SpringerLink
Log in

A novel computational approach to the local fractional Lonngren wave equation in fractal media

  • Original Research
  • Published:
Mathematical Sciences Aims and scope Submit manuscript

Abstract

The main purpose of this paper is to investigate the local fractional Lonngren wave equation, which is a generalization of Lonngren wave equation in fractal media. Firstly, an extremely effective approach is presented to obtain the fractal travelling wave solution of the local fractional Lonngren wave equation; secondly, the characteristics of fractal travelling wave solution are illustrated by some 3D graphs; finally, the comparative results of the local fractional Lonngren wave equation and the classical Lonngren wave equation are discussed. This proposed new method is simple and efficient and provides a novel idea for the study of fractal-fractional wave models in fractal media.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Aguilar, J.F.G., Kumar, S., Kumar, A., Osman, M.S., Samet, B.: A chaos study of tumor and effector cells in fractional tumor-immune model for cancer treatment. Chaos. Soliton. Fract. 141, 110321 (2020)

    MathSciNet  MATH  Google Scholar 

  2. Dineshkumar, C., Nisar, K.S., Udhayakumar, R., Vijayakumar, V.: A discussion on the approximate controllability of Hilfer fractional neutral stochastic integro-differential systems. Chaos. Soliton. Fract. 142, 110472 (2021)

    MathSciNet  MATH  Google Scholar 

  3. Alshabanat, A., Jleli, M., Kumar, S., Samet, B.: Generalization of Caputo-Fabrizio fraftional derivative and applications to electrical circuits. Front. Phys. 8, 64 (2020)

    Google Scholar 

  4. Liu, J.G., Yang, X.J., Feng, Y.Y., Cui, P.: On group analysis of the time fractional extended (2+1)-dimensional Zakharov-Kuznetsov equation in quantum magneto-plasmas. Math. Comput. Simulat. 178(2020), 407–421 (2020)

    MathSciNet  MATH  Google Scholar 

  5. Shaikh, A., Nisar, K.S., Jadhav, V., Elagan, S.K., Zakarya, M.: Dynamical behaviour of HIV/AIDS model using fractional derivative with Mittag-Leffler Kernel. Alex. Eng. J. 61(4), 2601–2610 (2022)

    Google Scholar 

  6. Khan, D., Khan, A., Khan, I., Chu, Y.M., Nisar, K.S.: A new idea of Fractal-Fractional derivative with power law Kernel for Free Convection Heat Transfer in a Channel Flow between two static upright parallel plates. CMC 65(2), 1237–1251 (2020)

    Google Scholar 

  7. Xu, C.J., Farman, M., Akgul, A., Nisar, K.S., Ahmad, A.: Modeling and analysis fractal order cancer model with effects of chemotherapy. Chaos. Soliton. Fract. 161, 112325 (2022)

    MathSciNet  Google Scholar 

  8. Wang, K.L.: Exact travelling wave solution for the fractal Riemann wave model arising in ocean science. Fractals 30(7), 2250143 (2022)

    MATH  Google Scholar 

  9. Srivastava, H.M., Torres, D.F.M., Yang, X.J., Zhang, Y.D.: Non-differential solutions for local fractional nonlinear Riccati differential equations. Fund. Inform. 145, 55–63 (2016)

    Google Scholar 

  10. Kumar, D., Nieto, J.J., Singh, J.: A reliable algorithm for a local fractional Tricomi equation arising in fractal Transonic flow. Entropy 18(6), 2061–2068 (2016)

    Google Scholar 

  11. Wang, K.L.: Exact travelling wave solution for the local fractional Camassa-Holm-Kadomtsev-Petviashvili equation. Alex. Eng. J. 63(1), 371–376 (2023)

    Google Scholar 

  12. Wang, K.L.: Exact travelling wave solutions for the local fractional Kadomtsov-Petviashvili-Benjamin-Bona-Mahony model by variational perspective. Fractals 30(6), 2250101 (2022)

    MATH  Google Scholar 

  13. He, J.H., He, C.H., Qie, N.: Solitary waves travelling along an unsmooth boundary. Results. Phys. 24, 104104 (2021)

    Google Scholar 

  14. Cheng, P., Yu, B.M.: A fractal permeability model for bi-dispersed porous media. Int. J. Heat. Mass. Tran. 45(14), 2983–2993 (2002)

