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Journal of Scientific Computing

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01-01-2020

An Efficient Formulation of Chebyshev Tau Method for Constant Coefficients Systems of Multi-order FDEs

Authors: A. Faghih, P. Mokhtary

Published in: Journal of Scientific Computing | Issue 1/2020

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Abstract

The objective of the present work is to introduce a computational approach employing Chebyshev Tau method for approximating the solutions of constant coefficients systems of multi-order fractional differential equations. For this purpose, a series representation for the exact solutions in a neighborhood of the origin is obtained to monitor their smoothness properties. We prove that some derivatives of the exact solutions of the underlying problem often suffer from discontinuity at the origin. To fix this drawback and design a high order approach a regularization procedure is developed. In addition to avoid high computational costs, a suitable strategy is implemented such that approximate solutions are obtained by solving some triangular algebraic systems. Complexity and convergence analysis of the proposed scheme are provided. Various practical test problems are presented to exhibit capability of the given approach.

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Abdulaziz, O., Hashim, I., Momani, S.: Solving systems of fractional differential equations by homotopy-perturbation method. Phys. Lett. A 372(4), 451–459 (2008)MathSciNetMATHCrossRef

Atabakzadeh, M.H., Akrami, M.H., Erjaee, G.H.: Chebyshev operational matrix method for solving multi-order fractional ordinary differential equations. Appl. Math. Model. 37(20–21), 8903–8911 (2013)MathSciNetMATHCrossRef

Bagley, R.L., Torvik, P.J.: On the fractional calculus model of viscoelastic behavior. J. Rheol. 30(1), 133–155 (1986)MATHCrossRef

Bataineh, A.S., Alomari, A.K., Noorani, M.S.M., Hashim, I., Nazar, R.: Series solutions of systems of nonlinear fractional differential equations. Acta Appl. Math. 105(2), 189–198 (2009)MathSciNetMATHCrossRef

Bhrawy, A., Alhamed, Y., Baleanu, D., Al-Zahrani, A.: New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions. Fract. Calc. Appl. Anal. 17(4), 1137–1157 (2014)MathSciNetMATHCrossRef

Bhrawy, A.H., Zaky, M.A.: Shifted fractional-order Jacobi orthogonal functions: application to a system of fractional differential equations. Appl. Math. Model. 40(2), 832–845 (2016)MathSciNetMATHCrossRef

Biazar, J., Farrokhi, L., Islam, M.R.: Modeling the pollution of a system of lakes. Appl. Math. Comput. 178(2), 423–430 (2006)MathSciNetMATH

Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods. Fundamentals in Single Domains. Springer, Berlin (2006)MATHCrossRef

Cardoso, L.C., Dos Santos, F.L.P., Camargo, R.F.: Analysis of fractional-order models for hepatitis B. Comput. Appl. Math. 37(4), 4570–4586 (2018)MathSciNetMATHCrossRef

10.

Changpin, L., Zeng, F.: Numerical Methods for Fractional Calculus. Chapman and Hall/CRC, Boca Raton (2015) MATH

11.

Chen, W.C.: Nonlinear dynamics and chaos in a fractional-order financial system. Chaos Solitons Fractals 36(5), 1305–1314 (2008)CrossRef

12.

Chen, Y., Ke, X., Wei, Y.: Numerical algorithm to solve system of nonlinear fractional differential equations based on wavelets method and the error analysis. Appl. Math. Comput. 251, 475–488 (2015)MathSciNetMATH

13.

Daftardar-Gejji, V., Jafari, H.: Adomian decomposition: a tool for solving a system of fractional differential equations. J. Math. Anal. Appl. 301(2), 508–518 (2005)MathSciNetMATHCrossRef

14.

Demirci, E., Unal, A., Özalp, N.: A fractional order SEIR model with density dependent death rate. J. Math. Stat. 40(2), 287–295 (2011)MathSciNetMATH

15.

