Ad hoc Exact Techniques for Nonlinear Partial Differential Equations

Abstract

This chapter discusses ad hoc procedures including physical applications and discusses the concept of equation splitting. In specific nonlinear examples, the ad hoc techniques of the linear theory may prove useful. However, one must recognize that their great utility rests primarily upon the principle of superposition. In accordance with this principle, the elementary solutions of the pertinent mathematical equations can be combined to yield more flexible ones, namely, the ones that could satisfy the auxiliary conditions that arise from the particular physical phenomena. In nonlinear problems, this principle no longer holds. Its loss and the lack of an effective replacement constitute the major hurdle in the present chaotic state. In concept of equation splitting, one disregards the inviolate nature of the equation(s) to be solved. It is then decomposed into parts, which are equated to a common factor, in such a manner that a general solution (containing the appropriate number of arbitrary functions) can be constructed for at least one part. The form of the arbitrary function(s) is then obtained by means of the requirement that the other part be satisfied. The chapter also discusses the application of equation splitting concept in Navier–Stokes equations.