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Solving a system of ODE with multiple 'time' variables

  1. Jan 2, 2012 #1
    1. The problem statement, all variables and given/known data
    Hi everyone,

    Consider the following system of (first order) differential equations:
    [itex]\dot{x}=f(t_1,x,y,z) [/itex]
    [itex]\dot{y}=g(t_2,x,y,z) [/itex]
    [itex]\dot{z}=h(t_3,x,y,z) [/itex]

    where [itex]\dot{x}=\frac{\partial x}{\partial t_1}[/itex], [itex]\dot{y}=\frac{\partial y}{\partial t_2}[/itex], and [itex]\dot{z}=\frac{\partial z}{\partial t_3}[/itex].


    2. Relevant equations

    All existence theorems I know (picard and peano) are formulated such that [itex]t_1=t_2=t_3[/itex], but I'd like to know how to extend these results to the cae shown above.

    3. The attempt at a solution

    I've tried reading the proofs to see if I can figure out a way to apply them to this problem, but I can't see how...Does someone knows whether these theorems hold true when [itex]t_1 \neq t_2 \neq t_3[/itex]? Any help/reference where to look for such theorem would be greatly appreciate!!!
     
  2. jcsd
  3. Jan 2, 2012 #2
    I'm no ODE guru but it seems to me you need some type of relation between the three t variables.
     
  4. Jan 2, 2012 #3
    Thanks for your answer. These variables are independent....what I want are conditions of
    the [itex] f [/itex]'s functions to ensure existence and uniqueness without having to solve the system...any idea how I should proceed???

     
  5. Jan 2, 2012 #4
    Using different differentiation rules, and given a relations [itex]t_2=t_2(t_1)[/itex] and [itex]t_3=t_3(t_1)[/itex], you should be able to reduce the problem to a set of (possibly nonlinear nonautonomous) delay differential equations.

    EDIT:
    Sorry, I missed your last comment.
     
  6. Jan 2, 2012 #5
    No problem...should I infer that without this relation between the t variables I cannot proceed any further???
     
  7. Jan 3, 2012 #6
    Not neccesarily. You should still be able to solve the system. Just keep in mind that since each f, g and h are implicitly dependent on all the t's, so all of x, y and z will also be dependent an all the t's. Also, you won't get neccesarily get constants as initial contitions, but functions. For example, the initial condition for x might be a function of t2 and t3.
     
  8. Jan 3, 2012 #7
    ok...understood. As initila conditions I have [itex]f(0,x,y,z)=c_1[/itex], [itex]g(0,x,y,z)=c_2[/itex], and [itex]h(0,x,y,z)=c_1[/itex], where [itex]c[/itex]'s are constants. I don't want to actually solve the system, I want conditions on the f,g, and h functions that ensures existence (and uniqueness) of some solution. Is there any theorem that might help me?
     
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