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

There is given an overview of methods of pseudo-analysis 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.