    MATH  Google Scholar 

  15. **ao, B.Q., Huang, Q.W., Hanxin Chen, H.X., Xubing Chen, X.B., Long, G.B.: A fractal model for capillary flow through a single tortuous capillary with roughened surfaces in fibrous porous media. Fractals 29(1), 2150017 (2021)

    MATH  Google Scholar 

  16. Duran, S.: Travelling wave solutions and simulation of the Lonngren wave equation for tunnel diode. Opt. Quant. Electron. 53(8), 458 (2021)

    Google Scholar 

  17. Machado, J.A.T., Yang, X.J.: A new insight into complexity from the local fractional calculus view point: modelling growths of populations. Math. Method. Appl. Sci. 40(17), 6070–6075 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Baleanu, D., Jafari, H., Jassim, H.K., Qurahi, M.A.: On the existence and uniqueness of solutions for local fractional differential equations. Entropy 18(11), 4201–4209 (2016)

    Google Scholar 

  19. Ames, W.F., Hsuan, H.C.S., Lonngren, K.E.: On the soliton, invariant, and shock solutions of a fourth-order nonlinear equation. J. Math. Anal. Appl. 52, 538–545 (1975)

    MathSciNet  MATH  Google Scholar 

  20. Baskonus, H.M., Bulut, H., Sulaiman, T.A.: New complex Hyperbolic structures to the Lonngren-Wave equation by using sineGordon expansion method. Appl. Math. Nonlinear. Sci. 4(1), 129–138 (2019)

    Google Scholar 

  21. Akcagil, S., Aydemir, T.: Comparison between the -expansion method and the modified extended tanh method. Open. Phys. 14, 88–94 (2016)

    MATH  Google Scholar 

  22. Yokus, A.: Simulation of bright-dark soliton solutions of the Lonngren wave equation arising the model of transmission lines. Mod. Phys. Lett. B. 35(32), 2150484 (2021)

    MathSciNet  Google Scholar 

  23. Wang, K.L.: A novel variational approach to fractal Swift-Hohenberg model arising in fluid dynamics. Fractals 30(7), 2250156 (2022)

    MATH  Google Scholar 

  24. Ahmad, H., Li, F.Q., Nadeem, M.: Modified Laplace variational iteration method for solving fourth-order parabolic partial differential equation with variable coefficients. Comput. Math. Appl. 78(6), 2052–2062 (2019)

    MathSciNet  MATH  Google Scholar 

  25. Inc, M.: The approximate and exact solutions of the space-and time-fractional Burgers equations with initial conditions by variational iteration method. J. Math. Anal. Appl. 345(1), 476–484 (2008)

    MathSciNet  MATH  Google Scholar 

  26. Kumar, S., Kumar, R., Osman, M.S., Samet, B.: A wavelet based numerical scheme for fractional order SEIR epidemic of measles by using Genocchi polynomials. Mumer. Meth. Part. D.E. 37, 1250–1268 (2021)

    MathSciNet  Google Scholar 

  27. Agarwal, R.P., Kumar, S., Kumar, R., Samet, B.: A study of fractional Lotka-Volterra population model using Haar wavelet and Adams-Bashforth-Moulton methods. Math. Method. Appl. Sci. 43(8), 5564–5578 (2020)

    MathSciNet  MATH  Google Scholar 

  28. Kumar, S.: A new analytical modelling for fractional telegraph equation via Laplace transform. Appl. Math. Model. 38, 3154–3163 (2014)

    MathSciNet  MATH  Google Scholar 

  29. El-Dib, Y.O., He, J.H.: Homotopy perturbation method with three expansions. J. Math. Chem. 59, 1139–1150 (2021)

    MathSciNet  MATH  Google Scholar 

  30. El-Dib, Y.O., He, J.H.: The enhanced Homotopy perturbation method for axial vibration of strings. Facta. Univ-ser. Mech. 19(4), 735–750 (2021)

    Google Scholar 

  31. He, J.H., Jiao, M.L., Gepreel, K.A., Khan, Y.: Homotopy perturbation method for strongly nonlinear oscillators. Math. Comput. Simulat. 204, 243–258 (2022)

