An $ {\varepsilon} $-approximation solution of time-fractional diffusion equations based on Legendre polynomials

Yingchao Zhang; Yingzhen Lin; Yingchao Zhang; Yingzhen Lin

doi:10.3934/math.2024813

AIMS Mathematics

2024, Volume 9, Issue 6: 16773-16789. doi: 10.3934/math.2024813

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Research article

An $ {\varepsilon} $-approximation solution of time-fractional diffusion equations based on Legendre polynomials

Yingchao Zhang ^,,
Yingzhen Lin

Zhuhai Campus, Beijing Institute of Technology, Zhuhai, Guangdong 519085, China

Received: 25 March 2024 Revised: 25 April 2024 Accepted: 06 May 2024 Published: 14 May 2024
MSC : 35K57, 65M12, 65M15

The purpose of this paper is to establish a numerical method for solving time-fractional diffusion equations. To obtain the numerical solution, a binary reproducing kernel space is defined, and the orthonormal basis is constructed based on Legendre polynomials in this space. In order to find the $ {\varepsilon} $-approximation solution of time-fractional diffusion equations, which is defined in this paper, the algorithm is designed using the constructed orthonormal basis. Some numerical examples are analyzed to illustrate the procedure and confirm the performance of the proposed method. The results faithfully reveal that the presented method is considerably accurate and effective, as expected.
- Legendre polynomials,
- $ {\varepsilon} $-approximation solution,
- multiscale orthonormal basis,
- time-fractional diffusion equations
Citation: Yingchao Zhang, Yingzhen Lin. An $ {\varepsilon} $-approximation solution of time-fractional diffusion equations based on Legendre polynomials[J]. AIMS Mathematics, 2024, 9(6): 16773-16789. doi: 10.3934/math.2024813

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Abstract

The purpose of this paper is to establish a numerical method for solving time-fractional diffusion equations. To obtain the numerical solution, a binary reproducing kernel space is defined, and the orthonormal basis is constructed based on Legendre polynomials in this space. In order to find the $ {\varepsilon} $-approximation solution of time-fractional diffusion equations, which is defined in this paper, the algorithm is designed using the constructed orthonormal basis. Some numerical examples are analyzed to illustrate the procedure and confirm the performance of the proposed method. The results faithfully reveal that the presented method is considerably accurate and effective, as expected.

