Numerical solution of integral equation with parametric variable

Hello!

Please, how can I solve integral equation in Matlab

L = integral (f(b,t)) dt for third variable (parametric variable b)

Limits of integral are t0, t1.

[bym]

numerically or symbolically?

Answer 1

[Jarrod Rivituso]

OK, I'm gonna assume you want to do it numerically. Check out this (warning, it gets a little crazy with the function handles):

function solveIntegralForB

bfound = fsolve(@func2minimize,2)

      function output = func2minimize(b)
          t0 = 0;
          t1 = 3;
          L = 50;
          output = (L - quad(@myFunc,t0,t1))^2;
          function f = myFunc(t)
              f = exp(b*t);
          end
      end
end

Essentially, what it does is use the quad function to perform an integration for some value of b. Additionally, it uses the fsolve function to then minimize the "func2minimize" function, which performs the integral for some value of b and checks it against my desired solution.

Here, I've assumed a simple function (exp(b*t)) as f, but you could see how it could be changed. Of course, this is an iterative solution, so there's no guarantee of it finding a solution for all functions f.

Hope this helps!

[Pooyan]

How if L is a known function of t and b is an unknown function of t? Can anyone help me on that? and again I want to solve the equation for b(t). Let's assume L(t)= sin(t) and t=0:0.1:10;

Thanks.

Answer 2

[Matt Tearle]

A variation on Jarrod's approach, using function handles (because everyone loves function handles):

myFunc = @(t,b) exp(t*b); % or whatever
t0 = 0;
t1 = 3;
L = 50;
f = @(b) quad(@(t) myFunc(t,b),t0,t1);
bsolve = fzero(f,2);

Or fsolve instead of fzero if you have Optimization Toolbox.

Answer 3

[Walter Roberson]

If you have the symbolic toolbox,

syms x b
solve(int(f(x,b),x,t0,t1)-L,b)

In theory if it can be solved symbolically it will do so, and if not then MuPad should switch to numeric integrations, I think.