Skip to main content
SpringerLink
Log in

Analysis of Lie Symmetry, Explicit Series Solutions, and Conservation Laws for the Nonlinear Time-Fractional Phi-Four Equation in Two-Dimensional Space

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

The range of FPDEs is a leading bind between theoretical mathematics and its implementations. FPDEs administer a wide territory of physical real phenomena like binary fluid mixtures problems. This analysis intends to utilize results on Lie symmetry analysis, explicit series solutions, and conservation laws for the nonlinear TFPFE in two-dimensional space. Such a fractional equation yields the mathematical model which describes an appearance of stationary spatial stripe modalities in the framework of the theory of conservation laws. The utilized fractional model is autonomous of the kind of approximate approaches employed to resolve it and can be utilized for various real phenomena implementations inclusive magnetohydrodynamics and environmental flow processes. More specifically, the governing equation is solved analytically by the well-defined method, so-called power series such as the total derivative used is based on the Riemann–Liouville. Presented analyses show that the TFPFE and the utilized approach are conservative, delicate, and robust. Finally, some concluding remarks and future recommendations are utilized.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Data Availability

Enquiries about data availability should be directed to the authors.

Abbreviations

TFPFE:

Time-fractional Phi-four equation

FPDE:

Fractional partial differential equation

LPS:

Lie point symmetry

LS:

Lie symmetry

RLD:

Riemann–Liouville derivative

FDE:

Fractional differential equation

FDO:

Fractional differential operator

References

  1. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, UK (2010)

    Book  MATH  Google Scholar 

  2. Zaslavsky, G.M.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, UK (2005)

    MATH  Google Scholar 

  3. Podlubny, I.: Fractional Differential Equations. Academic Press, USA (1999)

    MATH  Google Scholar 

  4. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives Theory and Applications. Gordon and Breach, USA (1993)

    MATH  Google Scholar 

  5. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier, Netherlands (2006)

    MATH  Google Scholar 

  6. Ramos, H., Qureshi, S., Soomro, A.: Adaptive step-size approach for Simpson’s-type block methods with time efficiency and order stars. Comput. Appl. Math. 40, 1–20 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  7. Qureshi, S., Jan, R.: Modeling of measles epidemic with optimized fractional order under Caputo differential operator. Chaos Solitons Fractals 145, 110766 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  8. Atangana, A., Qureshi, S.: Modeling attractors of chaotic dynamical systems with fractal–fractional operators. Chaos Solitons Fractals 123, 320–337 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Qureshi, S., Atangana, A.: Mathematical analysis of dengue fever outbreak by novel fractional operators with field data. Phys. A 526, 121127 (2019)

    Article  MathSciNet  Google Scholar 

  10. Abu Arqub, O., Shawagfeh, N.: Application of reproducing kernel algorithm for solving Dirichlet time-fractional diffusion-Gordon types equations in porous media. J. Porous Media 22, 411–434 (2019)

    Article  Google Scholar 

  11. Abu Arqub, O.: Fitted reproducing kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and Neumann boundary conditions. Comput. Math. Appl. 73, 1243–1261 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Abu Arqub, O.: Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm. Int. J. Numer. Methods Heat Fluid Flow 28, 828–856 (2018)

    Article  Google Scholar 

  13. Abu Arqub, O.: The reproducing kernel algorithm for handling differential algebraic systems of ordinary differential equations. Math. Methods Appl. Sci. 39, 4549–4562 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Abu Arqub, O.: Solutions of time-fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert space. Numer. Methods Partial Differ. Equ. 34, 1759–1780 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Abu Arqub, O.: Numerical solutions of systems of first-order, two-point BVPs based on the reproducing kernel algorithm. Calcolo 55, 1–28 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Abu Arqub, O., Shawagfeh, N.: Solving optimal control problems of Fredholm constraint optimality via the reproducing kernel Hilbert space method with error estimates and convergence analysis. Math. Methods Appl. Sci. 2019, 1–18 (2019). https://doi.org/10.1002/mma.5530

    Article  MATH  Google Scholar 

  17. Abu Arqub, O.: Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential equations. Neural. Comput. Appl. 28, 1591–1610 (2017)

    Article  Google Scholar 

  18. Abu Arqub, O., Maayah, B.: Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana-Baleanu fractional operator. Chaos, Solitons Fractals 117, 117–124 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Abu Arqub, O., Maayah, B.: Fitted fractional reproducing kernel algorithm for the numerical solutions of ABC–Fractional Volterra integro-differential equations. Chaos Solitons Fractals 126, 394–402 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. Abu Arqub, O., Maayah, B.: Modulation of reproducing kernel Hilbert space method for numerical solutions of Riccati and Bernoulli equations in the Atangana-Baleanu fractional sense. Chaos Solitons Fractals 125, 163–170 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Beghami, W., Maayah, B., Bushnaq, S., Abu Arqub, O.: The Laplace optimized decomposition method for solving systems of partial differential equations of fractional order. Int. J. Appl. Comput. Math. 8, 1–18 (2022)

