Fractional multiwavelet methods for solving spatiotemporal fractional diffusion equations with non-smooth solutions
DOI: https://doi.org/10.3846/mma.2025.22650Abstract
This introduces a new method that effectively solves spatiotemporal fractional diffusion equation(FDE) using fractional Lagrange interpolation and fractional multiwavelets. The method effectively addresses situations with non-smooth solutions. The approach begins by discretizing the time variable t using the fractional piecewise parabolic Lagrange interpolation method. For the spatial variables, we construct fractional multiwavelets. Through the least residue method, we obtain approximate solutions, while also conducting convergence analysis. Numerical demonstrations validate the high accuracy achieved by the proposed method, notably showcasing the better approximation capability of fractional polynomials compared to their integer counterparts.
Keywords:
fractional calculus, fractional diffusion equation, least residue method, fractional multiwaveletsHow to Cite
Share
License
Copyright (c) 2025 The Author(s). Published by Vilnius Gediminas Technical University.
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
M. Ahmed, H.F. Ahmed and W.A. Hashem. A new operational matrices based-technique for fractional integro reaction-diffusion equation involving spatiotemporal variable-order derivative. Int. J. Anal. Appl., 22:209–209, 2024. https://doi.org/10.28924/2291-8639-22-2024-209
M.P. Alam, A. Khan and P. Roul. High-resolution numerical method for the timefractional fourth-order diffusion problems via improved quintic B-spline function. J. Appl. Math. Comput., pp. 1–39, 2024. https://doi.org/10.1007/s12190-024-02229-7
B. Alpert, G. Beylkin, D. Gines and L. Vozovoi. Adaptive solution of partial differential equations in multiwavelet bases. J. Comput. Phys., 182(1):149–190, 2002. https://doi.org/10.1006/jcph.2002.7160
A. Baihi, A. Kajouni, K. Hilal and H. Lmou. Laplace transform method for a coupled system of (p, q)-Caputo fractional differential equations. J. Appl. Math. Comput., pp. 1–20, 2024. https://doi.org/10.1007/s12190-024-02254-6
S. Bu and Y. Jeon. Higher-order predictor–corrector methods for fractional Benjamin–Bona–Mahony–Burgers’ equations. J. Appl. Math. Comput., pp. 1– 30, 2024. https://doi.org/10.1007/s12190-024-02223-z
Z. Chen, L. Wu and Y. Lin. Exact solution of a class of fractional integrodifferential equations with the weakly singular kernel based on a new fractional reproducing kernel space. Math. Meth. Appl. Sci., 41(10):3841–3855, 2018. https://doi.org/10.1002/mma.4870
W.H. Deng and J.S Hesthaven. Local discontinuous galerkin methods for fractional diffusion equations. ESAIM: Math. Model. Numer. Anal., 47(6):1845– 1864, 2013. https://doi.org/10.1051/m2an/2013091
H. Du and Z. Chen. A new method of solving the riesz fractional advection– dispersion equation with nonsmooth solution. Appl. Math. Lett., 152:109022, 2024. https://doi.org/10.1016/j.aml.2024.109022
H. Du, Z. Chen and T. Yang. A stable least residue method in reproducing kernel space for solving a nonlinear fractional integro-differential equation with a weakly singular kernel. Appl. Numer. Math., 157:210–222, 2020. https://doi.org/10.1016/j.apnum.2020.06.004
J. Duan and L. Jing. The solution of the time-space fractional diffusion equation based on the Chebyshev collocation method. Indian J. Pure Appl. Math., pp. 1–10, 2023. https://doi.org/10.1007/s13226-023-00495-y
Q. Fan, G.-Ch. Wu and H. Fu. A note on function space and boundedness of the general fractional integral in continuous time random walk. J. Nonlinear Math. Phys., pp. 1–8, 2022. https://doi.org/10.1007/s44198-021-00021-w
L.B Feng, P. Zhuang, F. Liu, I. Turner and Y. Gu. Finite element method for space-time fractional diffusion equation. Numer. Algorithms., 72:749–767, 2016. https://doi.org/10.1007/s11075-015-0065-8
R. Gupta and S. Kumar. Space-time pseudospectral method for the variableorder space-time fractional diffusion equation. Math. Sci., 18(3):419–436, 2024. https://doi.org/10.1007/s40096-023-00510-7
H. Jafari and V. Daftardar-Gejji. Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition. Appl. Math. Comput., 180(2):488–497, 2006. https://doi.org/10.1016/j.amc.2005.12.031
A.A. Kilbas, H.M. Srivastava and J.J Trujillo. Theory and Applications of Fractional Differential Equations, volume 204. Elsevier, Netherlands, 2006.