References

[1]	D. A. Benson, S. W. Wheatcraft, M. M. Meerschaert, The fractional-order governing equation of Levy motion, Water Resour. Res., 36 (2000), 1413–1423. https://doi.org/10.1029/2000WR900032 doi: 10.1029/2000WR900032
[2]	R. L. Magin, Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng., 32 (2004), 1–104. https://doi.org/10.1615/critrevbiomedeng.v32.i1.10 doi: 10.1615/critrevbiomedeng.v32.i1.10
[3]	M. Raberto, E. Scalas, F. Mainardi, Waiting-times and returns in high-frequency financial data: An empirical study, Phys. A, 314 (2002), 749–755. https://doi.org/10.1016/S0378-4371(02)01048-8 doi: 10.1016/S0378-4371(02)01048-8
[4]	A. M. Vargas, Finite difference method for solving fractional differential equations at irregular meshes, Math. Comput. Simul., 193 (2022), 204–216. https://doi.org/10.1016/j.matcom.2021.10.010 doi: 10.1016/j.matcom.2021.10.010
[5]	M. Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228 (2009), 7792–7804. https://doi.org/10.1016/j.jcp.2009.07.021 doi: 10.1016/j.jcp.2009.07.021
[6]	X. H. Yang, Q. Zhang, G. W. Yuan, Z. Q. Sheng, On positivity preservation in nonlinear finite volume method for multi-term fractional subdiffusion equation on polygonal meshes, Nonlinear Dyn., 92 (2018), 595–612. https://doi.org/10.1007/s11071-018-4077-5 doi: 10.1007/s11071-018-4077-5
[7]	B. T. Jin, R. Lazarov, Z. Zhou, Error estimates for a semidiscrete finite element method for fractional order parabolie equations, SIAM J. Numer. Anal., 51 (2013), 445–466. https://doi.org/10.1137/120873984 doi: 10.1137/120873984
[8]	A. Mardani, M. R. Hooshmandasl, M. H. Heydari, C. Cattani, A meshless method for solving the time fractional advection-diffusion equation with variable coefficients, Comput. Math. Appl., 75 (2018), 122–133. https://doi.org/10.1016/j.camwa.2017.08.038 doi: 10.1016/j.camwa.2017.08.038
[9]	M. Dehghan, M. Abbaszadeh, A. Mohebbi, Analysis of a meshless method for the time fractional diffusion-wave equation, Numer. Algor., 73 (2016), 445–476. https://doi.org/10.1007/s11075-016-0103-1 doi: 10.1007/s11075-016-0103-1
[10]	V. R. Hosseini, M. Koushki, W. N. Zou, The meshless approach for solving 2D variable-order time-fractional advection–diffusion equation arising in anomalous transport, Eng. Comput., 38 (2022), 2289–2307. https://doi.org/10.1007/s00366-021-01379-7 doi: 10.1007/s00366-021-01379-7
[11]	W. N. Zou, Y. Tang, V. R. Hosseini, The numerical meshless approachfor solving the 2D time nonlinear multiterm fractional cable equation in complex geometries, Fractals, 30 (2022), 2240170. https://doi.org/10.1142/S0218348X22401703 doi: 10.1142/S0218348X22401703
[12]	F. Zeng, F. Liu, C. Li, K. Burrage, I. Turner, V. Anh, Crank-Nicolson ADI spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation, SIAM J. Numer. Anal., 52 (2014), 2599–2622. https://doi.org/10.1137/130934192 doi: 10.1137/130934192
[13]	Z. Mao, G. E. Karniadakis, A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative, SIAM J. Numer. Anal., 56 (2018), 24–49. https://doi.org/10.1137/16M1103622 doi: 10.1137/16M1103622
[14]	Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533–1552. https://doi.org/10.1016/j.jcp.2007.02.001 doi: 10.1016/j.jcp.2007.02.001
[15]	E. H. Doha, A. H. Bhrawy, S. S. Ezz-Eldien, Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Appl. Math. Model., 35 (2011), 5662–5672. https://doi.org/10.1016/j.apm.2011.05.011 doi: 10.1016/j.apm.2011.05.011
[16]	N. Aronszain, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337–404. https://doi.org/10.1090/S0002-9947-1950-0051437-7 doi: 10.1090/S0002-9947-1950-0051437-7
[17]	M. Cui, B. Wu, Numerical analysis of reproducing kernel spaces, (Chinese), Beijing: Science Press, 2004.
[18]	Fazal-i-Haq, Siraj-ul-Islam, I. A. Tirmizi. A numerical technique for solution of the MRLW equation using quartic B-splines, Appl. Math. Model., 34 (2010), 4151–4160. https://doi.org/10.1016/j.apm.2010.04.012 doi: 10.1016/j.apm.2010.04.012
[19]	A. Iqbala, N. N. Abd Hamida, A. I. Md. Ismaila, Cubic B-spline Galerkin method for numerical solution of the coupled nonlinear Schrodinger equation, Math. Comput. Simula., 174 (2020), 32–44. http://dx.doi.org/10.1016/j.matcom.2020.02.017 doi: 10.1016/j.matcom.2020.02.017
[20]	A. Iqbala, N. N. Abd Hamida, A. I. Md. Ismaila, Soliton solution of Schrodinger equation using Cubic B-spline Galerkin method, Fluids, 4 (2019), 108. https://doi.org/10.3390/fluids4020108 doi: 10.3390/fluids4020108
[21]	Y. C. Zhang, H. B. Sun, Y. T. Jia, Y. Z. Lin, An algorithm of the boundary value problem based on multiscale orthogonal compact base, Appl. Math. Lett., 101 (2020), 106044. https://doi.org/10.1016/j.aml.2019.106044 doi: 10.1016/j.aml.2019.106044
[22]	Y. C. Zhang, L. C. Mei, Y. Z. Lin, A new method for high-order boundary value problems, Bound. Value Probl., 2021 (2021), 48. https://doi.org/10.1186/s13661-021-01527-4 doi: 10.1186/s13661-021-01527-4
[23]	Y. C. Zhang, L. C. Mei, Y. Z. Lin, A novel method for nonlinear boundary value problems based on multiscale orthogonal base, Int. J. Comp. Methods, 18 (2021), 2150036. https://doi.org/10.1142/S0219876221500365 doi: 10.1142/S0219876221500365
[24]	F. Liu, P. Zhuang, Q. Liu, Numerical methods of fractional partial differential equations and applications, (Chinese), Beijing: Science Press, 2015.
[25]	C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral methods: Fundamentals in single domains, Heidelberg: Springer Berlin, 2006. https://doi.org/10.1007/978-3-540-30726-6
[26]	B. P. Moghaddam, J. A. T. Machado, A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations, Comput. Math. Appl., 73 (2017), 1262–1269. https://doi.org/10.1016/j.camwa.2016.07.010 doi: 10.1016/j.camwa.2016.07.010
[27]	H. Liao, Y. Zhang, Y. Zhao, H. Shi, Stability and convergence of modified du fort-frankel schemes for solving time-fractional subdiffusion equations, J. Sci. Comput., 61 (2014), 629–648. https://doi.org/10.1007/s10915-014-9841-1 doi: 10.1007/s10915-014-9841-1
[28]	Y. Jiang, J. Ma, High-order finite element methods for time-fractional partial differential equations, J. Comput. Appl. Math., 235 (2011), 3285–3290. https://doi.org/10.1016/j.cam.2011.01.011 doi: 10.1016/j.cam.2011.01.011
[29]	J. Liu, J. Zhang, X. D. Zhang, Semi-discretized numerical solution for time fractional convection-diffusion equation by RBF-FD, Appl. Math. Lett., 128 (2022), 107880. https://doi.org/10.1016/j.aml.2021.107880 doi: 10.1016/j.aml.2021.107880

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