Research article

Numerical solution of the linear time fractional Klein-Gordon equation using transform based localized RBF method and quadrature

  • Received: 28 March 2020 Accepted: 10 June 2020 Published: 19 June 2020
  • MSC : 26A33, 65M12, 65R10, 81Q05

  • This work aims to approximate the solution of the linear time-fractional Klein-Gordon equations in Caputo's sense. The Laplace transform is applied to linear time fractional Klein-Gordon equation to eliminate the time variable and avoid the time stepping procedure. Application of the Laplace transform avoids the time instability issues which commonly occurs in time stepping methods and reduces the computational cost. The transform problem is then solved using local RBFs and finally the solution is obtained by the inverse Laplace transform. The solution is represented as an integral along a smooth curve in the complex plane which is then approximated by quadrature rule. The proposed method is capable of solving linear time fractional partial differential equations. The stability and convergence of the method are discussed. The efficiency of the method is demonstrated with the help of numerical experiments.

    Citation: Xiangmei Li, Kamran, Absar Ul Haq, Xiujun Zhang. Numerical solution of the linear time fractional Klein-Gordon equation using transform based localized RBF method and quadrature[J]. AIMS Mathematics, 2020, 5(5): 5287-5308. doi: 10.3934/math.2020339

    Related Papers:

  • This work aims to approximate the solution of the linear time-fractional Klein-Gordon equations in Caputo's sense. The Laplace transform is applied to linear time fractional Klein-Gordon equation to eliminate the time variable and avoid the time stepping procedure. Application of the Laplace transform avoids the time instability issues which commonly occurs in time stepping methods and reduces the computational cost. The transform problem is then solved using local RBFs and finally the solution is obtained by the inverse Laplace transform. The solution is represented as an integral along a smooth curve in the complex plane which is then approximated by quadrature rule. The proposed method is capable of solving linear time fractional partial differential equations. The stability and convergence of the method are discussed. The efficiency of the method is demonstrated with the help of numerical experiments.


