In this paper, an effective numerical method is introduced for the treatment of nonlinear two-dimensional Volterra-Fredholm integro-differential equations. Here, we use the so-called two-dimensional block-pulse functions.First, the two-dimensional block-pulse operational matrix of integration and differentiation has been presented. Then, by using this matrices, the nonlinear two-dimensional Vol...

Abstract We study robust regularity estimates for a class of nonlinear integro-differential operators with anisotropic and singular kernels. In this paper, we prove Sobolev-type inequality, weak Harnack local Hölder estimate.

In this paper, we present a comparative study between the modified variational iteration method (MVIM) and a hybrid of Fourier transform and variational iteration method (FTVIM). The study outlines the efficiencyand convergence of the two methods. The analysis is illustrated by investigating four singular partial differential equations with variable coefficients. The solution of singular partia...

We study a problem of optimal consumption and portfolio selection in a market where the logreturns of the uncertain assets are not necessarily normally distributed. The natural models then involve pure-jump L evy processes as driving noise instead of Brownian motion like in the Black and Scholes model. The state constrained optimization problem involves the notion of local substitution and is o...

In this paper, we propose and justify the quadrature-differences method for the full linear singular integro-differential equations with the Cauchy kernel on the interval (−1,1). We consider equations of zero, positive and negative indices. It is shown that the method converges to an exact solution, and the error estimation depends on the sharpness of derivative approximations and on the smooth...

In this work we consider a partial integro-differential equation. We reformulate it a functional integro-differential equation in a suitable Hilbert space. We apply the method of lines to establish the existence and uniqueness of a strong solution.

The non-local in space two-phase Stefan problem (a prototype phase change problems) can be formulated via a singular nonlinear parabolic integro-differential equation which admits unique weak solution. This formulation makes to part of the General Filtration Problems; class includes Porous Medium Equation. In this work, we prove that solutions both and Media problems are continuous.

We study a generalized extrapolated Crank–Nicolson scheme for the time discretization of a semilinear integro-differential equation with a weakly singular kernel, in combination with a space discretization by linear finite elements. The scheme uses variable grids in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assumin...