Technique for Solving the Bagley-Torvik Equation via Integer-Order Differential Equations
DOI:
https://doi.org/10.29304/jqcm.2021.13.3.960Keywords:
Caputo derivative, Fractional differential equations, Bagley-Torvik equationAbstract
This paper proposes an analytical technique for solving the Bagley-Torvik equation (BTE) in the Caputo sense. The main idea of this technique is based on reformulating the considered problem as a system of linear FDEs of half-order. Then the resulting system is transformed into a set of integer-order differential equations. In such a transformation, the singularity terms are removed from the FDE system. So, the solution of the BTE can be obtained via solving this system. Finally, two examples are given to demonstrate the efficiency of the proposed technique.
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References
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[2] S. L. Khalaf, A. R. Khudair, Particular Solution of Linear sequential fractional differential equation with constant coefficients by inverse fractional differential operators, Differ Equ Dyn Syst, 25(2017)373–383.
[3] R. Alchikh and S.A. Khuri. Numerical solution of a fractional differential equation arising in optics. Optik, 208 (2020) 163911–163919.
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[6] Zahra Salman and Bushra Taha. Cubic B-splines Method for Solving Singularly Perturbed Delay Partial Differential Equations. Journal of Al-Qadisiyah for computer science and mathematics, 13 (2021) 1–12.
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[9] S. L. Khalaf , K. K. Kassid, Approximate solution of quadratic time varying optimal control problems via differential transform method, Bas. J. Sci., 38(2020)26-47.
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[14] Mohamed El-Gamel and Mahmoud Abd El-Hady. Numerical solution of the Bagley-Torvik equation by Legendre-collocation method. SeMA Journal, 74 (2016) 371–383.
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[25] Markus Oeser and Terhi Pellinien. Computational framework for common visco-elastic models in engineering based on the theory of rheology. Computers and Geotechnics, 42 (2012) 145–156.
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[30] A.R. Khudair, S.A.M. Haddad, S.L. khalaf, Restricted fractional differential transform for solving irrational order fractional differential equations, Cha., Sol. & Frac.,101(2017)81-85.
[31] Abdul Rauf and Yasir Mahsud and Imran Siddique. Multi-layer flows of immiscible fractional Maxwell fluids in a cylindrical domain. Chinese Journal of Physics, 67 (2020) 265–282.
[32] S. Saha Ray. On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley Torvik equation. Applied Mathematics and Computation, 218 (2012) 5239–5248.
[33] S. Saha Ray and R.K. Bera. Analytical solution of the Bagley Torvik equation by Adomian decomposition method. Applied Mathematics and Computation, 168 (2005) 398–410.
[34] Mohammed Rashid and Ahmed Rashid. An Iterative Method to Solve Nonlinear Equation. Journal of Al-Qadisiyah for Computer Science and Mathematics, 13 (2021) 87–95.
[35] H. M. Srivastava and Rajarama Mohan Jena and Snehashish Chakraverty and Subrat Kumar Jena. Dynamic Response Analysis of Fractionally-Damped Generalized Bagley–Torvik Equation Subject to External Loads. Russian Journal of Mathematical Physics, 27 (2020) 254–268.
[36] H. M. Srivastava and F. A. Shah and R. Abass. An Application of the Gegenbauer Wavelet Method for the Numerical Solution of the Fractional Bagley-Torvik Equation. Russian Journal of Mathematical Physics, 26 (2019) 77–93.
[37] P. J. Torvik and R. L. Bagley. On the Appearance of the Fractional Derivative in the Behavior of Real Materials. Journal of Applied Mechanics, 51 (1984) 294–298.
[38] Petra Wittbold and Patryk Wolejko and Rico Zacher. Bounded weak solutions of time-fractional porous medium type and more general nonlinear and degenerate evolutionary integro-differential equations. Journal of Mathematical Analysis and Applications, 499 (2021) 125007–125027.
[39] Mohammed Rashid and Ahmed Rashid. Adomian Decomposition Method with different polynomials for nonlinear Klein Gordon equation and a system of nonlinear partial differential equations. Journal of Al-Qadisiyah for Computer Science and Mathematics, 3 (2011) 1-11.
[40] Suayip YAzbas. Numerical solution of the Bagley-Torvik equation by the Bessel collocation method. Mathematical Methods in the Applied Sciences, 36 (2012) 300–312.
[41] M. RASHEED, A. Rashid, A. Rashid, T. Rashid, S. Abed Hamed, and M. Jasim AL-Kinani, “On Solving Nonlinear Equation Via Numerical Analysis for Photovoltaic Cell”, JQCM, vol. 13, no. 3, pp. Math Page 61-, Sep. 2021.
[42] H. Al-Humedi and Z. jameel, “Combining Cubic B-Spline Galerkin Method with Quadratic Weight Function for Solving Partial Integro-Differential Equations”, JQCM, vol. 12, no. 1, pp. Math Page 9 -, Feb. 2020.
[43] S. Mahdi Muosa, N. Ahmed Mohammed Taha, and H. Abbas Dahham, “FindingCorrection Terms Accompaniment of Rule MMS of Triple Integrals Partial Derivative Singular or Singular Numerically”, JQCM, vol. 8, no. 2, pp. 1-15, Aug. 2017.
