Numerical analysis of nonlinear fractional Klein–Fock–Gordon equation arising in quantum field theory via Caputo–Fabrizio fractional operator

Abstract

The present article deals with the solution of nonlinear fractional Klein–Fock–Gordon equation which involved the newly developed Caputo–Fabrizio fractional derivative with non-singular kernel. We adopt fractional homotopy perturbation transform method in order to find the approximate solution of fractional Klein–Fock–Gordon equation in the form of rapidly convergent series. Existence and uniqueness analysis of the considered model is provided. We consider few numerical examples to validate the projected technique. The obtained results shows that this method is very efficient, simple in implementation and that it can be applied to solve other nonlinear problems.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

References

  1. 1.

    Khan, Y., Wu, Q.: Homotopy perturbation transform method for nonlinear equations using He’s polynomials. Comput. Math Appl. 61(8), 1963–1967 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Gupta, S., Kumar, D., Singh, J.: Analytical solutions of convection-diffusion problems by combining Laplace transform method and homotopy perturbation method. Alex. Eng. J. 54(3), 645–651 (2015)

    Article  Google Scholar 

  3. 3.

    Prakash, A., Kaur, H.: Analysis and numerical simulation of fractional Biswas–Milovic model. Math. Comput. Simul. 181, 298–315 (2021)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Jleli, M., Kumar, S., Kumar, R., Samet, B.: Analytical approach for time fractional wave equations in the sense of Yang–Abdel-Aty–Cattani via the homotopy perturbation transform method. Alex. Eng. J. 59(5), 2859–2863 (2020)

    Article  Google Scholar 

  5. 5.

    Abbasbandy, S.: Application of He’s homotopy perturbation method for Laplace transform. Chaos Solitons Fract. 30, 1206–1212 (2006)

    MATH  Article  Google Scholar 

  6. 6.

    Prakash, A.: Analytical method for space-fractional telegraph equation by Homotopy perturbation transform method. Nonlinear Eng.-Model. Appl. 5(2), 123–128 (2016)

    Google Scholar 

  7. 7.

    Golshan, A.N., Nourazar, S.S., Fard, H.G., Yildirim, A., Campo, A.: A modified homotopy perturbation method coupled with the Fourier transform for nonlinear and singular Lane–Emden equations. Appl. Math. Lett. 26(10), 1018–1025 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Shirkhani, M.R., Hoshyara, H.A., Rahimipetroudi, I., Akhavan, H., Ganji, D.D.: Unsteady time dependent incompressible Newtonian fluid flow between two parallel plates by homotopy analysis method (HAM), homotopy perturbation method (HPM) and collocation method (CM). Propuls. Power Res. 7(3), 247–256 (2018)

    Article  Google Scholar 

  9. 9.

    Guirao, J.L.G., Baskonus, H.M., Kumar, A., Rawat, M.S., Yel, G.: Complex patterns to the (3 + 1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation. Symmetry. 12(17), 1–10 (2020)

    Google Scholar 

  10. 10.

    Guirao, J.L.G., Baskonus, H.M., Kumar, A., Causanilles, F.S.V., Bermudez, G.R.: Complex mixed dark bright wave patterns to the modified α and modified Vakhnenko–Parkes equations. Alex. Eng. J. 59(4), 2149–2160 (2020)

    Article  Google Scholar 

  11. 11.

    Guirao, J.L.G., Baskonus, H.M., Kumar, A.: Regarding new wave patterns of the newly extended nonlinear (2 + 1)-dimensional Boussinesq equation with fourth order. Mathematics. 8(3), 341 (2020)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Baskonus, H.M., Kumar, A., Gao, W.: Deeper investigations of the (4 + 1)-dimensional Fokas and (2 + 1)-dimensional Breaking soliton equations. Int. J. Mod. Phys. B 2050152, 1–16 (2020)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Gupta, P.K.: Approximate analytical solutions of fractional Benney-Lin equation by reduced differential transform method and the homotopy perturbation method. Comput. Math Appl. 61(9), 2829–2842 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Prakash, A., Goyal, M., Baskonus, H.M., Gupta, S.: A reliable hybrid numerical method for a time-dependent vibration model of arbitrary order. AIMS Math. 5(2), 979–1000 (2020)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Prakash, A., Verma, V.: Two efficient computational technique for fractional nonlinear Hirota–Satsuma coupled KdV equations. Eng. Comput. (2020). https://doi.org/10.1108/ec-02-2020-0091

    Article  Google Scholar 

  16. 16.

    Goyal, M., Baskonus, H.M., Prakash, A.: An efficient technique for a time fractional model of lassa hemorrhagic fever spreading in pregnant women. Eur. Phys. J. Plus. 134(482), 1–10 (2019)

    Google Scholar 

  17. 17.

    Prakash, A., Goyal, M., Gupta, S.: Fractional variational iteration method for solving time-fractional Newell–Whitehead–Segel equation. Nonlinear Eng.-Model. Appl. 8, 164–171 (2019)

    Article  Google Scholar 

  18. 18.

