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Math Help - Runge Kutta method

  1. #1
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    Mar 2008
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    Runge Kutta method

    Hi everyone

    I am an engineering student and I am having a little difficulty with a bit of exam practice involving solving an ODE with Runge-Kutta. I am able to use both in most examples, but this one is a little different.

    Any help you can give me would be amazing, so thanks for anything you can help me with!

    The Question:

    \frac{d^2u}{dx^2}-\frac{1}{2}\frac{du}{dx}+\frac{17}{16}u=0

    over the interval (0,\pi)

    and with the boundary conditions u(0)=1, \frac{du}{dx}=0


    I have solved by using an auxhilary equation already, as outlined in the first part of the equation.

    The second part of the equation is where I have the trouble with:

    "For a system of first order equations on (0,\pi]

    \frac{du}{dx} = f(u, z, x)
    \frac{dz}{dx} = g(u,z,x)

    with initial conditions u(0)=u_0 and z(0)=z_0, deruve the second order Runge-Kutta method based on using the trapezium rule to give approximations u_k and z_k to u(x_k) and z(x_k) for k = 1,...N"

    I have devised two first order ODE's:

    \frac{du}{dx}=\frac{1}{2}u-\frac{17}{16}u

    and

    \frac{dz}{dx}=\frac{1}{2}z-\frac{17}{16}u

    I think these are correct, but I am unsure on how to use them correctly. Every example we have from class is using a two variable function and I cannot find anything on the internet about it ( maybe using the wrong names ).

    I look forward to what you guys have to say, cheers guys.
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  2. #2
    Newbie
    Joined
    Feb 2010
    Posts
    1
    Quote Originally Posted by farso View Post
    Hi everyone

    I am an engineering student and I am having a little difficulty with a bit of exam practice involving solving an ODE with Runge-Kutta. I am able to use both in most examples, but this one is a little different.

    Any help you can give me would be amazing, so thanks for anything you can help me with!

    The Question:

    \frac{d^2u}{dx^2}-\frac{1}{2}\frac{du}{dx}+\frac{17}{16}u=0

    over the interval (0,\pi)

    and with the boundary conditions u(0)=1, \frac{du}{dx}=0


    I have solved by using an auxhilary equation already, as outlined in the first part of the equation.

    The second part of the equation is where I have the trouble with:

    "For a system of first order equations on (0,\pi]

    \frac{du}{dx} = f(u, z, x)
    \frac{dz}{dx} = g(u,z,x)

    with initial conditions u(0)=u_0 and z(0)=z_0, deruve the second order Runge-Kutta method based on using the trapezium rule to give approximations u_k and z_k to u(x_k) and z(x_k) for k = 1,...N"

    I have devised two first order ODE's:

    \frac{du}{dx}=\frac{1}{2}u-\frac{17}{16}u

    and

    \frac{dz}{dx}=\frac{1}{2}z-\frac{17}{16}u

    I think these are correct, but I am unsure on how to use them correctly. Every example we have from class is using a two variable function and I cannot find anything on the internet about it ( maybe using the wrong names ).

    I look forward to what you guys have to say, cheers guys.

    Certainly looks like you've split the equation up into a first order system correctly. I think the method you are trying to find is called 'The Improved Euler Method'. It would take too long to derive the method here, but you'd be looking at using

     u_{k+1} = h(f(u_k,z_k,x_k)+f(u_k+g(u_k,z_k,x_k),z_k+f(u_k,z_  k,x_k),x_k))

    and

     z_{k+1} = h(g(u_k,z_k,x_k)+g(u_k+g(u_k,z_k,x_k),z_k+f(u_k,z_  k,x_k),x_k))

    If you want a bit of code to solve this, just let me have your email and I'll send one to you. Looks like you'll be doing well in that exam
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