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    [1] F. Yousef, M. Alquran, I. Jaradat, et al. Ternary-fractional differential transform schema: Theory and application, Adv. Differ. Equ., 2019 (2019), 1-13. doi: 10.1186/s13662-018-1939-6
    [2] S. Bhatter, A. Mathur, D. Kumar, et al. A new analysis of fractional Drinfeld-Sokolov-Wilson model with exponential memory, Physica A, 537 (2020), 122578.
    [3] A. Goswami, J. Singh, D. Kumar, et al. An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma, Physica A, 524 (2019), 563-575. doi: 10.1016/j.physa.2019.04.058
    [4] D. Kumar, F. Tchier, J. Singh, et al. An efficient computational technique for fractal vehicular traffic flow, Entropy, 20 (2018), 1-9.
    [5] D. Kumar, J. Singh, D. Baleanu, On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law, Math. Meth. Appl. Sci., 43 (2020), 443-457. doi: 10.1002/mma.5903
    [6] A. Yusuf, M. Inc, A. I. Aliyu, et al. Efficiency of the new fractional derivative with nonsingular Mittag-Leffler kernel to some nonlinear partial differential equations, Chaos Soliton Fract., 116 (2018), 220-226. doi: 10.1016/j.chaos.2018.09.036
    [7] D. Kumar, J. Singh, K. Tanwar, et al. A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler laws, Int. J. Heat Mass Tran., 138 (2019), 1222-1227. doi: 10.1016/j.ijheatmasstransfer.2019.04.094
    [8] D. Kumar, J. Singh, S. D. Purohit, et al. A hybrid analytical algorithm for nonlinear fractional wave-like equations, Math. Model. Nat. Pheno., 14 (2019), 1-13.
    [9] M. J. Ablowitz, M. A. Ablowitz, P. A. Clarkson, et al. Solitons, Nonlinear Evolution Equations and Inverse Scatterin, Cambridge university press, 1991.
    [10] A. M. Yang, Y. Z. Zhang, C. Cattani, et al. Application of local fractional series expansion method to solve Klein-Gordon equations on Cantor sets, Abstr. Appl. Anal., 2014 (2014), 1-6.
    [11] A. M. Wazwaz, New travelling wave solutions to the Boussinesq and the Klein-Gordon equations, Commun. Nonlinear Sci., 13 (2008), 889-901. doi: 10.1016/j.cnsns.2006.08.005
    [12] A. M. Wazwaz, The tanh and the sine-cosine methods for compact and noncompact solutions of the nonlinear Klein-Gordon equation, Appl. Math. Comput., 167 (2005), 1179-1195.
    [13] A. S. V. R. Kanth, K. Aruna, Differential transform method for solving the linear and nonlinear Klein-Gordon equation, Comput. Phys. Commun., 180 (2018), 708-711.
    [14] K. Hosseini, P. Mayeli, R. Ansar, Modified Kudryashov method for solving the conformable timefractional Klein-Gordon equations with quadratic and cubic nonlinearities, Optik, 130 (2017), 737-742. doi: 10.1016/j.ijleo.2016.10.136
    [15] K. Hosseini, P. Mayeli, D. Kuma, New exact solutions of the coupled sine-Gordon equations in nonlinear optics using the modified Kudryashov method, J. Mod. Optic., 65 (2018), 361-364. doi: 10.1080/09500340.2017.1380857
    [16] K. Hosseini, P. Mayeli, R. Ansar, Bright and singular soliton solutions of the conformable timefractional Klein-Gordon equations with different nonlinearities, Wave. Random Complex, 28 (2018), 426-434. doi: 10.1080/17455030.2017.1362133
    [17] K. Hosseini, Y. J. Xu, P. Mayeli, et al. A study on the conformable time-fractional Klein-Gordon equations with quadratic and cubic nonlinearitiesA study on the conformable time-fractional Klein-Gordon equations with quadratic and cubic nonlinearities, Optoelectron. Adv. Mat., 11 (2017), 423-429.
    [18] E. Yusufoğlu, The variational iteration method for studying the Klein-Gordon equation, Appl. Math. Lett., 21 (2008), 669-674. doi: 10.1016/j.aml.2007.07.023
    [19] M. Alaroud, M. Al-Smadi, O. A. Arqub, et al. Numerical Solutions of Linear Time-fractional Klein-Gordon Equation by Using Power Series Approach, SSRN Electron. J., 2018 (2018), 1-6.
    [20] M. Kurulay, Solving the fractional nonlinear Klein-Gordon equation by means of the homotopy analysis method, Adv. Differ, Equ., 2012 (2012), 1-8. doi: 10.1186/1687-1847-2012-1
    [21] K. A. Gepreel, M. S. Mohamed, Analytical approximate solution for nonlinear space-time fractional Klein-Gordon equation, Chinese Phys. B, 22 (2013), 010201.
    [22] A. K. Golmankhaneh, A. K. Golmankhaneh, D. Baleanu, On nonlinear fractional Klein-Gordon equation, Signal Process., 91 (2011), 446-451. doi: 10.1016/j.sigpro.2010.04.016
    [23] E. A. B. Abdel-Salam, E. A. Yousif, Solution of nonlinear space-time fractional differential equations using the fractional Riccati expansion method, Math. Probl. Eng., 2013 (2013), 1-6.
    [24] D. Kumar, J. Singh, D. Baleanu, A hybrid computational approach for Klein-Gordon equations on Cantor sets, Nonlinear Dynam., 87 (2017), 511-517. doi: 10.1007/s11071-016-3057-x
    [25] A. Mohebbi, M. Abbaszadeh, M. Dehghan, High-order difference scheme for the solution of linear time fractional Klein-Gordon equations, Numer. Meth. Part. Differ. Equ., 30 (2014), 1234-1253. doi: 10.1002/num.21867
