1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

IVP prob involving convolutions

  1. Jan 31, 2005 #1
    there's an example in the text that we're supposed to use to solve this problem. the example solves the ODE u" = f, & to find the fundamental solution F(x) we want to solve [tex]F"(x) = \delta (x)[/tex] where [tex]\delta (x)[/tex] is the Dirac delta function. the Heaviside function satisfies (H(x)+c)' = delta(x) for any c but for convenience use c = -1/2 & solve F(x) = 1/2 for x>0 or F(x) = -1/2 for x<0 & integrate to get the FUNDAMENTAL SOLUTION [tex]F(x) = \frac{1}{2}|x|[/tex]. assuming the function [tex]f \in L^1 (\mathbb{R})[/tex] has compact support then the integral [tex]\frac{1}{2} \int_{-\infty}^{\infty} |x-y|f(y) dy[/tex] converges and defines a solution of u" = f.

    now to the problem:
    a) use use the fundamental solution (above) to solve the initial value problem u" = f for x>0 with [tex]u(0) = u_0[/tex] and [tex]u'(0) = u'_0[/tex] where [tex]f \in C^{\infty}([0,\infty))[/tex] and [tex]f = \bigcirc (|x|^{-(2+\epsilon)}) [/tex] as [tex]|x| \rightarrow \infty[/tex]

    i've used the convolution property that [tex]F' * f(0) = F * f'(0) = u'(0)[/tex] on [tex]u(x) = F * f(x) = \int_{\mathbb{R}^n} F(x-y)f(y) dy[/tex] to get [tex]u_0 = u(0) = F * f(0) = \frac{1}{2} \int_{-\infty}^{\infty} |y|f(y) dy[/tex] & [tex]u'_0 = u'(0) = F' * f(0) = \frac{1}{2} \int_{-\infty}^{\infty}f(y) dy[/tex] (since x>0) but not sure how to pick an f so that [tex]f = \bigcirc (|x|^{-(2+\epsilon)}) [/tex] as [tex]|x| \rightarrow \infty[/tex]
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: IVP prob involving convolutions
  1. IVP unique solution (Replies: 6)

  2. Capacitor prob (Replies: 18)

  3. Thermodynamic prob (Replies: 3)

  4. Spring prob (Replies: 8)

Loading...