Group classification of nonlinear time-fractional dual-phase-lag heat equation with a small parameter

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{In this paper, we solve the group classification problem for a nonlinear one-dimensional time-fractional  heat conduction equation with full memory and dual-phase-lag, including thermal relaxation and thermal damping. The characteristic times of relaxation processes are assumed to be small enough and therefore a small parameter for fractional differential relaxation terms is introduced. All thermal properties of a medium are considered as functions of temperature. Group classification is performed with respect to groups of approximate point transformations (groups of approximate symmetries) admitted by the equation up to equivalence transformations. We prove that generally the equation admits five-parameter group of approximate transformations, and the cases of its extension to seven- and nine-parameters groups are found. Also, it is shown that the considered nonlinear equation has an infinite approximate symmetry group if the corresponding unperturbed equation is linear. We find that the equation in question always exactly inherits the symmetries of the unperturbed equation. The obtained results make it possible to construct approximately invariant solutions of equation under consideration. In particular, it follows from the classification found that the equation always has a traveling wave solution. The self-similar solutions can be constructed only if the medium thermal properties have power-law dependences on temperature. Ansatzes of these types of solutions are obtained and symmetry reductions of the equation under consideration to the corresponding ordinary fractional differential equations are performed.

About the authors

Veronika O. Lukashchuk

Ufa University of Science and Technology

Author for correspondence.
Email: voluks@gmail.com
ORCID iD: 0000-0002-3082-1446

Ph.D. in Phys. and Math., Associate Professor, Department of High Performance Computing Systems and Technologies
Russian Federation, 32 Zaki Validi St., Ufa 450076, Russia

Stanislav Yu. Lukashchuk

Ufa University of Science and Technology

Email: lsu@ugatu.su
ORCID iD: 0000-0001-9209-5155

Dr.Sci. in Phys. and Math., Professor, Department of High Performance Computing Systems and Technologies
Russian Federation, 32 Zaki Validi St., Ufa 450076, Russia

