A Legendre wavelet method is presented for numerical solutions of stochastic Volterra-Fredholm integral equations. The main characteristic of the proposed method is that it reduces stochastic Volterra-Fredholm integral equations into a linear system of equations. Convergence and error analysis of the Legendre wavelets basis are investigated. The efficiency and accuracy of the proposed method wa...

Moment vector equations (MVEs) are presented for use in approximating and analyzing multi-dimensional non-linear discreteand continuous-time equations. A non-linear equation is expanded into simultaneous equations of generalized moments and then reduced to an MVE of a coefficient matrix and a moment vector. The MVE can be used to analyze the statistical properties, such as the mean, variance, c...

The authors establish weighted decay rates for the energy associated with the Cauchy problem { utt − div(b(x)∇u) + a(x)ut = 0, x ∈ R, t > 0 u(0, x) = u0(x), ut(0, x) = u1(x), x ∈ R. Such a system appears in models for traveling waves in a non-homogeneous gas with damping that changes with the position. This problem has been studied intensively for the homogeneous medium (when b is constant), bu...

^ C O and Z)0(x) stand for ——^ ^ " ^ T » (bx ) . . . (o* ) y and Dz stands for all the derivatives Dz for i = \, . . . , N and 0 < |a| < mt (of course if |a | = 0 , Dz = z). Equations of the form (1 ) were discussed in my paper "Partial regularity theorems for elliptic systems" which appeared in the January 1968 issue of the Journal of Mathematics and Mechanics [17] where it was assumed that th...

The function u = u(t, a) usually represents the population density of a certain specie at time t > 0 and age a > 0, so that ū(t) in equation (1.4) is the (weighted) total population independent of age. The operator A[ū](t) in equation (1.1) acts for a fixed function ū and time t as a linear (and unbounded) operator on a Banach space E0. In concrete applications, A[ū](t) plays the role of non-li...

We investigate the symmetry properties of hierarchies of non-linear Schrödinger equations, introduced in [1], which describe non-interacting systems in which tensor product wave-functions evolve by independent evolution of the factors (the separation property). We show that there are obstructions to lifting symmetries existing at a certain number of particles to higher numbers. Such obstruction...

In this paper, Semi-orthogonal (SO) B-spline scaling functions and wavelets and their dual functions are presented to approximate the solutions of linear and non-linear second order Fredholm integro-differential equations. The B-spline scaling functions and wavelets, their properties and the operational matrices of derivative for this functions are presented to reduce the solution of linear and...

abstract. the main contribution of the current paper is to propose a new effective numerical method for solving the first-order linear matrix differential equations. properties of the legendre basis operational matrix of integration together with a collocation method are applied to reduce the problem to a coupled linear matrix equations. afterwards, an iterative algorithm is examined for solvin...