Skip to main content
SpringerLink
Log in

On a Generalized Bagley–Torvik Equation with a Fractional Integral Boundary Condition

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

The Bagley–Torvik equation is generalized by using variable coefficients and the fractional order \(0<\alpha <2.\) A fractional integral boundary condition is proposed to investigate the nonlocal behavior of the generalized Bagely–Torvik equation. Based on the concept of Riemann–Liouville fractional derivative, the Fredholm integral equations of the second kind are derived for \(0<\alpha <1\) and \(1\le \alpha <2\), respectively. Moreover, the generalized piecewise Taylor-series expansion method is proposed to find the approximate solution, and its convergence and error estimate are analyzed. Numerical results are reported to illustrate the effects of the boundary-value conditions on the approximate solutions. The obtained results reveal that the boundary value conditions play an important role to appropriately model a practical process.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Ionkin, N.I., Moiceev, E.I.: Solution of boundary value problem in heat conduction theory with nonlocal boundary conditions. Differ. Equ. 13, 294–304 (1977)

    Google Scholar 

  2. Maccamy, R.C.: Nonlocal boundary conditions. In: Herdman, T.L., Stech, H.W., Rankin, S.M. (eds.) Integral and Functional Differential Equations. Springer, New York (1981)

  3. Amosov, A.A.: A positive solution of an elliptic equation with nonlinear integral boundary condition of the radiation type. Math. Notes 22, 555–561 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  4. Jankowski, T.: Differential equations with integral boundary conditions. J. Comput. Appl. Math. 147, 1–8 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Pečiulytė, S., Štikonas, A.: Sturm–Liouville problem for stationary differential operator with nonlocal integral boundary condition. Nonlinear Math. Modell. Anal. 10, 377–392 (2005)

    MATH  MathSciNet  Google Scholar 

  6. Wang, Y., Liu, G., Hu, Y.: Existence and uniqueness of solutions for a second order differential equation with integral boundary conditions. Appl. Math. Comput. 216, 2718–2727 (2010)

    MATH  MathSciNet  Google Scholar 

  7. Xu, Y.F., He, Z.M.: Existence of solutions for nonlinear high-order fractional boundary value problem with integral boundary condition. J. Appl. Math. Comput. 44, 417–435 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  8. Podlubny, I.: Fractional Differential Equations. Academic, New York (1993)

    MATH  Google Scholar 

  9. Das, S.: Functional Fractional Calculus for System Identification and Controls. Springer, New York (2008)

    MATH  Google Scholar 

  10. Teodor, M.A., Pilipovic, S.: Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes. Wiley, New York (2014)

    MATH  Google Scholar 

  11. Uchaikin, V.V.: Fractional Derivatives for Physicists and Engineers. Higher Education Press, Beijing (2013)

    Book  MATH  Google Scholar 

  12. Feng, H., Zhai, C.: Existence and uniqueness of positive solutions for a class of fractional differential equation with integral boundary conditions. Nonlinear Anal. Modell. Control 22, 160–172 (2017)

    MathSciNet  Google Scholar 

  13. Ahmad, B., Ntouyas, S.K., Alsaedi, A.: Existence of solutions for fractional differential equations with nonlocal and average type integral boundary conditions. J. Appl. Math. Comput. 53, 129–145 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  14. Torvik, P.J., Bagley, R.L.: On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51, 294–298 (1984)

    Article  MATH  Google Scholar 

  15. Dai, H.Z., Zheng, Z.B., Wang, W.: On generalized fractional vibration equation. Chaos Solitons Fract. 95, 48–51 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  16. Daftardar-Gejji, V., Jafari, H.: Adomian decomposition: a tool for solving a system of fractional differential equations. J. Math. Anal. Appl. 301, 508–518 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Wang, Z.H., Wang, X.: General solution of the Bagley–Torvik equation with fractional-order derivative. Comm. Nonlinear Sci. Numer. Simul. 15, 1279–1285 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Diethelm, K., Ford, N.J.: Numerical solution of the Bagley–Torvik equation. BIT Numer. Math. 42, 490–507 (2002)

    MATH  MathSciNet  Google Scholar 

  19. Momani, S., Odibat, Z.: Numerical comparison of methods for solving linear differential equations of fractional order. Chaos Solitons Fract. 31, 124–1255 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Arikoglu, A., Ozkol, I.: Solution of fractional differential equations by using differential transform method. Chaos Solitons Fract. 34, 1473–1481 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Atanackovic, T.M., Zorica, D.: On the Bagley–Torvik equation. J. Appl. Mech. 80, 369–384 (2013)

    Article  Google Scholar 

  22. Çenesiz, Y., Keskin, Yü, Kurnaz, A.: The solution of the Bagley–Torvik equation with the generalized Taylor collocation method. J. Frank. Inst. 347, 452–466 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  23. Jafari, H., Yousefi, S.A., Firoozjaee, M.A., Momani, S., Khalique, C.M.: Application of Legendre wavelets for solving fractional differential equations. Comput. Math. Appl. 62, 1038–1045 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  24. Ray, S.S.: On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley Torvik equation. Appl. Math. Comput. 218, 5239–5248 (2012)

