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Non-linear DE's

  1. Aug 21, 2012 #1
    Hey gang, how would you specifically define what a non-linear DE is? I mean, what specifically qualifies a DE to be nonlinear, and what does that mean? I am currently working with a team that models brain function as an array of coupled oscillators with complex feedback profiles and I understand the the DE's that are put in the Runge-Kutta solver are nonlinear, but looking at a symbolic equation on a chalkboard, I'm not sure if I could pick out a non-linear ODE from a linear one. Could someone help?
     
  2. jcsd
  3. Aug 21, 2012 #2
    Think on what you learned at school. A linear function (really a straight line) is of the form
    [itex] y= a\cdot x + b [/itex]
    Now a differential equation of the form
    [itex] x'(t) = A(t) \cdot x(t) + b(t) \qquad \qquad (*) [/itex]
    is a linear differential equation. (Note the dependency on t)

    Again back to school, a non-linear function is e.g. a parabola
    [itex] y=x^2 [/itex]
    Hence, a DE of the form
    [itex] x'(t) = x(t)^2 [/itex]
    is non-linear.

    Summarizing, every DE which could be written in the form (*) is a linear differential equation. All other equations are non-linear.

    Note that linear / non-linear has nothing to do with RK solver. But it is true that linear DEs could mostly easily be solved analytically, and therefore a RK solver is not necessary. But it is not true that DEs which are solved by an RK solver are non-linear.

    Check also https://www.physicsforums.com/showthread.php?t=628922
     
  4. Aug 21, 2012 #3

    Mute

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    Homework Helper

    If you have a coupled set of differential equations for some set of functions ##{y_1(t), y_2(t), \dots, y_n(t)}##, the equations are non-linear if any products of these functions or their derivatives appear.

    e.g.,

    $$y_1(t)^2,~y_{17}(t)y_2(t),~y_3(t)\dot{y}_4(t),~ \ddot{y}_9(t)y_8(t)y_2^{(400)}(t).$$

    It is also non-linear if you have a function of one of the y's, e.g.,

    $$\sin(y_{27}(t)),~\exp(-y_{12}(t)^2/2).$$

    Note that terms such as

    $$ty_5(t),~t^2 + y_2(t)$$

    are linear as far as the differential equation is concerned. Linear or non-linear in the context of DEs is referring only to the functions you are solving for, not to the variables the function is a function of.
     
  5. Aug 22, 2012 #4
    I guess what I am confused about is the notion of non-linear. I understand that squaring the coefficient defining the slope, or cubing it, etc., will lead to a nonlinear solution there, but there is another definition of non-linear, isn't there? One relating to feedback-dependent effects in the solution to the equations, such as the solutions to the Einstein field equations?
     
  6. Aug 22, 2012 #5
    Einstein… okay I am not sure what you mean. This guy made definitely some non-linear stuff.

    From your initial post I sought you have some problems to determine if an ordinary differential equation is linear or non-linear.

    Before you start with some fancy DEs you have to understand what a DE actually is.

    I repeat the simplest example again (this is actually a test for students in an oral examination if they got it or not, and believe me 20-25% will fail…)

    Solve

    [itex] x'(t) = -k \cdot x(t) \, , \qquad x(0)=x^0 [/itex]

    What is the solution?

    If you are not able to write down the solution immediately forget about Einstein....
     
    Last edited: Aug 22, 2012
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