Modeling integro-differential equations and a method for computing their symmetries and conservation laws

Четвериков, В. Н.; Kudryavtsev, A. G.

doi:10.1090/trans2/167/01

Cited by 14 publications

(7 citation statements)

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“…It is nontrivial to find a solution of the Cauchy problem for the non-local nonlinear homogeneous unperturbed Fisher-KPP Equation 10with the non-zero initial condition (14) and for the linear non-local inhomogeneous Equation 12for the first-order correction with the zero initial condition (16).…”

Section: Model Equations and Perturbation Theorymentioning

confidence: 99%

“…We call Equations (30)-(32) the Einstein-Ehrenfest system for (29), following [23,29]. Applying initial condition (14) to Equations (25)- (27) yields…”

Section: Semiclassical Solution Of the Non-local Fisher-kpp Equationmentioning

confidence: 99%

“…Extending the scope of the Lie-group methods to integro-differential equations by invoking various means of bringing IDEs to PDEs (see, e.g., [13][14][15][16][17]) made it possible to apply the group and symmetry analysis to non-local RD models with long-range interactions, including non-local generalizations of the Fisher-KPP equation [18].…”

Section: Introductionmentioning

confidence: 99%

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Approximate Solutions and Symmetry of a Two-Component Nonlocal Reaction-Diffusion Population Model of the Fisher–KPP Type

Шаповалов

Trifonov

2019

Symmetry

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We propose an approximate analytical approach to a ( 1 + 1 ) dimensional two-component system consisting of a nonlocal generalization of the well-known Fisher–Kolmogorov–Petrovskii– Piskunov (KPP) population equation and a diffusion equation for the density of the active substance solution surrounding the population. Both equations of the system have terms that describe the interaction effects between the population and the active substance. The first order perturbation theory is applied to the system assuming that the interaction parameter is small. The Wentzel–Kramers–Brillouin (WKB)–Maslov semiclassical approximation is applied to the generalized nonlocal Fisher–KPP equation with the diffusion parameter assumed to be small, which corresponds to population dynamics under certain conditions. In the framework of the approach proposed, we consider symmetry operators which can be used to construct families of special approximate solutions to the system of model equations, and the procedure for constructing the solutions is illustrated by an example. The approximate solutions are discussed in the context of the released activity effect variously debated in the literature.

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Section: Model Equations and Perturbation Theorymentioning

confidence: 99%

“…We call Equations (30)-(32) the Einstein-Ehrenfest system for (29), following [23,29]. Applying initial condition (14) to Equations (25)- (27) yields…”

Section: Semiclassical Solution Of the Non-local Fisher-kpp Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Approximate Solutions and Symmetry of a Two-Component Nonlocal Reaction-Diffusion Population Model of the Fisher–KPP Type

Шаповалов

Trifonov

2019

Symmetry

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“…The first method was used to calculate the Lie point symmetry group for the Vlasov-Maxwell equations in plasma theory [15] and for the Benney, Vlasov-type, and Boltzmann-type kinetic equations [16]. The covering method was developed [17] and applied to a coagulation kinetic equation.…”

Section: Introductionmentioning

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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“…Several groups, including Chen et. al., [5,6,7] [2,3,4], and others, [8,10,12,21,27], have proposed a foundation for such a theory. Perhaps the most promising is the KrasilshchikVinogradov theory of coverings, [18,19,20,28,29], but this has the disadvantage that their construction relies on the a priori specification of the underlying differential equation, and so, unlike local jet space, does not form a universally valid foundation for the theory.…”

Section: Introductionmentioning

confidence: 99%

Ghost Symmetries

Olver¹,

Sanders²,

Wang³

2002

JNMP

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We introduce the notion of a ghost characteristic for nonlocal differential equations. Ghosts are essential for maintaining the validity of the Jacobi identity for the characteristics of nonlocal vector fields.The local theory of symmetries of differential equations has been well-established since the days of Sophus Lie. Generalized, or higher order symmetries can be traced back to the original paper of Noether, [24], and received added importance after the discovery that they play a critical role in integrable (soliton) partial differential equations, cf. [25]. While the local theory is very well developed, the theory of nonlocal symmetries of nonlocal differential equations remains incomplete. Several groups, including Chen et. al., [5,6,7] [2,3,4], and others, [8,10,12,21,27], have proposed a foundation for such a theory. Perhaps the most promising is the KrasilshchikVinogradov theory of coverings, [18,19,20,28,29], but this has the disadvantage that their construction relies on the a priori specification of the underlying differential equation, and so, unlike local jet space, does not form a universally valid foundation for the theory.Recently, the second and third author made a surprising discovery that the Jacobi identity for nonlocal vector fields appears to fail for the usual characteristic computations! This observation arose during an attempt to systematically investigate the symmetry properties of the Kadomtsev-Petviashvili (KP) equation, previously studied in [6,7,9,22,23]. The observed violation of the naïve version of the Jacobi identity applies to all of the preceding nonlocal symmetry calculi, and, consequently, many statements about the "Lie algebra" of nonlocal symmetries of differential equations are, by in large, not valid as stated. This indicates the need for a comprehensive re-evaluation of all earlier results on nonlocal symmetry algebras.In this announcement, we show how to resolve the Jacobi paradox through the introduction of what we name "ghost characteristics". Ghost characteristics are genuinely

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Modeling integro-differential equations and a method for computing their symmetries and conservation laws

Cited by 14 publications

References 0 publications

Approximate Solutions and Symmetry of a Two-Component Nonlocal Reaction-Diffusion Population Model of the Fisher–KPP Type

Approximate Solutions and Symmetry of a Two-Component Nonlocal Reaction-Diffusion Population Model of the Fisher–KPP Type

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

Ghost Symmetries

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