    MathSciNet  MATH  Google Scholar 

  32. Wang, K.L.: Novel approach for fractal nonlinear oscillators with discontinuities by Fourier series. Fractals 30(1), 2250009 (2022)

    MATH  Google Scholar 

  33. Alshehri, A.M., Dubey, V.P., Dubey, S., Kumar, D., Singh, J.: A hybrid computational method for local fractional dissipative and damped wave equations in fractal media. Wave. Random. Complex. 2022, 2049395 (2022)

    Google Scholar 

  34. Jothimani, K., Nisar, K.S., Ravichandran, C., Subashini, R.: New results on nonlocal functional integro-differential equations via Hilfer fractional derivative. Alex. Eng. J. 59, 2891–2899 (2020)

    Google Scholar 

  35. Abro, K.A., Khan, I., Nisar, K.S.: Novel technique of Atangana and Baleanu for heat dissipation in transmission line of electrical circuit. Chaos. Soliton. Fract. 129, 40–45 (2019)

    MathSciNet  Google Scholar 

  36. Wang, K.L.: A novel perspective to the local fractional bidirectional wave model on Cantor sets. Fractals 30(6), 2250107 (2022)

    MATH  Google Scholar 

  37. Ghanbari, B., Nisar, K.S.: Some effective numerical techniques for chaotic systems involving fractal-fractional derivatives with different laws. Front. Phys.-Lausanne. 8, 192 (2020)

    Google Scholar 

  38. Anjum, N., He, C.H., He, J.H.: Two-scale fractal theory for the population dynamics. Fractals 29(7), 2150182 (2021)

    MATH  Google Scholar 

  39. Liu, J.G., Yang, X.J., Feng, Y.Y., Cui, P.: On the (N+1)-dimensional local fractional reduced differential transform method and its applications. Math. Method. Appl. Sci. 43(15), 8856–8866 (2020)

    MathSciNet  MATH  Google Scholar 

  40. Liu, J.G., Zhang, Y.F.: Analytical study of exact solutions of the nonlinear Korteweg–de Vries equation with space–time fractional derivatives. Mod. Phys. Lett. B. 32(2), 1850012 (2018)

    MathSciNet  Google Scholar 

  41. Liu, J.G., Yang, X.J., Feng, Y.Y., Cui, P.: A new perspective to study the third-order modified KDV equation on fractal sets. Fractals 6(28), 2050110 (2020)

    MATH  Google Scholar 

  42. Baleanu, D., He, J.H., Srivastava, H.M., Yang, X.J.: Cantor-type cylindrical method for differential equations with local fractional derivative. Phys. Lett. A. 377(28), 1696–1700 (2015)

    MATH  Google Scholar 

  43. Yang, X.J., Tenreiro Machado, J.A., Baleanu, D., Cattani, C.: On exact traveling-wave solutions for local fractional Korteweg-de Vries equation. Chaos 26(8), 084312 (2016)

    MathSciNet  MATH  Google Scholar 

  44. Baleanu, D., Srivastava, H.M., Yang, X.J.: Local fractional integral transforms and their applications. Academic Press, New York (2015)

    MATH  Google Scholar 

  45. Almohsen, B., Chu, Y.M., Inc, M., Korpinar, Z.: Some numerical solutions of local fractional tricomi equation in fractal transonic flow. Alex. Eng. J. 60(1), 1147–1153 (2021)

    Google Scholar 

  46. Rahman, G., Nisar, K.S., Golamankaneh, A.K.: The nonlocal fractal integral reverse Minkowski’s and other related inequalities on fractal sets. Math. Probl. Eng. 2021, 4764891 (2021)

    MathSciNet  Google Scholar 

  47. Chen, G.S., Chao, L., Liang, J.S., Srivastava, H.M.: Local fractional integral HÖlder-Type inequalities and some related results. Fractal. Fract. 6, 195 (2022)

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Kang-Le Wang

Authors
  1. Kang-Le Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kang-Le Wang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, KL. A novel computational approach to the local fractional Lonngren wave equation in fractal media. Math Sci (2023). https://doi.org/10.1007/s40096-023-00509-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40096-023-00509-0

Keywords

Search

Navigation