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FRACTAL VARIATIONAL PRINCIPLES FOR TWO DIFFERENT TYPES OF FRACTAL PLASMA MODELS WITH VARIABLE COEFFICIENTS

KANG-LE WANG

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, P. R. China

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and

HAO WANG

https://orcid.org/0000-0002-2875-9006

School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo, P. R. China

E-mail Address: hpu_wanghao@163.com

Corresponding author.

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https://doi.org/10.1142/S0218348X22500438Cited by:16 (Source: Crossref)

Abstract

In this paper, two different types of plasma models with variable coefficients are described by using the fractal derivative. This paper aims at establishing fractal variational principles for two different types of discontinuous plasma physics by the fractal semi-inverse method, so that it can be applied to numerical analysis of the effect of the discontinuous time and discontinuous space on the solution properties.

Keywords:

References

1. K. Jothimani, N. Vaslliammal and C. Ravichandran, Existence result for a neutral fractional integro-differential equation with state dependent delay, J. Appl. Nonlinear Dyn. 7(4) (2018) 371–381. Crossref, Google Scholar
2. C. Ravichandran and J. J. Trujillo, Controllability of impulsive fractional functional integro-differential equations in Banach spaces, J. Funct. Spaces Appl. 2013 (2013) 812501. Crossref, Google Scholar
3. B. Ghanbari, S. Kumar and R. Kumar, A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative, Chaos Solitons Fractals 133 (2020) 109619. Crossref, Web of Science, Google Scholar
4. P. Veeresha, D. G. Prakasha and S. Kumar, A fractional model for propagation of classical optical solitons by using nonsingular derivative, Math. Methods Appl. Sci. (2020), https://doi.org/10.1002/mma.6335. Crossref, Web of Science, Google Scholar
5. C. Ravichandran, K. Logeswari and F. Jarad, New results on existence in the framework of Atangana–Baleanu derivative for fractional integro-differential equations, Chaos Solitons Fractals 125 (2019) 194–200. Crossref, Web of Science, Google Scholar
6. S. Kumar, S. Ghosh, B. Samet and E. F. D. Goufo, An analysis for heat equations arises in diffusion process using new Yang–Abdel–Aty–Cattani fractional operator, Math. Methods Appl. Sci. 43(9) (2020) 6062–6080. Crossref, Web of Science, Google Scholar
7. N. Vaslliammal and C. Ravichandran, Results on fractional neutral integro-differential systems with state-dependent delay in Banach spaces, Nonlinear Stud. 25(1) (2018) 159–171. Google Scholar
8. E. F. D. Goufo, S. Kumar and S. B. Mugisha, Similarities in a fifth-order evolution equation with and with no singular kernel, Chaos Solitons Fractals 130 (2020) 109467. Crossref, Web of Science, Google Scholar
9. S. Kumar, R. Kumar, C. Cattani and B. Samet, Chaotic behaviour of fractional predator-prey dynamical system, Chaos Solitons Fractals 135 (2020) 109811. Crossref, Web of Science, Google Scholar
10. C. Ravichandran, N. Valliammal and J. J. Nieto, New results on exact controllability of a class of fractional neutral integro-differential system with state-dependent systems with state-dependent delay in Banach space, J. Franklin Inst. 356(3) (2019) 1535–1565. Crossref, Web of Science, Google Scholar
11. M. A. Alqudah, C. Ravichandran, T. Abdeljawad and N. Valliammal, New results on Caputo fractional-order neutral differential inclusions without compactness, Adv. Differ. Equ. 2019 (2019) 528. Crossref, Web of Science, Google Scholar
12. S. K. Panda, C. Ravichandran and B. Hazarika, Results on system of Atangana–Baleanu fractional order Willis aneurysm and nonlinear singularly perturbed boundary value problems, Chaos Solitons Fractals 142 (2021) 110390. Crossref, Web of Science, Google Scholar
13. K. S. Nisar, K. Jothimani, K. Kaliraj and C. Ravichandran, An analysis of controllability results for nonlinear Hilfer neutral fractional derivatives with non-dense domain, Chaos Solitons Fractals 146 (2021) 110915. Crossref, Web of Science, Google Scholar
14. J. H. He, A modified Li-He’s variational principle for plasma, Int. J. Numer. Methods Heat Fluid Flow (2021), https://doi.org/10.1108/HFF-06-2019-0523. Crossref, Web of Science, Google Scholar
15. H. M. Sedighi and F. Daneshmand, Nonlinear transversely vibrating beams by the homotopy perturbation method with an auxiliary term, J. Appl. Comput. Mech. 1(1) (2015) 1–9. Web of Science, Google Scholar
16. S. Kumar, A new fractional modeling arising in engineering sciences and its analytical approximate solution, Alex. Eng. J. 52 (2013) 813–819. Crossref, Google Scholar
17. M. Nadeem and F. Q. Li, He-Laplace method for nonlinear vibration systems and nonlinear wave equation, J. Low Freq. Noise Vib. Act. Control 38(3–4) (2019) 1060–1074. Crossref, Web of Science, Google Scholar