    Article  MathSciNet  Google Scholar 

  22. Berredjem, N., Maayah, B., Abu Arqub, O.: A numerical method for solving conformable fractional integrodifferential systems of second-order, two-points periodic boundary conditions. Alex. Eng. J. 61, 5699–5711 (2022)

    Article  Google Scholar 

  23. Djennadi, S., Shawagfeh, N., Abu Arqub, O.: A numerical algorithm in reproducing kernel-based approach for solving the inverse source problem of the time-space fractional diffusion equation. Partial Differ. Equ. Appl. Math. 4, 100164 (2021)

    Article  MATH  Google Scholar 

  24. Djennadi, S., Shawagfeh, N., Abu Arqub, O.: A fractional Tikhonov regularization method for an inverse backward and source problems in the time-space fractional diffusion equations. Chaos Solitons Fractals 150, 111127 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  25. Abu Arqub, O., Hayat, T., Alhodaly, M.: Reproducing kernel Hilbert pointwise numerical solvability of fractional Sine-Gordon model in time-dependent variable with Dirichlet condition. Phys. Scripta 96, 104005 (2021)

    Article  Google Scholar 

  26. Djennadi, S., Shawagfeh, N., Osman, M.S., Gómez-Aguilar, J.F., Abu Arqub, O.: The Tikhonov regularization method for the inverse source problem of time fractional heat equation in the view of ABC-fractional technique. Phys. Scripta 96, 094006 (2021)

    Article  Google Scholar 

  27. Abu Arqub, O., Singh, J., Maayah, B., Alhodaly, M.: Reproducing kernel approach for numerical solutions of fuzzy fractional initial value problems under the Mittag-Leffler kernel differential operator. Math. Methods Appl. Sci. 2021, 1–22 (2021)

    Google Scholar 

  28. Abu Arqub, O., Singh, J., Alhodal, M.: Adaptation of kernel functions-based approach with Atangana–Baleanu–Caputo distributed order derivative for solutions of fuzzy fractional Volterra and Fredholm integrodifferential equations. Math. Methods Appl. Sci. 2021, 1–28 (2021)

    Google Scholar 

  29. Momani, S., Abu Arqub, O., Maayah, B.: Piecewise optimal fractional reproducing kernel solution and convergence analysis for the Atangana-Baleanu-Caputo model of the Lienard’s equation. Fractals 28, 2040007 (2020)

    Article  MATH  Google Scholar 

  30. Momani, S., Maayah, B., Abu Arqub, O.: The reproducing kernel algorithm for numerical solution of Van der Pol damping model in view of the Atangana-Baleanu fractional approach. Fractals 28, 2040010 (2020)

    Article  MATH  Google Scholar 

  31. Shawagfeh, N.: Omar Abu Arqub, Shaher Momani, Analytical solution of nonlinear second-order periodic boundary value problem using reproducing kernel method. J. Comput. Anal. Appl. 16, 750–762 (2014)

    MathSciNet  Google Scholar 

  32. Alquran, M., Jaradat, H.M., Syam, M.I.: Analytical solution of the time-fractional Phi-4 equation by using modified residual power series method. Nonlinear Dyn. 90, 2525–2529 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Akter, J., Akbar, M.A.: Exact solutions to the Benney-Luke equation and the Phi-4 equations by using modified simple equation method. Results Phys. 5, 125–130 (2015)

    Article  Google Scholar 

  34. Shahen, N.H.M., Ali, M.S., Rahman, M.M.: Interaction among lump, periodic, and kink solutions with dynamical analysis to the conformable time-fractional Phi-four equation. Partial Differ. Equ. Appl. Math. 4, 100038 (2021)

    Article  Google Scholar 

  35. Younis, M., Zafar, A.: The modified simple equation method for solving nonlinear Phi-Four equation. Int. J. Innov. Appl. Stud. 2, 661–664 (2013)

    Google Scholar 

  36. Rezazadeh, H., Tariq, H., Eslami, M., Mirzazadeh, M., Zhou, Q.: New exact solutions of nonlinear conformable time-fractional Phi-4 equation. Chin. J. Phys. 56, 2805–2816 (2018)

    Article  Google Scholar 

  37. Newell, A., Whitehead, J.: Finite bandwidth finite amplitude convection. J. Fluid Mech. 38, 279–303 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  38. Bluman, G., Kumei, S.: Symmetries and Differential Equations. Springer, USA (1989)

    Book  MATH  Google Scholar 

  39. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, USA (1993)

