Computational and Applied Mathematics

, Volume 37, Issue 4, pp 5456–5475 | Cite as

Generalized Jacobi–Galerkin method for nonlinear fractional differential algebraic equations

  • F. Ghanbari
  • K. Ghanbari
  • P. Mokhtary


In this paper, we provide an approximate approach based on the Galerkin method to solve a class of nonlinear fractional differential algebraic equations. The fractional derivative operator in the Caputo sense is utilized and the generalized Jacobi functions are employed as trial functions. The existence and uniqueness theorem as well as the asymptotic behavior of the exact solution are provided. It is shown that some derivatives of the solutions typically have singularity at origin dependence on the order of the fractional derivative. The influence of the perturbed data on the exact solutions along with the convergence analysis of the proposed scheme is also established. Some illustrative examples provided to demonstrate that this novel scheme is computationally efficient and accurate.


Fractional differential algebraic equation Generalized Jacobi–Galerkin method Regularity Convergence analysis 

Mathematics Subject Classification

34A09 65L05 65L20 65L60 65L80 



The authors cordially thank anonymous referees for their valuable comments that improved the quality of this paper.


  1. Atkinson KE, Han W (2009) Theoretical numerical analysis, a functional analysis framework, 3rd edn, Texts in Applied Mathematics, vol 39. Springer, DordrechtGoogle Scholar
  2. Babaei A, Banihashemi S (2017) A stable numerical approach to solve a time-fractional inverse heat conduction problem. Iran J Sci Technol Trans A Sci. Google Scholar
  3. Bhrawy AH, Zaky MA (2016a) Shifted fractional-order Jacobi orthogonal functions: application to a system of fractional differential equations. Appl Math Model 40(2):832–845Google Scholar
  4. Bhrawy AH, Zaky MA (2016b) A fractional-order Jacobi–Tau method for a class of time-fractional PDEs with variable coefficients. Math Methods Appl Sci 39(7):1765–1779Google Scholar
  5. Canuto C, Hussaini MY, Quarteroni A, Zang TA (2006) Spectral methods. Fundamentals in single domains. Springer, BerlinzbMATHGoogle Scholar
  6. Chen S, Shen J, Wang LL (2016) Generalized Jacobi functions and their applications to fractional differential equations. Math Comput 85:1603–1638MathSciNetCrossRefzbMATHGoogle Scholar
  7. Dabiri A, Butcher EA (2016) Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods. Appl Math Model 56:424–448MathSciNetCrossRefGoogle Scholar
  8. Dabiri A, Butcher EA (2017a) Efficient modified Chebyshev differentiation matrices for fractional differential equations. Commun Nonlinear Sci Numer Simul 50:284–310Google Scholar
  9. Dabiri A, Butcher EA (2017b) Stable fractional Chebyshev differentiation matrix for the numerical solution of multi-order fractional differential equations. Nonlinear Dyn 90(1):185–201Google Scholar
  10. Dabiri A, Nazari M, Butcher EA (2016) Optimal fractional state feedback control for linear fractional periodic time-delayed systems. In: 2016 American control conference (ACC).
  11. Dabiri A, Moghaddam BP, Tenreiro Machadoc JA (2018) Optimal variable-order fractional PID controllers for dynamical systems. J Comput Appl Math 339:40–48MathSciNetCrossRefzbMATHGoogle Scholar
  12. Damarla SK, Kundu M (2015) Numerical solution of fractional order differential algebraic equations using generalized triangular function operational matrices. J Fract Calc Appl 6(2):31–52MathSciNetzbMATHGoogle Scholar
  13. Diethelm K (2010) The analysis of fractional differential equations. Springer, BerlinCrossRefzbMATHGoogle Scholar
  14. Ding XL, Jiang YL (2014) Waveform relaxation method for fractional differential algebraic equations. Fract Calc Appl Anal 17(3):585–604MathSciNetCrossRefzbMATHGoogle Scholar
  15. Fletcher R (1987) Practical methods of optimization, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  16. Ghoreishi F, Mokhtary P (2014) Spectral collocation method for multi-order fractional differential equations. Int J Comput Methods 11:23. MathSciNetCrossRefzbMATHGoogle Scholar
  17. Gear CW (1990) Differential algebraic equations, indices, and integral algebraic equations. SIAM J Numer Anal 27(6):1527–1534MathSciNetCrossRefzbMATHGoogle Scholar
  18. Hairer E, Lubich C, Roche M (1989) The numerical solution of differential-algebraic systems by Runge–Kutta methods. Springer, BerlinCrossRefzbMATHGoogle Scholar
  19. Hesthaven JS, Gottlieb S, Gottlieb D (2007) Spectral methods for time-dependent problems, Cambridge Monographs on Applied and Computational Mathematics, vol 21. Cambridge University Press, CambridgezbMATHGoogle Scholar