Demirci, E., Ozalp, N.: A method for solving differential equations of fractional order. J. Comput. Appl. Math. 236(11), 2754–2762 (2012)MathSciNetMATHCrossRef

16.

Diethelm, K., Siegmund, S., Tuan, H.T.: Asymptotic behavior of solutions of linear multi-order fractional differential systems. Fract. Calc. Appl. Anal. 20(5), 1165–1195 (2017)MathSciNetMATHCrossRef

17.

Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)MATHCrossRef

18.

Dragomir, S.S.: Some Gronwall Type Inequalities and Applications. Nova Science Publishers, New York (2003)MATH

19.

Ertürk, V.S., Momani, S.: Solving systems of fractional differential equations using differential transform method. J. Comput. Appl. Math. 215(1), 142–151 (2008)MathSciNetMATHCrossRef

20.

Ferrás, L.L., Ford, N.J., Morgado, M.L., Rebelo, M.: A hybrid numerical scheme for fractional-order systems. In: International Conference on Innovation, Engineering and Entrepreneurship, Vol. 505, pp. 735–742. Springer, Cham (2018)

21.

Fitt, A.D., Goodwin, A.R.H., Ronaldson, K.A., Wakeham, W.A.: A fractional differential equation for a MEMS viscometer used in the oil industry. J. Comput. Appl. Math. 229(2), 373–381 (2009)MathSciNetMATHCrossRef

22.

Ghanbari, F., Ghanbari, K., Mokhtary, P.: High-order Legendre collocation method for fractional order linear semi explicit differential algebraic equations. Electron. Trans. Numer. Anal. 48, 387–409 (2018)MathSciNetMATHCrossRef

23.

Ghanbari, F., Ghanbari, K., Mokhtary, P.: Generalized Jacobi Galerkin method for nonlinear fractional differential algebraic equations. Comput. Appl. Math. 37, 5456–5475 (2018)MathSciNetMATHCrossRef

24.

Ghanbari, F., Mokhtary, P., Ghanbari, K.: On the numerical solution of a class of linear fractional integro-differential algebraic equations with weakly singular kernels. Appl. Numer. Math. 144, 1–20 (2019)MathSciNetMATHCrossRef

25.

Ghanbari, F., Mokhtary, P., Ghanbari, K.: Numerical solution of a class of fractional order integro-differential algebraic equations using Müntz–Jacobi Tau method. J. Comput. Appl. Math. 362, 172–184 (2019)MathSciNetMATHCrossRef

26.

Hamri, N.E., Houmor, T.: Chaotic dynamics of the fractional order nonlinear Bloch system. Electron. J. Theor. Phys. 8(25), 233–244 (2011)

27.

Hille, E.: Lectures on Ordinary Differential Equations. Addison-Wesley, Reading (1969)MATH

28.

Kaczorek, T.: Positive linear systems consisting of n subsystems with different fractional orders. IEEE Trans. Circuits Syst. I. Regul. Pap. 58(6), 1203–1210 (2011)MathSciNetCrossRef

29.

Khader, M.M., El Danaf, T.S., Hendy, A.S.: A computational matrix method for solving systems of high order fractional differential equations. Appl. Math. Model. 37(6), 4035–4050 (2013)MathSciNetMATHCrossRef

30.

Khader, M.M., Sweilam, N.H., Mahdy, A.M.S.: Two computational algorithms for the numerical solution for system of fractional differential equations. Arab J. Math. Sci. 21(1), 39–52 (2015)MathSciNetMATH

31.

Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)MATH

32.

Liu, W., Chen, K.: Chaotic behavior in a new fractional-order love triangle system with competition. J. Appl. Anal. Comput. 5(1), 103–113 (2015)MathSciNetMATH

33.

Magin, R., Feng, X., Baleanu, D.: Solving the fractional order Bloch equation. Concepts Magn. Reson. 34(1), 16–23 (2009)CrossRef

34.

