# Numerical solution of certain Volterra-Fredholm integral equations

I am dealing with Volterra-Fredholm integral equations of the following form: $$\tag{1} \phi(x) = f(x) + n \int_a^x K(x,t) \phi(t) dt + \int_a^b K(x,t)\phi(t)dt,$$ 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 (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?

• I didn't even know "Volterra-Fredholm" equations were a thing; most of the time it's either-or. Not much of the literature develops this theory, I guess. Thanks for posting. – SZN Nov 17 '16 at 5:21

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 $\tilde{K}(x,t) = K(x,t)(1+n1_{\{x\ge t\}})$.