A Novel Lagrange Operational Matrix and Tau-Collocation Method for Solving Variable-Order Fractional Differential Equations

  • S. Sabermahani
  • Y. OrdokhaniEmail author
  • P. M. Lima
Research Paper
Part of the following topical collections:
  1. Mathematics


The main result achieved in this paper is an operational Tau-Collocation method based on a class of Lagrange polynomials. The proposed method is applied to approximate the solution of variable-order fractional differential equations (VOFDEs). We achieve operational matrix of the Caputo’s variable-order derivative for the Lagrange polynomials. This matrix and Tau-Collocation method are utilized to transform the initial equation into a system of algebraic equations. Also, we discuss the numerical solvability of the Lagrange-Tau algebraic system in the case of a variable-order linear equation. Error estimates are presented. Some examples are provided to illustrate the accuracy and computational efficiency of the present method to solve VOFDEs. Moreover, one of the numerical examples is concerned with the shape-memory polymer model.


Variable-order fractional differential equation Lagrange polynomial Tau-Collocation method 



P.M. Lima acknowledges support from FCT, within project SFRH/BSAB/135130/2017. Also, the authors would like to thank the referees for their valuable comments and suggestions that improved the paper.


  1. Atkinson KE, Han W (2009) Theoretical numerical analysis. A functional analysis framework, 3 edn. Texts in applied mathematics, vol 39. Springer, DordrechtGoogle Scholar
  2. Bagley RL, Torvik PJ (1985) Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA J 23:918–925CrossRefGoogle Scholar
  3. Bahaa GM (2017) Fractional optimal control problem for variable-order differential systems. Fract Calc Appl Anal 20(6):1447–1470MathSciNetCrossRefGoogle Scholar
  4. Baillie RL (1996) Long memory processes and fractional integration in econometrics. J Econom 73:5–59MathSciNetCrossRefGoogle Scholar
  5. Bhrawyi AH, Zaky MA, Abdel-Aty M (2017) A fast and precise numerical algorithm for a class of variable-order fractional differential equations. Proc Romanian Acad Ser Math Phys Tech Sci Inf Sci 18(1):17–24MathSciNetGoogle Scholar
  6. Bota C, Căruntu B (2017) Analytic approximate solutions for a class of variable order fractional differential equations using the polynomial least squares method. Fract Calc Appl Anal 20(4):1043–1050MathSciNetCrossRefGoogle Scholar
  7. Canuto C, Hussaini MY, Quarteroni A, Zang TA (2006) Spectral methods. Springer, BerlinCrossRefGoogle Scholar
  8. Chen YM, Liu LQ, Liu D, Boutat D (2016) Numerical study of a class of variable order nonlinear fractional differential equation in terms of Bernstein polynomials. Ain Shams Eng J. CrossRefGoogle Scholar
  9. Chen YM, Wei YQ, Liu DY, Yu H (2015) Numerical solution for a class of nonlinear variable order fractional differential equations with Legendre wavelets. Appl Math Lett 46:83–88MathSciNetCrossRefGoogle Scholar
  10. Doha EH, Abdelkawy AM, Amin AZM, Baleanu D (2017) Spectral technique for solving variable-order fractional Volterra integro-differential equations. Numer Methods Partial Differ Equ. CrossRefzbMATHGoogle Scholar
  11. Gomez-Aguilar JF (2018) Analytical and Numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations. Phys A 494:52–75MathSciNetCrossRefGoogle Scholar
  12. Hassani H, Dahaghin MS, Heydari H (2017) A new optimized method for solving variable-order fractional differential equations. J. Math. Ext. 11:85–98MathSciNetzbMATHGoogle Scholar
  13. Hosseininia M, Heydari MH, Ghaini FM, Avazzadeh Z (2019) A wavelet method to solve nonlinear variable-order time fractional 2D Klein-Gordon equation. Comput Math Appl 78(12):3713–3730MathSciNetCrossRefGoogle Scholar
  14. Hosseininia M, Heydari MH (2019) Legendre wavelets for the numerical solution of nonlinear variable-order time fractional 2D reaction-diffusion equation involving Mittag–Leffler non-singular kernel. Chaos Solitons Fractals 127:400–407MathSciNetCrossRefGoogle Scholar
  15. Hosseininia M, Heydari MH (2019) Meshfree moving least squares method for nonlinear variable-order time fractional 2D telegraph equation involving Mittag–Leffler non-singular kernel. Chaos Solitons Fractals 127:389–399MathSciNetCrossRefGoogle Scholar
  16. Heydari MH, Avazzadeh Z, Yang Y (2019) A computational method for solving variable-order fractional nonlinear diffusion-wave equation. Appl Math Comput 352:235–248MathSciNetGoogle Scholar
  17. Heydari MH, Avazzadeh Z, Farzi Haromi M (2019) A wavelet approach for solving multi-term variable-order time fractional diffusion-wave equation. Appl Math Comput 341:215–228MathSciNetGoogle Scholar
  18. Ingman D, Suzdalnitsky J, Zeifman M (2000) Constitutive dynamic-order model for nonlinear contact phenomena. J Appl Mech 67:383–390CrossRefGoogle Scholar
