# 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

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

Bruno Luong on 13 Nov 2020
These equation does not make sense.
In order for
int (c*x+b)/(a*x+d)
to converge at -inf; the, limits must be 0, meaning c/a is 0 so c==0. So the indefinite integral is 1/a*log(..), and it diverges.
These integrals you write has no limiting value. Period. No need to write an equation involving something that is not defined.
SACHIN VERMA on 13 Nov 2020
sorry for wrong equations. These are only sample equation as actual equation is quite complicated. I think Changing the limit from 0-1 fix the problem. I only need rough idea that how to solve four self-consistent integral equations in matlab. I have done it for two equations already by applying iterration.
SACHIN VERMA on 13 Nov 2020
Bjorn Gustavsson thanks for the answer your code works well for any finite limit. I solved two equations iteratively but i don't know how to iterate four equations. I will try to solve the actual equations by using above code.