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Numerical Solution of Space-Time Fractional Klein-Gordon Equation by Radial Basis Functions and Chebyshev Polynomials

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Abstract

The current study is conducted to evaluate the numerical solution of the space-time fractional Klein-Gordon equation. This equation is obtained by adopting the generalized Caputo fractional derivative instead of the classical integer order derivative. The significant benefit of fractional derivatives is their nonlocal property and the capability to provide detailed and precise analysis of the model. The novel collocation technique, which is capable of dealing with space-time fractional derivatives simultaneously, has been implemented to evaluate the numerical solution for the fractional Klein Gordon equation. In this technique, the discretizations for time and space do not depend on each other, which is its most significant advantage. Hence, we have the liberty to employ two different basis functions, namely radial basis functions and the Chebyshev polynomials in space and time, respectively. The convergence, stability, and consistency of the method are also discussed. The results acquired by the implemented technique have been compared with other approaches to prove the efficiency and applicability of the proposed method.

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References

  1. Li, W., Li, C.: Second-order explicit difference schemes for the space fractional advection-diffusion equation. Appl. Math. Comput. 257, 446–457 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Xu, Y., He, Z., Xu, Q.: Numerical solutions of fractional advection-diffusion equations with a kind of new generalized fractional derivative. Int. J. Comput. Math. 91, 588–600 (2014)

    Article  MathSciNet  Google Scholar 

  3. Garg, M., Manohar, P.: Numerical solution of fractional diffusion-wave equation with two space variables by matrix method. Fract. Calculus Appl. Anal. 13, 191–207 (2010)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. El-Sayed, S.M.: The decomposition method for studying the Klein-Gordon equation. Chaos, Solitons Fractals 18, 1025–1030 (2003)

    Article  MathSciNet  Google Scholar 

  7. Chowdhury, M.S.H., Hashim, I.: Application of homotopy-perturbation method to Klein-Gordon and sine-Gordon equations. Chaos, Solitons Fractals 39, 1928–1935 (2009)

    Article  MathSciNet  Google Scholar 

  8. Hosseinzadeh, K., Mardani, M.R., Salehi, S., Paikar, M., Waqas, M., Ganji, D.D.: Entropy generation of three-dimensional Bödewadt flow of water and hexanol base fluid suspended by Fe\(_3\)O\(_4\) and MoS\(_2\) hybrid nanoparticles. Pramana J. Phys. 95, 1–14 (2021)

    Article  Google Scholar 

  9. Gholinia, M., Hosseinzadeh, K., Ganji, D.D.: Investigation of different base fluids suspend by CNTs hybrid nanoparticle over a vertical circular cylinder with sinusoidal radius. Case Stud. Thermal Eng. 21, 100666 (2020)

    Article  Google Scholar 

  10. Ziane, D., Cherif, M.: A new analytical solution of Klein-Gordon equation with local fractional derivative. Asian-Eur. J. Math. 14, 2150029 (2021)

    Article  MathSciNet  Google Scholar 

  11. Wang, K., Wang, K.: A new analysis for Klein-Gordon model with local fractional derivative. Alex. Eng. J. 59, 3309–3313 (2020)

    Article  Google Scholar 

  12. Alderremy, A.A., Elzaki, T.M., Chamekh, M.: New transform iterative method for solving some Klein-Gordon equations. Res. Phys. 10, 655–659 (2018)

    Google Scholar 

  13. Mustafa, T.: Effective computation of solutions for nonlinear heat transfer problems in fins. J. Heat Transfer 136, 0919011–0919016 (2014)

    Google Scholar 

  14. Mustafa, T.: Solution of initial and boundary value problems by an effective accurate method. Int. J. Comput. Methods 14, 1750069 (2017)

    Article  MathSciNet  Google Scholar 

  15. Mustafa, T.: High-order nonlinear Volterra-Fredholm-Hammerstein integro-differential equations and their effective computation. Appl. Math. Comput. 247, 410–416 (2014)

