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    I have been asked to solve the following heat conduction equation:

    \displaystyle 2\frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t}

    over 0<x<3 and t>0 for the boundary conditions

    u(0,t) = u(3,t) = 0

    and the initial condition

    u(x,0) = 5sin(4*pi*x)

    Hint: The general solution of the ODE

    \frac{d^2 Y}{dx^2} = -lander^2Y

    is Bcos(lander.x) + Csin(lander.x) where lander, B and C are constants.

    I have done the first few steps and have found that either Y(0) = Y(3) = 0 and G(t) does not equal 0.

    I am stuck now, I am unsure how to progress further with this question. How do i find Y(x) and G(t)?
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    (Original post by sarah_vickers)
    I have been asked to solve the following heat conduction equation:

    \displaystyle 2\frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t}

    over 0<x<3 and t>0 for the boundary conditions

    u(0,t) = u(3,t) = 0

    and the initial condition

    u(x,0) = 5sin(4*pi*x)

    Hint: The general solution of the ODE

    \frac{d^2 Y}{dx^2} = -lander^2Y

    is Bcos(lander.x) + Csin(lander.x) where lander, B and C are constants.

    I have done the first few steps and have found that either Y(0) = Y(3) = 0 and G(t) does not equal 0.

    I am stuck now, I am unsure how to progress further with this question. How do i find Y(x) and G(t)?
    How do you normally solve these types of questions? I use separation of variable. Let

    U(x,t)=T(t)X(x) and substitue into the heat equation to give 2X"(1/x)=T'(1/t) then let this to equal lamda^2 i.e. a constant and solve for the 2 equations,

    2X"(1/x)=lamda^2 which you can use the hint to solve

    T'(1/t)=lamda^2 which is a standard fode

    sub in the contraints and you are done.

    By the way what year if this material 1st or 2nd year?
 
 
 
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