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2nd Order Differential Equation with Improved Euler Method (Heun's)

  1. May 1, 2014 #1
    1. The problem statement, all variables and given/known data
    I would like to solve a 2nd Order Differential Equation using the Improved Euler Method. The 2nd ODE is a Mass-Spring-Damper equation. I tried coming up with an solution for the Improved Euler Method, but not entirely sure. Can you help me and have a look if this is correct?
    This solution assumes that inital conditions for x and u are known.

    2. Relevant equations
    Forced Mass-Spring-Damper equation:

    [itex]m\ddot{x} + b\dot{x}+kx = F[/itex]


    3. The attempt at a solution

    Converting the equation to a pair of first order differential equations:
    [itex]u=\dot{x}[/itex]
    [itex]\dot{u}=\ddot{x}[/itex]

    And thus we have:
    [itex]\dot{u}=\frac{1}{m}\left[F-bu-kx\right][/itex]

    The forward Euler solution would result in:
    [itex]x_{n+1}=x_{n}+dt*u_{n}[/itex]
    [itex]u_{n+1}=u_{n}+dt*\dot{u_{n}}=u_{n}+dt*\frac{1}{m}\left[F_{n}-bu_{n}-kx_{n}\right][/itex]

    And the Improved Euler solution would be:
    [itex]x_{n+1}=x_{n}+dt*u_{n}[/itex]
    [itex]u_{n+1}=u_{n}+dt*\dot{u_{n}}=u_{n}+\frac{dt}{2} \left[ \frac{1}{m} \left[F_{n}-bu_{n}-kx_{n}\right]+\frac{1}{m}\left[F_{n+1}-bu_{n+1}-kx_{n+1}\right] \right][/itex]

    In the last equation, does the last [itex]F[/itex] needs to be [itex]F_{n+1}[/itex] like I have it?

    Thanks a bunch, if you have time to have a look at it!
     
  2. jcsd
  3. May 1, 2014 #2

    Zondrina

    User Avatar
    Homework Helper

    Your improved Euler solution looks fine to me. You indeed want the 'n+1'.
     
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