    MATH  MathSciNet  Google Scholar 

  25. Setia, A., Liu, Y., Vatsala, A.S.: The solution of the Bagley–Torvik equation by using second kind Chebyshev wavelet. In: Conference on Information Technology New Generations, pp 443–446 (2014)

  26. Gülsu, M., Öztürk, Y., Anapali, A.: Numerical solution of the fractional Bagley–Torvik equation arising in fluid mechanics. Int. J. Comput. Math. 94, 173–184 (2017)

    Article  MATH  Google Scholar 

  27. Wei, H.M., Zhong, X.C., Huang, Q.A.: Uniqueness and approximation of solution for fractional Bagley–Torvik equations with variable coefficients. Int. J. Comput. Math. 94, 1542–1561 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  28. Zahra, W.K., Daele, M.V.: Discrete spline methods for solving two point fractional Bagley–Torvik equation. Appl. Math. Comput. 296, 42–56 (2017)

    MathSciNet  Google Scholar 

  29. Al-Mdallal, Q.M., Syam, M.I., Anwar, M.N.: A collocation-shooting method for solving fractional boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 15, 3814–3822 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. Yuzbas, S.: Numerical solution of the Bagley–Torvik equation by the Bessel collocation method. Math. Methods Appl. Sci. 36, 300–312 (2013)

    Article  MathSciNet  Google Scholar 

  31. Stanek, S.: Two-point boundary value problems for the generalized Bagley–Torvik fractional differential equation. Central Eur. J. Math. 11, 574–593 (2013)

    MATH  MathSciNet  Google Scholar 

  32. Huang, Q.A., Zhong, X.C., Guo, B.L.: Approximate solution of Bagley–Torvik equations with variable coefficients and three-point boundary-value conditions. Int. J. Appl. Comput. Math. 2, 327–347 (2016)

    Article  MathSciNet  Google Scholar 

  33. Karaaslan, M.F., Celiker, F., Kurulay, M.: Approximate solution of the Bagley–Torvik equation by hybridizable discontinuous Galerkin methods. Appl. Math. Comput. 285, 51–58 (2016)

    MathSciNet  Google Scholar 

  34. Krishnasamy, V.S., Razzaghi, M.: The numerical solution of the Bagley–Torvik equation with fractional Taylor method. J. Comput. Nonlinear Dyn. 11, 051010 (2016)

    Article  Google Scholar 

  35. Kress, R.: Linear Integral Equations. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  36. Kythe, P.K., Puri, P.: Computational Methods for Linear Integral Equations. Birkhauser, Boston (2002)

    Book  MATH  Google Scholar 

  37. Wazwaz, A.: Linear and Nonlinear Integral Equations: Methods and Applications. High Education Press, Beijing (2011)

    Book  MATH  Google Scholar 

  38. Ren, Y., Zhang, B., Qiao, H.: A simple Taylor-series expansion method for a class of second kind integral equations. J. Comput. Appl. Math. 110, 15–24 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  39. Li, X.F.: Approximate solution of linear ordinary differential equations with variable coefficients. Math. Comput. Simu. 75, 113–125 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  40. Huabsomboona, P., Novaprateep, B., Kaneko, H.: On Taylor-series expansion methods for the second kind integral equations. J. Comput. Appl. Math. 234, 1466–1472 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  41. Zhong, X.C., Huang, Q.A.: Approximate solution of three-point boundary value problems for second order ordinary differential equations with variable coefficients. Appl. Math. Comput. 247, 18–29 (2014)

    MATH  MathSciNet  Google Scholar 

  42. Lang, S.: Real and Functional Analysis. Springer, New York (1993)

    Book  MATH  Google Scholar 

  43. Householder, A.S.: The Theory of Matrices in Numerical Analysis. Blaisdell, New York (1964)

    MATH  Google Scholar 

Download references

Acknowledgements

The work was supported by the National Natural Science Foundation of China (No. 11362002), the Guangxi Natural Science Foundation (No. 2016GXNSFAA380261), the Innovation Project of Guangxi Graduate Education (No. YCSW2017048), 2017 Guangxi high school innovation team and outstanding scholars plan, and the Project of outstanding young teachers’ training in higher education institutions of Guangxi.

Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Guangxi University, Nanning, 530004, Guangxi, People’s Republic of China

    Xian-Ci Zhong, Xue-Ling Liu & Shan-Li Liao

Authors
  1. Xian-Ci Zhong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xue-Ling Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shan-Li Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xian-Ci Zhong.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhong, XC., Liu, XL. & Liao, SL. On a Generalized Bagley–Torvik Equation with a Fractional Integral Boundary Condition. Int. J. Appl. Comput. Math 3 (Suppl 1), 727–746 (2017). https://doi.org/10.1007/s40819-017-0379-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40819-017-0379-4

Keywords

Search

Navigation