Computational and Applied Mathematics

, Volume 37, Issue 3, pp 3525–3538 | Cite as

A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations

  • Mahmoud A. ZakyEmail author


In this study, a Legendre spectral tau method is revisited to handle the multi-term time-fractional diffusion equations (MTT-FDEs). An error estimate and rigorous convergence analysis are carried out using some critical theorems. The applicability and accuracy of the solution method are demonstrated by a numerical example to verify the theoretical analysis.


Spectral tau method Multi-term time-fractional diffusion equation Caputo fractional derivative Convergence analysis 

Mathematics Subject Classification

65M70 34A08 33C45 11B83 


  1. Abd-Elhameed WM, Youssri YH (2017) Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations. Comput Appl Math.
  2. Abdelkawy MA, Amin AZ, Bhrawy AH, Machado JAT, Lopes AM (2017) Jacobi collocation approximation for solving multi-dimensional volterra integral equations. Int J Nonlinear Sci Numer Simul.
  3. Atangana A (2015) On the stability and convergence of the time-fractional variable order telegraph equation. J Comput Phys 293:104–114MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bhrawy AH, Zaky MA (2015) A method based on the Jacobi tau approximation for solving multi-term time–space fractional partial differential equations. J Comput Phys 281:876–895MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bhrawy AH, Zaky MA (2015) Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation. Nonlinear Dyn 80:101–116MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bhrawy AH, Zaky MA (2016) Shifted fractional-order Jacobi orthogonal functions: application to a system of fractional differential equations. Appl Math Model 40:832–845MathSciNetCrossRefGoogle Scholar
  7. Bhrawy AH, Zaky MA (2017) Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations. Comput Math Appl 73:1100–1117MathSciNetCrossRefGoogle Scholar
  8. Bhrawy AH, Zaky MA (2017) Numerical simulation of multi-dimensional distributed-order generalized Schrödinger equations. Nonlinear Dyn 89:1415–1432CrossRefzbMATHGoogle Scholar
  9. Bhrawy AH, Zaky MA (2017) An improved collocation method for multi-dimensional space–time variable-order fractional Schrödinger equations. Appl Numer Math 111:197–218MathSciNetCrossRefzbMATHGoogle Scholar
  10. Bhrawy AH, Zaky MA, Machado JAT (2016) Efficient Legendre spectral tau algorithm for solving two-sided space–time Caputo fractional advection–dispersion equation. J Vib Control 22(8):2053–2068MathSciNetCrossRefzbMATHGoogle Scholar
  11. Bhrawy AH, Zaky MA, Van Gorder RA (2016) A space–time Legendre spectral tau method for the two-sided space–time Caputo fractional diffusion-wave equation. Numer Algorithms 71:151–180MathSciNetCrossRefzbMATHGoogle Scholar
  12. Canuto C, Hussaini MY, Quarteroni A, Zang TA (2006) Spectral methods: fundamentals in single domains. Springer, New YorkzbMATHGoogle Scholar
  13. Coimbra CFM (2003) Mechanics with variable-order differential operators. Ann Phys 12:692–703MathSciNetCrossRefzbMATHGoogle Scholar
  14. Dabiri A, Butcher EA (2017) Efficient modified Chebyshev differentiation matrices for fractional differential equations. Commun Nonlinear Sci Numer Simul 50:284–310MathSciNetCrossRefGoogle Scholar
  15. Dabiri A, Butcher EA (2017) Stable fractional Chebyshev differentiation matrix for the numerical solution of multi-order fractional differential equations. Nonlinear Dyn 90:185–201MathSciNetCrossRefzbMATHGoogle Scholar
  16. Dabiri A, Butcher EA, Nazari M (2017) Coefficient of restitution in fractional viscoelastic compliant impacts using fractional Chebyshev collocation. J Sound Vib 388:230–244CrossRefGoogle Scholar
  17. Dabiri A, Butcher EA (2016) The spectral parameter estimation method for parameter identification of linear fractional order systems. American control conference (ACC)Google Scholar
  18. Dehghan M, Safarpoor M, Abbaszadeh M (2015) Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equations. J Comput Appl Math 290:174–195MathSciNetCrossRefzbMATHGoogle Scholar
