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Integrals of PDEs (help needed to interpret theorem)

  1. Nov 20, 2007 #1
    Hello friends,

    I'm reading about PDEs and my textbook lists 'integrals' of the pde

    [tex]f(x,y,z,p,q) = 0[/tex]

    where [itex]p = \partial z/\partial x[/itex] and [itex]q = \partial z/\partial y[/itex], as

    1. Complete Integral
    2. General Integral
    3. Singular Integral
    4. Special Integral (solution that can't be classified into the above three categories...and can't be obtained from the general integral).

    Specifically, I have the following questions:

    1. What is the geometrical/physical significance of each of these 'integral' solutions, esp the singular solution of the PDE?

    2. Let [itex]z = F(x,y,a)[/itex] be a one parameter family of solutions of the above PDE, parametrized by [itex]a[/itex]. Then the envelope of this family, if it exists, also satisfies the PDE. What is the geometrical significance of this theorem?

    Thanks.
     
  2. jcsd
  3. Dec 8, 2007 #2
    I wouldn't sweat the geometric significance of this. You're dealing with 5 dimensions. What helped for me was to get my geometric intuition from the quasi-linear case, and just know that the analysis works for the fully non-linear case.
     
  4. Jan 7, 2008 #3

    epenguin

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    Part answer and new question

    It is not 5-D it is 3-D surely?

    But it would be good to get it straight for 2D firstly, then AFAIK there are not big differences of principle for other D but I stand to be corrected.

    It would be pretty obvious to you if you found them illustrated, or if you use a graphic plotter and plot a family [itex]F(x,y,a) = 0[/itex]
    For 2D and equation f(x, y, dy/dx) = 0, with a family of solutions g(x, y, c) = 0 depending on a parameter a, you may have a continuous curve (called the envelope) whose slope at every point is equal to that in the parameter dependent solution. Therefore this curve also satisfies the differential equation f = 0. Better an example: for instance to the equation

    dy/dx = y^(1/2)

    there is the family of solutions y = (x – a)^2 . Each of these solutions is a curve with a minimum at y = 0, i.e. dy/dx = 0 at y = 0 so satisfying also the d.e. just given. y = 0 is a singular solution of the given d.e.


    I always found singular solutions appealing because of their unlooked-for character.:surprised I wonder if there are any examples of where they are of any interest in general dynamics? A point travelling along a general solution and hopping onto a singular one when it mneets it? It should only be able to hop off if and when it gets back to the original point. Imagine there are no examples.
     
  5. Jan 7, 2008 #4
    For a general first-order equation in two variables, you have to solve a system of 5 ODEs.
     
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