517.9 Transformations, reductions and exact solutions of a wide class of nonstationary equations with nonlinearity of the Monge–Ampere type
Polyanin A. D. (Ishlinsky Institute for Problems in Mechanics)
PARABOLIC MONGE–AMPERE EQUATIONS, HIGHLY NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, EXACT SOLUTIONS, ONE-DIMENSIONAL REDUCTIONS, TWO-DIMENSIONAL REDUCTIONS, GENERALIZED SEPARABLE SOLUTIONS, SELF-SIMILAR SOLUTIONS
doi: 10.18698/2309-3684-2024-1-124142
Rather general nonstationary strongly nonlinear partial differential equations with three independent variables are investigated, which contain the first time derivative and a quadratic combination of the second derivatives with respect to spatial variables of the Monge–Ampere type (such equations are often called parabolic Monge–Ampere equations). Some equations of this type are found in differential geometry and electron magnetohydrodynamics. This paper describes multiparameter transformations that preserve the form of the considered class of nonlinear equations, which is given by an arbitrary function. Two-dimensional and one-dimensional reductions leading to simpler partial differential equations with two independent variables or ordinary differential equations are also considered. Using methods of generalized separation of variables, a number of exact solutions have been constructed, many of which can be represented in elementary functions. The obtained results and exact solutions can be used to assess the accuracy and analyze the adequacy of numerical methods for solving problems described by strongly nonlinear partial differential equations.
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Работа выполнена по теме государственного задания (№ госрегистрации 123021700057-0).
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