    Assas, L.M.B.: Variational iteration method for solving coupled-KdV equations. Chaos Solitons Fract. 38(4), 1225–1228 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Gupta, S., Goyal, M., Prakash, A.: Numerical treatment of Newell–Whitehead–Segel equation. TWMS J. App. Eng. Math. 10(2), 312–320 (2020)

    Google Scholar 

  20. 20.

    Goyal, M., Baskonus, H.M., Prakash, A.: Regarding new positive, bounded and convergent numerical solution of nonlinear time fractional HIV/AIDS transmission model. Chaos Solitons Fract. 139, 1–12 (2020)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Prakash, A., Kumar, M.: Numerical solution of two-dimensional time fractional order biological population model. Open Phys. 14, 177–186 (2016)

    Article  Google Scholar 

  22. 22.

    Prakash, A., Kumar, M., Sharma, K.K.: Numerical method for solving coupled Burgers equation. Appl. Math. Comput. 260, 314–320 (2015)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Verma, V., Prakash, A., Kumar, D., Singh, J.: Numerical study of fractional model of multi-dimensional dispersive partial differential equation. J. Ocean Eng. Sci. 4, 338–351 (2019)

    Article  Google Scholar 

  24. 24.

    Prakash, A., Verma, V.: Numerical solution of nonlinear fractional Zakharov–Kuznetsov equation arising in ion-acoustic waves. Pramana-J. Phys. 93(66), 1–19 (2019)

    Google Scholar 

  25. 25.

    Prakash, A., Kaur, H.: q-homotopy analysis transform method for space and time-fractional KdV-Burgers equation. Nonlinear Sci. Lett. A. 9(1), 44–61 (2018)

    Google Scholar 

  26. 26.

    Padmavathi, V., Prakash, A., Alagesan, K., Magesh, N.: Analysis and numerical simulation of novel coronavirus (COVID-19) model with Mittag–Leffler Kernel. Math. Methods Appl. Sci. (2020). https://doi.org/10.1002/mma.6886

    Article  Google Scholar 

  27. 27.

    Prakash, A., Kumar, M.: Numerical solution of time-fractional order Fokker–Planck equation. TWMS J. App. Eng. Math. 9(3), 446–454 (2019)

    Google Scholar 

  28. 28.

    He, J.H.: Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178(3–4), 257–262 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Momani, S., Odibat, Z.: Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys. Lett. A 365(5–6), 345–350 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Gupta, P.K., Singh, M.: Homotopy perturbation method for fractional Fornberg–Whitham equation. Comput. Math Appl. 61(2), 250–254 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Algahtani, O.J.J.: Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model. Chaos Solitons Fract. 89, 552–559 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Zhenga, X., Wanga, H., Fu, H.: Well-posedness of fractional differential equations with variable-order Caputo–Fabrizio derivative. Chaos Solitons Fract. 138(109966), 1–7 (2020)

    MathSciNet  Google Scholar 

  33. 33.

    Gong, X., Khan, M.A.: A new numerical solution of the competition model among bank data in Caputo–Fabrizio derivative. Alex. Eng. J. 59(4), 2251–2259 (2020)

    Article  Google Scholar 

  34. 34.

    Yépez-Martínez, H., Gómez-Aguilar, J.F.: A new modified definition of Caputo–Fabrizio fractional order derivative and their applications to the multi-step homotopy analysis method. J. Comput. Appl. Math. 346(15), 247–260 (2020)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Saelao, J., Yokchoo, N.: The solution of Klein–Gordon equation by using modified Adomian decomposition method. Math. Comput. Simul. 171, 94–102 (2020)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Baleanu, D., Aydogn, S.M., Mohammadi, H., Rezapour, S.: On modelling of epidemic childhood diseases with the Caputo–Fabrizio derivative by using the Laplace Adomian decomposition method. Alex. Eng. J. 59(5), 3029–3039 (2020)

    Article  Google Scholar 

  37. 37.

    Zhang, J.L., Wang, M.L., Feng, F.D.: The improved F-expansion method and its applications. Phys. Lett. A 350(1–2), 103–109 (2006)

    Article  Google Scholar 

  38. 38.

    Craddock, M., Platen, E.: Symmetry group methods for fundamental solutions. J. Differ. Equ. 207(2), 285–302 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Yokus, A., Durur, H., Ahmad, H., Thounthong, P., Zhang, Y.: Construction of exact traveling wave solutions of the Bogoyavlenskii equation by (G′/G, 1/G)-expansion and (1/G′)-expansion techniques. Results Phys. 103409(19), 1–8 (2020)

    Google Scholar 

  40. 40.

