Abstract
We develop a numerical scheme for finding the approximate solution for one- and two-dimensional multi-term time fractional diffusion and diffusion-wave equations considering smooth and nonsmooth solutions. The concept of multi-term time fractional derivatives is conventionally defined in the Caputo view point. In the current research, the convergence analysis of Legendre collocation spectral method was carried out. Spectral collocation method is consequently tested on several benchmark examples, to verify the accuracy and to confirm effectiveness of proposed method. The main advantage of the method is that only a small number of shifted Legendre polynomials are required to obtain accurate and efficient results. The numerical results are provided to demonstrate the reliability of our method and also to compare with other previously reported methods in the literature survey.
Similar content being viewed by others
References
Aghdam YE, Mesgrani H, Javidi M, Nikan O (2020) A computational approach for the space-time fractional advection–diffusion equation arising in contaminant transport through porous media 1–13. https://doi.org/10.1007/s00366-020-01021-y
Ahmadi N, Vahidi A, Allahviranloo T (2017) An efficient approach based on radial basis functions for solving stochastic fractional differential equations. Math Sci 11(2):113–118
Arshed S (2017) Quintic b-spline method for time-fractional superdiffusion fourth-order differential equation. Math Sci 11(1):17–26
Bhrawy A, Doha EH, Baleanu D, Ezz-Eldien SS (2015) A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations. J Comput Phys 293:142–156
Burrage K, Hale N, Kay D (2012) An efficient implicit FEM scheme for fractional-in-space reaction–diffusion equations. SIAM J Sci Comput 34(4):A2145–A2172
Canuto C, Hussaini MY, Quarteroni A, Zang TA (2007) Spectral methods: fundamentals in single domains. Springer, Berlin
Chen H, Lü S, Chen W (2018) A unified numerical scheme for the multi-term time fractional diffusion and diffusion-wave equations with variable coefficients. J Comput Appl Math 330:380–397
Costa FS, Pereira MR (2018) Fractional space-time nonlinear reaction-convection-diffusion. Comput Appl Math 37(4):4357–4375
Cui M (2012) Compact alternating direction implicit method for two-dimensional time fractional diffusion equation. J Comput Phys 231(6):2621–2633
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 290:174–195
Dehghan M, Abbaszadeh M, Mohebbi A (2016) Analysis of a meshless method for the time fractional diffusion-wave equation. Num Algorithms 73(2):445–476
Doha E, Bhrawy A, Ezz-Eldien S (2012) A new Jacobi operational matrix: an application for solving fractional differential equations. Appl Math Model 36(10):4931–4943
Ford NJ, Xiao J, Yan Y (2011) A finite element method for time fractional partial differential equations. Fract Calculus Appl Anal 14(3):454–474
Ghehsareh HR, Zaghian A, Zabetzadeh SM (2018) The use of local radial point interpolation method for solving two-dimensional linear fractional cable equation. Neural Comput Appl 29(10):745–754
Golbabai A, Nikan O, Nikazad T (2019) Numerical analysis of time fractional Black–Scholes European option pricing model arising in financial market. Comput Appl Math 38(4):173
Heydari M, Hooshmandasl M, Ghaini FM, Cattani C (2015) Wavelets method for the time fractional diffusion-wave equation. Phys Lett A 379(3):71–76
Hosseini VR, Shivanian E, Chen W (2016) Local radial point interpolation (MLRPI) method for solving time fractional diffusion-wave equation with damping. J Comput Phys 312:307–332
Jiang H, Liu F, Turner I, Burrage K (2012) Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain. Comput Math Appl 64(10):3377–3388
Jiang Y, Ma J (2011) High-order finite element methods for time-fractional partial differential equations. J Comput Appl Math 235(11):3285–3290
Kanth AR, Garg N (2019) An implicit numerical scheme for a class of multi-term time-fractional diffusion equation. Eur Phys J Plus 134(6):312
Karatay I, Bayramoğlu ŞR, Şahin A (2011) Implicit difference approximation for the time fractional heat equation with the nonlocal condition. Appl Num Math 61(12):1281–1288
Kargar Z, Saeedi H (2017) B-spline wavelet operational method for numerical solution of time-space fractional partial differential equations. Int J Wavelets Multiresolut Inf Process 15(04):1750034
Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations, vol 204. Elsevier, New York
Li M, Huang C, Ming W (2018a) Mixed finite-element method for multi-term time-fractional diffusion and diffusion-wave equations. Comput Appl Math 37(2):2309–2334
Li MZ, Chen LJ, Xu Q (2018) Ding XH (2018b) An efficient numerical algorithm for solving the two-dimensional fractional cable equation. Adv Diff Equ 1:424
Li X, Yang X (2017) Error estimates of finite element methods for stochastic fractional differential equations. J Comput Math 35(3):346–362
Liu F, Meerschaert M, McGough R, Zhuang P, Liu Q (2013) Numerical methods for solving the multi-term time-fractional wave-diffusion equation. Fract Calculus Appl Anal 16(1):9–25
Mirzaee F, Samadyar N (2018) Application of hat basis functions for solving two-dimensional stochastic fractional integral equations. Comput Appl Math 37(4):4899–4916
Nikan O, Golbabai A, Machado JT, Nikazad T (2020a) Numerical solution of the fractional Rayleigh–Stokes model arising in a heated generalized second-grade fluid. Eng Comput 1:1–14. https://doi.org/10.1007/s00366-019-00913-y