[44] M. RASHEED, S. SHIHAB, A. Rashid, T. Rashid, S. Abed Hamed, and F. Hamed Alshebeeb, “Multistep Iterative Algorithms for Solving Nonlinear Equation”, JQCM, vol. 13, no. 2, pp. Math Page 123-, May 2021.
[45] H. J. AL-Nasrawy, A. K. Al-Jubor, and K. Hussain, “On The Higher-Order Pantograph Type Delay Differential Equation Via Orthonormal Bernstein Polynomials”, JQCM, vol. 12, no. 3, pp. Math Page 1-, Sep. 2020.
[2] S. L. Khalaf, A. R. Khudair, Particular Solution of Linear sequential fractional differential equation with constant coefficients by inverse fractional differential operators, Differ Equ Dyn Syst, 25(2017)373–383.
[3] R. Alchikh and S.A. Khuri. Numerical solution of a fractional differential equation arising in optics. Optik, 208 (2020) 163911–163919.
[4] Sanaa L. Khalaf and Zainab A. Lazim. Stability Analysis of Caputo Fractional Model for HIV Infection. Journal of Al-Qadisiyah for computer science and mathematics, 12) 2020( 1–16.
[5] Omar Abu Arqub and Banan Maayah. Solutions of Bagley–Torvik and Painlevé equations of fractional order using iterative reproducing kernel algorithm with error estimates. Neural Computing and Applications, 29 (2016) 1465–1479.
[6] Zahra Salman and Bushra Taha. Cubic B-splines Method for Solving Singularly Perturbed Delay Partial Differential Equations. Journal of Al-Qadisiyah for computer science and mathematics, 13 (2021) 1–12.
[7] Mohammed Rashid and Ahmed Rashid. Numerical Solving of Nonlinear Equation Using Iterative Algorithms. Journal of Al-Qadisiyah for Computer Science and Mathematics,13 (2021) 51–60.
[8] A. R. Khudair, On solving non-homogeneous fractional differential equations of Euler type, Comp. Appl. Math., 32(2013)577–584.
[9] S. L. Khalaf , K. K. Kassid, Approximate solution of quadratic time varying optimal control problems via differential transform method, Bas. J. Sci., 38(2020)26-47.
[10] M. Caputo. Linear Models of Dissipation whose Q is almost Frequency Independent–II. Geophysical Journal International, 13 (1967) 529–539.
[11] Caputo, Michele. The splitting of the free oscillations of the Earth caused by the rheology. Rendiconti Lincei, 1 (1990) 119–125.
[12] YAcel Cenesiz and Yldray Keskin and Aydn Kurnaz. The solution of the Bagley–Torvik equation with the generalized Taylor collocation method. Journal of the Franklin Institute, 347 (2010) 452–466.
[13] Jacky Cresson. Inverse problem of fractional calculus of variations for partial differential equations. Communications in Nonlinear Science and Numerical Simulation, 15 (2010) 987–996.
[14] Mohamed El-Gamel and Mahmoud Abd El-Hady. Numerical solution of the Bagley-Torvik equation by Legendre-collocation method. SeMA Journal, 74 (2016) 371–383.
[15] Shahrokh Esmaeili and M. Shamsi. A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations. Communications in Nonlinear Science and Numerical Simulation, 16 (2011) 3646–3654.
[16] X.Q. He and M. Rafiee and S. Mareishi and K.M. Liew. Large amplitude vibration of fractionally damped viscoelastic CNTs fiber polymer multiscale composite beams. Composite Structures, 131 (2015) 1111–1123.
[17] Bilal Jamil and Muhammad Shoaib Anwar and Amer Rasheed and Muhammad Irfan. MHD Maxwell flow modeled by fractional derivatives with chemical reaction and thermal radiation. Chinese Journal of Physics, 67 (2020) 512–533.
[18] Mehmet Fatih Karaaslan and Fatih Celiker and Muhammet Kurulay. Approximate solution of the Bagley–Torvik equation by hybridizable discontinuous Galerkin methods. Applied Mathematics and Computation, 285 (2016) 51–58.
[19] Kilbas, A. A. Theory and applications of fractional differential equations. Elsevier, Amsterdam Boston, 2006.
[20] Vipin Kumar and Muslim Malik and Amar Debbouche. Stability and controllability analysis of fractional damped differential system with non-instantaneous impulses. Applied Mathematics and Computation, 391 (2021) 125633–125650.
[21] Changpin Li and Weihua Deng. Remarks on fractional derivatives. Applied Mathematics and Computation, 187 (2007) 777–784.
[22] Waleed Kh. Alzubaidi and Shaimaa H. Shaker. Methods of Secure Routing Protocol in Wireless Sensor Networks. Journal of AL-Qadisiyah for computer science and mathematics, 10 (2018) 38–55.
[23] Somayeh Mashayekhi and Mohsen Razzaghi. Numerical solution of the fractional Bagley-Torvik equation by using hybrid functions approximation. Mathematical Methods in the Applied Sciences, 39 (2015) 353–365.