18. H. Ahmad, A. R. Seadawy and T. A. Khan, Study on numerical solution of dispersive water wave phenomena by using a reliable modification of variational iteration algorithm, Math. Comput. Simul. 17(17) (2020) 13–23. Crossref, Web of Science, Google Scholar
19. M. Nadeem, F. Q. Li and H. Ahmad, Modified Laplace variational iteration method for solving fourth-order parabolic partial differential equation with variable coefficients, Comput. Math. Appl. 78(6) (2019) 2052–2062. Crossref, Web of Science, Google Scholar
20. H. Ahmad and T. A. Khan, Variational iteration algorithm-I with an auxiliary parameter for wave-like vibration equations, J. Low Freq. Noise Vib. Act. Control 38(3) (2019) 1113–1124. Crossref, Web of Science, Google Scholar
21. M. Nadeem and J. H. He, He-Laplace variational iteration method for solving the nonlinear equations arising in chemical kinetics and population dynamics, J. Math. Chem. 59 (2021) 1234–1245. Crossref, Web of Science, Google Scholar
22. S. Kumar, R. Kumar, R. P. Agarwal and B. Samet, A study of fractional Lotka–Volterra population model using Haar wavelet and Adams–Bashforth–Moulton methods, Math. Methods Appl. Sci. (2020), https://doi.org/10.1002/mma.6297. Crossref, Web of Science, Google Scholar
23. S. Kumar, A. Ahmadian, R. Kumar, D. Kumar and J. Singh, An efficient numerical method for fractional SIR epidemic model of infectious disease by using Bernstein wavelets, Mathematics 8(4) (2020) 558. Crossref, Web of Science, Google Scholar
24. S. Kumar, R. Kumar, S. Momani and S. Hadid, A study on fractional COVID-19 disease model by using Hermite wavelets, Math. Methods Appl. Sci. (2021), https://doi.org/10.1002/mma.7065. Crossref, Web of Science, Google Scholar
25. J. Singh et al., An efficient computational approach for local fractional Poisson equation in fractal media, Numer. Methods Partial Differ. Equ. 37(2) (2021) 1439–1448. Crossref, Web of Science, Google Scholar
26. X. J. Yang, M. Tenreiro and D. Baleanu, Exact traveling-wave solution for local fractional Boussinesq equation in fractal domain, Fractals 25(4) (2017) 1740006. Link, Web of Science, Google Scholar
27. I. Jaradat et al., An Avant-Garde handling of temporal-spatial fractional physical models, Int. J. Nonlinear Sci. Numer. Simul. 21(2) (2020) 183–194. Crossref, Web of Science, Google Scholar
28. J. Singh, On the local fractional wave equation in fractal strings, Math. Methods Appl. Sci. 42(5) (2019) 1588–1595. Crossref, Web of Science, Google Scholar
29. K. L. Wang, He’s frequency formulation for fractal nonlinear oscillator arising in a microgravity space, Numer. Methods Partial Differ. Equ. 37(2) (2020) 1374–1384. Crossref, Web of Science, Google Scholar
30. K. L. Wang, A new fractal model for the soliton motion in a microgravity space, Int. J. Numer. Methods Heat Fluid Flow (2020), https://doi.org/10.1108/HFF-05-2020-0247. Crossref, Web of Science, Google Scholar
31. J. H. He, A fractal variational theory for one-dimensional compressible flow in a microgravity space, Fractals 28(2) (2020) 2050024. Link, Web of Science, Google Scholar
32. J. H. He, Fractal calculus and its geometrical explanation, Result Phys. 10 (2018) 272–276. Crossref, Web of Science, Google Scholar
33. J. H. He and F. Y. Ji, Two-scale mathematics and fractional calculus for thermodynamics, Therm. Sci. 23(4) (2019) 2131–2133. Crossref, Web of Science, Google Scholar
34. J. H. He, A tutorial review on fractal spacetime and fractional calculus, Int. J. Theor. Phys. 53 (2014) 3698–3718. Crossref, Web of Science, Google Scholar
35. K. L. Wang and C. F. Wei, New analytical approach for fractal $K (p, q)$ $K (p, q)$ model, Fractals 29(5) (2021) 2150116. Link, Web of Science, Google Scholar
36. K. L. Wang, Novel approach for fractal nonlinear oscillators with discontinuities by Fourier series, Fractals 30(1) (2022) 2250009. Link, Web of Science, Google Scholar
37. K. J. Wang, A new fractional nonlinear singular heat conduction model for the human head considering the effect of febrifuge, Eur. Phys. J. Plus 135 (2020) 871. Crossref, Web of Science, Google Scholar
38. K. L. Wang, H. Wang and H. Muhammad, A new perspective for two different types of Fractal Zakharov–Kuznetsov models, Fractals 29(6) (2021) 2150168. Link, Web of Science, Google Scholar
39. J. H. He, N. Qie, C. H. He and T. Saeed, On a strong minimum condition of a fractal variational principle, Appl. Math. Lett. 2021 (2021) 107199. Crossref, Web of Science, Google Scholar
40. J. H. He, Q. Na and C. H. He, Solitary waves travelling along an unsmooth boundary, Results Phys. 24 (2021) 104104. Crossref, Web of Science, Google Scholar
41. K. J. Wang and G. D. Wang, Periodic solution of the (2+1)-dimensional nonlinear electrical transmission line equation via variational method, Results Phys. 20 (2021) 103666. Crossref, Web of Science, Google Scholar
42. K. J. Wang and G. D. Wang, Solitary and periodic wave solutions of the generalized fourth order Boussinesq equation via He’s variational methods, Math. Methods Appl. Sci. 44(7) (2021) 5617–5625. Crossref, Web of Science, Google Scholar