Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Fully Petrov–Galerkin spectral method for the distributed-order time-fractional fourth-order partial differential equation

  • 131 Accesses


Distributed fractional derivative operators can be used for modeling of complex multiscaling anomalous transport, where derivative orders are distributed over a range of values rather than being just a fixed integer number. In this paper, we consider the space-time Petrov–Galerkin spectral method for a two-dimensional distributed-order time-fractional fourth-order partial differential equation. By applying a proper Gauss-quadrature rule to discretize the distributed integral operator, the problem is converted to a multi-term time-fractional equation. Then, the proposed method for solving the obtained equation is based on using Jacobi polyfractonomial, which are eigenfunctions of the first kind fractional Sturm–Liouville problem (FSLP), as temporal basis and Legendre polynomials for the spatial discretization. The eigenfunctions of the second kind FSLP are used as temporal basis in test space. This approach leads to finding the numerical solution of the problem through solving a system of linear algebraic equations. Finally, we provide some examples with smooth solutions and finite regular solutions to numerically demonstrate the efficiency, accuracy, and exponential convergence of the proposed method.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4


  1. 1.

    Abbaszadeh M, Dehghan M (2017) An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate. Numer Algorithms 75(1):173–211

  2. 2.

    Ainsworth M, Glusa C (2017) Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver. Comput Methods Appl Mech Eng 327:4–35

  3. 3.

    Ammi MRS, Jamiai I (2018) Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete Contin Dyn Syst Ser S 11(1)

  4. 4.

    Ardakani AG (2016) Investigation of Brewster anomalies in one-dimensional disordered media having Lévy-type distribution. Eur Phys J B 89(3):76

  5. 5.

    Armour KC, Marshall J, Scott JR, Donohoe A, Newsom ER (2016) Southern ocean warming delayed by circumpolar upwelling and equatorward transport. Nat Geosci 9(7):549

  6. 6.

    Atanackovic T M, Pilipovic S, Zorica D (2009) Time distributed-order diffusion-wave equation. I. Volterra-type equation. Proc R Soc A Math Phys Eng Sci 465(2106):1869–1891

  7. 7.

    Benson DA, Wheatcraft SW, Meerschaert MM (2000) Application of a fractional advection–dispersion equation. Water Resour Res 36(6):1403–1412

  8. 8.

    Bu W, Xiao A, Zeng W (2017) Finite difference/finite element methods for distributed-order time fractional diffusion equations. J Sci Comput 72(1):422–441

  9. 9.

    Chechkin A, Gorenflo R, Sokolov I (2002) Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys Rev E 66(4):046129

  10. 10.

    Chen H, Lü S, Chen W (2016) Finite difference/spectral approximations for the distributed order time fractional reaction–diffusion equation on an unbounded domain. J Comput Phys 315:84–97

  11. 11.

    Chen S, Shen J, Wang L-L (2018) Laguerre functions and their applications to tempered fractional differential equations on infinite intervals. J Sci Comput 74(3):1286–1313

  12. 12.

    Cheng A, Wang H, Wang K (2015) A Eulerian–Lagrangian control volume method for solute transport with anomalous diffusion. Numer Methods Part Differ Equ 31(1):253–267

  13. 13.

    Coronel-Escamilla A, Gómez-Aguilar J, Torres L, Escobar-Jiménez R (2018) A numerical solution for a variable-order reaction–diffusion model by using fractional derivatives with non-local and non-singular kernel. Phys A 491:406–424

  14. 14.

    Diethelm K, Ford NJ (2001) Numerical solution methods for distributed order differential equations

  15. 15.

    Diethelm K, Ford NJ (2009) Numerical analysis for distributed-order differential equations. J Comput Appl Math 225(1):96–104

  16. 16.

    Duan J-S, Baleanu D (2018) Steady periodic response for a vibration system with distributed order derivatives to periodic excitation. J Vib Control 24(14):3124–3131

  17. 17.

    Edery Y, Dror I, Scher H, Berkowitz B (2015) Anomalous reactive transport in porous media: experiments and modeling. Phys Rev E 91(5):052130

  18. 18.

    Fei M, Huang C (2019) Galerkin–Legendre spectral method for the distributed-order time fractional fourth-order partial differential equation. Int J Comput Math 1–14

  19. 19.

