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Original Articles

Uniqueness and approximation of solution for fractional Bagley–Torvik equations with variable coefficients

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Pages 1542-1561 | Received 08 Nov 2015, Accepted 01 Jun 2016, Published online: 27 Jul 2016

References

  • Q.M. Al-Mdallal, M.I. Syam, and M.N. Anwar, A collocation-shooting method for solving fractional boundary value problems, Commun. Nonlinear Sci. Numer. Simul. 15 (2010), pp. 3814–3822. doi: 10.1016/j.cnsns.2010.01.020
  • A. Arikoglu and I. Ozkol, Solution of fractional differential equations by using differential transform method, Chaos, Solitons Fract. 34 (2007), pp. 1473–1481. doi: 10.1016/j.chaos.2006.09.004
  • T.M. Atanackovic and D. Zorica, On the Bagley–Torvik equation, ASME J. Appl. Mech. 80 (2013), pp. 369–384. doi: 10.1115/1.4007850
  • Y. Çenesiz, Y. Keskin, and A. Kurnaz, The solution of the Bagley–Torvik equation with the generalized Taylor collocation method, J. Franklin Inst. 347 (2010), pp. 452–466. doi: 10.1016/j.jfranklin.2009.10.007
  • V. Daftardar-Gejji and H. Jafari, Adomian decomposition: A tool for solving a system of fractional differential equations, J. Math. Anal. Appl. 301 (2005), pp. 508–518. doi: 10.1016/j.jmaa.2004.07.039
  • S. Das, Functional Fractional Calculus for System Identification and Controls, Springer-Verlag, New York, 2008.
  • K. Diethelm and N.J. Ford, Numerical solution of the Bagley–Torvik equation, BIT Numer. Math. 42 (2002), pp. 490–507.
  • E. Goursat, A course in mathematical analysis, Vol III Part 2: Integral Equations, Calculus of Variations, Dover Publications Inc, New York, 1964.
  • M. Gülsu, Y. Öztürk, and A. Anapali, Numerical solution of the fractional Bagley–Torvik equation arising in fluid mechanics, Int. J. Comput. Math. (2015), doi:10.1080/00207160.2015.1099633.
  • Q.A. Huang, X.C. Zhong, and B.L. Guo, Approximate solution of Bagley–Torvik equations with variable coefficients and three-point boundary-value conditions, Int. J. Appl. Comput. Math., doi:10.1007/s40819-015-0063-5.
  • 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 (2011), pp. 1038–1045. doi: 10.1016/j.camwa.2011.04.024
  • S. Lang, Real and Functional Analysis, 3rd ed., Springer-Verlag, New York, 1993.
  • X.-F. Li, Approximate solution of linear ordinary differential equations with variable coefficients, Math. Comput. Simul. 75 (2007), pp. 113–125. doi: 10.1016/j.matcom.2006.09.006
  • J.K. Lu and S.G. Zhong, On the Theory of Integral Equations (in Chinese), Wuhan University Press, Wuhan, 2008.
  • S. Momani and Z. Odibat, Numerical comparison of methods for solving linear differential equations of fractional order, Chaos, Solitons Fract. 31 (2007), pp. 1248–1255. doi: 10.1016/j.chaos.2005.10.068
  • K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
  • M.D. Ortigueira, Fractional Calculus for Scientists and Engineers, Springer-Verlag, New York, 2011.
  • H.M. Ozaktas and Z. Zalevsky, The Fractional Fourier Transform, John Wiley and Sons Inc, New York, 2001.
  • I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1993.
  • S.S. Ray, On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley–Torvik equation, Appl. Math. Comput. 218 (2012), pp. 5239–5248.
  • A. Setia, Y. Liu, and A.S. Vatsala, The solution of the Bagley–Torvik equation by using second kind Chebyshev wavelet, Proceedings of the 2014 11th International Conference on Information Technology – New Generations (ITNG '14), pp. 443–446.
  • S. Staněk, Two-point boundary value problems for the generalized Bagley–Torvik fractional differential equation, Central Eur. J. Math. 11 (2013), pp. 574–593.
  • M.A. Teodor and S. Pilipovic, Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes, John Wiley and Sons Inc, New York, 2014.
  • P.J. Torvik and R.L. Bagley, On the appearance of the fractional derivative in the behavior of real materials, ASME J. Appl. Mech. 51 (1984), pp. 294–298. doi: 10.1115/1.3167615
  • V.V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Higher Education Press, Beijing, 2013.
  • Z.H. Wang and X. Wang, General solution of the Bagley–Torvik equation with fractional-order derivative, Commun. Nonlinear Sci. Numer. Simul. 15 (2010), pp. 1279–1285. doi: 10.1016/j.cnsns.2009.05.069
  • S. Yüzbas, Numerical solution of the Bagley–Torvik equation by the Bessel collocation method, Math. Methods Appl. Sci. 36 (2013), pp. 300–312. doi: 10.1002/mma.2588
  • X.C. Zhong and Q.A. Huang, Approximate solution of three-point boundary value problems for second order ordinary differential equations with variable coefficients, Appl. Math. Comput. 247 (2014), pp. 18–29.

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