CHAPTER 1 - Introduction

Abstract

This chapter discusses the need for realistic solutions of the nonlinear stochastic equations arising in the modeling of frontier problems in every area of science. This includes linear or deterministic models or both as special cases. What makes this methodology different is the avoidance of perturbation, linearization, truncation, discretization, or the assumption of unphysical processes. In the stochastic case, the method yields a natural statistical separability so that no truncations or closure approximations become necessary. The method is nonperturbative and does not resort to linearization or assumptions of small or weak nonlinearity, small fluctuations, or special processes; therefore, the solutions are not only general but more realistic as well. Dynamical models whether for solitons, population problems, control systems, or the national economy are nonlinear and stochastic in general. The chapter discusses deterministic linear systems, deterministic nonlinear systems, stochastic linear systems, and stochastic nonlinear systems in a unified framework. The method was evolved to achieve statistical separability and avoid truncations but is valuable in the deterministic case as well.