Nikan O, Machado JT, Golbabai A, Nikazad T (2020b) Numerical approach for modeling fractal mobile/immobile transport model in porous and fractured media. Int Commun Heat Mass Transfer 111:104443
Pedas A, Tamme E (2011) Spline collocation methods for linear multi-term fractional differential equations. J Comput Appl Math 236(2):167–176
Permoon M, Rashidinia J, Parsa A, Haddadpour H, Salehi R (2016) Application of radial basis functions and sinc method for solving the forced vibration of fractional viscoelastic beam. J Mech Sci Technol 30(7):3001–3008
Podlubny I (1999) An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Math Sci Eng 198:340
Povstenko Y (2015) Linear fractional diffusion-wave equation for scientists and engineers. Springer, Berlin
Qiao L, Xu D (2018) Orthogonal spline collocation scheme for the multi-term time-fractional diffusion equation. Int J Comput Math 95(8):1478–1493
Rashidinia J, Mohmedi E (2018) Convergence analysis of tau scheme for the fractional reaction–diffusion equation. Eur Phys J Plus 133(10):402
Ren J, Sun ZZ (2015) Efficient numerical solution of the multi-term time fractional diffusion-wave equation. East Asian J Appl Math 5(1):1–28
Ren L, Liu L (2019) A high-order compact difference method for time fractional Fokker–Planck equations with variable coefficients. Comput Appl Math 38(3):101
Ren L, Wang YM (2017) A fourth-order extrapolated compact difference method for time-fractional convection-reaction-diffusion equations with spatially variable coefficients. Appl Math Comput 312:1–22
Richard H (2014) Fractional calculus: an introduction for physicists. World Sci
Saffarian M, Mohebbi A (2019) The Galerkin spectral element method for the solution of two-dimensional multi term time fractional diffusion-wave equation. Math Methods Appl Sci
Sakar MG, Saldır O, Erdogan F (2018) An iterative approximation for time-fractional Cahn–Allen equation with reproducing kernel method. Comput Appl Math 37(5):5951–5964
Schiessel H, Metzler R, Blumen A, Nonnenmacher T (1995) Generalized viscoelastic models: their fractional equations with solutions. J Phys A: Math Gen 28(23):6567
Shen S, Liu F, Anh V (2011) Numerical approximations and solution techniques for the space-time Riesz–Caputo fractional advection-diffusion equation. Num Algorithms 56(3):383–403
Siddiqi SS, Arshed S (2015) Numerical solution of time-fractional fourth-order partial differential equations. Int J Comput Math 92(7):1496–1518
Soltani Sarvestani F, Heydari MH, Niknam A, Avazzadeh Z (2019) A wavelet approach for the multi-term time fractional diffusion-wave equation. Int J Comput Math 96(3):640–661
Sousa E (2012) A second order explicit finite difference method for the fractional advection diffusion equation. Comput Math Appl 64(10):3141–3152
Srivastava V, Rai K (2010) A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues. Math Comput Modell 51(5–6):616–624
Sunarto A, Sulaiman J, Saudi A (2014) Implicit finite difference solution for time-fractional diffusion equations using AOR method. In: Journal of Physics: Conference Series, IOP Publishing, vol 495, p 012032
Uchaikin VV (2013) Fractional derivatives for physicists and engineers, vol 2. Springer, Berlin
Wang YM (2015) A compact finite difference method for a class of time fractional convection–diffusion-wave equations with variable coefficients. Num Algorithms 70(3):625–651
Wang YM, Ren L (2019) Efficient compact finite difference methods for a class of time-fractional convection-reaction-diffusion equations with variable coefficients. Int J Comput Math 96(2):264–297
Yb Wei, Ym Zhao, Zg Shi, Fl Wang, Yf Tang (2018) Spatial high accuracy analysis of FEM for two-dimensional multi-term time-fractional diffusion-wave equations. Acta Math Appl Sin English Ser 34(4):828–841
Xing Y, Yan Y (2018) A higher order numerical method for time fractional partial differential equations with nonsmooth data. J Comput Phys 357:305–323
Yang X, Jiang X, Zhang H (2018) A time-space spectral tau method for the time fractional cable equation and its inverse problem. Appl Num Math 130:95–111
Yang Y, Chen Y, Huang Y, Wei H (2017) Spectral collocation method for the time-fractional diffusion-wave equation and convergence analysis. Comput Math Appl 73(6):1218–1232
Zaky MA (2018a) An improved tau method for the multi-dimensional fractional Rayleigh–Stokes problem for a heated generalized second grade fluid. Comput Math Appl 75(7):2243–2258
Zaky MA (2018b) A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations. Comput Appl Math 37(3):3525–3538
Zhang J, Yang X (2018) A class of efficient difference method for time fractional reaction-diffusion equation. Comput Appl Math 37(4):4376–4396
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(7–8):4970–4985
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by José Tenreiro Machado.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Rashidinia, J., Mohmedi, E. Approximate solution of the multi-term time fractional diffusion and diffusion-wave equations. Comp. Appl. Math. 39, 216 (2020). https://doi.org/10.1007/s40314-020-01241-4
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-020-01241-4
Keywords
- Multi-term time fractional diffusion and diffusion-wave equations
- Caputo derivative
- Legendre collocation method
- Convergence analysis