Canonical Equation K 56 for the Solution of the System of Linear Differential Equations with Random Coefficients

Abstract

In this chapter we consider a system of linear differential equations with random coefficients $$ \frac{{d{{\vec x}_n}\;(t)}}{{dt}} = \;{\Xi _{n\;{\rm{x }}m}}{\vec x_n}\;(t)\;,\;\;{\rm{0}} \le t \le T,\;\;{\vec x_n}\;{\rm{(0) = }}{\vec c_n}, $$ when the dimension of such a system is large and every random coefficient tends to a certain constant in probability i.e., no single coefficient may be influential enough to dominate the system of the equation as a whole when the dimension of this system tends to infinity. Self-averaging of the solutions of a system of linear differential equations with random coefficients means that the vector-solution \( {\vec x_n} \) converges to the solution of a certain nonrandom equation when the dimension n of a system of linear differential equations tends to infinity. The necessity for the solution of such systems arises in different problems of calculus, differential and integral equations, experiment design, etc. Unfortunately, in practical problems, it is very difficult to find the distribution functions of the random coefficients ξ ij such systems. For this reason, we have developed a new analysis in which these coefficients ξ ij have an arbitrary distribution function. It is natural in this case to use the methods of General Statistical Analysis (see [Gir96]). The system \( d{\vec x_m}\;{\rm{(}}t{\rm{)/}}dt = \;\left\{ {\left. {{n^{ - 1}}\sum\limits_{k = 1}^n {X_{m\;{\rm{x }}m}^{(k)}} } \right\}} \right.{\vec x_m}{\rm{ (}}t{\rm{) , 0 }} \le \;\;t \le \;\;T \) with random coefficients arises when instead of a nonrandom matrix \( {A_{m\;{\rm{x }}m}} = \;{\rm{(}}{a_{ij}}{\rm{)}}_{i,j = 1}^m \) of the system \( d{\vec x_m}\;{\rm{(}}t{\rm{)/}}dt = {A_{m\;{\rm{x }}m}}{\vec x_m}\;(t)\;{\rm{, 0 }} \le {\rm{ }}t \le \;T, {\vec x_m}\;{\rm{(0) = }}{\vec c_m} \) we use the standard estimator \( {n^{ - 1}}\sum\limits_{k = 1}^n {X_{m\;{\rm{x }}m}^{(k)}} \;{\rm{of matrix }}{A_{m\;{\rm{x }}m}},\;{\rm{where }}X_{m\;{\rm{x }}m}^{(k)}\;{\rm{are }}n observations X_{m\;{\rm{x }}m}^{(k)}{\rm{ of random matrix }}{\Xi _{m\;{\rm{x }}m}}. \)