Can chebfun solve nonlinear Volterra integral equations?

[Mike]

Hello Chebfun friends!

I am wondering whether or not if is possible to solve nonlinear integral equations of Volterra types using chebfun? For example

is it possible to solve the following integral equation:

u(x) = exp(x) + 1/3*x*(1-exp(3*x)) + integral(x*u(t)^{3} dt, t from 0 to x)

whose solution is u(x)=exp(x); 

Your help is greatly appreciated!

Kind wishes,

Babak

[Nick Hale]

Hi Babek

Yes, it can!

K = @(x,y) 1+0*x;
N = chebop(@(x,u) - u + exp(x) + 1/3*x*(1-exp(3*x)) + x*volt(K, u^3), [0, 1]);
u = N\0;
norm(u - chebfun(@exp, [0 1]))
ans =
   8.2789e-13

You could also use cumsum(u^3) rather than volt(K, u^3).

Note however that there's currently small bug in the master branch that prevents this code from working.

I issued a fix here:  https://github.com/chebfun/chebfun/pull/2361 and hopefully this can be merged soon.

Nick

[Calin Gheorghiu]

Hello CHEBFUN friends, hello Nick Hale,

 
Unfortunately, I cannot find the bug for the above code. I still get the following error:

Error using mat2cell (line 106)…

Error in linop/fitBCs>my partition(line 173)…

Error in linop/fitBCs (line 131)…

The code seems simple and clear but…

Any suggestions are welcome and appreciated.

Thanks un advance.

Best regards,

Calin

[Nick Hale]

Dear Calin 

Please provide a minimal working of your code that is not working.

However, if you are referring to the code I posted previously, that seems

to work fine for me using the current release:

>> K = @(x,y) 1+0*x;
N = chebop(@(x,u) - u + exp(x) + 1/3*x*(1-exp(3*x)) + x*volt(K, u^3), [0, 1]);
u = N\0;
norm(u - chebfun(@exp, [0 1]))
ans =

   6.1457e-13

Regards

Nick