A Qualitative Study of Random Differential Equations and Its Applications

Abstract

In mathematical sciences, mathematical equations play the newlineimportant role. To represents the any physical phenomenon or physical newlinesystem we use mathematical equations with some parameters or newlinecoefficients having some definite physical interpretation e.g. in diffusion newlinetheory, diffusion coefficient is used, in elasticity theory, the modulus of newlineelasticity is used. Parameters in these examples can be determined newlineexperimentally. While solving the corresponding mathematical equations newlineit is generally used the mean of the number of experimental newlinedeterminations is used as the coefficient or parameter. It provides the newlinereasonable description. sometimes most of the times the variance may be newlinelarge .Therefore in case of constants or parameters, many times we are newlinenot constants at all, but random whose values are calculated by some newlineprobability distribution variables. In classical physical theories, Initial newlineconditions are supposed to be known. In real position, only within the newlinecertain range of values, initial conditions are known. Therefore in the newlinemathematical expression of initial value problems, initial conditions or newlinedata is supposed to be random and in case of boundary value problems , newlineboundary conditions or data are random. newlineIn an applied mathematics ,the study of random equations is in the form newlineof family of random operator equations over a probability measure space newline(and#61527;,and#61505;,and#61549; ) . The probability measure and#61549; determines the probability of an newlineevent that is a subset of and#61527; and therefore the probability of the newlinecorresponding equation of the family. Above stated random equation will newlineappear in the forcing function and system of equations with random newlinecoefficients. Hence it is important for us to take into account the random newlineiii newlinesolutions which are obtained by random initial conditions and boundary newlineconditions within the theory of random equations. Many times such types newlineof problems can be transformed to random differential equations. newlineDevelopment in the Subject and Review of Literature newlineGenerally,Five basic classes of random equations are as Random newlineal