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Group Invariant Solutions and Local Conservation Laws of Heat Conduction Equation Arising Laser Heating Carbon Nanotubes Using Lie Group Analysis

Yıl 2022, Cilt: 10 Sayı: 2, 102 - 113, 01.06.2022
https://doi.org/10.36753/mathenot.926867

Öz

In this study, based on the continuous transformations of Lie groups, the exact analytic solutions of the laser heating carbon nanotubes formulated by using the classical heat conduction equation with various physical properties were constructed. These solutions are the type of group invariant solutions. The constructed solutions have expanded and enriched the solution forms of this new model existing in the literature. With the help of the Maple package program, 3D, density, and contour graphs were drawn for the special values of the parameters in the solutions, and the physical structures of the solutions obtained in this way were also observed. The solutions obtained can be used in the explanation of physical phenomena occurring in cancer investigations.

Destekleyen Kurum

Bursa Uludag University Scientific Research Project Unit (BAP)

Proje Numarası

KUAP(F)-2019/11

Teşekkür

This study was supported by the Scientific Research Project Unit (BAP) of Bursa Uludağ University, under project number: KUAP(F)-2019/11. The authors thank Bursa Uludag University.

Kaynakça

  • [1] Yaşar, E., Yıldırım, Y., & Khalique, C. M.: Lie symmetry analysis, conservation laws and exact solutions of the seventh-order time fractional Sawada–Kotera–Ito equation. Results in physics. 6, 322-328 (2016).
  • [2] Yaşar, E., Yıldırım, Y., & Adem, A. R.: Perturbed optical solitons with spatio-temporal dispersion in (2+ 1)-dimensions by extended Kudryashov method. Optik. 158, 1-14 (2018).
  • [3] Yaşar, E.: On the conservation laws and invariant solutions of the mKdV equation. Journal of mathematical analysis and applications, 363(1), 174-181 (2010).
  • [4] Yaşar, E., & Özer, T.: On symmetries, conservation laws and invariant solutions of the foam-drainage equation. International Journal of Non-Linear Mechanics, 46(2), 357-362 (2011).
  • [5] Yaşar, E., Yıldırım, Y., & Adem, A.R.: Extended transformed rational function method to nonlinear evolution equations. International Journal of Nonlinear Sciences and Numerical Simulation, 20(6), 691-701 (2019).
  • [6] Hirota, R.: The direct method in soliton theory (No. 155). Cambridge University Press (2004).
  • [7] Fuchssteiner, B., & Fokas, A. S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Physica D: Nonlinear Phenomena, 4(1), 47-66 (1981).
  • [8] Guo, B., Ling, L., & Liu, Q. P.: Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions. Physical Review E, 85(2), 026607 (2012).
  • [9] Ramani, A., Grammaticos, B., & Bountis, T.: The Painlevé property and singularity analysis of integrable and non-integrable systems. Physics Reports, 180(3), 159-245 (1989).
  • [10] Bluman, G., & Anco, S.: Symmetry and integration methods for differential equations (Vol. 154). Springer Science & Business Media (2008).
  • [11] He, J. H.: Variational iteration method–a kind of non-linear analytical technique: some examples. International journal of non-linear mechanics, 34(4), 699-708 (1999).
  • [12] Malfliet, W., & Hereman, W.: The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Physica Scripta, 54(6), 563 (1996).
  • [13] Ma, W. X.: A refined invariant subspace method and applications to evolution equations. Science China Mathematics, 55(9), 1769-1778 (2012).
  • [14] Olver, P. J.: Applications of Lie groups to differential equations (Vol. 107). Springer Science & Business Media (2000).
  • [15] Bluman, G. W., Cheviakov, A. F., & Anco, S. C.: Applications of symmetry methods to partial differential equations (Vol. 168, pp. xx+-398). New York: Springer (2010).
  • [16] Siregar, S., Oktamuliani, S., & Saijo, Y.: A theoretical model of laser heating carbon nanotubes. Nanomaterials, 8(8), 580 (2018).
  • [17] Nakamiya, T., Ueda, T., Ikegami, T., Ebihara, K., & Tsuda, R.: Thermal analysis of carbon nanotube film irradiated by a pulsed laser. Current Applied Physics, 8(3-4), 400-403(2008).
  • [18] Nakamiya, T., & Ebihara, K.: The Finite Element Thermal Analysis of Amorphous Silicon Thin Films Irradiated by a Pulsed Laser. The transactions of the Institute of Electrical Engineers of Japan. A, 108(10), 443-450 (1988).
  • [19] Younis, M., & Rizvi, S. T. R.: Optical soliton like-pulses in ring-cavity fiber lasers of carbon nanotubes. Journal of Nanoelectronics and Optoelectronics, 11(3), 276-279 (2016).
  • [20] Wang, G., Kara, A. H., Buhe, E., & Fakhar, K.: Group analysis and conservation laws of a coupled system of partial differential equations describing the carbon nanotubes conveying fluid. Romanian Journal in physics, 60(7-8), 952-960 (2015).
  • [21] Muriel, C.; Romero, J.L.: First integrals, integrating factors and λ-Symmetries of second-order differential equations. J. Phys. A Math. Theor. 42, 365207 (2009).
  • [22] Bai, Y. S., Pei, J. T., & Ma, W. X.: Symmetry and-Symmetry Reductions and Invariant Solutions of Four Nonlinear Differential Equations. Mathematics, 8, 1138 (2020).
  • [23] A. Cheviakov.: GeM software package for computation of symmetries and conservation laws of differential equations. Comp. Phys. Comm. 176 (2007), 48-61.
  • [24] Gaeta, G. C., & Morando, P.: On the relation between standard and μ-symmetries for PDEs. J. Phys. A: Math. Gen., Vol. 37, 9467-9486 (2004).
Yıl 2022, Cilt: 10 Sayı: 2, 102 - 113, 01.06.2022
https://doi.org/10.36753/mathenot.926867

