Symbolic Computation of Local Symmetries of Nonlinear and Linear Partial and Ordinary Differential Equations

Cheviakov, Alexei F.

doi:10.1007/s11786-010-0051-4

Cited by 45 publications

(43 citation statements)

References 17 publications

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“…where f = f (t, x), g = g(t, x), h = h(t, x) and k = k(t, x). We further substitute the derived forms of ξ and η into the rest two equations of system (9) and split the resulting equations with respect to variable u. This leads to a system of nine determining equations involving operator coefficients f , g, h, and k as well as the arbitrary element c(t) of class (9).…”

Section: Nonclassical Methodsmentioning

confidence: 99%

Classification of reduction operators and exact solutions of variable coefficient Newell–Whitehead–Segel equations

Vaneeva

Boyko

Zhalij

et al. 2019

Journal of Mathematical Analysis and Applications

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A class of the Newell-Whitehead-Segel equations (also known as generalized Fisher equations and Newell-Whitehead equations) is studied with Lie and "nonclassical" symmetry points of view. The classifications of Lie reduction operators and of regular nonclassical reduction operators are performed. The set of admissible transformations (the equivalence groupoid) of the class is described exhaustively. The criterion of reducibility of variable coefficient Newell-Whitehead-Segel equations to their constant coefficient counterparts is derived. Wide families of exact solutions for such variable coefficient equations are constructed.

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Section: Nonclassical Methodsmentioning

confidence: 99%

Classification of reduction operators and exact solutions of variable coefficient Newell–Whitehead–Segel equations

Vaneeva

Boyko

Zhalij

et al. 2019

Journal of Mathematical Analysis and Applications

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“…The entropy inequality on the solution set is consequently computed, using the substitution (32). It is straightforward to verify that the result involves neither the leading derivatives nor their differential consequences.…”

Section: Symbolic Computations For the Solution Set Approachmentioning

confidence: 99%

A solution set-based entropy principle for constitutive modeling in mechanics

Heß

Cheviakov

2018

Continuum Mech. Thermodyn.

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Entropy principles based on thermodynamic consistency requirements are widely used for constitutive modeling in continuum mechanics, providing physical constraints on a priori unknown constitutive functions. The well-known Müller-Liu procedure is based on Liu's lemma for linear systems. While the Müller-Liu algorithm works well for basic models with simple constitutive dependencies, it cannot take into account nonlinear relationships that exist between higher derivatives of the fields in the cases of more complex constitutive dependencies.The current contribution presents a general solution set-based procedure, which, for a model system of differential equations, respects the geometry of the solution manifold, and yields a set of constraint equations on the unknown constitutive functions, which are necessary and sufficient conditions for the entropy production to stay nonnegative for any solution of the model. Similarly to the Müller-Liu procedure, the solution set approach is algorithmic, its output being a set of constraint equations and a residual entropy inequality. The solution set method is applicable to virtually any physical model, allows for arbitrary initially postulated forms of the constitutive dependencies, and does not use artificial constructs like Lagrange multipliers. A Maple implementation makes the solution set method computationally straightforward and useful for the constitutive modeling of complex systems.Several computational examples are considered, in particular, models of gas, anisotropic fluid, and granular flow dynamics. The resulting constitutive function forms are analyzed, and comparisons are provided. It is shown how the solution set entropy principle can yield classification problems, leading to several complementary sets of admissible constitutive functions; such classification problems have not previously appeared in the constitutive modeling literature.

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“…The symbolic software package GeM for Maple contains routines for point and local (higher-order) symmetry analysis of systems of ordinary and partial differential equations, routines for local conservation law analysis, and other related functions. For details on the available routines, additional options, methods, background, and examples, see [10,12,34]. The GeM package uses an efficient representation of the DEs and resulting symmetry/conservation law determining equations, through the treatment of dependent variables and their derivatives as Maple symbols, rather than functions or expressions: ∂B 1 /∂x ≡ B1x, etc.…”

Section: Symbolic Computation Of Equivalence Transformationsmentioning

confidence: 99%

“…The rifsimp routine is capable of the solution space dimension prediction (when used with mindim = 1 option), and case splitting/classification (when used with casesplit option); for details, see [10,12,34] or Maple help.…”

Section: Simplified_eqs := Detools[rifsimp](det_eqs_all Sym_componenmentioning

confidence: 99%

Symbolic computation of equivalence transformations and parameter reduction for nonlinear physical models

Cheviakov

2017

Computer Physics Communications

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An efficient systematic procedure is provided for symbolic computation of Lie groups of equivalence transformations and generalized equivalence transformations of systems of differential equations that contain arbitrary elements (arbitrary functions and/or arbitrary constant parameters), using the software package GeM for Maple. Application of equivalence transformations to the reduction of the number of arbitrary elements in a given system of equations is discussed, and several examples are considered. First computational example of a generalized equivalence transformation where the transformation of the dependent variable involves the arbitrary constitutive function is presented.As a detailed physical example, a three-parameter family of nonlinear wave equations describing finite anti-plane shear displacements of an incompressible hyperelasic fiber-reinforced medium is considered. Equivalence transformations are computed and employed to radically simplify the model for an arbitrary fiber direction, invertibly reducing the model to a simple form that corresponds to a special fiber direction, and involves no arbitrary elements.The presented computation algorithm is applicable to wide classes of systems of differential equations containing arbitrary elements.

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Symbolic Computation of Local Symmetries of Nonlinear and Linear Partial and Ordinary Differential Equations

Cited by 45 publications

References 17 publications

Classification of reduction operators and exact solutions of variable coefficient Newell–Whitehead–Segel equations

Classification of reduction operators and exact solutions of variable coefficient Newell–Whitehead–Segel equations

A solution set-based entropy principle for constitutive modeling in mechanics

Symbolic computation of equivalence transformations and parameter reduction for nonlinear physical models

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