The Numerical Solution of Volterra Integral Equations

Abstract

Recently we studied a collocation–quadrature method in weighted ${\mathbf{L}}^2$ spaces as well as in the space of continuous functions for a Volterra-like integral equation of the form $u\left(x\right)-{\int }_{\alpha x}^{1}h(x-\alpha y)u\left(y\right)dy=f\left(x\right),0<x<1,$ where $h\left(x\right)$ (with a possible singularity at $x=0$) and $f\left(x\right)$ are given (in general complex-valued) functions, and $\alpha \in (0,1)$ is a fixed parameter. Here, we want to investigate the same method for the case when $\alpha =1.$ More precisely, we consider (in general weakly singular) Volterra integral equations of the form $u\left(x\right)-{\int }_0^{x}h(x,y){(x-y)}^{\kappa }u\left(y\right)dy=f\left(x\right),0<x<1,$ where $\kappa >-1$, and $h:D⟶\mathbb{C}$ is a continuous function, $D=\left\{(x,y)\in {\mathbb{R}}^2:0<y<x<1\right\}.$ The passage from $0<\alpha <1$ to $\alpha =1$ and the consideration of more general kernel functions $h(x,y)$ make the studies more involved. Moreover, we enhance the family of interpolation operators defining the approximating operators, and, finally, we ask if, in comparison to collocation–quadrature methods, the application of the Nyström method together with the theory of collectively compact operator sequences is possible.