Taking the standard zero curvature approach we derive an infinite set of inte-grable equations, which taken together form the negative Volterra hierarchy. The resulting equations turn out to be nonlocal, which is usual for the negative flows. However, in some cases the nonlocality can be eliminated. Studying the combined action of both positive (classical) and negative Volterra flows, i.e. cons...

In this paper, stochastic Volterra equations, particularly fractional, in Hilbert space are studied. Sufficient conditions for mild solutions to be strong solutions are provided. Several examples of Volterra equations having strong solutions are given, as well.

In this paper stochastic Volterra equations admitting exponentially bounded resolvents are studied. After obtaining convergence of resolvents, some properties for stochastic convolutions are studied. Our main results provide sufficient conditions for strong solutions to stochastic Volterra equations.

In this paper, we investigate periodic solutions of linear and nonlinear discrete Volterra equations of convolution or non-convolution type with unbounded memory. For linear discrete Volterra equations of convolution type, we establish Fredholm’s alternative theorem and for equations of non-convolution type, and we prove that a unique periodic solution exists for a particular bounded initial fu...

Over the last decade, the field of geometric integration has rapidly established itself as one of the core research areas in numerical ordinary differential equations. Geometric integrators are numerical methods which preserve some of the mathematical or physical properties of the system they are approximating. In the case of the Lotka-Volterra equations, which are a Poisson system, a good geom...

Abstract A new numerical method based on Haar wavelet is proposed for two-dimensional nonlinear Fredholm, Volterra and Volterra-Fredholm integral equations of first and second kind. The proposed method is an extension of the Haar wavelet method [1–3] from one-dimensional nonlinear integral equations (Fredholm and Volterra) to twodimensional nonlinear integral equations (Fredholm, Volterra and V...

In this paper, the successive overrelaxation iteration and the variation of parameters formula are used to discuss the stability of linear and nonlinear Volterra integrodifferential equations. Some sufficient conditions are obtained, such that the trivial solution of the Volterra integrodifferential equations is asymptotically stable; these stability conditions are given directly from the coeff...

This paper presents a numerical matrix method based on Bernstein polynomials (BPs) for approximate the solution of a system of m-th order nonlinear Volterra integro-differential equations under initial conditions. The approach is based on operational matrices of BPs. Using the collocation points,this approach reduces the systems of Volterra integro-differential equations associated with the giv...