Numerical investigation of a new difference scheme on a graded mesh for solving the time-space fractional sub-diffusion equations with nonsmooth solutions

Document Type : Original Paper


1 Department of Applied Mathematics, Faculty of Mathematical Science, Shahrekord University, Shahrekord, P. O. Box 115, Iran.

2 Department of Mathematics, Payme Noor University, P. O. Box 19395-4697 Tehran, IRAN.


‎In this paper‎, ‎we provide a new difference scheme on a graded mesh for solving the time-space fractional diffusion problem‎. ‎In ‎this ‎equation ‎the ‎time ‎derivative ‎is ‎the ‎Caputo ‎of ‎order ‎‎$‎‎gammain(0,1)$ ‎and ‎the ‎space ‎derivative ‎is ‎the ‎Riesz ‎of ‎order ‎‎$‎‎alphain(1,2]$.‎ ‎The stability and convergence of the difference scheme are discussed which provides the theoretical basis of the proposed‎ ‎schemes‎. W‎e prove that the new difference scheme is unconditionally stable‎. Also, ‎we find that the difference scheme is convergent with order $min{2-gamma,rgamma}$ in time for all $gammain (0,1)$ and $alpha in (1,2]$‎. ‎A test example is given to verify the efficiency and accuracy of the difference scheme‎.