[24] J.A. Mier and R. Sánchez and D.E. Newman. Tracer particle transport dynamics in the diffusive sandpile cellular automaton. Chaos, Solitons & Fractals, 140 (2020) 110117-110126.
[25] Markus Oeser and Terhi Pellinien. Computational framework for common visco-elastic models in engineering based on the theory of rheology. Computers and Geotechnics, 42 (2012) 145–156.
[26] Keith B. Oldham. Fractional differential equations in electrochemistry. Advances in Engineering Software, 41 (2010) 9–12.
[27] Oldham, Keith and Spanier, Jerome. The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier, 1974.
[28] Rutuparna Panda and Madhumita Dash. Fractional generalized splines and signal processing. Signal Processing, 86 (2006) 2340–2350.
[29] Podlubny, Igor. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, 1998.
[30] A.R. Khudair, S.A.M. Haddad, S.L. khalaf, Restricted fractional differential transform for solving irrational order fractional differential equations, Cha., Sol. & Frac.,101(2017)81-85.
[31] Abdul Rauf and Yasir Mahsud and Imran Siddique. Multi-layer flows of immiscible fractional Maxwell fluids in a cylindrical domain. Chinese Journal of Physics, 67 (2020) 265–282.
[32] S. Saha Ray. On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley Torvik equation. Applied Mathematics and Computation, 218 (2012) 5239–5248.
[33] S. Saha Ray and R.K. Bera. Analytical solution of the Bagley Torvik equation by Adomian decomposition method. Applied Mathematics and Computation, 168 (2005) 398–410.
[34] Mohammed Rashid and Ahmed Rashid. An Iterative Method to Solve Nonlinear Equation. Journal of Al-Qadisiyah for Computer Science and Mathematics, 13 (2021) 87–95.
[35] H. M. Srivastava and Rajarama Mohan Jena and Snehashish Chakraverty and Subrat Kumar Jena. Dynamic Response Analysis of Fractionally-Damped Generalized Bagley–Torvik Equation Subject to External Loads. Russian Journal of Mathematical Physics, 27 (2020) 254–268.
[36] H. M. Srivastava and F. A. Shah and R. Abass. An Application of the Gegenbauer Wavelet Method for the Numerical Solution of the Fractional Bagley-Torvik Equation. Russian Journal of Mathematical Physics, 26 (2019) 77–93.
[37] P. J. Torvik and R. L. Bagley. On the Appearance of the Fractional Derivative in the Behavior of Real Materials. Journal of Applied Mechanics, 51 (1984) 294–298.
[38] Petra Wittbold and Patryk Wolejko and Rico Zacher. Bounded weak solutions of time-fractional porous medium type and more general nonlinear and degenerate evolutionary integro-differential equations. Journal of Mathematical Analysis and Applications, 499 (2021) 125007–125027.
[39] Mohammed Rashid and Ahmed Rashid. Adomian Decomposition Method with different polynomials for nonlinear Klein Gordon equation and a system of nonlinear partial differential equations. Journal of Al-Qadisiyah for Computer Science and Mathematics, 3 (2011) 1-11.
[40] Suayip YAzbas. Numerical solution of the Bagley-Torvik equation by the Bessel collocation method. Mathematical Methods in the Applied Sciences, 36 (2012) 300–312.
[41] M. RASHEED, A. Rashid, A. Rashid, T. Rashid, S. Abed Hamed, and M. Jasim AL-Kinani, “On Solving Nonlinear Equation Via Numerical Analysis for Photovoltaic Cell”, JQCM, vol. 13, no. 3, pp. Math Page 61-, Sep. 2021.
[42] H. Al-Humedi and Z. jameel, “Combining Cubic B-Spline Galerkin Method with Quadratic Weight Function for Solving Partial Integro-Differential Equations”, JQCM, vol. 12, no. 1, pp. Math Page 9 -, Feb. 2020.
[43] S. Mahdi Muosa, N. Ahmed Mohammed Taha, and H. Abbas Dahham, “FindingCorrection Terms Accompaniment of Rule MMS of Triple Integrals Partial Derivative Singular or Singular Numerically”, JQCM, vol. 8, no. 2, pp. 1-15, Aug. 2017.
[44] M. RASHEED, S. SHIHAB, A. Rashid, T. Rashid, S. Abed Hamed, and F. Hamed Alshebeeb, “Multistep Iterative Algorithms for Solving Nonlinear Equation”, JQCM, vol. 13, no. 2, pp. Math Page 123-, May 2021.
[45] H. J. AL-Nasrawy, A. K. Al-Jubor, and K. Hussain, “On The Higher-Order Pantograph Type Delay Differential Equation Via Orthonormal Bernstein Polynomials”, JQCM, vol. 12, no. 3, pp. Math Page 1-, Sep. 2020.
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Published
2022-07-02
How to Cite
Abduljaleel, A. F., & Khudair, A. R. (2022). Technique for Solving the Bagley-Torvik Equation via Integer-Order Differential Equations. Journal of Al-Qadisiyah for Computer Science and Mathematics, 13(3), Math Page 107–117. https://doi.org/10.29304/jqcm.2021.13.3.960
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