In this work, we prove the existence and uniqueness of the conditional expectation through the geometry of Lebesgue spaces.
This paper proposes to model fractional behaviors using Volterra equations. As fractional differentiation-based models that are commonly used to model such behaviors exhibit several drawbacks and are particular cases of Volterra equations... more
In this article, we investigate the method of upper and lower solutions for Volterra integral equation of the first kind on arbitrary time scale T. We establish some existence results in a certain sector. Moreover, monotone iterative... more
In this paper, it is proved that the Hofbauer-So-Takeuchi conjecture for the global stability of a Lotka-Volterra system with discrete diffusion holds true in the case of n = 5. By using the computer algebra system Maple, and based on the... more
In this paper, by applying the geometric criterion and time average property to Lotka-Volterra systems, some results for the global asymptotic stability of the systems are obtained. Furthermore, we consider Li-Wang Conjecture for a... more
In this paper, by applying the geometric criterion and time average property to Lotka-Volterra systems, some results for the global asymptotic stability of the systems are obtained. Furthermore, we consider Li-Wang Conjecture for a... more
In this paper, it is proved that the Hofbauer-So-Takeuchi conjecture for the global stability of a Lotka-Volterra system with discrete diffusion holds true in the case of n = 5. By using the computer algebra system Maple, and based on the... more
Precognitive dreams-vivid previews of future events-suggest consciousness transcends linear time. This paper presents an in-depth, physics-based model integrating quantum mechanics, post-quantum mechanics (PQM), and Stafford's... more
The almost sure rate of exponential-polynomial growth or decay of affine stochastic Volterra and affine stochastic finite-delay equations is investigated. These results are achieved under suitable smallness conditions on the intensities... more
The paper studies the subexponential convergence of solutions of scalar Itô-Volterra equations. First, we consider linear equations with an instantaneous multiplicative noise term with intensity σ. If the kernel obeys lim t→∞ k (t)/k(t) =... more
In this paper, we are interested in comparing solutions to stochastic Volterra equations for the convex order on the space of continuous R d -valued paths and for the monotonic convex order when d = 1. Even if in general these solutions... more
This paper mainly focuses on the numerical solution of the two-dimensional second-kind Fredholm integral equation by the Jacobi-Legendre Spectral-Galerkin method. Based on the Gauss points related to the Jacobi weight function as the... more
In this paper, stochastic Volterra equations, particularly fractional, in Hilbert space are studied. Sufficient conditions for existence of strong solutions are provided.
A new algorithm for determining the output frequency range and the frequency components of Volterra models under multiple inputs is introduced for nonlinear system analysis. For a given Volterra model, the output frequency components... more
A new algorithm for determining the output frequency range and the frequency components of Volterra models under multiple inputs is introduced for nonlinear system analysis. For a given Volterra model, the output frequency components... more
This paper deals with the controllability, observability, and stability of the solution of time-varying Volterra integro-dynamic system on time scales. We obtain new results about controllability and observability and generalize to a time... more
We prove the existence and uniqueness, local in time, of the solution of a one-phase Stefan problem for a non-classical heat equation for a semi-infinite material with a heat flux boundary condition at the fixed face x = 0. Here the heat... more
A direct method is implemented for the numerical solution of second-order Volterra integrodifferential equations (VIDEs). The formulation of two-point hybrid block method will be discussed in this paper to solve second order VIDEs... more
In this study, we develop and implement a numerical approach for solving first-order Volterra integro-differential equations. We derive the integral form of the problem, which is then transformed into an algebraic equation system using... more
The gravitational interaction between two bodies is a fundamental problem in classical mechanics, traditionally solved using energy conservation or differential equations. This paper explains a novel approach to find the velocities of two... more
By using the theory of resolvent families, fixed point theorems and measures of noncompactness, we prove the existence of mild solutions on a compact interval for a semilinear Volterra equation with state-dependent delay. An example is... more
Given a ∈ L 1 (R) and A the generator of an L 1-integrable family of bounded and linear operators defined on a Banach space X, we prove the existence of almost automorphic mild solution to the semilinear integral equation u(t) = ∫ t −∞... more
In order to reveal the relationship between system time domain model parameters and system frequency response functions, new magnitude bounds of frequency response functions for nonlinear Volterra systems described by NARX model are... more