Öz

Proje Numarası

KUAP(F)-2019/11

Kaynakça

  • [1] Yaşar, E., Yıldırım, Y., & Khalique, C. M.: Lie symmetry analysis, conservation laws and exact solutions of the seventh-order time fractional Sawada–Kotera–Ito equation. Results in physics. 6, 322-328 (2016).
  • [2] Yaşar, E., Yıldırım, Y., & Adem, A. R.: Perturbed optical solitons with spatio-temporal dispersion in (2+ 1)-dimensions by extended Kudryashov method. Optik. 158, 1-14 (2018).
  • [3] Yaşar, E.: On the conservation laws and invariant solutions of the mKdV equation. Journal of mathematical analysis and applications, 363(1), 174-181 (2010).
  • [4] Yaşar, E., & Özer, T.: On symmetries, conservation laws and invariant solutions of the foam-drainage equation. International Journal of Non-Linear Mechanics, 46(2), 357-362 (2011).
  • [5] Yaşar, E., Yıldırım, Y., & Adem, A.R.: Extended transformed rational function method to nonlinear evolution equations. International Journal of Nonlinear Sciences and Numerical Simulation, 20(6), 691-701 (2019).
  • [6] Hirota, R.: The direct method in soliton theory (No. 155). Cambridge University Press (2004).
  • [7] Fuchssteiner, B., & Fokas, A. S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Physica D: Nonlinear Phenomena, 4(1), 47-66 (1981).
  • [8] Guo, B., Ling, L., & Liu, Q. P.: Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions. Physical Review E, 85(2), 026607 (2012).
  • [9] Ramani, A., Grammaticos, B., & Bountis, T.: The Painlevé property and singularity analysis of integrable and non-integrable systems. Physics Reports, 180(3), 159-245 (1989).
  • [10] Bluman, G., & Anco, S.: Symmetry and integration methods for differential equations (Vol. 154). Springer Science & Business Media (2008).
  • [11] He, J. H.: Variational iteration method–a kind of non-linear analytical technique: some examples. International journal of non-linear mechanics, 34(4), 699-708 (1999).
  • [12] Malfliet, W., & Hereman, W.: The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Physica Scripta, 54(6), 563 (1996).
  • [13] Ma, W. X.: A refined invariant subspace method and applications to evolution equations. Science China Mathematics, 55(9), 1769-1778 (2012).
  • [14] Olver, P. J.: Applications of Lie groups to differential equations (Vol. 107). Springer Science & Business Media (2000).
  • [15] Bluman, G. W., Cheviakov, A. F., & Anco, S. C.: Applications of symmetry methods to partial differential equations (Vol. 168, pp. xx+-398). New York: Springer (2010).
  • [16] Siregar, S., Oktamuliani, S., & Saijo, Y.: A theoretical model of laser heating carbon nanotubes. Nanomaterials, 8(8), 580 (2018).
  • [17] Nakamiya, T., Ueda, T., Ikegami, T., Ebihara, K., & Tsuda, R.: Thermal analysis of carbon nanotube film irradiated by a pulsed laser. Current Applied Physics, 8(3-4), 400-403(2008).
  • [18] Nakamiya, T., & Ebihara, K.: The Finite Element Thermal Analysis of Amorphous Silicon Thin Films Irradiated by a Pulsed Laser. The transactions of the Institute of Electrical Engineers of Japan. A, 108(10), 443-450 (1988).
  • [19] Younis, M., & Rizvi, S. T. R.: Optical soliton like-pulses in ring-cavity fiber lasers of carbon nanotubes. Journal of Nanoelectronics and Optoelectronics, 11(3), 276-279 (2016).
  • [20] Wang, G., Kara, A. H., Buhe, E., & Fakhar, K.: Group analysis and conservation laws of a coupled system of partial differential equations describing the carbon nanotubes conveying fluid. Romanian Journal in physics, 60(7-8), 952-960 (2015).
  • [21] Muriel, C.; Romero, J.L.: First integrals, integrating factors and λ-Symmetries of second-order differential equations. J. Phys. A Math. Theor. 42, 365207 (2009).
  • [22] Bai, Y. S., Pei, J. T., & Ma, W. X.: Symmetry and-Symmetry Reductions and Invariant Solutions of Four Nonlinear Differential Equations. Mathematics, 8, 1138 (2020).
  • [23] A. Cheviakov.: GeM software package for computation of symmetries and conservation laws of differential equations. Comp. Phys. Comm. 176 (2007), 48-61.
  • [24] Gaeta, G. C., & Morando, P.: On the relation between standard and μ-symmetries for PDEs. J. Phys. A: Math. Gen., Vol. 37, 9467-9486 (2004).
Toplam 24 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Articles
Yazarlar

