Numerical Solution of the Bagley-Torvik Equation Using the Integer-Order Derivatives Expansion
DOI:
https://doi.org/10.25271/2018.6.2.437Keywords:
Bagley-Torvik Equation, Riemann–Liouville Fractional Integral and Derivative, Numerical Solution, Infinite Series of Integer-Order Derivative ExpansionAbstract
Numerical solution of the well-known Bagley-Torvik equation is considered. The fractional-order derivative in the equation is converted, approximately, to ordinary-order derivatives up to second order. Approximated Bagley-Torvik equation is obtained using finite number of terms from the infinite series of integer-order derivatives expansion for the Riemann–Liouville fractional derivative. The Bagley-Torvik equation is a second-order differential equation with constant coefficients. The derived equation, by considering only the first three terms from the infinite series to become a second-order ordinary differential equation with variable coefficients, is numerically solved after it is transformed into a system of first-order ordinary differential equations. The approximation of fractional-order derivative and the order of the truncated error are illustrated through some examples. Comparison between our result and exact analytical solution are made by considering an example with known analytical solution to show the preciseness of our proposed approach.
References
Atanacković, T. M., Janev, M., Konjik, S., Pilipović, S., & Zorica, D. (2014). Expansion formula for fractional derivatives in variational problems. Journal of Mathematical Analysis and Applications, 409(2), 911-924.
Atanackovic, T., & Stankovic, B. (2004). An expansion formula for fractional derivatives and its application. Fractional Calculus and Applied Analysis, 7(3), 365-378.
Barkai E., Metzler R., & Klafter J. (2000). From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E, 61, 132-138.
Battaglia J.L., (2002). A modal approach to solve inverse heat conduction problems, Inverse Probl. Eng., 10 41-63.
Benson D.A., Wheatcraft S.W., & Meerschaert M.M., (2000). Application of a fractional advection dispersion equation, Water Resources Res. 36, 1403-1412.
Bhrawy A.H., Doha E.H., Baleanu D., Ezz-Eldien S.S., & Abdelkawy M.A., (2015). An accurate numerical technique for solving fractional optimal control problems, Proc. Romanian Acad. A, 16, 47-54.
Chatterjee A. (2005). Statistical origins of fractional derivatives in viscoelasticity, J. Sound Vibr. 284, 1239-1245.
Diethelm K., & Ford N., (2002). Numerical Solution of the Bagley-Torvik Equation. BIT Numerical Mathematics 42(3), 490-507.
Diethelm, K., & Freed, A. D., (1999). On the solution of nonlinear fractional differential equations used in the modeling of viscoplasticity. Kiel, F., Mackens, W., Voí¿, H., Werther, J.(Eds.), Scientific Computing in Chemical Engineering II-Computional Fluid Dynamics, Reaction Enineering, and Molecular Properties, Springer, Heidlberg, 217-224.
Gladkina, A., Shchedrin, G., Khawaja, U. A., & Carr, L. D. (2017). Expansion of fractional derivatives in terms of an integer derivative series: physical and numerical applications. arXiv preprint, arXiv:1710.06297.
Henry B., & Wearne S. (2000). Fractional reaction-diffusion, Phys. A, 276, pp. 448-455.
Hilfer R., (2000). Fractional calculus and regular variation in thermodynamics, Applications of fractional calculus in physics, World Sci. Publ., River Edge, NJ.
Ionescu C.M., Hodrea R., & Keyser R.D., (2011). Variable time-delay estimation for anesthesia control during intensive care, IEEE T. Bio-Med. Eng. 58, 363-369.
Kilbas A. A., Srivastava H. M. & Trujillo J. J., (2006). Theory and Applications of Fractional Differential Equations, Elsevier.
Leszczyński, J., & Ciesielski, M. (2001). A numerical method for solution of ordinary differential equations of fractional order. In International Conference on Parallel Processing and Applied Mathematics, 695-702. Springer, Berlin, Heidelberg.
Luchko Y. & Gorenflo R., (1998). The initial value problem for some fractional differential equations with the Caputo derivatives, Preprint Serie A 08-98, Fachbereich Mathematik und Informatik, Freie Universit¨at Berlin, Berlin.
Metzler R., & Klafter J. (2000). The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339, 1-77.
Mittag-Leffler, G. (1903). Sur la nouvelle function, CR Acad. Sic. Paris.(Ser. II), 137, 554-558.
Nouri, K., Elahi-Mehr, S., & Torkzadeh, L., (2016). Investigation of the behavior of the fractional Bagley-Torvik and Basset equations via numerical inverse Laplace transform. Rom. Rep. Phys, 68, 503-514.
Podlubny I., (1999). Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 198.
Ray, S. S., & Bera, R. K. (2005). Analytical solution of the Bagley-Torvik equation by Adomian decomposition method. Applied Mathematics and Computation, 168(1), 398-410.
Rousan A.A., Malkawi E., Rabei E. M., & Widyan H. (2002). Applications of fractional calculus to gravity, Fract. Calc. Appl. Anal. 5 155-168.
