Symmetry analysis of differential equations with Mathematica

Baumann, Gerd

doi:10.1016/s0895-7177(97)00056-3

Cited by 69 publications

(87 citation statements)

References 15 publications

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“…, y m−1 , the algorithm just described, for identification of admitted transformations, can be fully automated [c.f. Baumann, 2013]. Indeed, from Def.…”

Section: From An Ode To Its Admitted Transformationsmentioning

confidence: 93%

A Role for Symmetry in the Bayesian Solution of Differential Equations

Wang¹,

Cockayne²,

Oates³

2020

Bayesian Anal.

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The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators suggests that formal uncertainty quantification can also be performed in this context. Competing statistical paradigms can be considered and Bayesian probabilistic numerical methods (PNMs) are obtained when Bayesian statistical principles are deployed. Bayesian PNM have the appealing property of being closed under composition, such that uncertainty due to different sources of discretisation in a numerical method can be jointly modelled and rigorously propagated. Despite recent attention, no exact Bayesian PNM for the numerical solution of ordinary differential equations (ODEs) has been proposed. This raises the fundamental question of whether exact Bayesian methods for (in general nonlinear) ODEs even exist. The purpose of this paper is to provide a positive answer for a limited class of ODE. To this end, we work at a foundational level, where a novel Bayesian PNM is proposed as a proof-of-concept. Our proposal is a synthesis of classical Lie group methods, to exploit underlying symmetries in the gradient field, and non-parametric regression in a transformed solution space for the ODE. The procedure is presented in detail for first and second order ODEs and relies on a certain strong technical condition -existence of a solvable Lie algebra -being satisfied. Numerical illustrations are provided.

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“…, y m−1 , the algorithm just described, for identification of admitted transformations, can be fully automated [c.f. Baumann, 2013]. Indeed, from Def.…”

Section: From An Ode To Its Admitted Transformationsmentioning

confidence: 93%

A Role for Symmetry in the Bayesian Solution of Differential Equations

Wang¹,

Cockayne²,

Oates³

2020

Bayesian Anal.

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“…Definition 1 (Conservation laws). By a conservation law of a differential system ∆ a (x, u, u (1) , u (2) , . .…”

Section: Symmetries and Conservation Laws Variational Systemsmentioning

confidence: 99%

“…(the symmetries (6.6) were obtained with the use of the program MathLie TM in [2]). Applying X f to the alternative Lagrangians A = ∆ i , 1 ≤ i ≤ 3 and using the Noether identity (2.11), these infinite symmetries lead to the following infinite sets of conservation laws:…”

Section: Symmetries Of the Euler Vorticity Systemmentioning

confidence: 99%

Second Noether theorem for quasi-Noether systems

Shankar¹

2016

J. Phys. A: Math. Theor.

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Quasi-Noether differential systems are more general than variational systems and are quite common in mathematical physics. They include practically all differential systems of interest, at least those that have conservation laws. In this paper, we discuss quasi-Noether systems that possess infinite-dimensional (infinite) symmetries involving arbitrary functions of independent variables. For quasi-Noether systems admitting infinite symmetries with arbitrary functions of all independent variables, we state and prove an extension of the Second Noether Theorem. In addition, we prove that infinite sets of conservation laws involving arbitrary functions of all independent variables are trivial and that the associated differential system is under-determined. We discuss infinite symmetries and infinite conservation laws of two important examples of non-variational quasi-Noether systems: the incompressible Euler equations and the Navier-Stokes equations in vorticity formulation, and we show that the infinite sets of conservation laws involving arbitrary functions of all independent variables are trivial. We also analyze infinite symmetries involving arbitrary functions of not all independent variables, prove that the fluxes of conservation laws in these cases are total divergences on solutions, and demonstrate examples of this situation.

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“…Before we begin, it is worth mentioning that there are powerful and fully automated software routines to obtain symmetries that are not approximate, commonly referred to as exact symmetries, for example [13,14,15]. Eliminating all the perturbed terms in the Lagrangian (1), leads to the derivation of the oscillation equation…”

Section: Introductionmentioning

confidence: 99%

Approximate conditions admitted by classes of the Lagrangian L=12(−u′2+u2
Jamal
¹
,
Mnguni
²

2018
Applied Mathematics and Computation
10
0
8
0
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We investigate a class of Lagrangians that admit a type of perturbed harmonic oscillator which occupies a special place in the literature surrounding perturbation theory. We establish explicit and generalized geometric conditions for the symmetry determining equations. The explicit scheme provided can be followed and specialized for any concrete perturbed differential equation possessing the Lagrangian. A systematic solution of the conditions generate nontrivial approximate symmetries and transformations. Detailed cases are discussed to illustrate the relevance of the conditions, namely (a) G 1 as a quadratic polynomial, (b) the Klein-Gordon equation of a particle in the context of Generalized Uncertainty Principle and (c) an orbital equation from an embedded Reissner-Nordström black hole.

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Symmetry analysis of differential equations with Mathematica

Cited by 69 publications

References 15 publications

A Role for Symmetry in the Bayesian Solution of Differential Equations

A Role for Symmetry in the Bayesian Solution of Differential Equations

Second Noether theorem for quasi-Noether systems

Approximate conditions admitted by classes of the Lagrangian L=12(−u′2+u2
Jamal
¹
,
Mnguni
²

2018
Applied Mathematics and Computation
10
0
8
0
View full text Add to dashboard Cite

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Symmetry analysis of differential equations with Mathematica

Cited by 69 publications

References 15 publications

A Role for Symmetry in the Bayesian Solution of Differential Equations

A Role for Symmetry in the Bayesian Solution of Differential Equations

Second Noether theorem for quasi-Noether systems

Approximate conditions admitted by classes of the Lagrangian L=12(−u′2+u2Jamal1, Mnguni2 2018Applied Mathematics and Computation10080View full textAdd to dashboardCite

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Approximate conditions admitted by classes of the Lagrangian L=12(−u′2+u2
Jamal
¹
,
Mnguni
²

2018
Applied Mathematics and Computation
10
0
8
0
View full text Add to dashboard Cite