Results 1 to 6 of 6

Math Help - Solving system of nonlinear ODEs

  1. #1
    Newbie
    Joined
    May 2010
    Posts
    2

    Solving system of nonlinear ODEs

    Hello everyone,

    I have a system on the form

     x''_1 + \lambda x'_1 + C(x_1)x_1 = A(x_1)x_3 + B(x_1)x_3^2

     x'_3 + D(x_1)x_3 + E(x_1)x'_1x_3 = F(x_1)x'_1

    where A,B,C,D,E,F are all nonlinear functions, and both x_1, x_3 depend on time.

    Does anyone know a good book (if any) which covers methods for the solution of this type of system of nonlinear ODEs? Or will I have to use numerical integration?

    Thanks

    Update:
    Indeed, I need an analytical solution to solve my original problem...
    Last edited by thetouristbr; March 16th 2011 at 01:32 PM. Reason: update
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Rebesques's Avatar
    Joined
    Jul 2005
    From
    At my house.
    Posts
    538
    Thanks
    11
    Ok... First of all, let x_3=x_2 and call the original equations (1), (2) respectively.

    Now, (1) can be written as x_1''+\lambda x_1'=W(x_1,x_2)
    where W is nonlinear in its arguments. By integrating, we obtain x_1'=F_1(x_1,x_2), for a function F_1. Substitute this in (2), to end up with x_2'=F_2(x_1,x_2), for a nonlinear function F_2.

    So, the original system (1)+(2) is only an autonomous system of nonlinear first order ODEs. There is a rich theory behind autonomous systems, and you can get information about the qualitative behaviour of solutions in any good book on ODEs. Now, for obtaining solutions near x_1=0,x_2=0 (as I bet is the case) you can turn to a good book on asymptotic methods.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Quote Originally Posted by Rebesques View Post
    Ok... First of all, let x_3=x_2 and call the original equations (1), (2) respectively.

    Now, (1) can be written as x_1''+\lambda x_1'=W(x_1,x_2)
    where W is nonlinear in its arguments. By integrating, we obtain x_1'=F_1(x_1,x_2), for a function F_1. Substitute this in (2), to end up with x_2'=F_2(x_1,x_2), for a nonlinear function F_2.

    So, the original system (1)+(2) is only an autonomous system of nonlinear first order ODEs. There is a rich theory behind autonomous systems, and you can get information about the qualitative behaviour of solutions in any good book on ODEs. Now, for obtaining solutions near x_1=0,x_2=0 (as I bet is the case) you can turn to a good book on asymptotic methods.
    Is the fact that you've turned the second-order system of ODE's into a system of integro-differential equations important? I mean, "integrating the first equation" sounds nice, but the fact is that you don't yet know what either function is, so your F_{1} function is going to have integrals in it that can't be resolved yet.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member Rebesques's Avatar
    Joined
    Jul 2005
    From
    At my house.
    Posts
    538
    Thanks
    11
    integro-differential equations


    Not really. The functions involved actually define Nemytskii maps. It is a standard trick that I didn't find important to point out.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Hmm. Well, you're well beyond my understanding, I'm afraid. Maybe Danny knows something about that.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,347
    Thanks
    30
    Nemytskii maps - beyond me too! If you let x_1' = x_2 then you have a system of 3 first order ODEs. However, the OP want an analytical solution and without knowing the arbitrary functions A-F, I don't think we can do that.
    Last edited by Jester; March 20th 2011 at 05:33 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Solving a nonlinear system
    Posted in the Trigonometry Forum
    Replies: 4
    Last Post: December 5th 2011, 04:02 PM
  2. Replies: 1
    Last Post: April 2nd 2011, 04:31 AM
  3. Nonlinear System of two ODEs
    Posted in the Differential Equations Forum
    Replies: 25
    Last Post: July 9th 2010, 05:50 AM
  4. solving system of ODEs - defective matrix
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: November 29th 2009, 08:58 PM
  5. solving a system of nonlinear equations
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: November 16th 2008, 11:39 AM

Search Tags


/mathhelpforum @mathhelpforum