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Russian Journal of Mathematical Physics

, Volume 26, Issue 1, pp 77–93 | Cite as

An Application of the Gegenbauer Wavelet Method for the Numerical Solution of the Fractional Bagley-Torvik Equation

  • H. M. SrivastavaEmail author
  • F. A. ShahEmail author
  • R. AbassEmail author
Article
  • 13 Downloads

Abstract

In this paper, a potentially useful new method based on the Gegenbauer wavelet expansion, together with operational matrices of fractional integral and block-pulse functions, is proposed in order to solve the Bagley–Torvik equation. The Gegenbauer wavelets are generated here by dilation and translation of the classical orthogonal Gegenbauer polynomials. The properties of the Gegenbauer wavelets and the Gegenbauer polynomials are first presented. These functions and their associated properties are then employed to derive the Gegenbauer wavelet operational matrices of fractional integrals. The operational matrices of fractional integrals are utilized to reduce the problem to a set of algebraic equations with unknown coefficients. Illustrative examples are provided to demonstrate the validity and applicability of the method presented here.

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References

  1. 1.
    S. Alkan, “Approximate Solutions of Boundary Value Problems of Fractional Order by Using Sinc–Galerkin Method,” New Trends Math. Sci. 2, 1–11 (2014).Google Scholar
  2. 2.
    R. L. Bagley and P. J. Torvik, “On the Appearance of the Fractional Derivative in the Behavior of Real Materials,” ASME J. Appl. Mech. 51, 294–308 (1984).CrossRefzbMATHGoogle Scholar
  3. 3.
    M. Caputo, “Linear Models of Dissipation Whose Q Is Almost Frequency Independent. II,” J. Roy. Austral. Soc. 13, 529–539 (1967).ADSCrossRefGoogle Scholar
  4. 4.
    Y. Cenesiz, Y. Keskin, and A. Kurnaz, “The Solution of the Bagley–Torvik Equation with the Generalized Taylor Collocation Method,” J. Franklin Inst. 347, 452–466 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    J. Cermák and T. Kisela, “Exact and Discretized Stability of the Bagley–Torvik Equation,” J. Comput. Appl. Math. 269, 53–67 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    L. Debnath and F. A. Shah, Wavelet Transforms and Their Applications (Birkhäuser, Basel and New York, 2015).zbMATHGoogle Scholar
  7. 7.
    K. Diethelm and N. J. Ford, “Numerical Solution of the Bagley–Torvik Equation,” BIT Numer. Math. 42, 490–507 (2002).MathSciNetzbMATHGoogle Scholar
  8. 8.
    M. El–Gamel and M. El–Hady, “Numerical Solution of the Bagley–Torvik Equation by Legendre–Collocation Method,” SeMA J. 74, 371–383 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    W. M. Elhameed and Y. H. Youssri, “New Ultraspherical Wavelets Spectral Solutions for Fractional Riccati Differential Equations,” Abstr. Appl. Anal. 2014, Article ID 626275, 1–8 (2014).zbMATHGoogle Scholar
  10. 10.
    A. Ghorbani and A. Alavi, “Applications of He’s Variational Iteration Method to Solve Semi–Differential Equations of nth Order,” Math. Prob. Engrg. 2008, 1–9 (2008).CrossRefzbMATHGoogle Scholar
  11. 11.
    M. H. Heydari, M. R. Hooshmandas, F. M. Ghaini, and F. Mohammadi, “Wavelet Collocation Method for Solving Multi–Order Fractional Differential Equations,” J. Appl. Math. 2012, Article ID 542401, 1–19 (2012).Google Scholar
  12. 12.
    M. A. Iqbal, M. Shakeel, S. T. Mohyud–Din, and M. Rafiq, “Modified Wavelets Based Algorithm for Nonlinear Delay Differential Equations of Fractional Order,” Adv. Mech. Engrg. 9 (4), 1–8 (2017).Google Scholar
  13. 13.
    H. Jafari, S. A. Yousefi, M. A. Firoozjaee, S. Momani, and C. M. Khalique, “Application of Legendre Wavelets for Solving Fractional Differential Equations,” Comput. Math. Appl. 62, 1038–1045 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    S. Kazem, “An Integral Operational Matrix Based on Jacobi Polynomials for Solving Fractional–Order Differential Equations,” Appl. Math. Model. 37, 1126–1136 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    E. Keshavarz, Y. Ordokhani, and R. Razzaghi, “Bernoulli Wavelet Operational Matrix of Fractional–Order Integration and Its Applications in Solving the Fractional–Order Differential Equations,” Appl. Math. Model. 38, 6038–6051 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations (North–Holland Mathematical Studies, Vol. 204, Elsevier, North–Holland, Science Publishers, Amsterdam, London and New York, 2006).zbMATHGoogle Scholar
  17. 17.
