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Original Articles

Legendre wavelets method for approximate solution of fractional-order differential equations under multi-point boundary conditions

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Pages 998-1014 | Received 03 Nov 2016, Accepted 26 Feb 2017, Published online: 27 Mar 2017

References

  • B. Ahmad, A. Alsaedi, and D. Garout, Existence results for Liouville-Caputo type fractional differential equations with nonlocal multi-point and sub-strips boundary conditions, to appear in Comput. Math. Appl. Available at http://dx.doi.org/10.1016/j.camwa.2016.04.015.
  • Z. Avazzadeh and M. Heydari, The application of block pulse functions for solving higher-order dierential equations with multi-point boundary conditions, Adv. Differential Equ. 2016(1) (2016), pp. 1–16. doi: 10.1186/s13662-015-0739-5
  • I. Aziz and N.M. Siraj-ul-Islam, An efficient numerical algorithm based on Haar wavelet for solving a class of linear and nonlinear nonlocal boundary-value problems, Calcolo 53(4) (2016), pp. 621–633. doi: 10.1007/s10092-015-0165-9
  • Z. Bai and Y. Zhang, The existence of solutions for a fractional multi-point boundary value problem, Comput. Math. Appl. 60(8) (2010), pp. 2364–2372. doi: 10.1016/j.camwa.2010.08.030
  • A.H. Bhrawy and M.M. Al-Shomrani, A shifted Legendre spectral method for fractional-order multi-point boundary value problems, Adv. Differential Equ. 2012(1) (2012), pp. 1–19. doi: 10.1186/1687-1847-2012-1
  • M. Dehghan and F. Shakeri, A semi-numerical technique for solving the multi-point boundary value problems and engineering applications, Int. J. Numer. Method H. 21(7) (2011), pp. 794–809. doi: 10.1108/09615531111162783
  • E.H. Doha, A.H. Bhrawy, and R.M. Hafez, On shifted Jacobi spectral method for high-order multi-point boundary value problems, Commun. Nonlinear Sci. Numer. Simul. 17(10) (2012), pp. 3802–3810. doi: 10.1016/j.cnsns.2012.02.027
  • F. Geng and M. Cui, A reproducing kernel method for solving nonlocal fractional boundary value problems, Appl. Math. Lett. 25(5) (2012), pp. 818–823. doi: 10.1016/j.aml.2011.10.025
  • M.H. Heydari, M.R. Hooshmandasl, and F. Mohammadi, Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions, Appl. Math. Comput. 234 (2014), pp. 267–276.
  • H. Khalil, R.A. Khan, D. Baleanu, and S.H. Saker, Approximate solution of linear and nonlinear fractional differential equations under m-point local and nonlocal boundary conditions, Adv. Differential Equ. 2016(1) (2016), pp. 1–28. doi: 10.1186/s13662-015-0739-5
  • H. Khalil, M. Al-Smadi, K. Moaddy, R.A. Khan, and I. Hashim, Toward the approximate solution for fractional order nonlinear mixed derivative and nonlocal boundary value problems, Discrete. Dyn. Soc. (2016). doi: 10.1155/2016/5601821.
  • H. Khalil, M.M. Rashidi, and R.A. Khan, Application of fractional order Legendre polynomials: A new procedure for solution of linear and nonlinear fractional differential equations under m-point nonlocal boundary conditions, Comm. Numer. Anal. 2016(2) (2016), pp. 144–166. doi: 10.5899/2016/cna-00245
  • Y. Li, Solving a nonlinear fractional differential equation using Chebyshev wavelets, Commun. Nonlinear. Sci. 15(9) (2010), pp. 2284–2292. doi: 10.1016/j.cnsns.2009.09.020
  • X. Li and B. Wu, Approximate analytical solutions of nonlocal fractional boundary value problems, Appl. Math. Model. 39(5) (2015), pp. 1717–1724. doi: 10.1016/j.apm.2014.09.035
  • Z.Y. Li, Y.L. Wang, F.G. Tan, X.H. Wan, Y. Hao, and J.S. Duan, Solving a class of linear nonlocal boundaryvalue problems using the reproducing kernel, Appl. Math. Comput. 265 (2015), pp. 1098–1105.
  • S. Liang, J. Zhang, S. Liang, and J. Zhang, Existence and uniqueness of strictly nondecreasing and positive solution for a fractional three-point boundary value problem, Comput. Math. Appl. 62(3) (2011), pp. 1333–1340. doi: 10.1016/j.camwa.2011.03.073
  • N. Liu and E.-B. Lin, Legendre wavelet method for numerical solutions of partial differential equations, Numer. Methods Partial Differential Equations 26(1) (2010), pp. 81–94. doi: 10.1002/num.20417
  • Z. Meng, L. Wang, H. Li, and W. Zhang, Legendre wavelets method for solving fractional integro-differential equations, Int. J. Comput. Math. 92(6) (2015), pp. 1–17. doi: 10.1080/00207160.2014.932909
  • I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
  • M.U. Rehman and R.A. Khan, The Legendre wavelet method for solving fractional differential equations, Commun. Nonlinear. Sci. 16(11) (2011), pp. 4163–4173. doi: 10.1016/j.cnsns.2011.01.014
  • A. Saadatmandi and M. Dehghan, The use of Sinc-collocation method for solving multi-point boundary value problems, Commun. Nonlinear Sci. 17(2) (2012), pp. 593–601. doi: 10.1016/j.cnsns.2011.06.018
  • P.K. Sahu and S.S. Ray, Legendre spectral collocation method for Fredholm integro-differential-difference equation with variable coefficients and mixed conditions, Appl. Math. Comput. 268 (2015), pp. 575–580.
  • S.C. Shiralashetti and A.B. Deshi, An efficient Haar wavelet collocation method for the numerical solution of multi-term fractional differential equations, Nonlinear Dynam. 83(1–2) (2016), pp. 293–303. doi: 10.1007/s11071-015-2326-4
  • M. Tatari and M Dehghan, The use of the Adomian decomposition method for solving multipoint boundary value problems, Phys. Scripta 73(6) (2006), pp. 672–676. doi: 10.1088/0031-8949/73/6/023
  • M. Tatari and M Dehghan, An efficient method for solving multi-point boundary value problems and applications in physics, J. Vib. Control 18(18) (2012), pp. 1116–1124. doi: 10.1177/1077546311408467
  • S.P. Timoshenko and J.M. Gere, Theory of Elastic Stability, Courier Corporation, New York, 2009.
  • B.Y. Wu and X.Y. Li, A new algorithm for a class of linear nonlocal boundary value problems based on the reproducing kernel method, Appl. Math. Lett. 24(2) (2011), pp. 156–159. doi: 10.1016/j.aml.2010.08.036
  • L.-J. Xie, C.-L. Zhou, and S. Xu, A new algorithm based on differential transform method for solving multi-point boundary value problems, Int. J. Comput. Math. 93(6) (2016), pp. 981–994. doi: 10.1080/00207160.2015.1012070
  • Q. Zhang, Z. Feng, Q. Tang, and Y. Zhang, An adaptive wavelet collocation method for solving optimal control problem, Proc. Inst. Mech. Eng. G, J. Aerosp. Eng. 229(9) (2015), pp. 1640–1649. doi: 10.1177/0954410014558317

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