0
Research Papers

New Operational Matrix for Solving Multiterm Variable Order Fractional Differential Equations

[+] Author and Article Information
A. M. Nagy

Department of Mathematics,
Faculty of Science,
Benha University,
Benha 13518, Egypt
e-mail: abdelhameed_nagy@yahoo.com

N. H. Sweilam

Department of Mathematics,
Faculty of Science,
Cairo University,
Giza 12613, Egypt
e-mail: nsweilam@sci.cu.edu.eg

Adel A. El-Sayed

Department of Mathematics,
Faculty of Science,
Fayoum University,
Fayoum 63514, Egypt
e-mail: aaa18@fayoum.edu.eg

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 22, 2016; final manuscript received September 7, 2017; published online October 9, 2017. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 13(1), 011001 (Oct 09, 2017) (7 pages) Paper No: CND-16-1450; doi: 10.1115/1.4037922 History: Received September 22, 2016; Revised September 07, 2017

The multiterm fractional variable-order differential equation has a massive application in physics and engineering problems. Therefore, a numerical method is presented to solve a class of variable order fractional differential equations (FDEs) based on an operational matrix of shifted Chebyshev polynomials of the fourth kind. Utilizing the constructed operational matrix, the fundamental problem is reduced to an algebraic system of equations which can be solved numerically. The error estimate of the proposed method is studied. Finally, the accuracy, applicability, and validity of the suggested method are illustrated through several examples.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Baleanu, D. , Golmankhaneh, A. K. , Golmankhaneh, A. K. , and Nigmatullin, R. R. , 2010, “ Newtonian Law With Memory,” Nonlinear Dyn., 60(1–2), pp. 81–86. [CrossRef]
Engheta, N. , 1996, “ On Fractional Calculus and Fractional Multipoles in Electromagnetism,” IEEE Trans. Antennas Propag., 44(4), pp. 554–566. [CrossRef]
Golmankhaneh, A. K. , Golmankhaneh, A. K. , and Baleanu, D. , 2011, “ On Nonlinear Fractional Klein-Gordon Equation,” Signal Process., 91(3), pp. 446–451. [CrossRef]
Magin, R. L. , 2010, “ Fractional Calculus Models of Complex Dynamics in Biological Tissues,” Comput. Math. Appl., 59(5), pp. 1586–1593. [CrossRef]
Nagy, A. M. , and Sweilam, N. H. , 2014, “ An Efficient Method for Solving Fractional Hodgkin-Huxley Model,” Phys. Lett. A, 378(30–31), pp. 1980–1984. [CrossRef]
Sweilam, N. H. , Nagy, A. M. , and El-Sayed, A. A. , 2015, “ Second Kind Shifted Chebyshev Polynomials for Solving Space Fractional Order Diffusion Equation,” Chaos, Solitons Fractals, 73, pp. 141–147. [CrossRef]
Sweilam, N. H. , Nagy, A. M. , and El-Sayed, A. A. , 2016, “ On the Numerical Solution of Space Fractional Order Diffusion Equation Via Shifted Chebyshev Polynomials of the Third Kind,” J. King Saud Univ. Sci., 28(1), pp. 41–47. [CrossRef]
Samko, S. G. , and Ross, B. , 1993, “ Integration and Differentiation to a Variable Fractional Order,” Integr. Transfer Spec. Funct., 1(4), pp. 277–300. [CrossRef]
Bhrawy, A. H. , and Zaky, M. A. , 2016, “ Numerical Algorithm for the Variable-Order Caputo Fractional Functional Differential Equation,” Nonlinear Dyn., 85(3), pp. 1815–1823. [CrossRef]
Coimbra, C. F. M. , 2003, “ Mechanics With Variable-Order Differential Operators,” Ann. Phys., 12(11), pp. 692–703. [CrossRef]
Soon, C. M. , Coimbra, C. F. M. , and Kobayashi, M. H. , 2005, “ The Variable Viscoelasticity Oscillator,” Ann. Phys., 14(6), pp. 378–389. [CrossRef]
Bhrawy, A. H. , and Alshomrani, M. , 2012, “ A Shifted Legendre Spectral Method for Fractional-Order Multi-Point Boundary Value Problems,” Adv. Differ. Equations, 8, pp. 1–19.
Sweilam, N. H. , Nagy, A. M. , and El-Sayed, A. A. , 2016, “ Solving Time-Fractional Order Telegraph Equation Via Sinc-Legendre Collocation Method,” Mediterr. J. Math., 13(6), pp. 5119–5133. [CrossRef]
Babolian, E. , and Hosseini, M. M. , 2002, “ A Modified Spectral Method for Numerical Solution of Ordinary Differential Equations With Non-Analytic Solution,” Appl. Math. Comput., 132(2–3), pp. 341–351.
El-Mesiry, A. , El-Sayed, A. , and El-Saka, H. , 2005, “ Numerical Methods for Multi-Term Fractional (Arbitrary) Orders Differential Equations,” Appl. Math. Comput., 160(3), pp. 683–699.
Esmaeili, S. , and Shamsi, M. , 2011, “ A Pseudo-Spectral Scheme for the Approximate Solution of a Family of Fractional Differential Equations,” Commun. Nonlinear Sci. Numer. Simul., 16(9), pp. 3646–3654. [CrossRef]