Main Subjects

[1] D. A. Benson, R. Schumer, M. M. Meerschaert and S. W. Wheatcraft, Fractional Dispersion Levy
Motion and the MADE Tracer Tests, Transport in Porous Media, 42(2001), 211–240.
[2] A. Benson, W. David, Stephen Wheatcraft and Mark M. Meerschaert, The fractional-order governing
equation of Levy motion, Water resources research, 36(2000), 1413-1423.
[3] V. Gafiychuk, B. Datsko and V. Meleshko, Mathematical modeling of time fractional reactiondiffusion
systems, Journal of Computational and Applied Mathematics, 220(2008), 215-225.
[4] R. Hilfer, Applications of fractional calculus in physics, Singapore: World Scientific. 2000.
[5] R. L. Magin, Fractional calculus in bioengineering, New York: Begell., 2021.
[6] K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations,
New York, NY: Wiley., 1993.
[7] O. Bavi, M. Hosseininia, M. Heydari and N. Bavi, SARS-CoV-2 rate of spread in and across tissue,
groundwater and soil: A meshless algorithm for the fractional diffusion equation, Engineering Analysis
with Boundary Elements, 138 (2022), 108–117.
[8] S. Traytak, The use of fractional-order derivatives for determination of the time-dependent rate constant,
Chemical Physics Letters, 173(1990), 63-66.
[9] C. Friedrich, H. Schiesse and A. Blumen, Constitutive behavior modeling and fractional derivatives,
Rheology Series, 8(1999), 429-466.
[10] K. A. Diethelm, A fractional calculus based model for the simulation of an outbreak of Dengue fever,
Nonlinear Dyn. 71(2013), 613-619.
[11] L. Vinnett, M. Alvarez-Silva, A. Jaques, F. Hinojosa, and J. Yianatos, Batch flotation kinetics: Fractional
calculus approach, Minerals Engineering, 77(2015), 167-171.
[12] S. Abo-Dahab, A. Kilany, E. A. Abdel-Salam and A. Hatem, Fractional derivative order analysis and
temperature-dependent properties on p- and SV-waves reflection under initial stress and three-phaselag
model, Results in Physics, 18(2020), 103-270.
[13] E. El-Zahar, A. Alotaibi, A. Ebaid, A. Aljohani and J. G. Aguilar, The Riemann–Liouville fractional
derivative for Ambartsumian equation, Results in Physics, 19(2020), 103-551.
[14] S. Butera and M. D. Paola, A physically based connection between fractional calculus and fractal
geometry, Annals of Physics, 350(2014), 146-158.
[15] N. Faraz, M. Sadaf, G. Akram, I. Zainab and Y. Khan, Effects of fractional order time derivative
on the solitary wave dynamics of the generalized ZK–Burgers equation, Results in Physics, 25(2021),
[16] X. Li and P. J. Wong, High order approximation to new generalized Caputo fractional derivatives and
its applications, Journal of Computational Physics, 281(2018), 787-805.
[17] W. Tian, H. Zhou and W. Deng, A class of second order difference approximations for solving space
fractional diffusion equations, Mathematics of Computation, 84(2015), 1703-1727.
[18] Z. Hao and W. Cao, An Improved Algorithm Based on Finite Difference Schemes for Fractional
Boundary Value Problems with Nonsmooth Solution, Journal of Scientific Computing, 73(2017), 395-
[19] H. Wang and T. S. Basu, A Fast Finite Difference Method for Two-Dimensional Space-Fractional
Diffusion Equations, SIAM Journal on Scientific Computing, 34 (5) (2012) , A2444-A2458.
[20] Z. Hao, M. Park, G. Lin and Z. Cai, Finite Element Method for Two-Sided Fractional Differential
Equations with Variable Coefficients: Galerkin Approach, Journal of Scientific Computing, 79(2018),
[21] W. Zeng, A Space-Time Petrov-Galerkin Spectral Method for Time Fractional Fokker-Planck Equation
with Nonsmooth Solution, East Asian Journal on Applied Mathematics, 10(2020), 89-105.
[22] M. Habibli and M. Noori Skandari, Fractional Chebyshev pseudospectral method for fractional optimal
control problems, Optimal Control Applications and Methods, 40(3) (2019), 558–572.
[23] Y. Huang, F. Mohammadi Zadeh, M. Hadi Noori Skandari, H. Ahsani Tehrani and E .Tohidi, Space–
time Chebyshev spectral collocation method for nonlinear time‐fractional Burgers equations based on
efficient basis functions, Mathematical Methods in the Applied Sciences, 44(5) (2020), 4117–4136.
[24] N. Peykrayegan, M. Ghovatmand and M. H. Noori Skandari, On the convergence of Jacobi‐Gauss
collocation method for linear fractional delay differential equations, Mathematical Methods in the
Applied Sciences, 44(2) (2020), 2237–2253.
[25] N. Peykrayegan, M. Ghovatmand and M. H. N. Skandari, An efficient method for linear fractional
delay integro-differential equations. Computational and Applied Mathematics, 40(7)(2021).
[26] P. Xiaobing, X. Yang, M. H. Noori Skandari, E. Tohidi and S. Shateyi, A new high accurate approximate
approach to solve optimal control problems of fractional order via efficient basis functions.
Alexandria Engineering Journal, 61(8)(2022), 5805–5818.
[27] Y. Yang and H. M. Noori Skandari, Pseudospectral method for fractional infinite horizon optimal
control problems. Optimal Control Applications and Methods, 41(6)(2020), 2201–2212.
[28] Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang and M. Xiao, Propagation Dynamics of a Light
Beam in a Fractional Schrödinger Equation, Physical Review Letters, 115(18)(2015), 180403.
[29] D. del Castillo-Negrete and L. Chac on, Parallel heat transport in integrable and chaotic magnetic
fields, Phys. Plasmas, 19(5) (2012), 056112.
[30] Y. Zhang, M. M. Meerschaert and R. M. Neupauer, Backward fractional advection dispersion model
for contaminant source prediction, Water Resour. Res., 52(2016), 2462–2473.
[31] L. P lociniczak, Analytical studies of a time-fractional porous medium equation. Derivation, approximation
and applications, Commun. Nonlinear Sci. Numer. Simul., 24(2015), 169–183.
[32] R. Metzler, J. H. Jeon, A. G. Cherstvy and E. Barkai, Anomalous diffusion models and their properties:
non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking, Phys.
Chem. Chem. Phys., 16(2014), 24128–24164.
[33] M. Yamamoto, Asymptotic expansion of solutions to the dissipative equation with fractional Laplacian,
SIAM J. Math. Anal., 44(2012), 3786–3805.
[34] P. Constantin and J. Wu, Behavior of solutions of 2D Quasi-geostrophic equations, SIAM J. Math.
Anal., 30(1999), 937–948.
[35] A. de Pablo, F. Quirós, A. Rodrıguez, & J. L. Vazquez, A fractional porous medium equation, Adv.
Math., 226(2011), 1378–1409.
[36] S. Duo, L. Ju and Y. Zhang, A fast algorithm for solving the space–time fractional diffusion equation.
Computers & Mathematics with Applications, 75(6)(2018), 1929–1941.
[37] H. Ye, F. Liu, V. Anh and I. Turner, Maximum principle and numerical method for the multi-term
time–space Riesz–Caputo fractional differential equations, Applied Mathematics and Computation,
227 (2014), 531-540.
[38] M. Stynes, E. Oriordan and J. L. Gracia, Error Analysis of a Finite Difference Method on Graded
Meshes for a Time-Fractional Diffusion Equation, SIAM Journal on Numerical Analysis, 55(2017),
[39] C. Li and F. Zeng, Numerical methods for fractional calculus, Boca Raton, FL: CRC Press, Taylor &
Francis Group, 2015.
[40] M. Chen, W. Deng and Y. Wu, Superlinearly convergent algorithms for the two-dimensional spacetime
Caputo-Riesz fractional diffusion equation, Applied Numerical Mathematics, 70(2013), 22-41.