Exact and Approximate Solutions of A Fractional Diffusion Problem with Fixed Space Memory Length

Malgorzata Klimek; Tomasz Blaszczyk

doi:10.61822/amcs-2025-0022

.blurhash-client-img { display: none !important; }

Exact and Approximate Solutions of A Fractional Diffusion Problem with Fixed Space Memory Length

International Journal of Applied Mathematics and Computer Science

Volume 35 (2025): Issue 2 (June 2025)

By: Malgorzata Klimek and Tomasz Blaszczyk

Open Access

|Jun 2025

Abstract

We study a fractional differential diffusion equation, where the spatial derivative is expressed by the fractional differential operator with a fixed space memory length. The exact solution of the considered problem is presented, taking into account the homogeneous Dirichlet boundary conditions. Additionally, since the solution is in the form of a trigonometric series, we also present approximate solutions in the form of the truncated series. The accuracy of the approximation is controlled by the derived bound of a approximation error.

References

Alaroud, M., Aljarrah, H., Alomari, A.-K., Ishak, A. and Darus, M. (2024). Explicit and approximate series solutions for nonlinear fractional wave-like differential equations with variable coefficients, Partial Differential Equations in Applied Mathematics 10: 100680.
Search in Google Scholar Back to article
Bekir, A., Aksoy, E. and Cevikel, A.C. (2015). Exact solutions of nonlinear time fractional partial differential equations by sub-equation method, Mathematical Methods in the Applied Sciences 38(13): 2779–2784.
Search in Google Scholar Back to article
Blaszczyk, T., Bekus, K., Szajek, K. and Sumelka, W. (2021). On numerical approximation of the Riesz–Caputo operator with the fixed/short memory length, Journal of King Saud University-Science 33(1): 101220.
Search in Google Scholar Back to article
Chechkin, A.V., Seno, F., Metzler, R. and Sokolov, I.M. (2017). Brownian yet non-Gaussian diffusion: From superstatistics to subordination of diffusing diffusivities, Physical Review X 7: 021002.
Search in Google Scholar Back to article
Ciesielski, M. and Leszczynski, J. (2006). Numerical treatment of an initial-boundary value problem for fractional partial differential equations, Signal Processing 86(10): 2619–2631.
Search in Google Scholar Back to article
Das, S. (2009). Analytical solution of a fractional diffusion equation by variational iteration method, Computers & Mathematics with Applications 57(3): 483–487.
Search in Google Scholar Back to article
dos Santos, M.A. (2019). Analytic approaches of the anomalous diffusion: A review, Chaos, Solitons & Fractals 124: 86–96.
Search in Google Scholar Back to article
Echchaffani, Z., Aberqi, A., Karite, T. and Leiva, H. (2024). The existence of mild solutions and approximate controllability for nonlinear fractional neutral evolution systems, International Journal of Applied Mathematics and Computer Science 34(1): 15–2.
Search in Google Scholar Back to article
Elkott, I., Latif, M.S.A., El-Kalla, I.L. and Kader, A.H.A. (2023). Some closed form series solutions for the time-fractional diffusion-wave equation in polar coordinates with a generalized Caputo fractional derivative, Journal of Applied Mathematics and Computational Mechanics 22(2): 5–14.
Search in Google Scholar Back to article
Evangelista, L.R. and Lenzi, E.K. (2018). Fractional Diffusion Equations and Anomalous Diffusion, Cambridge University Press, Cambridge.
Search in Google Scholar Back to article
Gu, X.-M., Sun, H.-W., Zhao, Y.-L. and Zheng, X. (2021). An implicit difference scheme for time-fractional diffusion equations with a time-invariant type variable order, Applied Mathematics Letters 120: 107270.
Search in Google Scholar Back to article
Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.
Search in Google Scholar Back to article
Klimek, M. and Blaszczyk, T. (2024). Exact solutions of fractional oscillator eigenfunction problem with fixed memory length, Journal of Applied Mathematics and Computational Mechanics 23(1): 45–58.
Search in Google Scholar Back to article
Ledesma, C.T., Baca, J.V. and Sousa, J.V.d.C. (2022). Properties of fractional operators with fixed memory length, Mathematical Methods in the Applied Sciences 45(1): 49–76.
Search in Google Scholar Back to article
Ledesma, C.T., Rodríguez, J.A. and Sousa, J.V.d.C. (2023). Differential equations with fractional derivatives with fixed memory length, Rendiconti del Circolo Matematico di Palermo Series 2 72(1): 635–653.