    Yokus, A.: On the exact and numerical solutions to the FitzHugh-Nagumo equation. Int. J. Mod. Phys. B 2050149, 1–12 (2020). https://doi.org/10.1142/S0217979220501490

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Yokus, A., Kuzu, B., Demiroglu, U.: Investigation of solitary wave solutions for the (3 + 1)-dimensional Zakharov–Kuznetsov equation. Int. J. Mod. Phys. B 33(29), 1–19 (2019)

    MathSciNet  Article  Google Scholar 

  42. 42.

    Yokus, A., Durur, H., Ahmad, H., Yao, S.: Construction of different types analytic solutions for the Zhiber–Shabat equation. Mathematics. 8(908), 1–16 (2020). https://doi.org/10.3390/math8060908

    Article  Google Scholar 

  43. 43.

    Chen, Y., Tian, B., Qu, Q., Li, H., Zhao, X., Tian, H., Wang, M.: Ablowitz–Kaup Newell–Segur system, conservation laws and Backlund transformation of a variable-coefficient Korteweg-de Vries equation in plasma physics, fluid dynamics or atmospheric science. Int. J. Mod. Phys. B 2050226, 1–8 (2020). https://doi.org/10.1142/S0217979220502264

    Article  MATH  Google Scholar 

  44. 44.

    Yavuz, M., Yokus, A.: Analytical and numerical approaches to nerve impulse model of fractional-order. Numer Methods Partial Differ. Equ. 5, 1–21 (2020). https://doi.org/10.1002/num.22476

    MathSciNet  Article  Google Scholar 

  45. 45.

    Yusufoglu, E.: The variational iteration method for studying the Klein–Gordon equation. Appl. Math. Lett. 21, 669–674 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Khan, N.A., Rasheed, S.: Analytical solutions of linear and nonlinear Klein–Fock–Gordon equation. Nonlinear Eng.-Model. Appl. 4(1), 43–48 (2015)

    Google Scholar 

  47. 47.

    Ravi Kanth, A.S.V., Aruna, K.: Differential transform method for solving the linear and nonlinear Klein–Gordon equation. Comput. Phys. Commun. 180, 708–711 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    Aruna, K., Ravi Kanth, A.S.V.: Two-dimensional differential transform method and modified differential transform method for solving nonlinear fractional Klein–Gordon equation. Nat. Acad. Sci. Lett. 37(2), 163–171 (2014)

    MathSciNet  Article  Google Scholar 

  49. 49.

    Kumar, D., Singh, J., Kumar, S.: Numerical computation of Klein–Gordon equations arising in quantum field theory by using homotopy analysis transform method. Alex. Eng. J. 53, 469–474 (2014)

    Article  Google Scholar 

  50. 50.

    Veeresha, P., Prakasha, D.G., Kumar, D.: An efficient technique for nonlinear time-fractional Klein–Fock–Gordon equation. Appl. Math. Comput. 364(1), 124637 (2020)

    MathSciNet  MATH  Google Scholar 

  51. 51.

    Singh, H., Kumar, D., Singh, J., Singh, C.S.: A reliable numerical algorithm for the fractional Klein–Gordon equation. Eng. Trans. 67(1), 21–34 (2019)

    Google Scholar 

  52. 52.

    Abuteen, E., Freihat, A., Al-Smadi, M., Khalil, H., Khan, R.A.: Approximate series solution of nonlinear fractional Klein–Gordon equations using fractional reduced differential transform method. J. Math. Stat. 12(1), 23–33 (2016)

    Article  Google Scholar 

  53. 53.

    Kochetov, B.A.: Lie group symmetries and Riemann function of Klein–Gordon–Fock equation with central symmetry. Commun. Nonlinear Sci. Numer. Simul. 19(6), 1723–1728 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Aero, E.L., Bulygin, A.N., Pavlov, YuV: Functionally invariant solutions of nonlinear Klein–Fock–Gordon equation. Appl. Math. Comput. 223(15), 160–166 (2013)

    MathSciNet  MATH  Google Scholar 

  55. 55.

    Fainberg, V.Y., Pimentel, B.M.: Duffin–Kemmer–Petiau and Klein–Gordon–Fock equations for electromagnetic, Yang–Mills and external gravitational field interactions: proof of equivalence. Phys. Lett. A 271, 16–25 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Caputo, M.: Elasticita e Dissipazione. ZaniChelli, Bologna (1969)

    Google Scholar 

  57. 57.

    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    Google Scholar 

  58. 58.

    Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2004)

    Google Scholar 

  59. 59.

    Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 73–85 (2015)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Amit Prakash.

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

Verify currency and authenticity via CrossMark

Cite this article

Prakash, A., Kumar, A., Baskonus, H.M. et al. Numerical analysis of nonlinear fractional Klein–Fock–Gordon equation arising in quantum field theory via Caputo–Fabrizio fractional operator. Math Sci (2021). https://doi.org/10.1007/s40096-020-00365-2

Download citation

Keywords

  • Fractional Klein–Fock–Gordon equation
  • Fractional Homotopy perturbation transform method
  • Laplace transform
  • Caputo–Fabrizio fractional operator