    MathSciNet  Google Scholar 

  16. Marasi, H., Karimi, S.: Convergence of the variational iteration method for solving fractional Klein-Gordon equation. J. Math. Comput. Sci. 4, 257–266 (2014)

    Google Scholar 

  17. Singh, H., Kumar, D., Pandey, R.K.: An efficient computational method for the time-space fractional Klein-Gordon equation. Front. Phys. 8, 281 (2020)

    Article  Google Scholar 

  18. Khader, M., Kumar, S.: An accurate numerical method for solving the linear fractional Klein-Gordon equation. Math. Methods Appl. Sci. 37, 2972–2979 (2014)

    Article  MathSciNet  Google Scholar 

  19. Hosseinzadeh, K., Roghani, S., Mogharrebi, A.R., Asadi, A., Ganji, D.D.: Optimization of hybrid nanoparticles with mixture fluid flow in an octagonal porous medium by effect of radiation and magnetic field. J. Therm. Anal. Calorim. 143, 1413–1424 (2021)

    Article  Google Scholar 

  20. Hosseinzadeh, K., Montazer, E., Shafii, M.B., Ganji, D.D.: Heat transfer hybrid nanofluid (1-Butanol/MoS\(_2\)-Fe\(_3\)O\(_4\)) through a wavy porous cavity and its optimization. Int. J. Numer. Methods Heat Fluid Flow 31, 1547–1567 (2021)

    Article  Google Scholar 

  21. Jafari, H.: Numerical solution of time-fractional Klein-Gordon equation by using the decomposition methods. J. Comput. Nonlinear Dyn. 11, 1–5 (2016)

    Google Scholar 

  22. Mustafa, T.: Parametrized adomian decomposition method with optimum convergence. ACM Trans. Model. Comput. Simul. 27, 1–22 (2017)

    Article  MathSciNet  Google Scholar 

  23. Mohammad, A., Mohammed, A.S., Omar, A.A., Rokiah, R.A., Mamon, A.H., Shaher, M.: Numerical solutions of linear time-fractional Klein-Gordon equation by using power series approach. SSRN Electron. J. 2018, 1–6 (2018)

    Google Scholar 

  24. Li, X., Kamran, A.U., Haq, X.: Zhang, Numerical solution of the linear time fractional Klein-Gordon equation using transform based localized RBF method and quadrature. AIMS Math. 5, 5287–5308 (2020)

    Article  MathSciNet  Google Scholar 

  25. Ganji, R., Jafari, H., Kgarose, M.: Numerical solutions of time-fractional Klein-Gordon equations by clique polynomials. Alex. Eng. J. 60, 4563–4571 (2021)

    Article  Google Scholar 

  26. Dehghan, M., Mohebbi, A., Asgari, Z.: Fourth-order compact solution of the nonlinear Klein-Gordon equation. Numer. Algorithms 52, 523–540 (2009)

    Article  MathSciNet  Google Scholar 

  27. Hosseinzadeh, K., Roghani, S., Mogharrebi, A.R., Asadi, A., Waqas, M., Ganji, D.D.: Investigation of cross-fluid flow containing motile gyrotactic microorganisms and nanoparticles over a three-dimensional cylinder. Alex. Eng. J. 59, 3297–3307 (2020)

    Article  Google Scholar 

  28. Hosseinzadeh, K., Salehi, S., Mardani, M.R., Mahmoudi, F.Y., Waqas, M., Ganji, D.D.: Investigation of nano-Bioconvective fluid motile microorganism and nanoparticle flow by considering MHD and thermal radiation. Inform. Med. Unlocked 21, 100462 (2020)

    Article  Google Scholar 

  29. Mogharrebi, A. R., Reza, A., Ganji, D. D., Hosseinzadeh, K., Roghani, S., Asadi, A., Fazlollahtabar, A.: Investigation of magnetohydrodynamic nanofluid flow contain motile oxytactic microorganisms over rotating cone, International Journal of Numerical Methods for Heat & Fluid Flow, https://doi.org/10.1108/HFF-08-2020-0493