    [26] M. Dehghan, M. Abbaszadeh, A. Mohebbi, An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein-Gordon equations, Eng. Anal. Bound. Elem., 50 (2015), 412-434. doi: 10.1016/j.enganabound.2014.09.008
    [27] M. M. Khader, An efficient approximate method for solving linear fractional Klein-Gordon equation based on the generalized Laguerre polynomials, Int. J. Comput. Math., 90 (2013), 1853-1864. doi: 10.1080/00207160.2013.764994
    [28] G. Hariharan, Wavelet method for a class of fractional Klein-Gordon equations, J. Comput. Nonlinear Dynam., 8 (2013), 1-6.
    [29] H. Singh, D. Kumar, J. Singh, et al. A reliable numerical algorithm for the fractional klein-gordon equation, Eng. Trans., 67 (2019), 21-34.
    [30] S. Vong, Z. Wang, A compact difference scheme for a two dimensional fractional Klein-Gordon equation with Neumann boundary conditions, Entropy, 274 (2014), 268-282.
    [31] M. Dehghan, A. Shokri, Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions, J. Comput. Appl. Math., 230 (2009), 400-410. doi: 10.1016/j.cam.2008.12.011
    [32] M. Dehghan, A. Shokri, A method of solution for certain problems of transient heat conduction, AIAA J., 8 (1968), 2004-2009.
    [33] G. J. Moridis, D. L. Reddell, The Laplace transform finite difference method for simulation of flow through porous media, Water Resour. Res., 27 (1991), 1873-1884. doi: 10.1029/91WR01190
    [34] G. J. Moridis, D. L. Reddell, The Laplace transform boundary element (LTBE) method for the solution of diffusion-type equations, In: Boundary Elements XIII, Springer, Dordrecht, 1991, 83-97.
    [35] G. J. Moridis, D. L. Reddell, The Laplace transform finite element (LTFE) numerical method for the solution of the ground water equations, paper H22C-4, AGU 91 Spring Meeting, Baltimore, May 28-31, 1991, EOS Trans. of the AGU, 72 (1991), 17.
    [36] E. A. Sudicky, R. G. McLaren, The Laplace Transform Galerkin Technique for large-scale simulation of mass transport in discretely fractured porous formations, Water Resour. Res., 28 (1992), 499-514. doi: 10.1029/91WR02560
    [37] G. J. Moridis, E. J. Kansa, The Laplace transform multiquadric method: A highly accurate scheme for the numerical solution of partial differentia, J. Appl. Sci. Comput., (1993), 55910181.
    [38] Q. T. L. Gia, W. Mclean, Solving the heat equation on the unit sphere via Laplace transforms and radial basis functions, Adv. Comput. Math., 40 (2014), 353-375. doi: 10.1007/s10444-013-9311-6
    [39] W. McLean, V. Thomee, Time discretization of an evolution equation via Laplace transforms, IMA J. Numer. Anal., 24 (2004), 439-463. doi: 10.1093/imanum/24.3.439
    [40] M. L. Fernandez, C. Palencia, On the numerical inversion of the Laplace transform of certain holomorphic mappings, Appl. Numer. Math., 51 (2004), 289-303. doi: 10.1016/j.apnum.2004.06.015
    [41] W. McLean, V. Thomee, Numerical solution via Laplace transforms of a fractional order evolution equation, J. Integral Equ. Appl., 22 (2010), 57-94. doi: 10.1216/JIE-2010-22-1-57
    [42] J. A. C. Weideman, L. N. Trefethen, Parabolic and Hyperbolic contours for computing the Bromwich integral, Math. Comput., 76 (2007), 1341-1356. doi: 10.1090/S0025-5718-07-01945-X
    [43] D. Sheen, I. H. Shaon, V. Thomee, A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature, IMA J. Numer.l Anal., 23 (2003), 269-299. doi: 10.1093/imanum/23.2.269
    [44] Z. J. Fu, W. Chen, H. T. Yang, Boundary particle method for Laplace transformed time fractional diffusion equations, J. Comput. Phys., 235 (2013), 52-66. doi: 10.1016/j.jcp.2012.10.018
    [45] M. Uddin, A. Ali, On the approximation of time-fractional telegraph equations using localized kernel-based method, Adv. Differ. Equ., 2018 (2018), 1-14. doi: 10.1186/s13662-017-1452-3
    [46] M. Uddin, A. Ali, A localized transform-based meshless method for solving time fractional wave- diffusion equation, Eng. Anal. Bound. Elem., 92 (2018), 108-113. doi: 10.1016/j.enganabound.2017.10.021
    [47] K. B. Oldham, J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order, Academic Press, New York, London, 1974.
    [48] M. Uddin, On the selection of a good value of shape parameter in solving time dependent partial differential equations using RBF approximation method, Appl. Math. Model., 38 (2014), 135-144. doi: 10.1016/j.apm.2013.05.060
    [49] R. E. Carlson, T. A. Foley, The parameter r2 in multiquadric interpolation, Comput. Math. Appl., 21 ((1991), 29-42.
    [50] R. E. Carlson, T. A. Foley, Near optimal parameter selection for multiquadric interpolation, Manuscript, Computer Science and Engineering Department, Arizona State University, Tempe., 20 (1994).
    [51] D. Kumar, F. Tchier, J. Singh, et al. Error estimates and condition numbers for radial basis function interpolation, Adv. Comput. Math., 3 (1995), 251-264. doi: 10.1007/BF02432002
    [52] L. N. Trefethen, D. Bau, Numerical Linear Algebra, SIAM, 1997.
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