How can I solve an integral equation with an unknown kernel?
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The equation I am trying to solve is:
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/170497/image.gif)
where f(x) and h(x) are both complex and known, and g(x) is an unknown function. Presumably, the result should be a function g(x), however, it is not to be excluded that g(x) could actually be an operator instead. Can this be solved for either cases in MATLAB?
Thanks!
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Torsten
2017년 12월 1일
g is not unique - it can be of any function type you like (we already had this discussion).
g(x)=1/integral_{x=0}^{x=2*pi} f(x)*h(x)dx
or
g(x)=1/(f(x)*h(x)*2*pi)
or
...
Best wishes
Torsten.
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Torsten
2017년 12월 12일
Why don't you start from a solution that worked ?
syms L C x
assume (L>0);
h = 1;
g = 5;
y = C-exp(2*g*1i*x/h);
z = C-exp(-2*g*1i*x/h);
prod = y*z*(1+x^2);
Csol = solve(int(prod,x,0,L)-1==0,C);
Best wishes
Torsten.
추가 답변 (2개)
John D'Errico
2017년 12월 1일
If g(x) is unknown, then if all you have is a single equation equal to a constant, then there is no simple solution. Or, you can look at it as if there are infinitely many solutions, one of them being a constant function.
Just compute the integral of h(x)*f(x). Take the reciprocal. That is the value of the constant g that will make int(h*f*g) equal 1. So as long as int(h*f) over [0,2*pi] is not identically 0, then A solution is trivial. Yes there may be infinitely many other solutions, but they cannot be found unless you have information as to the functional form of g(x).
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John D'Errico
2017년 12월 4일
I don't see why not. Integration is just a linear operator. If g is a constant, then it can be pulled outside the integral.
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