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.(adsbygoogle = window.adsbygoogle || []).push({});

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]

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: IVP prob involving convolutions

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**