Vestnik St. Petersburg University, Mathematics

, Volume 51, Issue 1, pp 87–94

# A Multidimensional Nonautonomous Equation Containing a Product of Powers of Partial Derivatives

• I. V. Rakhmelevich
Mathematics

## Abstract

A class of multidimensional differential equations containing products of powers of partial derivatives of any order is considered. Solutions with additive, multiplicative, and combined separation of variables are obtained. A family of pseudopolynomial solutions expressed in terms of polynomials in independent variables with arbitrary coefficients and functions being solutions of certain ordinary differential equations is also obtained. Solutions of the type of traveling wave and self-similar solutions, as well as families of solutions having the form of a sum or a product of solutions of the type of a traveling wave and self-similar solutions, are found. Finally, solutions that can be represented as functions of more complicated arguments expressed in terms of linear combinations and products of the initial independent variables are found. For all of the obtained solutions, conditions on the righthand side of the equation and its parameters under which these solutions exist are determined.

## Keywords

partial differential equation separation of variables power-law nonlinearity solution of traveling wave type self-similar solution pseudopolynomial solution

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