Samko S. G., Kilbas A. A. & Marichev O. I., (1993). Fractional Integrals and Derivatives:Theory and Applications, Gordon and Breach, Yverdon.
Torvik P. J., & Bagley R. L. (1983). Fractional calculus - a different approach to the analysis of viscoelastically damped structures, AIAA Journal, 21(5), 741-748.
Torvik P. J., & Bagley R. L. (1984). On the appearance of the fractional derivative in the behavior of real materials, Journal of Applied Mechanics, Transactions of ASME, 51(2), 294-298.
Atanackovic, T., & Stankovic, B. (2004). An expansion formula for fractional derivatives and its application. Fractional Calculus and Applied Analysis, 7(3), 365-378.
Barkai E., Metzler R., & Klafter J. (2000). From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E, 61, 132-138.
Battaglia J.L., (2002). A modal approach to solve inverse heat conduction problems, Inverse Probl. Eng., 10 41-63.
Benson D.A., Wheatcraft S.W., & Meerschaert M.M., (2000). Application of a fractional advection dispersion equation, Water Resources Res. 36, 1403-1412.
Bhrawy A.H., Doha E.H., Baleanu D., Ezz-Eldien S.S., & Abdelkawy M.A., (2015). An accurate numerical technique for solving fractional optimal control problems, Proc. Romanian Acad. A, 16, 47-54.
Chatterjee A. (2005). Statistical origins of fractional derivatives in viscoelasticity, J. Sound Vibr. 284, 1239-1245.
Diethelm K., & Ford N., (2002). Numerical Solution of the Bagley-Torvik Equation. BIT Numerical Mathematics 42(3), 490-507.
Diethelm, K., & Freed, A. D., (1999). On the solution of nonlinear fractional differential equations used in the modeling of viscoplasticity. Kiel, F., Mackens, W., Voí¿, H., Werther, J.(Eds.), Scientific Computing in Chemical Engineering II-Computional Fluid Dynamics, Reaction Enineering, and Molecular Properties, Springer, Heidlberg, 217-224.
Gladkina, A., Shchedrin, G., Khawaja, U. A., & Carr, L. D. (2017). Expansion of fractional derivatives in terms of an integer derivative series: physical and numerical applications. arXiv preprint, arXiv:1710.06297.
Henry B., & Wearne S. (2000). Fractional reaction-diffusion, Phys. A, 276, pp. 448-455.
Hilfer R., (2000). Fractional calculus and regular variation in thermodynamics, Applications of fractional calculus in physics, World Sci. Publ., River Edge, NJ.
Ionescu C.M., Hodrea R., & Keyser R.D., (2011). Variable time-delay estimation for anesthesia control during intensive care, IEEE T. Bio-Med. Eng. 58, 363-369.
Kilbas A. A., Srivastava H. M. & Trujillo J. J., (2006). Theory and Applications of Fractional Differential Equations, Elsevier.
Leszczyński, J., & Ciesielski, M. (2001). A numerical method for solution of ordinary differential equations of fractional order. In International Conference on Parallel Processing and Applied Mathematics, 695-702. Springer, Berlin, Heidelberg.
Luchko Y. & Gorenflo R., (1998). The initial value problem for some fractional differential equations with the Caputo derivatives, Preprint Serie A 08-98, Fachbereich Mathematik und Informatik, Freie Universit¨at Berlin, Berlin.
Metzler R., & Klafter J. (2000). The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339, 1-77.
Mittag-Leffler, G. (1903). Sur la nouvelle function, CR Acad. Sic. Paris.(Ser. II), 137, 554-558.
Nouri, K., Elahi-Mehr, S., & Torkzadeh, L., (2016). Investigation of the behavior of the fractional Bagley-Torvik and Basset equations via numerical inverse Laplace transform. Rom. Rep. Phys, 68, 503-514.
Podlubny I., (1999). Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 198.
Ray, S. S., & Bera, R. K. (2005). Analytical solution of the Bagley-Torvik equation by Adomian decomposition method. Applied Mathematics and Computation, 168(1), 398-410.
Rousan A.A., Malkawi E., Rabei E. M., & Widyan H. (2002). Applications of fractional calculus to gravity, Fract. Calc. Appl. Anal. 5 155-168.
Samko S. G., Kilbas A. A. & Marichev O. I., (1993). Fractional Integrals and Derivatives:Theory and Applications, Gordon and Breach, Yverdon.
Torvik P. J., & Bagley R. L. (1983). Fractional calculus - a different approach to the analysis of viscoelastically damped structures, AIAA Journal, 21(5), 741-748.
Torvik P. J., & Bagley R. L. (1984). On the appearance of the fractional derivative in the behavior of real materials, Journal of Applied Mechanics, Transactions of ASME, 51(2), 294-298.
Downloads
Published
2018-06-30
How to Cite
Hasan, A. S. (2018). Numerical Solution of the Bagley-Torvik Equation Using the Integer-Order Derivatives Expansion. Science Journal of University of Zakho, 6(2), 64–69. https://doi.org/10.25271/2018.6.2.437
Issue
Section
Science Journal of University of Zakho
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License [CC BY-NC-SA 4.0] that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work, with an acknowledgment of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online.
