Pseudo-analysis approach to nonlinear partial differential equations

An overview of methods of pseudo-analysis in applications on important classes of nonlinear partial differential equations, occurring in different fields, is given. Hamilton-Jacobi equations, specially important in the control theory, are for important models usually with non-linear Hamiltonian H which is also not smooth, e.g., the absolute value, min or max operations, where it can not apply 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 classes of nonlinear equations, where there are 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 (Navier-Stokes equation) and image processing (Perona and Malik equation).