References

  1. L. V. Ovsyannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982, 416 p.
  2. N. H. Ibragimov, Transformation groups applied to mathematical physics, Reidel, Dordrecht, 1985, 394 p.
  3. P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986, 497 p.
  4. CRC Handbook of Lie Group Analysis of Differential Equations. Vol. 1, ed. N. H. Ibragimov, CRC Press, Boca Raton, FL, 1994, 448 p.
  5. CRC Handbook of Lie Group Analysis of Differential Equations. Vol. 2, ed. N. H. Ibragimov, CRC Press, Boca Raton, FL, 1995, 576 p.
  6. CRC Handbook of Lie Group Analysis of Differential Equations. Vol. 3, ed. N. H. Ibragimov, CRC Press, Boca Raton, FL, 1996, 560 p.
  7. Yu. N. Grigoriev, N. H. Ibragimov, V. F. Kovalev, S. V. Meleshko, Symmetries of integro-differential equations: with applications in mechanics and plasma physics, Springer, Dordrecht, 2010, 305 p.
  8. Yu. N. Grigoriev, V. F. Kovalev, S. V. Meleshko, Symmetries of non-local equations: Theory and Applications, Nauka, Novosibirsk, 2018 (In Russ.), 436 p.
  9. R. K. Gazizov, A. A. Kasatkin, S. Yu. Lukashchuk, "Symmetries and group invariant solutions of fractional ordinary differential equations", Volume 2 Fractional Differential Equations, eds. A. Kochubei, Yu. Luchko, De Gruyter, Boston, 2019, 65–90 doi: 10.1515/9783110571660-004.
  10. R. K. Gazizov, A. A. Kasatkin, S. Yu. Lukashchuk, "Symmetries, conservation laws and group invariant solutions of fractional PDEs", Volume 2 Fractional Differential Equations, eds. A. Kochubei, Yu. Luchko, De Gruyter, Boston, 2019, 353–382 doi: 10.1515/9783110571660-016.
  11. V. A. Baikov, R. K. Gazizov, N. Kh. Ibragimov, "Approximate symmetries", Math. USSR-Sb., 64:2 (1989), 427–441.
  12. V. A. Baikov, R. K. Gazizov, N. Kh. Ibragimov, "Perturbation methods in group analysis", J. Soviet Math., 55:1 (1991), 1450–1490.
  13. V. A. Baikov, R. K. Gazizov, N. Kh. Ibragimov, "Approximate symmetries and preservation laws", Proc. Steklov Inst. Math., 200 (1993), 35–47.
  14. V. A. Baikov, R. K. Gazizov, N. H. Ibragimov, "Approximate groups of transformations", Differ. Equ., 29:10 (1993), 1487–1504.
  15. R. K. Gazizov, S. Yu. Lukashchuk, "Approximations of Fractional Differential Equations and Approximate Symmetries", IFAC-PapersOnLine, 50:1 (2017), 14022–14027. doi: 10.1016/j.ifacol.2017.08.2426.
  16. 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.
  17. S. Yu. Lukashchuk, "Approximate conservation laws for fractional differential equations", Commun. Nonlinear Sci. Numer. Simul., 68 (2019), 147–159. doi: 10.1016/j.cnsns.2018.08.011.
  18. S. Lie, "Classification und integration von gewohnlichen differential-gleichungen
  19. zwischen x, y, die gruppe von transformationen gestatten", Arch. Math. Natur. Christiania., 9 (1883), 371–393.
  20. L. V. Ovsyannikov, "Group classification of equations of the form y'' = f(x, y)", J. Appl. Mech. Tech. Phys., 45:2 (2004), 153–157.
  21. L. V. Ovsyannikov, "Group properties of nonlinear heat-conduction equations", Dokl. Akad. Nauk SSSR, 125:3 (1959), 492–495 (In Russ.).
  22. V. A. Dorodnitsyn, "On invariant solutions of the equation of non-linear heat conduction with a source", U.S.S.R. Comput. Math. Math. Phys., 22:6 (1982), 115–122.
  23. S. Yu. Lukashchuk, A. V. Makunin, "Group classification of nonlinear time-fractional diffusion equation with a source term", Appl. Math. Comput., 257 (2015), 335–343.
  24. 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.
  25. S. Yu. Lukashchuk, "An approximate group classification of a perturbed subdiffusion equation", J. Samara State Tech. Univ., Ser. Phys. & Math. Sci., 20:4 (2016), 603–619. doi: 10.14498/vsgtu1520 (In Russ.).
  26. A. V. Luikov, Analytical Heat Diffusion Theory, Academic Press, New York, 1968, 685 p.
  27. A. V. Kostanovskiy, M. E. Kostanovskaya, "A criterion of applicability of the parabolic heat conduction equation", Tech. Phys. Lett., 34:6 (2008), 500–502.
  28. T. Q. Qiu, C. L. Tien, "Heat Transfer Mechanisms During Short-Pulse Laser Heating of Metals", J. Heat Transfer., 115:4 (1993), 835–841. doi: 10.1115/1.2911377.
  29. H. D. Wang, B. Y. Cao, Z. Y. Guo, "Non-Fourier Heat Conduction in Carbon Nanotubes", J. Heat Transfer, 134:5 (2012), 051004.
  30. W. Roetzel, N. Putra, S. K. Das, "Experiment and analysis for non-Fourier conduction in materials with non-homogeneous inner structure", Int. J. Therm. Sci., 42:6 (2003), 541–552.
  31. A. I. Zhmakin, "Heat Conduction Beyond the Fourier Law", Tech. Phys., 66:1 (2021), 1–22. doi: 10.21883/JTF.2021.01.50267.207-20.
  32. C. Cattaneo, "A form of heat equation which eliminates the paradox of instantaneous propagation", Comptes Rendus de l'Academie des Sciences, 247 (1958), 431–433.
  33. P. Vernotte, "Paradoxes in the continuous theory of the heat equation", Comptes Rendus de l'Academie des Sciences, 246 (1958), 3154–3155.
  34. D. Y. Tzou, "A unified field approach for heat conduction from macro to micro-scales", ASME J. Heat Transfer., 117 (1995), 8–16. doi: 10.1115/1.2822329.
  35. H.-Y. Xu, X.-Y. Jiang, "Time fractional dual-phase-lag heat conduction equation", Chin. Phys. B., 24:3 (2015), 034401. doi: 10.1088/1674-1056/24/3/034401.
  36. H. Sobhani, A. Azimi, A. Noghrehabadi, M. Mozafarifard, "Numerical study and parameters estimation of anomalous diffusion process in porous media based on variable-order time fractional dual-phase-lag model", Numerical Heat Transfer, Part A: Applications, 83:7 (2023), 679–710. doi: 10.1080/10407782.2022.2157915.
  37. Q. Zhuang, B. Yu, X. Jiang, "An inverse problem of parameter estimation for time-fractional heat conduction in a composite medium using carbon–carbon experimental data", Phys. B: Condens. Matter., 456 (2015), 9–15. doi: 10.1016/j.physb.2014.08.011.
  38. M. H. Fotovvat, Z. Shomali, "A time-fractional dual-phase-lag framework to investigate transistors with TMTC channels (TiS3, In4Se3) and size-dependent properties", Micro and Nanostructures, 168 (2022), 207304. doi: 10.48550/arXiv.2203.06523.
  39. S. Yu. Lukashchuk, "A semi-explicit algorithm for parameters estimation in a timefractional dual-phase-lag heat conduction model", Modelling, 5:3 (2024), 776–796. doi: 10.3390/modelling5030041.
  40. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006, 523 p.
  41. S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: theory and applications, Gordon and Breach, New York, 1993, 976 p.
  42. I. Sh. Akhatov, R. K. Gazizov, N. Kh. Ibragimov, "Nonlocal symmetries. Heuristic approach", J. Soviet Math., 55:1 (1991), 1401–1450.

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