Existence and Uniqueness Results for Random Differential Equations

Abstract

The nondeterministic nature of phenomena in the general areas of newlinebiological, engineering, oceanographic and Physical Sciences the newlinemathematical descriptions of such phenomena frequently result in random newlineequations. It is to present theoretical results concerning certain classes of newlinerandom equations and then to apply those results to problems that arise in newlinethe general areas. Usually, the mathematical models or the equations that newlinehave been used to describe a particular phenomenon or process of the newlineuniverse contains a parameter, which has some specific physically newlineinterpretations but whose values is not known. If any such phenomena newlineinvolving mentioned parameters and which satisfy certain probabilistic newlineand measure theoretic behavior with respect to this parameter, then we newlinesay it is a random phenomena. For example, in the theory of diffusion newlineor heat conduction, we have the diffusion coefficient or the coefficient newlineof conductivity that play the prominent role. Similarly in the case of newlinewave theory, the propagation coefficient and in the theory of newlineelasticity, the modulus of elasticity play the significant role in the newlinebehavior of the underlined processes. Thus, the coefficients or parameters newlinethat have an important role to play in the natural and physical phenomena newlinelike above are called random parameters . newlineThe mathematical equations are solved using as the value of the parameter newlineor coefficients, the mean value of the set of observations experimentally newlineobtained. However, if the experiment is performed repeatedly, then the mean newlinevalues found will vary ,and if the variation is large. The mean value actually newlineused may be quite unsatisfactory. Thus, in practice the physical constant is newlinenot really a constant, but a random variable whose behavior is governed by newlinesome probability distribution. It is thus advantegeous to view these equations newline(iii) newlineas being random rather than deterministic ,to search random solution and to newlinestudy its statistical properties. we take about some parameters or newlinecoefficients, the random analysis of the random equa