Results 1 to 2 of 2

Math Help - Advanced Numerical solution of differential equations

  1. #1
    Junior Member
    Joined
    Mar 2014
    From
    uk
    Posts
    53

    Advanced Numerical solution of differential equations

    Fint the modified equation when the implicit Euler method is applied to y'= f(y). If f(y)=λy, where λ is negative. what is the effect on the amplication factor?
    =>

    y ' = λ * y

    dy / dx = λ * y
    dy / y = λ dx
    ln y = λ* x + C
    y = Ae^( λ* x ), the constant factor does not depend on λ.
    i SOLVE THIS FOR THE ACTUAL SOLUTIONS
    NOW,
    The implicit Euler scheme is given by:
    y_(n+1)= y_n +hf(y_n+1 , t_n+1 )
    For f(y)=λ y, we have:
    y_(n+1)= y_n +hf(y_n+1 , t_n+1 )= y_n + h λ y_n+1
    Solving this for y_n+1 (in general, this is not possible), we arrive at:
    y_n+1 = y_n / (1-h λ)..............(eqn 1)

    From (eqn 1), we can see that if |1−h λ|≥1, the solution is decaying (stable). Compare this to the actual solution of y(x)=Ae^(λ x).
    If we have λ being negative, we would have:
    y_n+1 = y_n / (1+h λ)..............(eqn 2)

    Compare this to the actual solution of y(x)=Ae^(−λ x). What conclusion can i draw?
    Trying it for λ= 1 , what happens to stability.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,826
    Thanks
    669

    Re: Advanced Numerical solution of differential equations

    Hey grandy.

    I think you will have to use some kind of norm or error result, but intuitively the larger that lambda gets, the higher the error of the numerical solutions will be in contrast to the exact solution. If lambda is negative then this means that you have an upper bound of 0 as a result which means that in terms of stability things should be relatively stable.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Numerical solution of partial differential equation
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: October 27th 2013, 09:58 AM
  2. solution of equations by numerical analysis
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: April 30th 2010, 11:05 PM
  3. Numerical methods to solve higher order differential equations
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: August 18th 2009, 05:18 AM
  4. Replies: 7
    Last Post: April 7th 2009, 05:57 PM
  5. Numerical solution of differential equations (Euler method)
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: October 27th 2008, 03:12 PM

Search Tags


/mathhelpforum @mathhelpforum