Numerical solutions of the space-time fractional diffusion equation via a gradient-descent iterative procedure
Authors
K. Tansri
- Department of Mathematics, School of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand.
A. Kittisopaporn
- Department of Mathematics, School of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand.
P. Chansangiam
- Department of Mathematics, School of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand.
Abstract
A one-dimensional space-time fractional diffusion equation describes anomalous diffusion on fractals in one dimension. In this paper, this equation
is discretized by finite difference schemes based on
the Grünwald-Letnikov approximation for Riemann-Liouville and Caputo's fractional derivatives. It turns out that the discretized equations can be put into a compact form, i.e., a linear system with a block lower-triangular coefficient matrix.
To solve the linear system, we formulate a matrix iterative algorithm based on gradient-descent technique.
In particular, we work out for the space fractional diffusion equation.
Theoretically, the proposed solver is always applicable with satisfactory convergence rate and error estimates. Simulations are presented numerically and graphically to illustrate the accuracy, the efficiency, and the performance of the algorithm, compared to other iterative procedures for linear systems.
Share and Cite
ISRP Style
K. Tansri, A. Kittisopaporn, P. Chansangiam, Numerical solutions of the space-time fractional diffusion equation via a gradient-descent iterative procedure, Journal of Mathematics and Computer Science, 31 (2023), no. 4, 353--366
AMA Style
Tansri K., Kittisopaporn A., Chansangiam P., Numerical solutions of the space-time fractional diffusion equation via a gradient-descent iterative procedure. J Math Comput SCI-JM. (2023); 31(4):353--366
Chicago/Turabian Style
Tansri, K., Kittisopaporn, A., Chansangiam, P.. "Numerical solutions of the space-time fractional diffusion equation via a gradient-descent iterative procedure." Journal of Mathematics and Computer Science, 31, no. 4 (2023): 353--366
Keywords
- Fractional diffusion equation
- Grünwald-Letnikov approximation
- gradient descent
- iterative method
- matrix norms
MSC
References
-
[1]
K. Assaleh, W. M. Ahmad, Modeling of speech signals using fractional calculus, 2007 9th International Symposium on Signal Processing and Its Applications, (2007), 1–4
-
[2]
S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, (2004)
-
[3]
A. V. Chechkin, R. Gorenflo, I. M. Sokolov, Fractional diffusion in inhomogeneous media, J. Phys. A, 38 (2005), L679– L684.
-
[4]
Z.-Q. Chen, M. M. Meerschaert, E. Nane, Space-time fractional diffusion on bounded domains, J. Math. Anal. Appl., 393 (2012), 479–488
-
[5]
L. Chen, R. H. Nochetto, E. Ot´arola, A. J. Salgado, A PDE approach to fractional diffusion: a posteriori error analysis, J. Comput. Phys., 293 (2015), 339–358
-
[6]
M. Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228 (2009), 7792–7804
-
[7]
K. Diethelm, J. M. Ford, N. J. Ford, M. Weilbeer, Pitfalls in fast numerical solvers for fractional differential equations, J. Comput. Appl. Math., 186 (2006), 482–503
-
[8]
K. Diethelm, N. J. Ford, A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31–52
-
[9]
F. Ding, T. Chen, Iterative least-squares solutions of coupled Sylvester matrix equations, Systems Control Lett., 54 (2005), 95–107
-
[10]
F. Ding, T. Chen, On iterative solutions of general coupled matrix equations, SIAM J. Control Optim., 44 (2006), 2269– 2284
-
[11]
