the first partial derivative of u with respect to t)
    for 0 < x < 2, t > 0,
    with the conditions
    u(0,t) = 0
    u(2,t) = 0
    u(x,0) = 2sin(pi*x/2) - sin(pi*x) + 4sin(2pi*x)

    Thanks.
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  2. #2
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    Quote Originally Posted by PvtBillPilgrim View Post
    I'm in an elementary PDE's class and I'm trying to do problems in my textbook. I'm having trouble starting. I was wondering if someone could show me how to do this example, so I can model similar solutions after it.

    Find the solution and give a physical interpretation of the problem uxx = 4ut (i.e. second partial derivative of u with respect to x = 4 times the first partial derivative of u with respect to t)
    for 0 < x < 2, t > 0,
    with the conditions
    u(0,t) = 0
    u(2,t) = 0
    u(x,0) = 2sin(pi*x/2) - sin(pi*x) + 4sin(2pi*x)

    Thanks.
    Do you know the technique of separation of variables?
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  3. #3
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    I remember it from ODE's, but I'm not sure how to apply it here.
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  4. #4
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    Quote Originally Posted by PvtBillPilgrim View Post
    I'm in an elementary PDE's class and I'm trying to do problems in my textbook. I'm having trouble starting. I was wondering if someone could show me how to do this example, so I can model similar solutions after it.

    Find the solution and give a physical interpretation of the problem uxx = 4ut (i.e. second partial derivative of u with respect to x = 4 times the first partial derivative of u with respect to t)
    for 0 < x < 2, t > 0,
    with the conditions
    u(0,t) = 0
    u(2,t) = 0
    u(x,0) = 2sin(pi*x/2) - sin(pi*x) + 4sin(2pi*x)

    Thanks.
    The solution to u_{xx} = 4u_t on [0,2]\times [0,\infty) with initial conditions u(x,0) = \sum_{k=0}^n a_k \sin \tfrac{\pi k x}{2} and boundary conditions u(0,t)=u(2,t)=0 is given by u(x,t) = \sum_{k=0}^n a_k e^{-4(\pi k t/2)^2} \sin \tfrac{\pi k x}{2}.
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  5. #5
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    Quote Originally Posted by ThePerfectHacker View Post
    The solution to u_{xx} = 4u_t on [0,2]\times [0,\infty) with initial conditions u(x,0) = \sum_{k=0}^n a_k \sin \tfrac{\pi k x}{2} and boundary conditions u(0,t)=u(2,t)=0 is given by u(x,t) = \sum_{k=0}^n a_k e^{-4(\pi k t/2)^2} \sin \tfrac{\pi k x}{2}.
    Not for all n - right, just those to recover the initial condition.
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