    Gao G-H, Alikhanov AA, Sun Z-Z (2017) The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations. J Sci Comput 73(1):93–121

  20. 20.

    Gao G-H, Sun H-W, Sun Z-Z (2015) Some high-order difference schemes for the distributed-order differential equations. J Comput Phys 298:337–359

  21. 21.

    Gardiner JD, Laub AJ, Amato JJ, Moler CB (1992) Solution of the sylvester matrix equation \({AX}{B}^{T}+ {CX}{D}^{ T}= {E}\). ACM Trans Math Softw (TOMS) 18(2):223–231

  22. 22.

    Gorenflo R, Luchko Y, Yamamoto M (2015) Time-fractional diffusion equation in the fractional Sobolev spaces. Fract Calculus Appl Anal 18(3):799–820

  23. 23.

    Guo S, Mei L, Zhang Z, Jiang Y (2018) Finite difference/spectral-Galerkin method for a two-dimensional distributed-order time-space fractional reaction–diffusion equation. Appl Math Lett 85:157–163

  24. 24.

    Iwayama T, Murakami S, Watanabe T (2015) Anomalous eddy viscosity for two-dimensional turbulence. Phys Fluids 27(4):045104

  25. 25.

    Ji C-C, Sun Z-Z, Hao Z-P (2016) Numerical algorithms with high spatial accuracy for the fourth-order fractional sub-diffusion equations with the first Dirichlet boundary conditions. J Sci Comput 66(3):1148–1174

  26. 26.

    Jin B, Lazarov R, Thomée V, Zhou Z (2017) On nonnegativity preservation in finite element methods for subdiffusion equations. Math Comput 86(307):2239–2260

  27. 27.

    Kharazmi E, Zayernouri M (2018) Fractional pseudo-spectral methods for distributed-order fractional PDEs. Int J Comput Math 95(6–7):1340–1361

  28. 28.

    Kharazmi E, Zayernouri M, Karniadakis GE (2017a) Petrov–Galerkin and spectral collocation methods for distributed order differential equations. SIAM J Sci Comput 39(3):A1003–A1037

  29. 29.

    Kharazmi E, Zayernouri M, Karniadakis GE (2017b) A Petrov–Galerkin spectral element method for fractional elliptic problems. Comput Methods Appl Mech Eng 324:512–536

  30. 30.

    Kilbas A, Marichev O, Samko S (1993) Fractional integral and derivatives: theory and applications. Gordon and Breach, Switzerland

  31. 31.

    Kilbas AAA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations, volume 204. Elsevier Science Limited

  32. 32.

    Klages R, Radons G, Sokolov IM (2008) Anomalous transport: foundations and applications. Wiley, Hoboken

  33. 33.

    Konjik S, Oparnica L, Zorica D (2019) Distributed-order fractional constitutive stress-strain relation in wave propagation modeling. Zeitschrift für angewandte Mathematik und Physik 70(2):51

  34. 34.

    Li X, Rui H, Liu Z (2018) Two alternating direction implicit spectral methods for two-dimensional distributed-order differential equation. Numer Algorithms 1–27

  35. 35.

    Li X, Wu B (2016) A numerical method for solving distributed order diffusion equations. Appl Math Lett 53:92–99

  36. 36.

    Li X, Xu C (2009) A space-time spectral method for the time fractional diffusion equation. SIAM Journal on Numerical Analysis 47(3):2108–2131

  37. 37.

    Liao H-L, Lyu P, Vong S, Zhao Y (2017) Stability of fully discrete schemes with interpolation-type fractional formulas for distributed-order subdiffusion equations. Numer Algorithms 75(4):845–878

  38. 38.

    Liu Y, Du Y, Li H, He S, Gao W (2015) Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction–diffusion problem. Comput Math Appl 70(4):573–591

  39. 39.

    Liu Y, Fang Z, Li H, He S (2014) A mixed finite element method for a time-fractional fourth-order partial differential equation. Appl Math Comput 243:703–717

  40. 40.

    Macías-Díaz J (2018) An explicit dissipation-preserving method for Riesz space-fractional nonlinear wave equations in multiple dimensions. Commun Nonlinear Sci Numer Simul 59:67–87

  41. 41.

    Mainardi F, Mura A, Gorenflo R, Stojanović M (2007) The two forms of fractional relaxation of distributed order. J Vib Control 13(9–10):1249–1268

  42. 42.

