2
$\begingroup$

Is there any close-form solution for a function $f(t)$ satisfied the below equation: $f(t)=g(t)+\frac{1}{t^2}(h(t)*f(t))$. Operator $*$ is convolution integral, and $g(t)$ and $h(t)$ are known functions.

$\endgroup$
6
  • 2
    $\begingroup$ Please provide more detail, for example what class of functions are you considering? Measurable? Smooth? C^1? Schwarz? etc $\endgroup$
    – David Roberts
    Jul 8, 2013 at 10:56
  • 1
    $\begingroup$ On that set $f$ is defined? $\endgroup$
    – Andrew
    Jul 8, 2013 at 11:11
  • $\begingroup$ $t>0$ and is a real value. At the first, we have no condition on $f$. $\endgroup$
    – Venous007
    Jul 8, 2013 at 12:10
  • $\begingroup$ And what is a "closed form"? A formula with Fourier transforms and convolutions is a "closed form" or not? $\endgroup$ Jul 8, 2013 at 22:13
  • 1
    $\begingroup$ I think you should give explicitly $g$ and $h$. For $t\ne 0$, a Fourier transform of this equation gives $f''(\omega)+h(\omega)f(\omega)=g''(\omega)$ that has known solutions just for some special cases of h. $\endgroup$
    – Jon
    Jul 9, 2013 at 12:41

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.