The a posteriori error estimate in fractional differential equations using generalized Jacobi functions

Bo Tang; Huasheng Wang; Bo Tang; Huasheng Wang

doi:10.3934/math.20231486

AIMS Mathematics

2023, Volume 8, Issue 12: 29017-29041. doi: 10.3934/math.20231486

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The a posteriori error estimate in fractional differential equations using generalized Jacobi functions

Bo Tang ^{1
,
,},
Huasheng Wang ²

1.
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, Guangdong, China
2.
School of Mathematics and Computational Science, Wuyi University, Jiangmen, 529020, Guangdong, China

Received: 05 August 2023 Revised: 08 October 2023 Accepted: 11 October 2023 Published: 26 October 2023
MSC : 34A08, 42C10, 65L05, 65L10, 65L70

In this work, we study a posteriori error analysis of a general class of fractional initial value problems and fractional boundary value problems. A Petrov-Galerkin spectral method is adopted as the discretization technique in which the generalized Jacobi functions are utilized as basis functions for constructing efficient spectral approximations. The unique solvability of the weak problems is established by verifying the Babuška-Brezzi inf-sup condition. Then, we introduce some residual-type a posteriori error estimators, and deduce their efficiency and reliability in properly weighted Sobolev space. Numerical examples are given to illustrate the performance of the obtained error estimators.
Citation: Bo Tang, Huasheng Wang. The a posteriori error estimate in fractional differential equations using generalized Jacobi functions[J]. AIMS Mathematics, 2023, 8(12): 29017-29041. doi: 10.3934/math.20231486

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Abstract

In this work, we study a posteriori error analysis of a general class of fractional initial value problems and fractional boundary value problems. A Petrov-Galerkin spectral method is adopted as the discretization technique in which the generalized Jacobi functions are utilized as basis functions for constructing efficient spectral approximations. The unique solvability of the weak problems is established by verifying the Babuška-Brezzi inf-sup condition. Then, we introduce some residual-type a posteriori error estimators, and deduce their efficiency and reliability in properly weighted Sobolev space. Numerical examples are given to illustrate the performance of the obtained error estimators.