  20. İbis B, Bayram M (2011) Numerical comparison of methods for solving fractional differential-algebraic equations (FDAEs). Comput Math Appl 62(8):3270–3278MathSciNetCrossRefzbMATHGoogle Scholar
  21. İbis B, Bayram M, Göksel Ağargün A (2011) Applications of fractional differential transform method to fractional differential-algebraic equations. Eur J Pure Appl Math 4(2):129–141MathSciNetzbMATHGoogle Scholar
  22. Jaradat HM, Zurigat M, Al-Sharan S (2014) Toward a new algorithm for systems of fractional differential algebraic equations. Ital J Pure Appl Math 32:579–594zbMATHGoogle Scholar
  23. Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, AmsterdamzbMATHGoogle Scholar
  24. Keshi FK, Moghaddam BP, Aghili A (2018) A numerical approach for solving a class of variable-order fractional functional integral equations. Comput Appl Math. Google Scholar
  25. Moghaddam BP, Aghili A (2012) A numerical method for solving linear non-homogeneous fractional ordinary differential equations. Appl Math Inf Sci 6(3):441–445MathSciNetGoogle Scholar
  26. Moghaddam BP, Machado JAT (2017a) A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels. Fract Calc Appl Anal 20(4):1305–1312.
  27. Moghaddam BP, Machado JAT (2017b) SM-Algorithms for approximating the variable-order fractional derivative of high order. Fundam Inform 151(1–4):293–311Google Scholar
  28. Moghaddam BP, Machado JAT, Behforooz HB (2017a) An integro quadratic spline approach for a class of variable order fractional initial value problems. Chaos Solitons Fractals 102:354–360Google Scholar
  29. Moghaddam BP, Machado JAT, Babaei A (2017b) A computationally efficient method for tempered fractional differential equations with application. Appl Math Comput.
  30. Mokhtary P (2015) Reconstruction of exponentially rate of convergence to Legendre collocation solution of a class of fractional integro-differential equations. J Comput Appl Math 279:145–158MathSciNetCrossRefzbMATHGoogle Scholar
  31. Mokhtary P (2016a) Discrete Galerkin method for fractional integro-differential equations. Acta Math Sci 36B(2):560–578Google Scholar
  32. Mokhtary P (2016b) Numerical treatment of a well-posed Chebyshev Tau method for Bagley–Torvik equation with high-order of accuracy. Numer Algorithms 72:875–891Google Scholar
  33. Mokhtary P (2017) Numerical analysis of an operational Jacobi Tau method forfractional weakly singular integro-differential equations. Appl Numer Math 121:52–67MathSciNetCrossRefzbMATHGoogle Scholar
  34. Mokhtary P, Ghoreishi F (2011) The \(L^2\)-convergence of the Legendre-spectral Tau matrix formulation for nonlinear fractional integro-differential equations. Numer Algorithms 58:475–496MathSciNetCrossRefzbMATHGoogle Scholar
  35. Mokhtary P, Ghoreishi F (2014a) Convergence analysis of the operational Tau method for Abel-type Volterra integral equations. Electron Trans Numer Anal 41:289–305Google Scholar
  36. Mokhtary P, Ghoreishi F (2014b) Convergence analysis of spectral Tau method for fractional Riccati differential equations. Bull Iran Math Soc 40(5):1275–1296Google Scholar
  37. Mokhtary P, Ghoreishi F, Srivastava HM (2016) The Müntz–Legendre Tau method for fractional differential equations. Appl Math Model 40(2):671–684MathSciNetCrossRefGoogle Scholar
  38. Pedas A, Tamme E, Vikerpuur M (2016) Spline collocation for fractional weakly singular integro-differential equations. Appl Numer Math 110:204–214MathSciNetCrossRefzbMATHGoogle Scholar
  39. Podlubny I (1999) Fractional differential equations. Academic Press, New YorkzbMATHGoogle Scholar
  40. Shen J, Tang T, Wang LL (2006) Spectral methods algorithms, analysis and applications. J Math Anal Appl 313:251–261MathSciNetCrossRefGoogle Scholar
  41. Taghavi A, Babaei A, Mohammadpour A (2017) A stable numerical scheme for a time fractional inverse parabolic equations. Inverse Probl Sci Eng 25(10):1474–1491MathSciNetCrossRefzbMATHGoogle Scholar
  42. Zaky MA (2017) A Legendre spectral quadrature Tau method for the multi-term time-fractional diffusion equations. Appl Math Comput. Google Scholar
  43. Zhang W, Ge SS (2011) A global implicit function theorem without initial point and its applications to control of non-affine systems of high dimensions. Springer, BerlinGoogle Scholar
  44. Zurigat M, Momani S, Alawneha A (2010) Analytical approximate solutions of systems of fractional algebraic differential equations by homotopy analysis method. Comput Math Appl 59(3):1227–1235MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Basic SciencesSahand University of TechnologyTabrizIran

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