This paper deals with the numerical solution of systems of nonlinear integro-parabolic problems of Volterra type. The numerical approach is based on the method of upper and lower solutions. A monotone iterative method is constructed. Existence and uniqueness of a solution to the nonlinear difference scheme are established. An analysis of convergence rates of the monotone iterative method is given. Construction of initial upper and lower solutions is discussed. Numerical experiments are presented.

We introduce the boundary integral operator induced from the fractional Laplace equation on the boundary of a bounded smooth domain. For~$\frac 12\lt \alpha \lt 1$, we show the bijectivity of the boundary integral operator~$S_{2\alpha }:L^p(\partial \Omega )\to H^{2\alpha -1}_p(\partial \Omega )$ for $1 \lt p \lt \infty $. As an application, we demonstrate the existence of the solution of the Dirichlet boundary value problem of the fractional Laplace equation.

Given a concave integro-differential operator $I$, we study regularity for solutions of fully nonlinear, nonlocal, parabolic equations of the form $u_t-Iu=0$. The kernels are assumed to be smooth but non necessarily symmetric, which accounts for a critical non-local drift. We prove a $C^{\sigma +\alpha }$ estimate in the spatial variable and $C^{1,\alpha }$ estimates in time assuming time regularity for the boundary data. The estimates are uniform in the order of the operator $I$, hence allowing us to extend the classical Evans-Krylov result for concave parabolic equations.

We present a comparison among the performance of solvers based on Nystr\"om discretizations of several well-posed boundary integral equation formulations of Helmholtz transmission problems in two-dimensional Lipschitz domains. Specifically, we focus on the following four classes of boundary integral formulations of Helmholtz transmission problems: (1)~the classical first kind integral equations for transmission problems~\cite {costabel-stephan}, (2)~the classical second kind integral equations for transmission problems~\cite {KressRoach}, (3)~the single integral equation formulations~\cite {KleinmanMartin}, and (4)~certain direct counterparts of recently introduced generalized combined source integral equations \cite {turc2, turc3}. The former two formulations were the only formulations whose well-posedness in Lipschitz domains was rigorously established \cite {costabel-stephan, ToWe:1993}. We establish the well-posedness of the latter two formulations in appropriate functional spaces of boundary traces of solutions of transmission Helmholtz problems in Lipschitz domains. We give ample numerical evidence that Nystr\"om solvers based on formulations (3) and (4) are computationally more advantageous than solvers based on the classical formulations (1) and (2), especially in the case of high-contrast transmission problems at high frequencies.