Applications of pseudo-analysis on models with nonlinear partial differential equations

Abstract

There is given an overview of methods of pseudoanalysis in applications on important classes of nonlinear partial differential equations occurring in different fields, see [7], [8], [12], [16], [18], [19], [20], [21], [22], [24]. Some obtained principles, e. g., the pseudo-linear superposition principle, allows us to transfer the methods of linear equations to the nonlinear equations. Hamilton-Jacobi equations are specially important in the control theory. Unfortunately, usually the interesting models are represented by Hamilton-Jacobi equations in which the nonlinear Hamiltonian H is not smooth, e.g., the absolute value, min or max operations, where it can not apply on such cases the classical mathematical analysis. Using the pseudo-analysis with generalized pseudo-convolution it is possible to obtain solutions which can be interpreted in the mentioned classical way. Another important class of nonlinear equations, where it is applied the pseudo-analysis, are the Burgers type equations and Black and Shole equation in option pricing. Very recent applications of pseudo-analysis are obtained on equations which model fluid mechanics and image processing. © 2007 IEEE.