Symmetries and group invariant solutions of fractional ordinary differential equations

Газизов, Р. К.; Kasatkin, Alexey A.; Lukashchuk, Stanislav Yu.

doi:10.1515/9783110571660-004

Cited by 18 publications

(22 citation statements)

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“…At present, the basic methods of classical Lie group analysis have been successfully adopted to investigation of symmetry properties of FDEs with different types of fractional derivatives. A detailed discussion of this branch of Lie group analysis can be found in [16][17][18]. By using these methods, numerous Lie point symmetries, invariant solutions and conservation laws had been found for different classes of FDEs (see, e.g., the overview given in the last section of [17]).…”

Section: Therein)mentioning

confidence: 99%

Higher-Order Symmetries of a Time-Fractional Anomalous Diffusion Equation

Газизов

Lukashchuk

2021

Mathematics

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Higher-order symmetries are constructed for a linear anomalous diffusion equation with the Riemann–Liouville time-fractional derivative of order α∈(0,1)∪(1,2). It is proved that the equation in question has infinite sequences of nontrivial higher-order symmetries that are generated by two local recursion operators. It is also shown that some of the obtained higher-order symmetries can be rewritten as fractional-order symmetries, and corresponding fractional-order recursion operators are presented. The proposed approach for finding higher-order symmetries is applicable for a wide class of linear fractional differential equations.

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Section: Therein)mentioning

confidence: 99%

Higher-Order Symmetries of a Time-Fractional Anomalous Diffusion Equation

Газизов

Lukashchuk

2021

Mathematics

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“…is called a linear autonomous one-parameter Lie group of point transformations [30] (see also [14,15]). The Riesz potential [7] in n-dimensional space is defined by…”

Section: Preliminariesmentioning

confidence: 99%

“…Note that, contrary to integer-order differential equations, the invariance condition (11) is a necessary but not sufficient condition for fractional differential equations (the detailed discussion of this fact can be found in [14]). Now we give some useful properties of the Riesz potential in two-dimensional space.…”

Section: Preliminariesmentioning

confidence: 99%

“…Then, we use the generalized Leibniz rule (14) and its consequences (15). Taking into account that R α u = I 0,0 2−α u, we represent all terms with R α u and their derivatives via integral operators (12).…”

Section: Group Classification Of the Nonlinear Space-fractional Poroumentioning

confidence: 99%

“…These properties can be studied by the methods of Lie group analysis of differential equations [11][12][13]. Recently, some basic Lie symmetry group methods have been extended to fractional differential equations with the Riemann-Liouville and Caputo fractional derivatives (see the survey papers [14,15] and references therein). In [16], the algorithm for finding the Lie point symmetry group of FDEs with the Riesz potential was firstly proposed, and symmetries of the linear two-dimensional space-fractional filtration equation with the Riesz potential were obtained.…”

Section: Introductionmentioning

confidence: 99%

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Symmetry Group Classification and Conservation Laws of the Nonlinear Fractional Diffusion Equation with the Riesz Potential

Belevtsov

Lukashchuk

2020

Symmetry

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Symmetry properties of a nonlinear two-dimensional space-fractional diffusion equation with the Riesz potential of the order α ∈ ( 0 , 1 ) are studied. Lie point symmetry group classification of this equation is performed with respect to diffusivity function. To construct conservation laws for the considered equation, the concept of nonlinear self-adjointness is adopted to a certain class of space-fractional differential equations with the Riesz potential. It is proved that the equation in question is nonlinearly self-adjoint. An extension of Ibragimov’s constructive algorithm for finding conservation laws is proposed, and the corresponding Noether operators for fractional differential equations with the Riesz potential are presented in an explicit form. To illustrate the proposed approach, conservation laws for the considered nonlinear space-fractional diffusion equation are constructed by using its Lie point symmetries.

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Invariant analysis, approximate solutions, and conservation laws for the time fractional higher order Boussinesq–Burgers system

Rahioui,

El Kinani,

Ouhadan

2024

Math Methods in App Sciences

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In this paper, we employ the time fractional higher order nonlinear Boussinesq–Burgers system to investigate shallow water waves based on the Riemann–Liouville and Caputo fractional partial derivatives. The Lie algebra of infinitesimal generators associated with the symmetry groups that leave the studied system invariant is systematically constructed, and the similarity reductions are established. Furthermore, conserved quantities of the system under consideration are formulated. Next, the power series solution, including the convergence analysis, is obtained. Based on the fractional Sumudu–Adomian decomposition method, some approximate solutions are derived. Finally, in order to show the efficiency of the considered methods, the effect of the fractional order, as well as the dynamical behavior of solutions, numerical validation and graphical representations are designed.

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Symmetries and group invariant solutions of fractional ordinary differential equations

Cited by 18 publications

References 0 publications

Higher-Order Symmetries of a Time-Fractional Anomalous Diffusion Equation

Higher-Order Symmetries of a Time-Fractional Anomalous Diffusion Equation

Symmetry Group Classification and Conservation Laws of the Nonlinear Fractional Diffusion Equation with the Riesz Potential

Invariant analysis, approximate solutions, and conservation laws for the time fractional higher order Boussinesq–Burgers system

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