Multi-dimensional non-autonomous equation contaning the production of powers of partial derivatives

Abstract

We consider a class of multidimensional partial differential equations, containing the production of powers of partial derivatives of arbitrary order. As a result, solutions with additive, multiplicative and combined separation of variables are received. The family of pseudo-polynomial solutions which are expressed through the polynomials on independent variables with arbitrary coefficients is received, as well as functions which are the solutions of some ordinary differential equations. The solutions of travelling wave type and selfsimilar solutions are founded. Also the families of solutions which have a form of the sum or the production of the solutions of travelling wave type and self-similar solutions are received. Furthermore, the solutions representing as the functions of more complicated arguments which are expressed through the linear combinations and the productions of initial independent variables are received. The conditions of existence have been formulated for all received solutions, and the right side of equation and its parameters must satisfy these conditions.