Shiralashetti, S. C. , and Deshi, A. B. , 2016, “ An Efficient Haar Wavelet Collocation Method for the Numerical Solution of Multi-Term Fractional Differential Equations,” Nonlinear Dyn., 83(1–2), pp. 293–303. [CrossRef]
Ford, N. J. , and Connolly, J. A. , 2009, “ Systems-Based Decomposition Schemes for the Approximate Solution of Multi-Term Fractional Differential Equations,” Comput. Appl. Math., 229(2), pp. 382–391. [CrossRef]
Chen, Y. M. , Wei, Y. Q. , Liu, D. Y. , and Yu, H. , 2015, “ Numerical Solution for a Class of Nonlinear Variable Order Fractional Differential Equations With Legendre Wavelets,” Appl. Math. Lett., 46, pp. 83–88. [CrossRef]
Doha, E. H. , Bhrawy, A. H. , and Ezz-Eldien, S. S. , 2011, “ Efficient Chebyshev Spectral Methods for Solving Multi-Term Fractional Orders Differential Equations,” Appl. Math. Model., 35(12), pp. 5662–5672. [CrossRef]
Doha, E. H. , Bhrawy, A. H. , and Ezz-Eldien, S. S. , 2011, “ A Chebyshev Spectral Method Based on Operational Matrix for Initial and Boundary Value Problems of Fractional Order,” Comput. Math. Appl., 62(5), pp. 2364–2373. [CrossRef]
Bhrawy, A. H. , Taha, T. M. , and Machado, J. A. T. , 2015, “ A Review of Operational Matrices and Spectral Techniques for Fractional Calculus,” Nonlinear Dyn., 81(3), pp. 1023–1052. [CrossRef]
Ghoreishi, F. , and Yazdani, S. , 2011, “ An Extension of the Spectral TAU Method for Numerical Solution of Multi-Order Fractional Differential Equations With Convergence Analysis,” Comput. Math. Appl., 61(1), pp. 30–43. [CrossRef]
Vanani, S. K. , and Aminataei, A. , 2011, “ Tau Approximate Solution of Fractional Partial Differential Equations,” Comput. Math. Appl., 62(3), pp. 1075–1083. [CrossRef]
Youssri, Y. H. , and Abd-Elhameed, W. M. , 2016, “ Spectral Solutions for Multi-Term Fractional Initial Value Problems Using a New Fibonacci Operational Matrix of Fractional Integration,” Prog. Fract. Differ. Appl., 2(2), pp. 141–151. [CrossRef]
Zhou, F. Y. , and Xu, X. , 2016, “ The Third Kind Chebyshev Wavelets Collocation Method for Solving the Time-Fractional Convection Diffusion Equations With Variable Coefficients,” Appl. Math. Comput., 280, pp. 11–29.
Keshavarz, E. , Ordokhani, Y. , and Razzaghi, M. , 2014, “ Bernoulli Wavelet Operational Matrix of Fractional Order Integration and Its Applications in Solving the Fractional Order Differential Equations,” Appl. Math. Model., 38(24), pp. 6038–6051. [CrossRef]
Tavares, D. , Almeida, R. , and Torres, D. F. M. , 2016, “ Caputo Derivatives of Fractional Variable Order: Numerical Approximations,” Commun. Nonlinear Sci. Numer. Simul., 35, pp. 69–87. [CrossRef]
Atanackovic, T. M. , Janev, M. , Pilipovic, S. , and Zorica, D. , 2013, “ An Expansion Formula for Fractional Derivatives of Variable Order,” Cent. Eur. J. Phys., 11(10), pp. 1350–1360.
Chen, Y. M. , Liu, L. Q. , Li, B. F. , and Sun, Y. , 2014, “ Numerical Solution for the Variable Order Linear Cable Equation With Bernstein Polynomials,” Appl. Math. Comput., 238, pp. 329–341.
Liu, J. , Li, X. , and Wu, L. , 2016, “ An Operational Matrix of Fractional Differentiation of the Second Kind of Chebyshev Polynomial for Solving Multiterm Variable Order Fractional Differential Equation,” Math. Probl. Eng., 2016, pp. 1–10.
Maleknejad, K. , Nouri, K. , and Torkzadeh, L. , 2016, “ Operational Matrix of Fractional Integration Based on the Shifted Second Kind Chebyshev Polynomials for Solving Fractional Differential Equations,” Mediterr. J. Math., 13(3), pp. 1377–1390. [CrossRef]
Wang, L. F. , Ma, Y. P. , and Yang, Y. Q. , 2014, “ Legendre Polynomials Method for Solving a Class of Variable Order Fractional Differential Equation,” Comput. Model. Eng. Sci., 101(2), pp. 97–111.
Shen, S. , Liu, F. , Chen, J. , Turner, I. , and Anh, V. , 2012, “ Numerical Techniques for the Variable Order Time Fractional Diffusion Equation,” Ann. Phys., 218(22), pp. 10861–10870.
Sweilam, N. H. , and AL-Mrawm, H. M. , 2011, “ On the Numerical Solutions of the Variable Order Fractional Heat Equation,” Stud. Nonlinear Sci., 2(1), pp. 31–36.
Mason, J. C. , and Handscomb, D. C. , 2003, Chebyshev Polynomials, Chapman and Hall, New York. [PubMed] [PubMed]
Sweilam, N. H. , Nagy, A. M. , and El-Sayed, A. A. , 2016, “ Numerical Approach for Solving Space Fractional Order Diffusion Equations Using Shifted Chebyshev Polynomials of the Fourth Kind,” Turk. J. Math., 40, pp. 1283–1297. [CrossRef]
Veselić, K. , 2011, Damped Oscillations of Linear Systems—A Mathematical Introduction, Springer, Berlin. [CrossRef]

Figures

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In