    Book  MATH  Google Scholar 

  40. Ibragimov, N.H.: Elementary Lie Group Analysis and Ordinary Differential Equations. Wiley, USA (1999)

    MATH  Google Scholar 

  41. Gazizov, R.K., Kasatkin, A.A., Lukashchuk, S.Y.: Symmetry properties of fractional diffusion equations. Phys. Scr. 2009, 014016 (2009)

    Article  Google Scholar 

  42. Yang, H.W., Guo, M., He, H.: Conservation laws of space-time fractional mZK equation for Rossby solitary waves with complete Coriolis force. Int. J. Nonlinear Sci. Numer. Simul. 20, 17–32 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  43. Abdullahi, R.A., Muatjetjeja, B.: Conservation laws and exact solutions for a 2D Zakharov-Kuznetsov equation. Appl. Math. Lett. 1, 109–117 (2015)

    MathSciNet  MATH  Google Scholar 

  44. El-Kalaawy, O.H.: Modulational instability: conservation laws and bright soliton solution of ion-acoustic waves in electron-positron-ion-dust plasmas. The European Physical Journal Plus 133, 58 (2018)

    Article  Google Scholar 

  45. Adem, A.R., Khalique, C.M.: Symmetry reductions exact solutions and conservation laws of a new coupled KdV system. Commun. Nonlinear Sci. Numer. Simul. 17, 3465–3475 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  46. Baleanu, D., Inc, M., Yusuf, A., Aliyu, A.I.: Lie symmetry analysis, exact solutionsand conservation laws for the time fractionalmodified Zakharov-Kuznetsov equation. Nonlinear Anal. Modell. Control 22, 861–876 (2017)

    Article  MATH  Google Scholar 

  47. Baleanu, D., Inc, M., Yusuf, A., Aliyu, A.I.: Lie symmetry analysis, exact solutions and conservation laws for the time fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation. Commun. Nonlinear Sci. Numer. Simul. 59, 222–234 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  48. Inc, M., Yusuf, A., Aliyu, A.I., Baleanu, D.: Time-fractional Cahn-Allen and time-fractional Klein-Gordon equations: Lie symmetry analysis, explicit solutions and convergence analysis. Physica A 493, 94–106 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  49. Baleanu, D., Inc, M., Yusuf, A., Aliyu, A.I.: Time fractional third-order evolution equation: symmetry analysis, explicit solutions, and conservation laws. J. Comput. Nonlinear Dyn. 13, 021011 (2018)

    Article  MATH  Google Scholar 

  50. Inc, M., Yusuf, A., Aliyu, A.I., Baleanu, D.: Lie symmetry analysis, explicit solutions and conservation laws for the space–time fractional nonlinear evolution equations. Phys. A 496, 371–383 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  51. Wang, G.W., Liu, X.Q., Zhang, Y.Y.: Lie symmetry analysis to the time fractional generalized fifth-order KdV equation. Commun. Nonlinear Sci. Numer. Simul. 18, 2321–2326 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  52. Abu Arqub, O.: Application of residual power series method for the solution of time-fractional Schrödinger equations in one-dimensional space. Fund. Inf. 166, 87–110 (2019)

    MATH  Google Scholar 

  53. Abo-Hammour, Z., Abu Arqub, O., Alsmadi, O., Momani, S., Alsaedi, A.: An optimization algorithm for solving systems of singular boundary value problems. Appl. Math. Inf. Sci. 8, 2809–2821 (2014)

    Article  MathSciNet  Google Scholar 

  54. Abu Arqub, O., Abo-Hammour, Z.: Numerical solution of systems of second-order boundary value problems using continuous genetic algorithm. Inf. Sci. 279, 396–415 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  55. Abu Arqub, O.: Series solution of fuzzy differential equations under strongly generalized differentiability. J. Adv. Res. Appl. Math. 5, 31–52 (2013)

    Article  MathSciNet  Google Scholar 

  56. Ibragimov, N.H.: A new conservation theorem. J. Math. Anal. Appl. 28, 311–333 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express their gratitude to the unknown referees for carefully reading the paper and their helpful comments.

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt, 19117, Jordan

    Omar Abu Arqub

  2. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Omar Abu Arqub, Tasawar Hayat & Mohammed Alhodaly

  3. Department of Mathematics, Faculty of Science, Quaid-I-Azam University, Islamabad, 45320, Pakistan

    Tasawar Hayat

Authors
  1. Omar Abu Arqub
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tasawar Hayat
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohammed Alhodaly
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Omar Abu Arqub.

Ethics declarations

Conflict of interest

The authors declare that they have no conflicts 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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arqub, O.A., Hayat, T. & Alhodaly, M. Analysis of Lie Symmetry, Explicit Series Solutions, and Conservation Laws for the Nonlinear Time-Fractional Phi-Four Equation in Two-Dimensional Space. Int. J. Appl. Comput. Math 8, 145 (2022). https://doi.org/10.1007/s40819-022-01334-0

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40819-022-01334-0

Keywords

Search

Navigation