Solving a system of integral equations self-consistently

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

Answer 1

[Bjorn Gustavsson]

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]

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]

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]

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.