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Section B

An efficient approximate method for solving linear fractional Klein–Gordon equation based on the generalized Laguerre polynomials

Pages 1853-1864 | Received 23 Jun 2012, Accepted 10 Dec 2012, Published online: 02 Apr 2013

References

  • Abramowitz , M. and Stegun , I. A. 1964 . Handbook of Mathematical Functions , New York : Dover .
  • Askey , R. and Gasper , G. 1977 . Convolution structures for Laguerre polynomials . J. d'Anal. Math. , 31 : 48 – 68 . (doi:10.1007/BF02813297)
  • Canuto , C. , Hussaini , M. Y. , Quarteroni , A. and Zang , T. A. 2006 . Spectral Methods , New York : Springer-Verlag .
  • Cheng-Long , X. and Ben-Yu , G. 2002 . Laguerre pseudospectral method for non-linear partial differential equations . J. Comput. Math. , 20 : 413 – 428 .
  • Das , S. 2008 . Functional Fractional Calculus for System Identification and Controls , New York : Springer .
  • Hashim , I. , Abdulaziz , O. and Momani , S. 2009 . Homotopy analysis method for fractional IVPs . Commun. Nonlinear Sci. Numer. Simul. , 14 : 674 – 684 . (doi:10.1016/j.cnsns.2007.09.014)
  • He , J. H. 1999 . Variation iteration method-a kind of non-linear analytical technique: Some examples . Internat. J. Non-Linear Mech. , 34 : 699 – 708 . (doi:10.1016/S0020-7462(98)00048-1)
  • Jafari , H. and Daftardar-Gejji , V. 2006 . Solving linear and non-linear fractional diffusion and wave equations by ADM . Appl. Math. Comput. , 180 : 488 – 497 . (doi:10.1016/j.amc.2005.12.031)
  • Khabibrakhmanov , I. Z. and Summers , D. 1998 . The use of generalized Laguerre polynomials in spectral methods for non-linear differential equations . Comput. Math. Appl. , 36 : 65 – 70 . (doi:10.1016/S0898-1221(98)00117-5)
  • Khader , M. M. 2011 . On the numerical solutions for the fractional diffusion equation . Commun. Nonlinear Sci. Numer. Simul. , 16 : 2535 – 2542 . (doi:10.1016/j.cnsns.2010.09.007)
  • Khader , M. M. 2012 . Introducing an efficient modification of the homotopy perturbation method by using Chebyshev polynomials . Arab J. Math. Sci. , 18 : 61 – 71 . (doi:10.1016/j.ajmsc.2011.09.001)
  • Khader , M. M. and Hendy , A. S. 2012 . The approximate and exact solutions of the fractional-order delay differential equations using Legendre pseudospectral method . Int. J. Pure Appl. Math. , 74 ( 3 ) : 287 – 297 .
  • Khader , M. M. and Hendy , A. S. 2012 . A numerical technique for solving fractional variational problems . Math. Methods Appl. Sci. , doi: 10.1002/mma.2681.
  • Lewandowski , Z. and Szynal , J. 1998 . An upper bound for the Laguerre polynomials . J. Comput. Appl. Math. , 99 : 529 – 533 . (doi:10.1016/S0377-0427(98)00181-2)
  • Michalska , M. and Szynal , J. 2001 . A new bound for the Laguerre polynomials . J. Comput. Appl. Math. , 133 : 489 – 493 . (doi:10.1016/S0377-0427(00)00670-1)
  • Samko , S. , Kilbas , A. and Marichev , O. 1993 . Fractional Integrals and Derivatives: Theory and Applications , London : Gordon and Breach .
  • Sassaman , R. and Biswas , A. 2011 . Soliton solutions of the generalized Klein-Gordon equation by semi-inverse variational principle . Math. Eng. Sci. Aerospace , 2 ( 1 ) : 99 – 104 .
  • Smith , G. D. 1965 . Numerical Solution of Partial Differential Equations , Oxford : Oxford University Press .
  • Sweilam , N. H. and Khader , M. M. 2010 . A Chebyshev pseudo-spectral method for solving fractional integro-differential equations . ANZIAM , 51 : 464 – 475 . (doi:10.1017/S1446181110000830)
  • Sweilam , N. H. , Khader , M. M. and Al-Bar , R. F. 2007 . Numerical studies for a multi-order fractional differential equation . Phys. Lett. A , 371 : 26 – 33 . (doi:10.1016/j.physleta.2007.06.016)
  • Sweilam , N. H. , Khader , M. M. and Al-Bar , R. F. 2008 . Homotopy perturbation method for linear and nonlinear system of fractional integro-differential equations . Int. J. Comput. Math. Numer. Simul. , 1 ( 1 ) : 73 – 87 .
  • Sweilam , N. H. , Khader , M. M. and Nagy , A. M. 2011 . Numerical solution of two-sided space-fractional wave equation using finite difference method . J. Comput. Appl. Math. , 235 : 2832 – 2841 . (doi:10.1016/j.cam.2010.12.002)
  • Sweilam , N. H. , Khader , M. M. and Mahdy , A. M.S. 2012 . Crank-Nicolson finite difference method for solving time-fractional diffusion equation . J. Fract. Calc. Appl. , 2 ( 2 ) : 1 – 9 .
  • Sweilam , N. H. , Khader , M. M. and Mahdy , A. M.S. 2012 . Numerical studies for solving fractional-order Logistic equation . Int. J. Pure Appl. Math. , 78 ( 8 ) : 1199 – 1210 .
  • Sweilam , N. H. , Khader , M. M. and Mahdy , A. M.S. 2012 . Numerical studies for fractional-order logistic differential equation with two different delays . J. Appl. Math. , 2012 : 1 – 14 . (doi:10.1155/2012/764894)
  • Talay Akyildiz , F. 2009 . Laguerre spectral approximation of Stokes' first problem for third-grade fluid . J. Comput. Math. , 10 : 1029 – 1041 .
  • Wang , L. and Guo , Benyu . 2006 . Stair Laguerre pseudospectral method for differential equations on the half line . Adv. Comput. Math. , 25 : 305 – 322 . (doi:10.1007/s10444-003-7608-6)
  • Wazwaz , A. M. 2006 . Compacton solitons and periodic solutions for some forms of nonlinear Klein-Gordon equations . Chaos Solitons Fractals , 28 ( 4 ) : 1005 – 1013 . (doi:10.1016/j.chaos.2005.08.145)
  • Yusufoglu , E. 2008 . The variational iteration method for studying the Klein-Gordon equation . Appl. Math. Lett. , 21 ( 7 ) : 669 – 674 . (doi:10.1016/j.aml.2007.07.023)
  • Zhuanga , Q. and Xub , C. 2010 . Legendre Laguerre coupled spectral element methods for second-and fourth-order equations on the half line . J. Comput. Anal. Appl. , 235 : 615 – 630 .

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