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 TMF, 2015, Volume 184, Number 2, Pages 179–199 (Mi tmf8833)

Constructing conservation laws for fractional-order integro-differential equations

S. Yu. Lukashchuk

Ufa State Aviation Technical University, Ufa, Russia

Abstract: In a class of functions depending on linear integro-differential fractional-order variables, we prove an analogue of the fundamental operator identity relating the infinitesimal operator of a point transformation group, the Euler–Lagrange differential operator, and Noether operators. Using this identity, we prove fractional-differential analogues of the Noether theorem and its generalizations applicable to equations with fractional-order integrals and derivatives of various types that are Euler–Lagrange equations. In explicit form, we give fractional-differential generalizations of Noether operators that gives an efficient way to construct conservation laws, which we illustrate with three examples.

Keywords: integro-differential fractional-order equation, symmetry, conservation law, fundamental operator identity, Noether theorem

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 11.G34.31.0042

DOI: https://doi.org/10.4213/tmf8833

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English version:
Theoretical and Mathematical Physics, 2015, 184:2, 1049–1066

Bibliographic databases:

PACS: 11.10.Lm, 11.30.-j
MSC: 45K05, 70S10, 70G65
Revised: 03.03.2015

Citation: S. Yu. Lukashchuk, “Constructing conservation laws for fractional-order integro-differential equations”, TMF, 184:2 (2015), 179–199; Theoret. and Math. Phys., 184:2 (2015), 1049–1066

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tmf8833
• https://doi.org/10.4213/tmf8833
• http://mi.mathnet.ru/eng/tmf/v184/i2/p179

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. Yu. Lukashchuk, “Symmetry reduction and invariant solutions for nonlinear fractional diffusion equation with a source term”, Ufa Math. J., 8:4 (2016), 111–122
2. D.-W. Ding, J. Yan, N. Wang, D. Liang, “Adaptive synchronization of fractional order complex-variable dynamical networks via pinning control”, Commun. Theor. Phys., 68:3 (2017), 366–374
3. M. Pan, L. Zheng, Ch. Liu, F. Liu, “Symmetry analysis and conservation laws to the space-fractional Prandtl equation”, Nonlinear Dyn., 90:2 (2017), 1343–1351
4. S. Yu. Lukashchuk, R. D. Saburova, “Approximate symmetry group classification for a nonlinear fractional filtration equation of diffusion-wave type”, Nonlinear Dyn., 93:2 (2018), 295–305
5. Lukashchuk S.Yu., “Approximate Conservation Laws For Fractional Differential Equations”, Commun. Nonlinear Sci. Numer. Simul., 68 (2019), 147–159
6. Habibi N., Lashkarian E., Dastranj E., Hejazi S.R., “Lie Symmetry Analysis, Conservation Laws and Numerical Approximations of Time-Fractional Fokker-Planck Equations For Special Stochastic Process in Foreign Exchange Markets”, Physica A, 513 (2019), 750–766
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