On a Class of Linearizable Monge-Ampère Equations

Arrigo, Daniel J.; Hill, James M.

doi:10.2991/jnmp.1998.5.2.1

Cited by 2 publications

(1 citation statement)

References 6 publications

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“…More complicated cases of linearization with contact transformations are considered elsewhere [48,49]. The methods of differential constraints, degenerate hodograph, and group analysis for constructing solutions of PDEs are considered comprehensively elsewhere [19].…”

Section: Discussionmentioning

confidence: 99%

Symmetries of the Fisher–Kolmogorov–Petrovskii–Piskunov equation with a nonlocal nonlinearity in a semiclassical approximation

Levchenko

Шаповалов

Trifonov

2012

Journal of Mathematical Analysis and Applications

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a b s t r a c tThe classical group analysis approach used to study the symmetries of integro-differential equations in a semiclassical approximation is considered for a class of nearly linear integrodifferential equations. In a semiclassical approximation, an original integro-differential equation leads to a finite consistent system of differential equations whose symmetries can be calculated by performing standard group analysis.The approach is illustrated by the calculation of the Lie symmetries in explicit form for a special case of the one-dimensional nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov population equation.

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Section: Discussionmentioning

confidence: 99%

Symmetries of the Fisher–Kolmogorov–Petrovskii–Piskunov equation with a nonlocal nonlinearity in a semiclassical approximation

Levchenko

Шаповалов

Trifonov

2012

Journal of Mathematical Analysis and Applications

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Unsteady Magnetohydrodynamics PDE of Monge–Ampère Type: Symmetries, Closed-Form Solutions, and Reductions

Polyanin,

Aksenov

2024

Mathematics

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The paper studies an unsteady equation with quadratic nonlinearity in second derivatives, that occurs in electron magnetohydrodynamics. In mathematics, such PDEs are referred to as parabolic Monge–Ampère equations. An overview of the Monge–Ampère type equations is given, in which their unusual qualitative features are noted. For the first time, the Lie group analysis of the considered highly nonlinear PDE with three independent variables is carried out. An eleven-parameter transformation is found that preserves the form of the equation. Some one-dimensional reductions allowing to obtain self-similar and other invariant solutions that satisfy ordinary differential equations are described. A large number of new additive, multiplicative, generalized, and functional separable solutions are obtained. Special attention is paid to the construction of exact closed-form solutions, including solutions in elementary functions (in total, more than 30 solutions in elementary functions were obtained). Two-dimensional symmetry and non-symmetry reductions leading to simpler partial differential equations with two independent variables are considered (including stationary Monge–Ampère type equations, linear and nonlinear heat type equations, and nonlinear filtration equations). The obtained results and exact solutions can be used to evaluate the accuracy and analyze the adequacy of numerical methods for solving initial boundary value problems described by highly nonlinear partial differential equations.

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On a Class of Linearizable Monge-Ampère Equations

Cited by 2 publications

References 6 publications

Symmetries of the Fisher–Kolmogorov–Petrovskii–Piskunov equation with a nonlocal nonlinearity in a semiclassical approximation

Symmetries of the Fisher–Kolmogorov–Petrovskii–Piskunov equation with a nonlocal nonlinearity in a semiclassical approximation

Unsteady Magnetohydrodynamics PDE of Monge–Ampère Type: Symmetries, Closed-Form Solutions, and Reductions

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