    Mao Z, Shen J (2016) Efficient spectral-Galerkin methods for fractional partial differential equations with variable coefficients. J Comput Phys 307:243–261

  43. 43.

    Mao Z, Shen J (2018) Spectral element method with geometric mesh for two-sided fractional differential equations. Adv Comput Math 44(3):745–771

  44. 44.

    Meerschaert MM (2012) Fractional calculus, anomalous diffusion, and probability. In: Fractional dynamics: recent advances. World Scientific, pp 265–284

  45. 45.

    Meerschaert MM, Sikorskii A (2011) Stochastic models for fractional calculus, vol 43. Walter de Gruyter, Berlin

  46. 46.

    Metzler R, Jeon J-H, Cherstvy AG, Barkai E (2014) Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys Chem Chem Phys 16(44):24128–24164

  47. 47.

    Metzler R, Klafter J (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339(1):1–77

  48. 48.

    Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, Hoboken

  49. 49.

    Naghibolhosseini M, Long GR (2018) Fractional-order modelling and simulation of human ear. Int J Comput Math 95(6–7):1257–1273

  50. 50.

    Perdikaris P, Karniadakis GE (2014) Fractional-order viscoelasticity in one-dimensional blood flow models. Ann Biomed Eng 42(5):1012–1023

  51. 51.

    Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol 198. Elsevier, New York

  52. 52.

    Ran M, Zhang C (2018) New compact difference scheme for solving the fourth-order time fractional sub-diffusion equation of the distributed order. Appl Numer Math 129:58–70

  53. 53.

    Samiee M, Kharazmi E, Zayernouri M, Meerschaert MM (2018) Petrov–Galerkin method for fully distributed-order fractional partial differential equations. arXiv preprint. arXiv:1805.08242

  54. 54.

    Samiee M, Zayernouri M, Meerschaert MM (2019) A unified spectral method for FPDEs with two-sided derivatives; part I: a fast solver. J Comput Phys 385:225–243

  55. 55.

    Shraiman BI, Siggia ED (2000) Scalar turbulence. Nature 405(6787):639

  56. 56.

    Siddiqi SS, Arshed S (2015) Numerical solution of time-fractional fourth-order partial differential equations. Int J Comput Math 92(7):1496–1518

  57. 57.

    Sokolov I, Chechkin A, Klafter J (2004) Distributed-order fractional kinetics. arXiv preprint. arXiv:cond-mat/0401146

  58. 58.

    Szegö G (1975) Orthogonal polynomials, vol. 23. In: American Mathematical Society Colloquium Publications

  59. 59.

    Tomovski Ž, Sandev T (2018) Distributed-order wave equations with composite time fractional derivative. Int J Comput Math 95(6–7):1100–1113

  60. 60.

    Wei L, He Y (2014) Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems. Appl Math Model 38(4):1511–1522

  61. 61.

    Zayernouri M, Karniadakis GE (2013) Fractional Sturm–Liouville eigen-problems: theory and numerical approximation. J Comput Phys 252:495–517

  62. 62.

    Zhang H, Yang X, Xu D (2019) A high-order numerical method for solving the 2D fourth-order reaction–diffusion equation. Numer Algorithms 80(3):849–877

  63. 63.

    Zhang P, Pu H (2017) A second-order compact difference scheme for the fourth-order fractional sub-diffusion equation. Numer Algorithms 76(2):573–598

  64. 64.

    Zhang Y, Meerschaert MM, Baeumer B, LaBolle EM (2015) Modeling mixed retention and early arrivals in multidimensional heterogeneous media using an explicit Lagrangian scheme. Water Resour Res 51(8):6311–6337

  65. 65.

    Zhang Y, Meerschaert MM, Neupauer RM (2016) Backward fractional advection dispersion model for contaminant source prediction. Water Resour Res 52(4):2462–2473

Download references


This work was supported by the Iran National Science foundation (INSF) (No. 98003770).

Author information

Correspondence to Farhad Fakhar-Izadi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fakhar-Izadi, F. Fully Petrov–Galerkin spectral method for the distributed-order time-fractional fourth-order partial differential equation. Engineering with Computers (2020). https://doi.org/10.1007/s00366-020-00968-2

Download citation


  • Fourth-order partial differential equation
  • Distributed-order fractional derivative
  • Jacobi polyfractonomials
  • Legendre polynomials
  • Spectral accuracy