D. Kumar, J. Singh and S. Kumar. Analytic and approximate solutions of spacetime fractional telegraph equations via Laplace transform. Walailak J. Sci. Tech., 11(8):711–728, 2014.
M. Kumar, A. Jhinga and J.T. Majithia. Solutions of time-space fractional partial differential equations using Picard’s iterative method. J. Comput. Nonlinear Dyn., 19(3):031006, 02 2024. ISSN 1555-1415. https://doi.org/10.1115/1.4064553
S. Kumar, V. Gupta and D. Zeidan. An efficient collocation technique based on operational matrix of fractional-order Lagrange polynomials for solving the space-time fractional-order partial differential equations. Appl. Numer. Math., 204:249–264, 2024. ISSN 0168-9274. https://doi.org/10.1016/j.apnum.2024.06.014
M. Lakestani, B.N. Saray and M. Dehghan. Numerical solution for the weakly singular fredholm integro-differential equations using legendre multiwavelets. J. Comput. Appl. Math., 235(11):3291–3303, 2011. https://doi.org/10.1016/j.cam.2011.01.043
M. Li. Theory of Fractional Engineering Vibrations, volume 9. Walter de Gruyter GmbH & Co KG, Germany, 2021. https://doi.org/10.1515/9783110726152
M.M. Meerschaert and C. Tadjeran. Finite difference approximations for fractional advection–dispersion flow equations. J. Comput. Appl. Math., 172(1):65– 77, 2004. https://doi.org/10.1016/j.cam.2004.01.033
M.M. Meerschaert and C. Tadjeran. Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math., 56(1):80–90, 2006. https://doi.org/10.1016/j.apnum.2005.02.008
C.A. Monje, Y. Chen, B.M. Vinagre, D. Xue and V. Feliu-Batlle. FractionalOrder Systems and Controls: Fundamentals and Applications. Springer Science & Business Media, Germany, 2010.
P. Pandey, S. Kumar, J.F. Go´mez-Aguilar and D. Baleanu. An efficient technique for solving the space-time fractional reaction-diffusion equation in porous media. Chin. J. Phys., 68:483–492, 2020. https://doi.org/10.1016/j.cjph.2020.09.031
R.K. Pandey, O.P. Singh and V.K. Baranwal. An analytic algorithm for the space–time fractional advection–dispersion equation. Comput. Phys. Commun., 182(5):1134–1144, 2011. https://doi.org/10.1016/j.cpc.2011.01.015
X. Qiu and X. Yue. Solving time fractional partial differential equations with variable coefficients using a spatio-temporal meshless method. AIMS Math., 9:27150–27166, 2024. https://doi.org/10.3934/math.20241320
E.A. Rawashdeh. Numerical solution of fractional integro-differential equations by collocation method. Appl. Math. Comput., 176(1):1–6, 2006. https://doi.org/10.1016/j.amc.2005.09.059
R Roohi, M.H. Heydari, M. Aslami and M.R. Mahmoudi. A comprehensive numerical study of space-time fractional bioheat equation using fractional-order Legendre functions. Eur. Phys. J. Plus., 133(10):412, 2018. https://doi.org/10.1140/epjp/i2018-12204-x
J. Singh, P.K. Gupta and K.N. Rai. Solution of fractional bioheat equations by finite difference method and HPM. Math. Comput. Modell., 54(9-10):2316–2325, 2011. https://doi.org/10.1016/j.mcm.2011.05.040
H. Sun, A.L. Chang, W. Chen and Y. Zhang. Anomalous diffusion: fractional derivative equation models and applications in environmental flows. Sci. Sin-Phys. Mech. Astron., 45:8–22, 2015. https://doi.org/10.1360/SSPMA2015-00313
C. Tadjeran and M.M. Meerschaert. A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys., 220(2):813–823, 2007. https://doi.org/10.1016/j.jcp.2006.05.030
P.J. Torvik and R.L. Bagley. On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech., 1984. https://doi.org/10.1115/1.3167615
View article in other formats
CrossMark check
Published
Issue
Section
Copyright
Copyright (c) 2025 The Author(s). Published by Vilnius Gediminas Technical University.
License
This work is licensed under a Creative Commons Attribution 4.0 International License.