Introduction of the MapDE algorithm for determination of mappings relating differential equations

Zahra, Mohammadi; Reid, Gregory J.; Huang, S. -L. Tracy

doi:10.48550/arxiv.1903.02180

Cited by 1 publication

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“…This paper is a sequel to [29] and is part of a series in which we explore algorithmic aspects of exact and approximate mappings of differential equations. We are interested in mapping less tractable differential equations into more tractable ones, in particular in this article focusing on mapping nonlinear systems to linear systems.…”

Section: Introductionmentioning

confidence: 99%

“…We are interested in mapping less tractable differential equations into more tractable ones, in particular in this article focusing on mapping nonlinear systems to linear systems. This builds on progress in [29] where we considered mappings from a specific differential system to a specific target system and mappings from a linear to a linear constant coefficient differential equation.…”

Section: Introductionmentioning

confidence: 99%

“…As in [29] we consider systems of (partial or ordinary) differential equations with n independent variables and m dependent variables which are local analytic functions of their arguments. Suppose R has independent variables x = (x 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…Our methods here and in [29] are based on the mapping approach initiated by Bluman and Kumei [19] which focuses on the interaction between such mappings and Lie symmetries via their infinitesimal form on the source and target. In particular, let G be the Lie group of transformations leaving R invariant.…”

Section: Introductionmentioning

confidence: 99%

“…In our previous work [29] we provided an algorithm to determine the existence of a mapping of a linear differential equation to the class of constant coefficient linear homogeneous differential equations. Key for this application was the exploitation of a commutative sub-algebra of symmetries of L corresponding to translations of the independent variables in the target.…”

Section: Introductionmentioning

confidence: 99%

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Symmetry-Based Algorithms for Invertible Mappings of Polynomially Nonlinear PDE to Linear PDE

2020

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This paper is a sequel to our previous work where we introduced the MapDE algorithm to determine the existence of analytic invertible mappings of an input (source) differential polynomial system (DPS) to a specific target DPS, and sometimes by heuristic integration an explicit form of the mapping. A particular feature was to exploit the Lie symmetry invariance algebra of the source without integrating its equations, to facilitate MapDE, making algorithmic an approach initiated by Bluman and Kumei. In applications, however, the explicit form of a target DPS is not available, and a more important question is, can the source be mapped to a more tractable class?We extend MapDE to determine if a source nonlinear DPS can be mapped to a linear differential system. MapDE applies differential-elimination completion algorithms to the various over-determined DPS by applying a finite number of differentiations and eliminations to complete them to a form for which an existence-uniqueness theorem is available, enabling the existence of the linearization to be determined among other applications. The methods combine aspects of the Bluman-Kumei mapping approach with techniques introduced by Lyakhov, Gerdt and Michels for the determination of exact linearizations of ODE. The Bluman-Kumei approach for PDE focuses on the fact that such linearizable systems must admit a usually infinite Lie sub-pseudogroup corresponding to the linear superposition of solutions in the target. In contrast, Lyakhov et al. focus on ODE and properties of the so-called derived subalgebra of the (finite) dimensional Lie algebra of symmetries of the ODE. Examples are given to illustrate the approach, and a heuristic integration method sometimes gives explicit forms of the maps. We also illustrate the

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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