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Differential equation

  1. Oct 5, 2015 #1
    Hi evry body
    i would like to have an help to resolve this exercice below
    the followin differential equation with its initial condition
    dy/dt=-lambda t y(t) t>=0
    avec y(0)=y0
    where lambda is damping coeficient strictly positive.
    -find the solution of this equation with Euler's explicite and implicite methode
    -find analytically the values of h in order to euler methode (explicite) being applicable and obviously stable ( lim IynI=0 where n --->infini .and find the superior borne of time lag h according lambda>0
    thanks
    warmest Regards
     
  2. jcsd
  3. Oct 5, 2015 #2

    BvU

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    Hello youcef, bienvenu a PF :smile: !

    $${dy\over dt } = - \lambda \, t \, y(t) \\
    y(0) = y_0$$ is what you want to solve ? Or have to solve (in that case it should be in the homework section!)

    Or is it ## -\lambda(t) \, y(t) ## or is it just ##- \lambda \, y(t)## ?

    What would make ##\lambda## a damping coefficient ? (I am used to damping coefficients in forms like ##{d^2y\over dt^2 } = - \lambda \, { dy\over dt}\ ## so I thought I'd better ask first.)
     
  4. Oct 5, 2015 #3
    $${dy\over dt } = - \lambda \, t \, y(t) \\
    y(0) = y_0$$
     
  5. Oct 5, 2015 #4

    BvU

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    OK, so let's get started on the first part: for Euler explicit you get $$ { y_{k+1}-y_k\over \Delta t} = -\lambda \, t \, y_k $$ and for Euler implicit you have to solve $$
    { y_{k+1}-y_k\over \Delta t} = -\lambda \, t \, y_{k+1}
    $$to get ##y_{k+1} ## as a function of ##y_k##, ## t##, and ##\Delta t##.

    Agree ?

    --
     
  6. Oct 5, 2015 #5
    Thanks BvU .I Agree.let's continue
     
  7. Oct 5, 2015 #6

    BvU

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    Well, where do you have a problem when you do continue ?
     
  8. Oct 5, 2015 #7

    BvU

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    Wow, I don't follow. Is this for explicit Euler ?
    So how do you come from $$
    { y_{k+1}-y_k\over \Delta t} = -\lambda \, t \, y_k
    $$ to your ......(1) ? I don't see a square appearing at all !
     
  9. Oct 5, 2015 #8
    sorry
    for implicite method yk+1=yk/(1+Δtλt)
    for explicit
    yk+1=yk(1-Δtλt)
     
  10. Oct 5, 2015 #9

    BvU

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    What happened to your post ? If you edit it away completely, no one else can follow the thread later on !

    Good. Any further problems ? If not then part one is ready ?
     
  11. Oct 5, 2015 #10
    you are very kind .yes no problem.let's go to second part
     
  12. Oct 5, 2015 #11

    BvU

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    IF part 1 is ready, then what does your solution look like ? Any differences between implicit and explicit methods ?
    Do you know the error both methods give when compared to the exact solution ?
    What choices of delta t and lambda did you make ? I tried lambda = 0.5 and delta t up to 0.5 (0.501 went bang for the explicit Euler...)

    But in fact the stability limit is exceeded a lot earlier. 0.32 also crashes
     
  13. Oct 5, 2015 #12
    i don't understand what do you mean.is that is wrong solution
     
  14. Oct 5, 2015 #13

    BvU

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    So far, I haven't seen your solution of the differential equation, so I don't know...
     
  15. Oct 6, 2015 #14
    good morning
    so any one can't resolve it???
     
  16. Oct 6, 2015 #15

    BvU

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    I don't understand. How far are you really with part 1? What results do you have to show ? See questions in post #11
     
  17. Oct 6, 2015 #16
    I have no idea if yes i do it by my self.
     
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