Mean Square Summability of Solution of Stochastic Difference Second-Kind Volterra Equation with Small Nonlinearity

Stochastic difference second-kind Volterra equation with continuous time and small nonlinearity is considered. Via the general method of Lyapunov functionals construction, sufficient conditions for uniform mean square summability of solution of the considered equation are obtained

Multistep collocation methods for Volterra Integral Equations

We introduce multistep collocation methods for the numerical integration of Volterra Integral Equations, which depend on the numerical solution in a fixed number of previous time steps. We describe the constructive technique, analyze the order of the resulting methods and their linear stability properties. Numerical experiments confirm the theoretical expectations

Stability of Equilibrium Points of Fractional Difference Equations with Stochastic Perturbations

It is supposed that the fractional difference equation , has an equilibrium point and is exposed to additive stochastic perturbations type of that are directly proportional to the deviation of the system state from the equilibrium point . It is shown that known results in the theory of stability of stochastic difference equations that were obtained via V. Kolmanovskii and L. Shaikhet general method of Lyapunov functionals construction can be successfully used for getting of sufficient conditions for stability in probability of equilibrium points of the considered stochastic fractional difference equation. Numerous graphical illustrations of stability regions and trajectories of solutions are plotted

Two-step almost collocation methods for Volterra integral equations

In this paper we construct a new class of continuous methods for Volterra integral equations. These methods are obtained by using a collocation technique and by relaxing some of the collocation conditions in order to obtain good stability properties

Exponentially fitted two-step hybrid methods for y''=f(x,y)

It is the purpose of this paper to derive two-step hybrid methods for y '' = f(x, y), with oscillatory or periodic solutions, specially tuned to the behaviour of the solution, through the usage of the exponential fitting technique. The construction of two-step exponentially fitted hybrid methods is shown and their properties are discussed. Some numerical experiments confirming the theoretical expectations are provided

Two classes of linearly implicit numerical methods for stiff problems: analysis and MATLAB software

The purpose of this work lies in the writing of efficient and optimized Matlab codes to implement two classes of promising linearly implicit numerical schemes that can be used to accurately and stably solve stiff Ordinary Differential Equations (ODEs), and also Partial Differential Equations (PDEs) through the Method Of Lines (MOL). Such classes of methods are the Runge-Kutta (RK) [28] and the Peer [17], and have been constructed using a variant of the Exponential-Fitting (EF) technique [27]. We carry out numerical tests to compare the two methods with each other, and also with the well known and very used Gaussian RK method, by the point of view of stability, accuracy and computational cost, in order to show their convenience

Some new uses of the η_m(Z) functions

We present a procedure and a MATHEMATICA code for the conversion of formulae expressed in terms of the trigonometric functions sin(omega x), cos(omega x) or hyperbolic functions sinh(lambda x), cosh(lambda x) to forms expressed in terms of eta(m)(Z) functions. The possibility of such a conversion is important in the evaluation of the coefficients of the approximation rules derived in the frame of the exponential fitting. The converted expressions allow, among others, a full elimination of the 0/0 undeterminacy, uniform accuracy in the computation of the coefficients, and an extended area of validity for the corresponding approximation formulae

Exponentially-fitted Gauss-Laguerre quadrature rule for integrals over an unbounded interval

New quadrature formulae are introduced for the computation of integrals over the whole positive semiaxis when the integrand has an oscillatory behavior with decaying envelope. The new formulae are derived by exponential fitting, and they represent a generalization of the usual Gauss-Laguerre formulae. Their weights and nodes depend on the frequency of oscillation in the integrand, and thus the accuracy is massively increased. Rules with one up to six nodes are treated with details. Numerical illustrations are also presented

Optimal control of system governed by nonlinear volterra integral and fractional derivative equations

AbstractThis work presents a novel formulation for the numerical solution of optimal control problems related to nonlinear Volterra fractional integral equations systems. A spectral approach is implemented based on the new polynomials known as Chelyshkov polynomials. First, the properties of these polynomials are studied to solve the aforementioned problems. The operational matrices and the Galerkin method are used to discretize the continuous optimal control problems. Thereafter, some necessary conditions are defined according to which the optimal solutions of discrete problems converge to the optimal solution of the continuous ones. The applicability of the proposed approach has been illustrated through several examples. In addition, a comparison is made with other methods for showing the accuracy of the proposed one, resulting also in an improved efficiency

Recommender systems: a novel approach based on singular value decomposition

Due to modern information and communication technologies (ICT), it is increasingly easier to exchange data and have new services available through the internet. However, the amount of data and services available increases the difficulty of finding what one needs. In this context, recommender systems represent the most promising solutions to overcome the problem of the so-called information overload, analyzing users' needs and preferences. Recommender systems (RS) are applied in different sectors with the same goal: to help people make choices based on an analysis of their behavior or users' similar characteristics or interests. This work presents a different approach for predicting ratings within the model-based collaborative filtering, which exploits singular value factorization. In particular, rating forecasts were generated through the characteristics related to users and items without the support of available ratings. The proposed method is evaluated through the MovieLens100K dataset performing an accuracy of 0.766 and 0.951 in terms of mean absolute error and root-mean-square error