Emrullah Yaşar 0000-0003-4732-5753

Yakup Yıldırım 0000-0003-4443-3337

Proje Numarası KUAP(F)-2019/11
Yayımlanma Tarihi 1 Haziran 2022
Gönderilme Tarihi 23 Nisan 2021
Kabul Tarihi 18 Kasım 2021
Yayımlandığı Sayı Yıl 2022 Cilt: 10 Sayı: 2

Kaynak Göster

APA Yaşar, E., & Yıldırım, Y. (2022). Group Invariant Solutions and Local Conservation Laws of Heat Conduction Equation Arising Laser Heating Carbon Nanotubes Using Lie Group Analysis. Mathematical Sciences and Applications E-Notes, 10(2), 102-113. https://doi.org/10.36753/mathenot.926867
AMA Yaşar E, Yıldırım Y. Group Invariant Solutions and Local Conservation Laws of Heat Conduction Equation Arising Laser Heating Carbon Nanotubes Using Lie Group Analysis. Math. Sci. Appl. E-Notes. Haziran 2022;10(2):102-113. doi:10.36753/mathenot.926867
Chicago Yaşar, Emrullah, ve Yakup Yıldırım. “Group Invariant Solutions and Local Conservation Laws of Heat Conduction Equation Arising Laser Heating Carbon Nanotubes Using Lie Group Analysis”. Mathematical Sciences and Applications E-Notes 10, sy. 2 (Haziran 2022): 102-13. https://doi.org/10.36753/mathenot.926867.
EndNote Yaşar E, Yıldırım Y (01 Haziran 2022) Group Invariant Solutions and Local Conservation Laws of Heat Conduction Equation Arising Laser Heating Carbon Nanotubes Using Lie Group Analysis. Mathematical Sciences and Applications E-Notes 10 2 102–113.
IEEE E. Yaşar ve Y. Yıldırım, “Group Invariant Solutions and Local Conservation Laws of Heat Conduction Equation Arising Laser Heating Carbon Nanotubes Using Lie Group Analysis”, Math. Sci. Appl. E-Notes, c. 10, sy. 2, ss. 102–113, 2022, doi: 10.36753/mathenot.926867.
ISNAD Yaşar, Emrullah - Yıldırım, Yakup. “Group Invariant Solutions and Local Conservation Laws of Heat Conduction Equation Arising Laser Heating Carbon Nanotubes Using Lie Group Analysis”. Mathematical Sciences and Applications E-Notes 10/2 (Haziran 2022), 102-113. https://doi.org/10.36753/mathenot.926867.
JAMA Yaşar E, Yıldırım Y. Group Invariant Solutions and Local Conservation Laws of Heat Conduction Equation Arising Laser Heating Carbon Nanotubes Using Lie Group Analysis. Math. Sci. Appl. E-Notes. 2022;10:102–113.
MLA Yaşar, Emrullah ve Yakup Yıldırım. “Group Invariant Solutions and Local Conservation Laws of Heat Conduction Equation Arising Laser Heating Carbon Nanotubes Using Lie Group Analysis”. Mathematical Sciences and Applications E-Notes, c. 10, sy. 2, 2022, ss. 102-13, doi:10.36753/mathenot.926867.
Vancouver Yaşar E, Yıldırım Y. Group Invariant Solutions and Local Conservation Laws of Heat Conduction Equation Arising Laser Heating Carbon Nanotubes Using Lie Group Analysis. Math. Sci. Appl. E-Notes. 2022;10(2):102-13.

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