  19. Jafari H, Yousefi SA, Firoozjaee MA, Momani S, Khalique CM (2011) Application of Legendre wavelets for solving fractional differential equations. Comput Math Appl 62(3):1038–1045MathSciNetCrossRefGoogle Scholar
  20. Jia YT, Xu MQ, Lin YZ (2017) A numerical solution for variable order fractional functional differential equation. Appl Math Lett 64:125–130MathSciNetCrossRefGoogle Scholar
  21. Lakestani M (2017) Numerical solutions of the KdV equation using B-Spline functions. Iran J Sci Technol Trans Sci 41:409. MathSciNetCrossRefzbMATHGoogle Scholar
  22. Li Z, Wang H, Xiao R, Yang S (2017) A variable-order fractional differential equation model of shape memory polymers. Chaos Solitons Fractals 102:473–485MathSciNetCrossRefGoogle Scholar
  23. Li X, Wu B (2017) A new reproducing kernel method for variable order fractional boundary value problems for functional differential equations. Comput Math Appl 311:387–393MathSciNetCrossRefGoogle Scholar
  24. Lin R, Liu F, Anh V, Turner I (2009) Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl Math Comput 212:435–45MathSciNetzbMATHGoogle Scholar
  25. Ma S, Xu Y, Yue W (2012) Numerical solutions of a variable-order fractional financial system. J Appl Math. MathSciNetCrossRefzbMATHGoogle Scholar
  26. Mainardi F (1997) Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri A, Mainardi F (eds) Fractals and fractional calculus in continuum mechanics. Springer, New York, pp 291–348CrossRefGoogle Scholar
  27. Omar AA, Mohammed OH (2017) Bernstein operational matrices for solving multi-term variable order fractional differential equations. Int J Curr Eng Technol 7(1):68–73Google Scholar
  28. Ortiz EL, El-Daou MK (1998) The Tau method as an analytic tool in the discussion of equivalence results across numerical methods. Computing 60:365. MathSciNetCrossRefzbMATHGoogle Scholar
  29. Pilate F, Toncheva A, Dubois P, Raquez JM (2016) Shape-memory polymers for multiple applications in the materials world. Eur Polym J 80:268–294CrossRefGoogle Scholar
  30. Podlubny I (1999) Fractional differential equations. Academic Press, San DiegozbMATHGoogle Scholar
  31. Rossikhin YA, Shitikova MV (1997) Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl Mech Rev 50:15–67CrossRefGoogle Scholar
  32. Sabermahani S, Ordokhani Y, Yousefi SA (2018) Numerical approach based on fractional-order Lagrange polynomials for solving a class of fractional differential equations. Comp Appl Math 37:3846–3868. MathSciNetCrossRefzbMATHGoogle Scholar
  33. Sabermahani S, Ordokhani Y, Yousefi SA (2019) Fractional-order general Lagrange scaling functions and their applications. BIT Numer Math. CrossRefzbMATHGoogle Scholar
  34. Samko SG (1995) Fractional integration and differentiation of variable order. Anal Math 21(3):213–236MathSciNetCrossRefGoogle Scholar
  35. Sedaghat S, Ordokhani Y, Dehghan M (2012) Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials. Commun Nonlinear Sci Numer Simul 17:4815–4830MathSciNetCrossRefGoogle Scholar
  36. Shekari Y, Tayebi A, Heydari MH (2019) A meshfree approach for solving 2D variable-order fractional nonlinear diffusion-wave equation. Comput Methods Appl Mech Eng 350:154–168MathSciNetCrossRefGoogle Scholar
  37. Stoer J, Bulirsch R (2013) Introduction to numerical analysis, 3 edn. (trans: Bartels R, Gautschi W, Witzgall C). SpringerGoogle Scholar
  38. Sun H, Chen W, Li C, Chen Y (2012) Finite difference schemes for variable-order time fractional diffusion equation. Int J Bifurc Chaos 22(04):1250085MathSciNetCrossRefGoogle Scholar
  39. Sun HG, Chen W, Sheng H, Chen YQ (2010) On mean square displacement behaviors of anomalous diffusions with variable and random orders. Phys Lett A 374:906–910CrossRefGoogle Scholar
  40. Sun HG, Zhang H, Chen W, Reeves DM (2014) Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media. J Contam Hydrol 157:47–58CrossRefGoogle Scholar
  41. Szegö G (1967) Orthogonal polynomials, 3rd edn. American Mathematical Society, ProvidencezbMATHGoogle Scholar
  42. Usman M, Hamid M, Haq RU, Wang W (2018) An efficient algorithm based on Gegenbauer wavelets for the solutions of variable-order fractional differential equations. Eur Phys J Plus 133(8):327. CrossRefGoogle Scholar
  43. Xiao R, Choi J, Lakhera N, Yakacki CM, Frick CP, Nguyen TD (2013) Modeling the glass transition of amorphous networks for shape-memory behavior. J Mech Phys Solids 61(7):1612–1635MathSciNetCrossRefGoogle Scholar
  44. Zhao X, Sun ZZ, Karniadakis GE (2015) Second-order approximations for variable order fractional derivatives: algorithms and applications. J Comput Phys 293:184–200MathSciNetCrossRefGoogle Scholar

Copyright information

© Shiraz University 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematical SciencesAlzahra UniversityTehranIran
  2. 2.Centro de Matemática Computacional e Estocástica, Instituto Superior TécnicoUniversidade de LisboaLisboaPortugal

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