In this paper we study singular perturbations of weakly coupled systems of elliptic equations. A model problem is given by small random perturbations of random evolution processes. In this setting we give a PDE proof of large deviation results, analogous to those studied in [5], [10, 11]. An essential tool in our approach is the weak KAM theory introduced in [15].

We describe a modified Nyström method for the discretization of the weakly singular boundary integral operators which arise from the formulation of linear elliptic boundary value problems as integral equations. Standard Nyström and collocation schemes proceed by representing functions via their values at a collection of quadrature nodes. Our method uses appropriately scaled function values in l...

In present paper we elaborated the numerical schemes of iterative methods for an approximate solution of weak singular integral equations with logarithmic Kernel. The equation is examined in a pair of spaces. The results obtained could be used for any pair of the functional spaces where the problem of finding the solution of weak singular integral equations is correctly formulated problem by Ti...

Let ū(t) be a control that satisfies the infinite-dimensional version of Pontryagin’s maximum principle for a linear control system, and let z(t) be the costate associated with ū(t). It is known that integrability of ‖z(t)‖ in the control interval [0, T ] guarantees that ū(t) is time and norm optimal. However, there are examples where optimality holds (or does not hold) when ‖z(t)‖ is not integ...

Abstract. To find the approximate solutions of a weakly singular integral equation by the collocation method it is necessary to solve linear systems whose coefficients are expressed as integrals. These integrals cannot usually be computed exactly. We get the fully discrete collocation method when we approximate the integrals by quadrature formulas on nonuniform grid. In this paper an appropriat...

‎In this paper‎, ‎a matrix based method is considered for the solution of a class of nonlinear Volterra integral equations with a kernel of the general form $s^{beta}(t-s)^{-alpha}G(y(s))$ based on the Tau method‎. ‎In this method‎, ‎a transformation of the independent variable is first introduced in order to obtain a new equation with smoother solution‎. ‎Error analysis of this method is also ...

In this study, we develop some novel Integral Inequalities (InIs) with weakly singular singularities that expand commonly known ones. Utilizing tempered ?-Fractional Differential Equations (FDEs), many applications for FDEs in the context of Caputo have been developed.