Mokhtary, P.: Discrete Galerkin method for fractional integro-differential equations. Acta Math. Sci. Ser. B Engl. Ed. 36(2), 560–578 (2016)MathSciNetMATHCrossRef

35.

Mokhtary, P.: Numerical analysis of an operational Jacobi Tau method for fractional weakly singular integro-differential equations. Appl. Numer. Math. 121, 52–67 (2017)MathSciNetMATHCrossRef

36.

Mokhtary, P.: Numerical treatment of a well-posed Chebyshev Tau method for Bagley-Torvik equation with high-order of accuracy. Numer. Algorithms 72, 875–891 (2016)MathSciNetMATHCrossRef

37.

Mokhtary, P., Ghoreishi, F.: Convergence analysis of spectral Tau method for fractional Riccati differential equations. Bull. Iran. Math. Soc. 40(5), 1275–1290 (2014)MathSciNetMATH

38.

Mokhtary, P.: Reconstruction of exponentially rate of convergence to Legendre collocation solution of a class of fractional integro-differential equations. J. Comput. Appl. Math. 279, 145–158 (2015)MathSciNetMATHCrossRef

39.

Mokhtary, P., Ghoreishi, F.: Convergence analysis of the operational Tau method for Abel-type Volterra integral equations. Elect. Trans. Numer. Anal. 41, 289–305 (2014)MathSciNetMATH

40.

Momani, S., Odibat, Z.: Numerical approach to differential equations of fractional order. J. Comput. Appl. Math. 207(1), 96–110 (2007)MathSciNetMATHCrossRef

41.

Odibat, Z.M.: Analytic study on linear systems of fractional differential equations. Comput. Math. Appl. 59(3), 1171–1183 (2010)MathSciNetMATHCrossRef

42.

Petráš, I.: Chaos in the fractional-order Volta’s system: modeling and simulation. Nonlinear Dyn. 57(1–2), 157–170 (2009)MATHCrossRef

43.

Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)MATH

44.

Qin, S., Liu, F., Turner, I., Vegh, V., Yu, Q., Yang, Q.: Multi-term time-fractional Bloch equations and application in magnetic resonance imaging. J. Comput. Appl. Math. 319, 308–319 (2017)MathSciNetMATHCrossRef

45.

Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, New York (2011)MATHCrossRef

46.

Sweilam, N.H., Khader, M.M., Al-Bar, R.F.: Numerical studies for a multi-order fractional differential equation. Phys. Lett. A 371(1–2), 26–33 (2007)MathSciNetMATHCrossRef

47.

Wang, J., Xu, T. Z., Wei, Y. Q., Xie, J. Q.: Numerical solutions for systems of fractional order differential equations with Bernoulli wavelets. Int. J. Comput. Math. 1–20 (2018)

48.

Xie, Z., Li, X., Tang, T.: Convergence analysis of spectral Galerkin methods for Volterra type integral equations. J. Sci. Comput. 53(2), 414–434 (2012)MathSciNetMATHCrossRef

49.

Yu, Y., Li, H.x, Wang, S., Yu, J.: Dynamic analysis of a fractional-order Lorenz chaotic system. Chaos Solitons Fractals 42(2), 1181–1189 (2008)MathSciNetMATHCrossRef

50.

Yu, Q., Liu, F., Turner, I., Burrage, K.: Numerical simulation of the fractional Bloch equations. J. Comput. Appl. Math. 255, 635–651 (2014)MathSciNetMATHCrossRef

51.

Zhu, H., Zhou, S., Zhang, J.: Chaos and synchronization of the fractional-order Chua’s system. Chaos Solitons Fractals 39(4), 1595–1603 (2009)MATHCrossRef

Title: An Efficient Formulation of Chebyshev Tau Method for Constant Coefficients Systems of Multi-order FDEs
Authors: A. Faghih
P. Mokhtary
Publication date: 01-01-2020
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2020
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-019-01104-z

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