Using ApproxFun.jl to solve a Volterra equation of the second kind

Specific Domains Numerics
1

I’m trying to learn how to solve Volterra equations of the second kind using ApproxFun.jl.

For example, consider the following equation, where u is an unknown function on [0,1/2].

volteq

The closest similar example I could find was this, but I don’t know how to include the information regarding the different limits of integration, so ApproxFun.jl is giving me a u that is different from the solution that I can solve for analytically.

Can anyone help? I’ve tried the documentation, but I don’t quite yet see how to adapt the examples there to the equation I have.

2

I’ll have a look and get back to you

1 Like
3

Just to confirm: it’s 1/(1-x) and not 1/(1-t), correct? So that the term can be pulled out? I think this works:

d = Segment(0.5,0)
x = Fun(d)
V = Volterra(Legendre(d))
u = (I - 1/(1-x) * V) \ 1

If you meant 1/(1-t) just put it to the right of the V, (using a bracket so it knows you don’t want to apply the operator to the function 1/(1-x)):

u = (I - V[1/(1-x)]) \ 1
2 Likes
4

Yes, my original post was correct, but the last line is good to know for more general versions of the problem I’m considering.

Thank you so much!

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(post deleted by author)

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Coming back to this, I am wondering what is the best way to represent

\int_0^1\exp(-|t-s|)u(s)ds

Indeed, if using a definite integral, the kernel is not smooth. We can cut it in two volterra kernels but the second one is \int_t^1 \exp(-s)u(s)ds et I dont know how to reverse Volterra