Search in Google Scholar Back to article
Lu, Z. and Fan, W. (2025). A fast algorithm for multi-term time-space fractional diffusion equation with fractional boundary condition, Numerical Algorithms 98: 1171–1194.
Search in Google Scholar Back to article
Magin, R.L. (2006). Fractional Calculus in Bioengineering, Begell House Publisher, Danbury.
Search in Google Scholar Back to article
Magin, R.L., Abdullah, O., Baleanu, D. and Zhou, X.J. (2008). Anomalous diffusion expressed through fractional order differential operators in the bloch-torrey equation, Journal of Magnetic Resonance 190(2): 255–270.
Search in Google Scholar Back to article
Malinowska, A.B., Odzijewicz, T. and Poskrobko, A. (2023). Applications of the fractional Sturm–Liouville difference problem to the fractional diffusion difference equation, International Journal of Applied Mathematics and Computer Science 33(3): 349–359, DOI: 10.34768/amcs-2023-0025.
Search in Google Scholar Back to article
Meerschaert, M.M. and Tadjeran, C. (2004). Finite difference approximations for fractional advection-dispersion flow equations, Journal of Computational and Applied Mathematics 172(1): 65–77.
Search in Google Scholar Back to article
Metzler, R. and Klafter, J. (2000). The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Physics Reports 339(1): 1–77.
Search in Google Scholar Back to article
Podlubny, I. (1999). Fractional Differential Equations, Academic Press, San Diego.
Search in Google Scholar Back to article
Stempin, P., Pawlak, T.P. and Sumelka, W. (2023). Formulation of non-local space-fractional plate model and validation for composite micro-plates, International Journal of Engineering Science 192: 103932.
Search in Google Scholar Back to article
Sumelka, W., Blaszczyk, T. and Liebold, C. (2015). Fractional euler-bernoulli beams: Theory, numerical study and experimental validation, European Journal of Mechanics A/Solids 54: 243–251.
Search in Google Scholar Back to article
Sumelka, W., Łuczak, B., Gajewski, T. and Voyiadjis, G.Z. (2020). Modelling of AAA in the framework of time-fractional damage hyperelasticity, International Journal of Solids and Structures 206: 30–42.
Search in Google Scholar Back to article
Tian, W., Zhou, H. and Deng, W. (2015). A class of second order difference approximations for solving space fractional diffusion equations, Mathematics of Computation 84(294): 1703–1727.
Search in Google Scholar Back to article
Tsallis, C. and Lenzi, E. (2002). Anomalous diffusion: Nonlinear fractional Fokker–Planck equation, Chemical Physics 284(1): 341–347.
Search in Google Scholar Back to article
Voyiadjis, G., Akbari, E., Łuczak, B. and Sumelka, W. (2023). Towards determining an engineering stress-strain curve and damage of the cylindrical lithium-ion battery using the cylindrical indentation test, Batteries 9(4): 233.
Search in Google Scholar Back to article
Wang, H. and Du, N. (2013). A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations, Journal of Computational Physics 240: 49–57.
Search in Google Scholar Back to article
Wei, Y., Chen, Y., Cheng, S. and Wang, Y. (2017). A note on short memory principle of fractional calculus, Fractional Calculus and Applied Analysis 20(6): 1382–1404.
Search in Google Scholar Back to article
Yang, X., Wu, L. and Zhang, H. (2023). A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity, Applied Mathematics and Computation 457: 128192.
Search in Google Scholar Back to article
Zaslavsky, G.M. (2000). Chaos, fractional kinetics, and anomalous transport, Physics Reports 371(6): 461–580.
Search in Google Scholar Back to article
Zhuang, P., Liu, F., Turner, I. and Anh, V. (2016). Galerkin finite element method and error analysis for the fractional cable equation, Numerical Algorithms 72: 447–466.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.61822/amcs-2025-0022 | Journal eISSN: 2083-8492 | Journal ISSN: 1641-876X

Journal RSS Feed

Language: English

Page range: 311 - 328

Submitted on: Aug 5, 2024

Accepted on: Jan 28, 2025

Published on: Jun 24, 2025

Published by: University of Zielona Góra

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

fractional diffusion problem,

Related subjects:

© 2025 Malgorzata Klimek, Tomasz Blaszczyk, published by University of Zielona Góra
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 35 (2025): Issue 2 (June 2025)