How to solve a system of integral equations?

Question

Pavel M 2020-2-14

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/505547-how-to-solve-a-system-of-integral-equations

评论： Star Strider 2020-2-14

I want to solve the system of integral equations, but limits on integrals contain an unknown ( x(2) ) which i want to find.

I try this:

function S = Integralsystem(x, t1, t2, n, a, b, Umax1, Umax2);
fun = @(T) x(2) - (Umax1/n)*(exp(a*(T*1e-6)) - exp(b*(T*1e-6)));
t01 = fzero(fun, 0.1);
fun = @(T) x(2) - (Umax2/n)*(exp(a*(T*1e-6)) - exp(b*(T*1e-6)));
t02 = fzero(fun, 1.1);
fun1 = @(T) ((Umax1/n)*(exp(a*(T*1e-6)) - exp(b*(T*1e-6)))) - x(2);
fun2 = @(T) ((Umax1/n)*(exp(a*(T*1e-6)) - exp(b*(T*1e-6)))) - x(2);
S(1) = x(1) - (integral(fun1,t01,t1));
S(2) = x(1) - (integral(fun2,t02,t2));
end
s = fsolve(@(x) Integralsystem(x, t1, t2, n, a, b, Umax1, Umax2),[100 1000])

but Matlab cant find solution.

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Answer 1

Star Strider 2020-2-14

1
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/505547-how-to-solve-a-system-of-integral-equations#answer_415623

在 MATLAB Online 中打开

It is probably best to use the more robust fsolve in the function instead of fzero.

Try this:

function S = Integralsystem(x, t1, t2, n, a, b, Umax1, Umax2);
fun1 = @(T) x(2) - (Umax1/n)*(exp(a*(T*1e-6)) - exp(b*(T*1e-6)));
t01 = fsolve(fun1, 0.1);
fun2 = @(T) x(2) - (Umax2/n)*(exp(a*(T*1e-6)) - exp(b*(T*1e-6)));
t02 = fsolve(fun2, 1.1);
fun3 = @(T) ((Umax1/n)*(exp(a*(T*1e-6)) - exp(b*(T*1e-6)))) - x(2);
fun4 = @(T) ((Umax1/n)*(exp(a*(T*1e-6)) - exp(b*(T*1e-6)))) - x(2);
S(1) = x(1) - (integral(fun3,t01,t1));
S(2) = x(1) - (integral(fun4,t02,t2));
end
s = fsolve(@(x) Integralsystem(x, t1, t2, n, a, b, Umax1, Umax2),[100 1000])

This slightly revised code (with random scalar values for the other agruments) ran without error and produced a (1x2) vector for ‘s’.

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Pavel M 2020-2-14

i use fzero because tmin - limit of integral has such condition t0 = f(x(2))

Star Strider 2020-2-14

With the random scalars I supplied to test your function, fzero threw errors. That was the reason I substituted fsolve. Use whatever works best in your application.

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