Numerical solution of certain Volterra-Fredholm integral equations

Question

I am dealing with Volterra-Fredholm integral equations of the following form: \begin{equation} \tag{1} \phi(x) = f(x) + n \int_a^x K(x,t) \phi(t) dt + \int_a^b K(x,t)\phi(t)dt, \end{equation} where $a,b \in \mathbb{R}, n \in \mathbb{N}$, $f$ is continuous, and $K \in L^2([a,b];\mathbb{R})$ is symmetric and positive semidefinite. $\phi$ is the unknown function to be solved for. I have already proved existence and uniqueness of a solution and would like to solve the equation numerically.

One could view (1) as a pure Fredholm equation with kernel $\tilde{K}(x,t) = K(x,t)(1 + n \, 1_{\{x \ge t\}}),$ but $\tilde{K}$ is neither symmetric nor continuous when $n \neq 0$. This appears overly complicated to me.

Most research papers on Volterra-Fredholm equations seem to deal with nonlinear or mixed equations. There is some nice theory about Fredholm equations with symmetric, positive semidefinite kernels, but I could not find similar results for Volterra-Fredholm equations.

Can you recommend literature that helps me solve (1) numerically?

Answer 1

It seems that viewing (1) as a pure Fredholm equation with asymmetric and discontinuous kernel is the method of choice.

This paper by K.E. Atkinson (1967) uses a quadrature method (also known as Nystrom method) to approximate Fredholm equations numerically. The only assumptions on the kernel $K$ are:

• $\sup_{x \in [a,b]} \int_a^b K(x,t) dt < \infty$ and
• $\int_a^b |K(x_1,t) - K(x_2,t)| dt \to 0$ uniformly for $x_1, x_2 \in [a,b]$ as $|x_1 - x_2| \to 0$.

If the kernel $K$ in (1) satisfies these assumptions, so does $\tilde{K}(x,t) = K(x,t)(1+n1_{\{x\ge t\}})$.

I am still grateful for anything you might know about Volterra-Fredholm equations. Judging from the age of Atkinson's paper, I guess there are more powerful methods available.