Solving a system of integral equations self-consistently
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SACHIN VERMA
on 13 Nov 2020
Edited: SACHIN VERMA
on 13 Nov 2020
Hi I have four integral equations with four unknowns I need to solve these equations self-consistently. Previously I solved two integral equations self-consistently in matlab and now trying solve four or multiple integral equations. The four sample equations with unknowns a,b,c and d are attached below. I don't need a complete code but a rough idea how solve such system of equations. (Note that the equations attached are not actual complex equations which I need to solve but these simpler equations gives an idea of the problem) Any help would be appreciated! Thanks
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Accepted Answer
Bjorn Gustavsson
on 13 Nov 2020
Why doesn't your solution to two (I assume similar) integral equations work for this case?
Here's a QD-cookup (that doesn't work too well for your possibly illposed example):
% Integration-function, ought to be adaptible, doesn't work too well in this example
Q1 = @(pars) integral(@(x) (pars(1)*x+pars(2))./(pars(3)*x+pars(4)),-inf,0);
% Sum-of-squared residuals function:
errfcn = @(pars) ( (pars(1) -Q1(pars([3 2 1 4])) )^2 + ...
(pars(2) -Q1(pars([1 4 2 3])) )^2 + ...
(pars(3) -Q1(pars([2 4 3 1])) )^2 + ...
(pars(4) -Q1(pars([1 2 4 3])) )^2 );
% Residual-function:
resfcn = @(pars) [(pars(1) -Q1(pars([3 2 1 4])) );
(pars(2) -Q1(pars([1 4 2 3])) );
(pars(3) -Q1(pars([2 4 3 1])) );
(pars(4) -Q1(pars([1 2 4 3])) )];
% Solution with fmnisearch:
par1 = fminsearch(@(pars) errfcn(pars),[1 1 1 1]);
% Solution with lsqnonlin:
par2 = lsqnonlin(@(pars) resfcn(pars),[1 1 1 1],[0 0 0 0],1e12*[1 1 1 1]);
Hopefully your integrals are at least nicely behaved enough that there are parameters that makes them all converge over the integration intervall.
HTH
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