Diffusion equation including a local fractional derivative and weighted inner product

• Autorzy: Cetinkaya Suleyman , Demir Ali

Identyfikatory

DOI

10.17512/jamcm.2022.1.02

• Języki publikacji: EN

Abstrakty

EN

In this research, we discuss the construction of the analytic solution of the homogenous initial boundary value problem including partial differential equations of fractional order. Since the homogenous initial boundary value problem involves a local fractional order derivative, it has classical initial and boundary conditions. By means of separation of the variables method and the inner product defined on L² [0, l], the solution is constructed in the form of a Fourier series including the exponential function. The illustrative examples present the applicability and influence of the separation of variables method on time fractional diffusion problems. Moreover, as the fractional order α tends to 1, the solution of the fractional diffusion problem tends to the solution of the diffusion problem which proves the accuracy of the solution.

Słowa kluczowe

EN

local fractional derivative; Dirichlet boundary conditions; spectral method; separation of variables; weighted inner product

PL

lokalna pochodna ułamkowa; warunek brzegowy Dirichleta; metoda spektralna; separacja zmiennych

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2022
• Tom: Vol. 21, nr 1
• Strony: 19--27
• Opis fizyczny: Bibliogr. 20 poz., rys.

Twórcy

autor

Cetinkaya Suleyman

• Faculty of Arts and Sciences, Kocaeli University, Kocaeli, Turkey

autor

Demir Ali

• Faculty of Arts and Sciences, Kocaeli University, Kocaeli, Turkey

Bibliografia

• [1] Baleanu, D., Fernandez, A., & Akgul, A. (2020). On a fractional operator combining proportional and classical differintegrals. Mathematics, 8(360).
• [2] Bisquert, J. (2005). Interpretation of a fractional diffusion equation with nonconserved probability density in terms of experimental systems with trapping or recombination. Physical Review E, 72, 011109.
• [3] Sene, N. (2019). Solutions of fractional diffusion equations and Cattaneo-Hristov diffusion model. International Journal of Analysis and Applications, 17(2), 191-207.
• [4] Aguilar, J.F.G., & Hern ́andez, M.M. (2014). Space-time fractional diffusion-advection equation with Caputo derivative. Abstract and Applied Analysis, 2014, Article ID 283019.
• [5] Naber, M. (2004). Distributed order fractional sub-diffusion. Fractals, 12(1), 23-32.
• [6] Nadal, E., Abisset-Chavanne, E., Cueto, E., & Chinesta, F. (2018). On the physical interpretation of fractional diffusion. Comptes Rendus Mecanique, 346, 581-589.
• [7] Zhang, W., & Yi, M. (2016). Sturm-Liouville problem and numerical method of fractional diffusion equation on fractals. Advances in Difference Equations, 2016(217).
• [8] Qureshi, S., Yusuf, A., & Aziz, S. (2021). Fractional numerical dynamics for the logistic population growth model under Conformable Caputo: a case study with real observations. Physica Scripta, 96(11).
• [9] Qureshi, S. (2020). Real life application of Caputo fractional derivative for measles epidemiological autonomous dynamical system. Chaos, Solitons & Fractals, 134, 109744.
• [10] Arqub, O.A., & Shawagfeh, N. (2019). Application of reproducing kernel algorithm for solving Dirichlet time-fractional diffusion-Gordon types equations in porous media. Journal of Porous Media, 22(4), 411-434.
• [11] Arqub, O.A., & Al-Smadi, M. (2020). An adaptive numerical approach for the solutions of fractional advection-diffusion and dispersion equations in singular case under Riesz’s derivative operator. Physica A: Statistical Mechanics and its Applications, 540, 123257(1-13).
• [12] Arqub, O.A., & Al-Smadi, M. (2020). Numerical solutions of Riesz fractional diffusion and advection-dispersion equations in porous media using iterative reproducing kernel algorithm. Journal of Porous Media, 23(8), 783-804.
• [13] Djennadi, S., Shawagfeh, N., & Arqub, O.A. (2021). A fractional Tikhonov regularization method for an inverse backward and source problems in the time-space fractional diffusion equations. Chaos, Solitons & Fractals, 150, 111127.
• [14] Yusuf, A., Aliyu, A.I. & Hashemi, M.S. (2018). Soliton solutions, stability analysis and conservation laws for the brusselator reaction diffusion model with time-and constant-dependent coefficients. The European Physical Journal Plus, 133(5), 1-11.
• [15] Yusuf, A., & Bayram, M. (2019). Invariant and simulation analysis to the time fractional Abrahams-Tsuneto reaction diffusion system. Physica Scripta, 94(12), 125005.
• [16] Aliyu, A.I., ̇Inc ̧, M., Yusuf, A., & Baleanu, D. (2018). Invariant subspace and lie symmetry analysis, exact solutions and conservation laws of a nonlinear reaction-diffusion Murray equation arising in mathematical biology. Journal of Advanced Physics, 7(2), 176-182.
• [17] Günerhan, H. (2020). Analytical and approximate solution of two-dimensional convection-diffusion problems. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 10(1), 73-77.
• [18] Cetinkaya, S., & Demir, A. (2019). The analytic solution of time-space fractional diffusion equation via new inner product with weighted function. Communications in Mathematics and Applications, 10(4), 865-873.
• [19] Cetinkaya, S., Demir, A., & Kodal Sevindir, H. (2020). The analytic solution of sequential space-time fractional diffusion equation including periodic boundary conditions. Journal of Mathematical Analysis, 11(1), 17-26.
• [20] Cetinkaya, S., Demir, A., & Kodal Sevindir, H. (2021). Solution of space-time-fractional problem by Shehu variational iteration method. Advances in Mathematical Physics, Article ID 5528928, (2021).