References

[1]	I. Babuška, W. C. Rheinboldt, A-posteriori error estimates for the finite element method, Int. J. Numer. Meth. Eng., 12 (1978), 1597–1615. https://doi.org/10.1002/nme.1620121010 doi: 10.1002/nme.1620121010
[2]	M. Benchohra, S. Hamani, S. K. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal.-Theor., 71 (2009), 2391–2396. https://doi.org/10.1016/j.na.2009.01.073 doi: 10.1016/j.na.2009.01.073
[3]	D. A. Benson, S. W. Wheatcraft, M. M. Meerschaert, The fractional-order governing equation of Lévy motion, Water Resour. Res., 36 (2000), 1413–1423. https://doi.org/10.1029/2000WR900032 doi: 10.1029/2000WR900032
[4]	C. Bernardi, Y. Maday, Spectral methods, In: Handbook numerical analysis, 5 (1997), 209–485. https://doi.org/10.1016/S1570-8659(97)80003-8
[5]	Y. Chen, X. Lin, Y. Huang, Error analysis of spectral approximation for space-time fractional optimal control problems with control and state constraints, J. Comput. Appl. Math., 413 (2022), 114293. https://doi.org/10.1016/j.cam.2022.114293 doi: 10.1016/j.cam.2022.114293
[6]	S. Chen, J. Shen, L. Wang, Generalized Jacobi functions and their applications to fractional differential equations, Math. Comput., 85 (2016), 1603–1638. http://doi.org/10.1090/mcom3035 doi: 10.1090/mcom3035
[7]	R. W. Ibrahim, Global controllability of a set of fractional differential equations, Miskolc Math. Notes, 12 (2011), 51–60. https://doi.org/10.18514/MMN.2011.259 doi: 10.18514/MMN.2011.259
[8]	R. Klages, G. Radons, I. M. Sokolov, Anomalous transport: Foundations and applications, Wiley, 2008.
[9]	D. Kusnezov, A. Bulgac, G. D. Dang, Quantum Lévy processes and fractional kinetics, Phys. Rev. Lett., 82 (1999), 1136–1139. https://doi.org/10.1103/PhysRevLett.82.1136 doi: 10.1103/PhysRevLett.82.1136
[10]	E. Lutz, Fractional transport equations for Lévy stable processes, Phys. Rev. Lett., 86 (2001), 2208–2211. https://doi.org/10.1103/PhysRevLett.86.2208 doi: 10.1103/PhysRevLett.86.2208
[11]	X. Li, C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM. J. Numer. Anal., 47 (2009), 2108–2131. https://doi.org/10.1137/080718942 doi: 10.1137/080718942
[12]	F. Mainardi, Y. Luchko, G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, 2007, arXiv: cond-mat/0702419v1. https://doi.org/10.48550/arXiv.cond-mat/0702419
[13]	B. B. Mandelbrot, J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422–437. https://doi.org/10.1137/1010093 doi: 10.1137/1010093
[14]	Z. Mao, S. Chen, J. Shen, Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations, Appl. Numer. Math., 106 (2016), 165–181. https://doi.org/10.1016/j.apnum.2016.04.002 doi: 10.1016/j.apnum.2016.04.002
[15]	W. Mao, Y. Chen, H. Wang, A-posteriori error estimations of the GJF-Petrov-CGalerkin methods for fractional differential equations, Comput. Math. Appl., 90 (2021), 159–170. https://doi.org/10.1016/j.camwa.2021.03.021 doi: 10.1016/j.camwa.2021.03.021
[16]	W. Mao, Y. Chen, H. Wang, A posteriori error estimations of the Petrov-Galerkin methods for fractional Helmholtz equations, Numer. Algor., 89 (2022), 1095–1127. https://doi.org/10.1007/s11075-021-01147-0 doi: 10.1007/s11075-021-01147-0
[17]	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
[18]	Z. Mao, J. Shen, Efficient spectral-Galerkin methods for fractional partial differential equations with variable coefficients, J. Comput. Phys., 307 (2016), 243–261. https://doi.org/10.1016/j.jcp.2015.11.047 doi: 10.1016/j.jcp.2015.11.047
[19]	R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1–77. https://doi.org/10.1016/S0370-1573(00)00070-3 doi: 10.1016/S0370-1573(00)00070-3
[20]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: Theory and applications, New York: Gordon and Breach Sciences Publishers, 1993.
[21]	L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pur. Appl. Math., 60 (2007), 67–112. https://doi.org/10.1002/cpa.20153 doi: 10.1002/cpa.20153
[22]	G. Su, L. Lu, B. Tang, Z. Liu, Quasilinearization technique for solving nonlinear Riemann-Liouville fractional-order problems, Appl. Math. Comput., 378 (2020), 125199. https://doi.org/10.1016/j.amc.2020.125199 doi: 10.1016/j.amc.2020.125199
[23]	G. Szegö, Orthogonal polynomials, American Mathematical Society, Providence, 1975.
[24]	B. Tang, J. Zhao, Z. Liu, Monotone iterative method for two-point fractional boundary value problems, Adv. Differ. Equ., 2018 (2018), 182. https://doi.org/10.1186/s13662-018-1632-9 doi: 10.1186/s13662-018-1632-9
[25]	B. Tang, Y, Chen, X. Lin, A posteriori error estimates of spectral Galerkin methods for multi-term time fractional diffusion equations, Appl. Math. Lett., 120 (2021), 107259. https://doi.org/10.1016/j.aml.2021.107259 doi: 10.1016/j.aml.2021.107259
[26]	H. Wang, Y. Chen, Y. Huang, W. Mao, A posteriori error estimates of the Galerkin spectral methods for space-time fractional diffusion equations, Adv. Appl. Math. Mech., 12 (2020), 87–100. https://doi.org/10.4208/aamm.OA-2019-0137 doi: 10.4208/aamm.OA-2019-0137
[27]	X. Ye, C. Xu, A posteriori error estimates of spectral method for the fractional optimal control problems with non-homogeneous initial conditions, AIMS Mathematics, 6 (2021), 12028–12050. https://doi.org/10.3934/math.2021697 doi: 10.3934/math.2021697
[28]	X. Ye, C. Xu, A posteriori error estimates for the fractional optimal control problems, J. Inequal. Appl., 2015 (2015), 141. https://doi.org/10.1186/s13660-015-0662-z doi: 10.1186/s13660-015-0662-z
[29]	M. Zayernouri, G. E. Karniadakis, Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation, J. Comput. Phys., 252 (2013), 495–517. https://doi.org/10.1016/j.jcp.2013.06.031 doi: 10.1016/j.jcp.2013.06.031
[30]	G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371 (2002), 461–580. https://doi.org/10.1016/S0370-1573(02)00331-9
[31]	F. Zeng, F. Liu, C. Li, K. Burrage, I. Turner, V. Anh, A Crank-Nicolson ADI spectral method for a 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
[32]	X. Zhao, L. Wang, Z. Xie, Sharp error bounds for Jacobi expansions and Gegenbauer-Gauss quadrature of analytic functions, SIAM J. Numer. Anal., 51 (2013), 1443–1469. https://doi.org/10.1137/12089421X doi: 10.1137/12089421X
[33]	M. Zheng, F. Liu, V. Anh, I. Turner, A high-order spectral method for the multi-term time-fractional diffusion equations, Appl. Math. Model., 40 (2016), 4970–4985. https://doi.org/10.1016/j.apm.2015.12.011 doi: 10.1016/j.apm.2015.12.011
[34]	M. Zheng, F. Liu, I. Turner, V. Anh, A novel high order space-time spectral method for the time fractional Fokker-Planck equation, SIAM J. Sci. Comput., 37 (2015), A701–A724. https://doi.org/10.1137/140980545 doi: 10.1137/140980545

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