  30. Hosseinzadeh, K., Asadi, A., Mogharrebi, A.R., Azari, M.E., Ganji, D.D.: Investigation of mixture fluid suspended by hybrid nanoparticles over vertical cylinder by considering shape factor effect. J. Therm. Anal. Calorim. 143, 1081–1095 (2021)

    Article  Google Scholar 

  31. Sajad, S., Amin, N., Hosseinzadeh, K., Ganji, D.D.: Hydrothermal analysis of MHD squeezing mixture fluid suspended by hybrid nanoparticles between two parallel plates. Case Stud. Therm. Eng. 21, 100650 (2020)

    Article  Google Scholar 

  32. Jain, M.K., Iyengar, S.R., Jain, R.K.: Numerical methods: problems and solutions. New Age International, New Delhi (2007)

    Google Scholar 

  33. El-Baghdady, G.I., El-Azab, M.: Chebyshev-Gauss-Lobatto pseudo-spectral method for one-dimensional advection-diffusion equation with variable coefficients. Sohag J. Math. 3, 1–8 (2016)

    Article  Google Scholar 

  34. Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations to methods of their solution and some of their applications. Academic Press, London (1998)

    MATH  Google Scholar 

  35. Schaback, R.: Improved error bounds for scattered data interpolation by radial basis functions. Math. Comput. 68, 201–216 (1999)

    Article  MathSciNet  Google Scholar 

  36. Powell, M.: The theory of radial basis function approximation in 1990. Clarendon Press, Oxford (1992)

    MATH  Google Scholar 

  37. Fornberg, B., Flyer, N.: Accuracy of radial basis function interpolation and derivative approximations on 1-d infinite grids. Adv. Comput. Math. 23, 5–20 (2005)

    Article  MathSciNet  Google Scholar 

  38. Bansu, H., Kumar, S.: Numerical solution of space and time fractional telegraph equation: a meshless approach. Int. J. Nonlinear Sci. Numer. Simul. 20, 325–337 (2019)

    Article  MathSciNet  Google Scholar 

  39. Bansu, H., Kumar, S.: Numerical solution of space and time fractional advection-diffusion equation by meshless approach, Proceedings of International Conference on Applied and Computational Mathematics, Kharagpur, Dec. 2018, Springer, Singapore

  40. Bansu, H., Kumar, S.: Meshless method for numerical solution of fractional Pennes bioheat equation, Proceedings of International Conference on Biomedical Engineering, Singapore 2019, Springer, Singapore

  41. Kumar, S., Piret, C.: Numerical solution of space-time fractional PDEs using RBF-QR and Chebyshev polynomials. Appl. Numer. Math. 143, 300–315 (2019)

    Article  MathSciNet  Google Scholar 

  42. Gasca, M., Sauer, T.: On the history of multivariate polynomial interpolation, Numerical Analysis: Historical Developments in the 20th Century, 2012, Elsevier, Amsterdam

  43. Mohebbi, A., Abbaszadeh, M., Dehghan, M.: High-order difference scheme for the solution of linear time fractional Klein-Gordon equations. Numer. Methods Partial Differ. Equ. 30, 1234–1253 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are thankful to anonymous reviewers for the fruitful comments and suggestions to improve the quality of the manuscript. The first author is also grateful to S. V. National Institute of Technology.

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  1. Applied Mathematics & Humanities Department, S. V. National Institute of Technology, Surat, 395007, India

    Hitesh Bansu

  2. Department of Mathematics & Humanities, S. V. National Institute of Technology, Surat, 395007, India

    Sushil Kumar

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  1. Hitesh Bansu
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Correspondence to Hitesh Bansu.

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Bansu, H., Kumar, S. Numerical Solution of Space-Time Fractional Klein-Gordon Equation by Radial Basis Functions and Chebyshev Polynomials. Int. J. Appl. Comput. Math 7, 201 (2021). https://doi.org/10.1007/s40819-021-01139-7

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