In this paper we consider a linear stochastic Volterra equation which has a stationary solution. We show that when the kernel of the fundamental solution is regularly varying at infinity with a log-convex tail integral, then the... more
We aim to give the numerical method for solving the fractional Volterra integral equations of first and second kinds. We here use the techniques based upon rational Chebyshev functions and Riemann-Liouville fractional integrals. Some... more
We continue the paper [Ts] on the boundedness of polynomials in the Volterra operator. This provides new ways of constructing power-bounded operators. It seems interesting to point out that a similar procedure applies to the operators... more
We continue the paper [Ts] on the boundedness of polynomials in the Volterra operator. This provides new ways of constructing power-bounded operators. It seems interesting to point out that a similar procedure applies to the operators... more
Non-stationary discrete time waveform relaxation methods for Abel systems of Volterra integral equations using fractional linear multistep formulae are introduced. Fully parallel discrete waveform relaxation methods having an optimal... more
Nonlinear problems for the one-dimensional heat equation in a bounded and homogeneous medium with temperature data on the boundaries x = 0 and x = 1, and a uniform spatial heat source depending on the heat flux (or the temperature) on the... more
In this paper we propose a method of piecewise constant approximation for the solution of ill-posed third kind Volterra equations p(t)z(t) + Z t 0 h(t) (t ;) 1; z()d = f(t) t 2 0 1] 0 < < 1: Here p(t) vanishes on some subset of t 1 t 2 ]... more
In this paper, we study a Lotka-Volterra model which contains two prey and one predator with the Beddington-DeAngelis functional responses. First, we establish a set of sufficient conditions for existence of positive periodic solutions.... more
Volterra integro-differential equations appear in many branches of engineering, physics, biology, astronomy, radiology and having many interesting applications such as process of glass forming, diffusion process, heat and mass transfer,... more
Oscillatory solutions play a pivotal role in understanding functional differential and integral equations, offering insights into the behaviour of these equations' solutions, and assisting in understanding their growth, stability, and... more
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with... more
In this paper, we study the graph theoretical polynomial known as the Hosoya polynomial obtained from one of the standard classes of graphs called path. Using this polynomial applied for the numerical solution of the nonlinear Fredholm... more
We study positive linear Volterra integro-differential systems with infinitely many delays. Positivity is characterized in terms of the system entries. A generalized version of the Perron-Frobenius Theorem is shown; this may be... more
In this paper, we generalize two types of Volterra integral equations given on time scales and examine their Hyers-Ulam and Hyers-Ulam-Rassias stabilities. We also prove these stability results for the non-homogeneous nonlinear Volterra... more
An exact real solution of linear Volterra – Fredholm and Volterra loaded integro-differential equation Bx = f is presented.
The purpose of this note is to generalize the Wazewski's Topological Method 11, originally stated for ordinary differential equations, to the integrodifferential equation of Volterra type (1), under suitable conditions on the functions... more
In this paper, we generalize two types of Volterra integral equations given on time scales and examine their Hyers-Ulam and Hyers-Ulam-Rassias stabilities. We also prove these stability results for the non-homogeneous nonlinear Volterra... more
In the present paper the authors propose two numerical methods to approximate Hadamard transforms of the type H p (f w β , t) = = f (x) (x − t) p+1 w β (x)d x, where p is a nonnegative integer and w β (x) = e −|x| β , β > 1, is a Freud... more
This paper deals with the controllability, observability, and stability of the solution of time-varying Volterra integro-dynamic system on time scales. We obtain new results about controllability and observability and generalize to a time... more
We study conditions under which the solutions of linear Volterra integrodynamic system of the formyΔt=Atyt+∫t0tKt,sysΔsare stable on certain time scales. We construct a number of Lyapunov functionals on time scales from which we obtain... more
The most famous models of A. Lotka and V. Volterra, "predator-prey" and "competition," are considered in relation to social systems. The peculiarity of the social systems that distinguishes them from the biological ones is a quick... more
Nonlinear problems for the one-dimensional heat equation in a bounded and homogeneous medium with temperature data on the boundaries x = 0 and x = 1, and a uniform spatial heat source depending on the heat flux (or the temperature) on the... more
Under the assumption that A is the generator of a twice integrated cosine family and K is a scalar valued kernel, we solve the singular perturbation problem (E) 2 u (t) + u (t) = Au (t) + (K * Au)(t) + f (t), (t ≥ 0)(> 0), when → 0 + ,... more