How to solve for unknown function in integral equation?

Hello,

I am trying to write code to solve the following integral equation:

Here, and are both known, and I would like to numerically find out what is. Can anyone help?

Thanks!

[Torsten]

In which form are h0 and g known ? As functions ? As discrete data ?

[Laila Voss]

@Torsten As functions, sorry for not specifying!

[John D'Errico]

You want to numerically solve for g(n), as a function? This can be sometimes difficult, but not always. The usual name given to the general problem is an inhomogeneous Fredholm integral equation (of the first kind).

https://en.wikipedia.org/wiki/Fredholm_integral_equation

A common class of solutions lies in the form of the Laplace transform, which you should recognize as a spcial case of the general problem you want to solve.

https://en.wikipedia.org/wiki/Laplace_transform

Another special case lies in the form of convolution/deconvolution problems, which should also be seen to be a special case.

You can even see similar problems arise from Fourier transforms, though over a different domain.

I would strongly recommend you do some reading before you even think about writing any code.

Answer 1

[Torsten]

One (maybe naive) approach:

Choose

0=n(1)<n(2)< ... <n(K)< oo

and

z(1),...,z(K)

Then use linsolve to solve the linear system of equations

h0(z(i)) - trapz(n,g(z(i),n).*f) = 0  (i=1,...,K)

for the unknown vector f=(f(1),...,f(K)).

[John D'Errico]

It might indeed have problems. Note that the integral goes to infinity, so trapz may have problems. As well, these problems are often ill-posed. So the approximation of a trapezoidal rule can be an issue.

[Torsten]

You are right, but it's a starting point.

In principle this is what

https://en.wikipedia.org/wiki/Integral_equation

suggests as numerical method (in general with a more complex quadrature rule than trapz, I guess).

Answer 2

[John D'Errico]

One idea can lie in the use of a family of functions the decompose the unknown f. Choose an orthogonal family over the chosen domain, so 0 to infinity in this case. That may mean Bessel functions. It may mean something like Laguerre polynomials. That is, we can write the integral in a form like

intt(K*f*dt) = int(K*exp(-t)*exp(t)*sum(a_i*p_i(t))*dt)

that is, if the polynomials P_i are Laguerre polynomials, they will be orthogonal with the inclusion of that exp(-t) in that Kernel. So now you write f(t) as a sum of the form

f(t) = sum(a_i*p_i(t))

This will work if that exp(t) term can be neatly absorbed into K.

How does this help? You will need to use a finite number of terms in that sum of course. But the integral and the sum can exchange places. And now you can try to solve for the unknown constants a_i.

The above are standard solution methods for this class of problems, though it has been many years since I worried about them.

Again, as I said before, you will need to do some reading, or spend some time talking to someone with expertise in these solutions.

[Laila Voss]

Thank you!

[Bjorn Gustavsson]

When chosing the basis-functions you should really think about the kernel g - that might give you a set of basis-functions that are ortogonal with that weighting. This would be a very practical set to use as far as I recall.

If you're lucky you might find some variable-transform that could convert your interval into a finite range and simultaneously get a kernel/weighting-function that has a known series of orthogonal basis-functions.