  19. Ertik H, Demirhan D, Şirin H, Büyükklç F (2010) Time fractional development of quantum systems. J Math Phys 51:082102MathSciNetCrossRefzbMATHGoogle Scholar
  20. Ezz-Eldien SS (2016) New quadrature approach based on operational matrix for solving a class of fractional variational problems. J Comput Phys 317:362–381MathSciNetCrossRefzbMATHGoogle Scholar
  21. Ezz-Eldien SS, Hafez RM, Bhrawy AH, Baleanu D, El-Kalaawy A (2017) New numerical approach for fractional variational problems using shifted Legendre orthonormal polynomials. J Optim Theory Appl 174:295–320MathSciNetCrossRefzbMATHGoogle Scholar
  22. Ganti V, Meerschaert M, Foufoula-Georgiou E, Viparelli E, Parker G (2010) Normal and anomalous diffusion of gravel tracer particles in rivers. J Geophys Res Earth Surf 115(F2):1–12CrossRefGoogle Scholar
  23. Glöckle WG, Nonnenmacher TF (1995) A fractional calculus approach to self-similar protein dynamics. Biophys J 68:46–53CrossRefGoogle Scholar
  24. Jiang H, Liu F, Turner I, Burrage K (2012) Analytical solutions for the multi-term time-space Caputo–Riesz fractional advection–diffusion equations on a finite domain. J Math Anal Appl 389:1117–1127MathSciNetCrossRefzbMATHGoogle Scholar
  25. Jiang H, Liu F, Meerschaert MM, McGough RJ (2013) The foundamental solutions for multi-term modified power law wave equations in a finite domain. Electr J Math Anal Appl 1:1Google Scholar
  26. Jin B, Lazarov R, Liu Y, Zhou Z (2015) The Galerkin finite element method for a multi-term time-fractional diffusion equation. J Comput Phys 281:825–843MathSciNetCrossRefzbMATHGoogle Scholar
  27. Li G, Sun C, Jia X, Du D (2016) Numerical solution to the multi-term time fractional diffusion equation in a finite domain. Numer Math Theory Methods Appl 9:337–357MathSciNetCrossRefzbMATHGoogle Scholar
  28. Li M, Huang C, Jiang F (2017) Galerkin finite element method for higher dimensional multi-term fractional diffusion equation on non-uniform meshes. Appl Anal 96:1269–1284MathSciNetCrossRefzbMATHGoogle Scholar
  29. Liu F, Meerschaert M, McGough R, Zhuang P, Liu Q (2013) Numerical methods for solving the multi-term time-fractional wave-diffusion equations. Fract Calculus Appl Anal 16:9–25MathSciNetzbMATHGoogle Scholar
  30. Lorenzo CF, Hartley TT (2002) Variable order and distributed order fractional operators. Nonlinear Dyn 29:57–98MathSciNetCrossRefzbMATHGoogle Scholar
  31. Luchko Y (2009) Boundary value problems for the generalized time-fractional diffusion equation of distributed order. Fract Calc Appl Anal 12:409–422MathSciNetzbMATHGoogle Scholar
  32. Luchko Y (2011) Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation. J Math Anal Appl 374:538–548MathSciNetCrossRefzbMATHGoogle Scholar
  33. Machado JAT, Kiryakova V (2017) The chronicles of fractional calculus. Fract Calc Appl Anal 20:307–336MathSciNetCrossRefzbMATHGoogle Scholar
  34. Mainardi F, Pagnini G, Gorenflo R (2007) Some aspects of fractional diffusion equations of single and distributed order. Appl Math Comput 187:295–305MathSciNetzbMATHGoogle Scholar
  35. Metzler R, Klafter J (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339:1–77MathSciNetCrossRefzbMATHGoogle Scholar
  36. Moghaddam BP, Machado JAT (2017) Extended algorithms for approximating variable order fractional derivatives with applications. J Sci Comput 71:1351–1374MathSciNetCrossRefzbMATHGoogle Scholar
  37. Moghaddam BP, Machado JAT (2017) A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels. Fract Calc Appl Anal 20:1023–1042MathSciNetzbMATHGoogle Scholar
  38. Moghaddam BP, Machado JAT (2017) A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations. Comput Math Appl 73:1262–1269MathSciNetCrossRefGoogle Scholar
  39. Moghaddam BP, Machado JAT, Behforooz H (2017) An integro quadratic spline approach for a class of variable-order fractional initial value problems. Chaos Soliton Fract 102:354–360MathSciNetCrossRefzbMATHGoogle Scholar
  40. Mokhtary P (2017) Numerical analysis of an operational Jacobi Tau method for fractional weakly singular integro-differential equations. Appl Numer Math 121:52–67MathSciNetCrossRefzbMATHGoogle Scholar