R. Du, W. R. Cao, Z. Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Appl. Math. Model., 34 (2010), 2998–3007
-
[12]
R. Gorenflo, F. Mainardi, Random walk models for space-fractional diffusion processes, Fract. Calc. Appl. Anal., 1 (1998), 167–191
-
[13]
R. Gorenflo, F. Mainardi, Approximation of L´evy-Feller diffusion by random walk, Z. Anal. Anwendungen, 18 (1999), 231–246
-
[14]
R. Gorenflo, F. Mainardi, D. Moretti, P. Paradisi, Time fractional diffusion: a discrete random walk approach, Nonlinear Dynam., 29 (2002), 129–143
-
[15]
R. Gorenflo, F. Mainardi, D. Moretti, G. Pagnini, P. Paradisi, Discrete random walk models for space-time fractional diffusion, Chem. Phys., 284 (2002), 521–541
-
[16]
R. Hilfer, Applications of fractional calculus in physics, World Scientific Publishing, River Edge (2000)
-
[17]
J. Huang, F. Liu, The space–time fractional diffusion equation with Caputo derivatives, J. Appl. Math. Comput., 19 (2005), 179–190
-
[18]
I. S. Jesus, J. A. T. Machado, J. B. Cunha, Fractional electrical impedances in botanical elements, J. Vib. Control, 14 (2008), 1389–1402
-
[19]
A. Kittisopaporn, P. Chansangiam, Gradient-descent iterative algorithm for solving a class of linear matrix equations with applications to heat and Poisson equations, Adv. Difference Equ., 2020 (2020), 24 pages
-
[20]
V. V. Kulish, J. L. Lage, Application of fractional calculus to fluid mechanics, J. Fluids Eng., 124 (2002), 803–806
-
[21]
P. Kumar, O. P. Agrawal, Numerical scheme for the solution of fractional differential equations of order greater than one, J. Comput. Nonlinear Dyn., 1 (2006), 178–185
-
[22]
H. L¨ utkepohl, Handbook of matrices, John Wiley & Sons, Chichester (1996)
-
[23]
R. L. Magin, O. Abdullah, D. Baleanu, X. J. Zhou, Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation, J. Magn. Reson., 190 (2008), 255–270
-
[24]
M. M. Meerschaert, D. A. Benson, H.-P. Scheffler, B. Baeumer, Stochastic solution of space-time fractional diffusion equations, Phys. Rev. E, 65 (2002), 4 pages
-
[25]
K. Miller, B. Roos, An Introduction to the Fractional calculus and Fractional differential Equations, John Wiley & Sons, New York (1993)
-
[26]
J. Mua, B. Ahmad, S. Huang, Existence and regularity of solutions to time-fractional diffusion equations, Comput. Math. Appl., 73 (2017), 985–996
-
[27]
R. H. Nochetto, E. Ot´arola, A. J. Salgado, A PDE approach to fractional diffusion in general domains: a prior error analysis, Found. Comput. Math., 15 (2015), 733–791
-
[28]
I. Podlubny, I. Petr´aˇs, B. M. Vinagre, P. O’Leary, L. Dorˇc´ak, Analogue realizations of fractional-order controllers, Nonlinear Dynam., 29 (2002), 281–296
-
[29]
F. Santamaria, S. Wils, E. De Schutter, G. J. Augustine, Anomalous diffusion in Purkinje cell dendrites caused by spines, Neuron, 52 (2006), 635–648
-
[30]
R. Scherer, S. L. Kalla, Y. Tang, J. Huang, The Gr¨unwald-Letnikov method for fractional differential equations, Comput. Math. Appl., 62 (2011), 902–917
-
[31]
Z.-Z. Sun, X.Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), 193–209
-
[32]
C. Tadjeran, M. M. Meerschaert, A second-order accurate numerical method for the two dimensional fractional diffusion equation, J. Comput. Phys., 220 (2007), 813–823
-
[33]
S. Umarov, S. Steinberg, Variable order differential equations with piecewise constant order-function and diffusion with changing modes, Z. Anal. Anwend., 28 (2009), 431–450
-
[34]
D. M. Young, Iterative Methods for solving partial differential equations of elliptic type, Trans. Amer. Math. Soc., 76 (1954), 92–111
-
[35]
D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York-London (1971)