    V. S. Krishnasamy and M. Razzaghi, “The Numerical Solution of the Bagley–Torvik Equation with Fractional Taylor Method,” J. Comput. Nonlinear Dyn. 11 (5), Article ID 051010, 1–6.Google Scholar
  18. 18.
    X. Li, “Numerical Solution of Fractional Differential Equations Using Cubic B–Spline Wavelet Collocation Method,” Commun. Nonlinear Sci. Numer. Simul. 17, 3934–3946 (2012).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Y. Li and W. Zhao, “Haar Wavelet Operational Matrix of Fractional–Order Integration and Its Applications in Solving the Fractional–Order Differential Equations,” Appl. Math. Comput. 216, 2276–2285 (2010).MathSciNetzbMATHGoogle Scholar
  20. 20.
    T. Mekkaoui and Z. Hammouch, “Approximate Analytic Solutions to the Bagley–Torvik Equation by the Fractional Iteration Method,” An. Univ. Craiova 39, 251–256 (2012).MathSciNetzbMATHGoogle Scholar
  21. 21.
    I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications (Mathematics in Science and Engineering, Vol. 198, Academic Press, New York, London, Sydney, Tokyo and Toronto, 1999).zbMATHGoogle Scholar
  22. 22.
    I. Podlubny, T. Skovranek, and B. M. Jara, “Matrix Approach to Discretization of Fractional Derivatives and to Solution of Fractional Differential Equations and Their Systems,” Proc. IEEE Conf. Emerg. Technol. Factory Automation 2009, 1–6 (2009).Google Scholar
  23. 23.
    A. G. Radwan, A. M. Soliman, A. S. Elwakil, and A. Sedeek, “On the Stability of Linear Systems with Fractional–Order Elements,” Chaos Solitons Fractals 40, 2317–2328 (2009).ADSCrossRefzbMATHGoogle Scholar
  24. 24.
    S. S. Ray, “On HaarWavelet Operational Matrix of General Order and Its Application for the Numerical Solution of Fractional Bagley–Torvik Equation,” Appl. Math. Comput. 218, 5239–5248 (2012).MathSciNetzbMATHGoogle Scholar
  25. 25.
    S. S. Ray, Fractional Calculus with Applications for Nuclear Reactor Dynamics (CRC Press, New York, 2016).zbMATHGoogle Scholar
  26. 26.
    S. S. Ray and R. Bera, “Analytical Solution of the Bagley–Torvik Equation by Adomian Decomposition Method,” Appl. Math. Comput. 168, 398–410 (2005).MathSciNetzbMATHGoogle Scholar
  27. 27.
    M. Rehman and R. A. Khan, “The Legendre Wavelet Method for Solving Fractional Differential Equations,” Commun. Nonlinear Sci. Numer. Simul. 16, 4163–4173 (2011).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    M. Rehman and U. Saeed, “GegenbauerWavelets Operational MatrixMethod for Fractional Differential Equations,” J. Korean Math. Soc. 52, 1069–1096 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    F. A. Shah and R. Abass, “Haar Wavelet Operational Matrix Method for the Numerical Solution of Fractional–Order Differential Equations,” Nonlinear Engrg. 4, 203–213 (2015).Google Scholar
  30. 30.
    F. A. Shah and R. Abass, “Generalized Wavelet Collocation Method for Solving Fractional Relaxation–Oscillation Equation Arising in Fluid Mechanics,” Internat. J. Comput. Mater. Sci. Engrg. 6 (2), 1–17 (2017).Google Scholar
  31. 31.
    F. A. Shah, R. Abass, and L. Debnath, “Numerical Solution of Fractional Differential Equations Using Haar Wavelet Collocation Method,” Internat. J. Appl. Comput. Math. 2016, 1–23 (2016).Google Scholar
  32. 32.
    Z.–H. Wang and X. Wang, “General Solution of the Bagley–Torvik Equation with Fractional–Order Derivative,” Commun. Nonlinear Sci. Numer. Simul. 15, 1279–1285 (2010).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    S. Yzbas, “Numerical Solution of the Bagley–Torvik Equation by the Bessel Collocation Method,” Math. Methods Appl. Sci. 36, 300–312 (2013).ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    M.–X. Yi, J. Huang, and J.–X. Wei, “Block–Pulse Operational Matrix Method for Solving Fractional Partial Differential Equation,” Appl. Math. Comput. 221, 121–131 (2013).MathSciNetzbMATHGoogle Scholar
  35. 35.
    M. A. Zahoor, J. A. Khan, and I. M. Qureshi, “Solution of Fractional–Order System of Bagley–Torvik Equation Using Evolutionary Computational Intelligence,” Math. Probl. Eng. 2011, Article ID 675075, 1–18 (2011).zbMATHGoogle Scholar
  36. 36.
    W. K. Zahra and S. M. Elkholy, “The Use of Cubic Splines in the Numerical Solution of Fractional Differential Equations,” Int. J. Math. Math. Sci. 2012, Article ID 638026, 1–16 (2012).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada
  2. 2.Department of Medical Research, China Medical University HospitalChina Medical UniversityTaichungTaiwan, Republic of China
  3. 3.Department of MathematicsUniversity of Kashmir (South Campus)AnantnagIndia

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