Author(s):  I.V. Rakhmelevich, Dr., associate Professor, Nizhny Novgorod State University, Nizhny Novgorod, Russia,

Issue:  Volume 51, № 2

Rubric:  Mathematics

Annotation:  The present work is devoted to the study of the class of two-dimensional determinant partial differential equations in which the left-hand side has a form of functional determinant. The elements of this determinant are expressed through some linear differential operators acting on separate independent variables. The most well-known equation related to this class is Monge – Ampere equation. There are investigated separately both homogeneous and inhomogeneous determinant equations in the given work. In particular, there is proved the theorem on the interconnection between the solutions of homogeneous determinant equation and some auxiliary linear differential-operator equation. It is supposed that the right-hand side of inhomogeneous equation can depend on independent variables, unknown function and its first derivatives. The received families of particular solutions are expressed through the eigenfunctions of differential operators acting on separate variables, and also through the functions which belong to the kernels of those operators. In the case of differential operators with constant coefficients there are investigated the solutions of the type of travelling wave both for homogeneous and inhomogeneous equations. The case when each of operators includes only one derivative is analyzed in detail. There are received the solutions of the type of power function for the case when the operators on separate variables are homogeneous and the right-hand side of equation has the power dependence on unknown function and its first derivatives. The dependence of received solutions on the parameters of equation is researched.

Keywords:  determinant equation, separation of variables, eigenfunction, linear differential operator, solution of travelling wave type

Full text (PDF):  Download

Downloads count:  331