  41. Nicolau DV, Hancock JF, Burrage K (2007) Sources of anomalous diffusion on cell membranes: a Monte Carlo study. Biophys J 92:1975–1987CrossRefGoogle Scholar
  42. Nigmatulin R (1986) The realization of the generalized transfer equation in a medium with fractal geometry. Phys Stat Sol B 133:425–430CrossRefGoogle Scholar
  43. Pimenov VG, Hendy AS, De Staelen RH (2017) On a class of non-linear delay distributed order fractional diffusion equations. J Comput Appl Math 318:433–443MathSciNetCrossRefzbMATHGoogle Scholar
  44. Podlubny I (1999) Fractional differential equations. Academic Press, San DiegozbMATHGoogle Scholar
  45. Qiao L, Xu D (2017) Orthogonal spline collocation scheme for the multi-term time-fractional diffusion equation. Int J Comput Math.
  46. Ren J, Sun Z (2014) Efficient and stable numerical methods for multi-term time-fractional sub-diffusion equations. East Asian J Appl Math 4:242–266MathSciNetCrossRefzbMATHGoogle Scholar
  47. Salehi R (2017) A meshless point collocation method for 2-D multi-term time fractional diffusion-wave equation. Numer Algorithms 74:1145–1168MathSciNetCrossRefzbMATHGoogle Scholar
  48. Scher H, Montroll EW (1975) Anomalous transit-time dispersion in amorphous solids. Phys Rev B 12:2455CrossRefGoogle Scholar
  49. Schneider W, Wyss W (1989) Fractional diusion and wave equations. J Math Phys 30:134–144MathSciNetCrossRefzbMATHGoogle Scholar
  50. Schumer R, Benson DA, Meerschaert MM, Baeumer B (2003) Fractal mobile/immobile solute transport. Water Res Res 39:1296Google Scholar
  51. Smit W, De Vries H (1970) Rheological models containing fractional derivatives. Rheol Acta 9:525–534CrossRefGoogle Scholar
  52. Song F, Zeng F, Cai W, Chen W, Karniadakis GE (2017) Efficient two-dimensional simulations of the fractional Szabo equation with different time-stepping schemes. Comput Math Appl 73(6):1286–1297MathSciNetCrossRefGoogle Scholar
  53. Tang X, Xu H (2016) Fractional pseudospectral integration matrices for solving fractional differential, integral, and integro-differential equations. Commun Nonlinear Sci Numer Simul 30:248–267MathSciNetCrossRefGoogle Scholar
  54. Tang X, Shi Y, Xu H (2017) Fractional pseudospectral schemes with equivalence for fractional differential equations. SIAM J Sci Comput 39(3):A966–A982MathSciNetCrossRefzbMATHGoogle Scholar
  55. Tayebi A, Shekari Y, Heydari MH (2017) A meshless method for solving two-dimensional variable-order time fractional advection–diffusion equation. J Comput Phys 340:655–669MathSciNetCrossRefzbMATHGoogle Scholar
  56. Vyawahare VA, Nataraj PSV (2013) Fractional-order modeling of neutron transport in a nuclear reactor. Appl Math Model 37:9747–9767MathSciNetCrossRefGoogle Scholar
  57. Wang L-L, Samson MD, Zhao X (2014) A well-conditioned collocation method using a pseudospectral integration matrix. SIAM J Sci Comput 36:A907–A929MathSciNetCrossRefzbMATHGoogle Scholar
  58. Wei L (2017) Stability and convergence of a fully discrete local discontinuous Galerkin method for multi-term time fractional diffusion equations. Numer Algorithms.
  59. Yaghoobi S, Moghaddam BP, Ivaz K (2017) An efficient cubic spline approximation for variable-order fractional differential equations with time delay. Nonlinear Dyn 87:815–826MathSciNetCrossRefzbMATHGoogle Scholar
  60. Zaky MA, Machado JAT (2017) On the formulation and numerical simulation of distributed-order fractional optimal control problems. Commun Nonlinear Sci Numer Simul 52:177–189MathSciNetCrossRefGoogle Scholar
  61. Zhaoa Y, Zhang Y, Liub F, Turner I, Tang Y, Anh V (2017) Convergence and superconvergence of a fully-discrete scheme for multi-term time fractional diffusion equations. Comput Math Appl 73:1087–1099MathSciNetCrossRefGoogle Scholar
  62. Zheng M, Liu F, Anh V, Turner I (2016) A high-order spectral method for the multi-term time-fractional diffusion equations. Appl Math Model 40:4970–4985MathSciNetCrossRefGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsNational Research CentreGizaEgypt

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