Numerical differentiation of fractional order derivatives based on inverse source problems of hyperbolic equations
Abstract
In this paper, we mainly study the numerical differentiation problem of computing the fractional order derivatives from noise data of a single variable function. Firstly, the numerical differentiation problem is reformulated into an inverse source problem of first order hyperbolic equation, and the corresponding solvability and the conditional stability are provided under suitable conditions. Then, four regularization methods are proposed to reconstruct the unknown source of hyperbolic equation which is the numerical derivative, and they are implemented by utilizing the finite dimensional expansion of source function and the superposition principle of hyperbolic equation. Finally, Numerical experiments are presented to show effectiveness of the proposed methods. It can be conclude that the proposed methods are very effective for small noise levels, and they are simpler and easier to be implemented than the previous PDEs-based numerical differentiation method based on direct and inverse problems of parabolic equations.
Keyword : numerical differentiation, fractional derivative, source inversion, hyperbolic equation, ill-posed problem
How to Cite
Wang, Z., Qiu, S., Rui, X., & Zhang, W. (2025). Numerical differentiation of fractional order derivatives based on inverse source problems of hyperbolic equations. Mathematical Modelling and Analysis, 30(1), 74–96. https://doi.org/10.3846/mma.2025.19339
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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T. Wei and Y.C. Hon. Numerical differentiation by radial basis functions approximation. Adv. Comput. Math., 27:247–272, 2007. https://doi.org/10.1007/s10444-005-9001-0
T. Wei and M. Li. High order numerical derivatives for one-dimensional scattered noisy data. Appl. Math. Comput., 175(2):1744–1759, 2006. https://doi.org/10.1016/j.amc.2005.09.018
Q. Yang, D. Chen, T. Zhao and Y. Chen. Fractional calculus in image processing: a review. Fract. Calc. Appl. Anal., 19(5):1222–1249, 2016. https://doi.org/10.1515/fca-2016-0063
W. Zhang, C. Wu, Z. Ruan and S. Qiu. A Jacobi spectral method for calculating fractional derivative based on mollification regularization. Asymptotic Analysis, 136(1):61–77, 2024. https://doi.org/10.3233/ASY-231869
Z. Zhao and L. You. A numerical differentiation method based on Legendre expansion with super order Tikhonov regularization. Appl. Math. Comput., 393:125811, 2021. https://doi.org/10.1016/j.amc.2020.125811
B. Berkowitz, H. Scher and S.E. Silliman. Anomalous transport in laboratoryscale heterogeneous porous media. Water Resources Research, 36(1):149–158, 2000. https://doi.org/10.1029/1999WR900295
F. Bozkurt, A. Yousef, T. Abdeljawad, A. Kalinli and Q.A. Mdallal. A fractionalorder model of COVID-19 considering the fear effect of the media and social networks on the community. Chaos, Solitons & Fractals, 152:111403, 2021. https://doi.org/10.1016/j.chaos.2021.111403
M. Cai and C. Li. Numerical approaches to fractional integrals and derivatives: A review. Mathematics, 8(1):43, 2020. https://doi.org/10.3390/math8010043
J. Cheng and M. Yamamoto. One new strategy for a priori choice of regularizing parameters in Tikhonov’s regularization. Inverse Problems, 16(4):L31–L38, 2000. https://doi.org/10.1088/0266-5611/16/4/101
J. Cullum. Numerical differentiation and regularization. SIAM Journal on Numerical Analysis, 8:254–265, 1971. https://doi.org/10.1137/0708026
V. Erturk, E. Godwe, D. Baleanu, P. Kumar, J. Asad and A. Jajarmi. Novel fractional-order Lagrangian to describe motion of beam on nanowire. Acta Physica Polonica A, 140(3):265–272, 2021. https://doi.org/10.12693/APhysPolA.140.265
Y. Ferdi, J.P. Herbeuval, A. Charef and B. Boucheham. R wave detection using fractional digital differentiation. ITBM-RBM, 24(5–6):273–280, 2003. https://doi.org/10.1016/j.rbmret.2003.08.002
K.O. Friedrichs. The identity of weak and strong extensions of differential operators. Trans. of the Amer. Math. Soc., 55:132–151, 1944. https://doi.org/10.1090/S0002-9947-1944-0009701-0
M. Ginoa, S. Cerbelli and H.E. Roman. Fractional diffusion equation and relaxation in complex viscoelastic materials. Phys. A, 191(1-4):449–453, 1992. https://doi.org/10.1016/0378-4371(92)90566-9
D.N. Hao, L.H. Chuong and D. Lesnic. Heuristic regularization methods for numerical differentiation. Comput. Math. Appl., 63(4):816–826, 2012. https://doi.org/10.1016/j.camwa.2011.11.047
Y. Hatano and N. Hatano. Dispersive transport of ions in column experiments: an explanation of long-tailed profiles. Water Resources Research, 34(5):1027–1034, 1998. https://doi.org/10.1029/98WR00214
C. Ionescu, A. Lopes, D. Copot, J.A.T. Machado and J.H.T. Bates. The role of fractional calculus in modeling biological phenomena: a review. Commun. Nonlinear Sci. Numer. Simul., 51:141–159, 2017. https://doi.org/10.1016/j.cnsns.2017.04.001
A. Kirsch. An introduction to the mathematical theory of inverse problems. Springer Science & Business Media, LLC, 2011. https://doi.org/10.1007/9781-4419-8474-6
I. Knowles and R. Wallace. A variational method for numerical differentiation. Numer. Math., 70:91–110, 1995. https://doi.org/10.1007/s002110050111
K. Kunisch and J. Zou. Iterative choices of regularization parameters in linear inverse problems. Inverse Problems, 14(5):1247–1264, 1998. https://doi.org/10.1088/0266-5611/14/5/010
M. Li, Y. Wang and L. Ling. Numerical Caputo differentiation by radial basis functions. J. Sci. Comput., 62(1):300–315, 2015. https://doi.org/10.1007/s10915-014-9857-6
Y. Li, C. Pan, X Meng, Y. Ding and H. Chen. A method of approximate fractional order differentiation with noise immunity. Chemometr. Intell. Lab., 144:31–38, 2015. https://doi.org/10.1016/j.chemolab.2015.03.009
J. Liu and M. Ni. A model function method for determining the regularizing parameter in potential approach for the recovery of scattered wave. Appl. Numer. Math., 58(8):1113–1128, 2008. https://doi.org/10.1016/j.apnum.2007.04.017
S. Lu and Y. Wang. The numerical differentiation of first and second order with Tikhonov regularization. Numer. Math. J. Chin. Univ., 26:62–74, 2004.
B.P. Moghaddam, A. Dabiri, Z.S. Mostaghim and Z. Moniri. Numerical solution of fractional dynamical systems with impulsive effects. International Journal of Modern Physics C, 34(1):2350013, 2023. https://doi.org/10.1142/S0129183123500134
B.P. Moghaddam, Z.S. Mostaghim, A.A. Pantelous and J. Machado. An integro quadratic spline-based scheme for solving nonlinear fractional stochastic differential equations with constant time delay. Commun. