Random Integral Equation of the Volterra Type with Applications (original) (raw)

On random solutions of Volterra-Fredholm integral equations

Pacific Journal of Mathematics, 1983

This paper is concerned with the existence, uniqueness and boundedness of random solutions of a random nonlinear mixed integral equation of Volterra-Fredholm type. The main tools for the study are the theory of admissibility of integral operators and the theory of random contractors.

On a random solution of a nonlinear perturbed stochastic integral equation of the Volterra type

Bulletin of the Australian Mathematical Society, 1973

where io € ft , the supporting set of the complete probability measure space (ft, A, \i). We are concerned with the existence and uniqueness of a random solution to the above equation. A random solution, x(t; u) , of the above equation is defined to be a vector random variable which satisfies the equation y almost everywhere.

Investigation of the Behaviour of Volterra Integral Equations with Random Effects

Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 2020

In this study, random Volterra integral equations obtained by transforming components of deterministic Volterra integral equations to random variables are analysed. Beta, Normal (Gaussian), Gamma, Geometric and Uniform distributions are used to investigate the random behaviour of the solutions for Volterra integral equations under random effects. The random version of Differential Transformation Method (RDTM) is used to obtain an approximation to the solution of the random Volterra integral equation. Using the approximate solutions, approximate expected values and approximate variances are calculated. Some integro-differential equations, obtained by using random components with the above mentioned distributions, are solved as numerical examples. Results are obtained in MAPLE and shown in graphs. It is seen that random Differential Transformation Method is effective for the examination of random Volterra integral equations. Comparison of the solutions is given to underline the accuracy of the method.

ABOUT THE SOLUTION STABILITY OF VOLTERRA INTEGRAL EQUATION WITH RANDOM KERNEL

For integral equation in n R with stochastic kernel of the special type, exact equations for first and second moments of the integral equation solution are derived. By means of these equations, the stability of the solution of the integral equation in mean and mean-square was analyzed. The analysis is based on reduction of integral equation to the equivalent differential equation with random coefficient, presenting two valued Markov process. An example is given.

A random Fredholm integral equation

Proceedings of the American Mathematical Society, 1972

The aim of this paper is the study of a random or stochastic integral equation of the Fredholm type given by x(t ; a>) = h(t; tt))+J'0x k0(t, t; cu)e(r, x{r%m))dr, />0, where coeQ, the supporting set of the probability measure space (Ü, A, P). The existence and uniqueness of a random solution to the above stochastic integral equation is considered. A random solution, x(t; co), of such a random equation is defined to be a random function which satisfies the equation almost surely. Several theorems and useful special cases are presented which give conditions such that a random solution exists.

Fredholm and Volterra nonlinear possibilistic integral equations

2021

In this paper we study the nonlinear functional equations obtained from the classical integral equations of Fredholm and of Volterra of second kind, by replacing there the linear Lebesgue integral with the nonlinear possibilistic integral. Mathematics Subject Classification (2010): 45B05, 47H10, 28E10, 28A99.

Random integral equations on time scales

Opuscula Mathematica, 2013

In this paper, we present the existence and uniqueness of random solution of a random integral equation of Volterra type on time scales. We also study the asymptotic properties of the unique random solution.

On a Stochastic Integro-Differential Equation of Volterra Type

SIAM Journal on Applied Mathematics, 1972

A nonlinear stochastic integro-differential equation of the form x'(t; w) = h(t, x(t; w)) + 3 k(t,zr; wo)f(x(z; w)) dz, where t > 0 (' = d/dt), and co E Q, the supporting set of a complete probability measure space (Q, A, P), is studied with respect to the existence of a unique random solution. Results are also given concerning the statistical behavior of the random solution as too , and an application to differential systems with random parameters is presented.

The Existence and Uniqueness of Random Solution to Itô Stochastic Integral Equation

Applied Mathematics, 2012

The objective of this paper is to attempt to apply the theoretical techniques of probabilistic functional analysis to answer the question of existence and Uniqueness of a Random Solution to Itô Stochastic Integral Equation. Another type of stochastic integral equation which has been of considerable importance to applied mathematicians and engineers is that involving the Itô or Itô-Doob form of stochastic integrals.

The Stochastic Volterra Equation

Barcelona Seminar on Stochastic Analysis, 1993

We study the stochastic (Skorohod) integral equation of the Volterra type t t Xt(w) = Yt(w) +I b(t, s)Xs(w)ds +I u(t, s)Xs(w)8Bs(w) 0 0 where Y, band u are given functions; band u are bounded, deterministic and yt is stochastic, not necessarily adapted. The stochastic integral (8B) is taken in the Skorohod sense. In general there need not exist a classical stochastic process Xt(w) satisfying this equation. However, we show that a unique solution exists in the following extended senses: (I) As a functional process (II) As a generalized white noise functional (Hida distribution). Moreover, in both cases we find explicit solution formulas. The formulas are similar to the formulas in the deterministic case (u = 0), but with Wick products in stead of ordinary (pointwise) products.