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Banach Fixed Point and Differential Equations

  1. Mar 26, 2012 #1
    1. The problem statement, all variables and given/known data
    Find the value of x, correct to three decimal places for which: [itex]\int^{x}_{0}\frac{t^{2}}{1+t^{2}}dt=\frac{1}{2}[/itex].

    2. Relevant equations
    Banach's Fixed Point Theorem
    Picard's Theorem?

    3. The attempt at a solution
    I'm not sure where to start with this type of problem.

    From other BFPT problems, I will need to show a contraction mapping into itself.

    Any pointers would be greatly appreciated.
     
  2. jcsd
  3. Mar 26, 2012 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Banach's fixed point theorem and Picard's method show that this problem has a unique solution. Actually finding the solution doesn't require that.

    You could, for example, actually integrate that:
    [tex]\int_0^x \frac{t^2}{1+ t^2}dt= \int_0^x 1- \frac{1}{t^2+ 1}dt[/itex]
    is pretty easy to integrate.
     
  4. Mar 27, 2012 #3
    [itex]\int^{x}_{0}\frac{t^{2}}{1+t^{2}}dt=\int^{x}_{0}(1-\frac{1}{1+t^{2}})dt[/itex]

    Let t=tanθ. Then dt=sec2θdθ.

    [itex]\int^{x}_{0}(1-\frac{1}{1+t^{2}})dt=\int^{x}_{0}dt-\int^{x}_{0}\frac{1}{sec^{2}\vartheta}[/itex]sec2θdθ=[itex]\int^{x}_{0}dt-\int^{x}_{0}[/itex]dθ=t-arctan(t) from 0 to x=x-arctan(x).

    x-arctan(x)=.5 at x=1.4743.

    Are you familiar with any approaches that use BFPT?
     
    Last edited: Mar 27, 2012
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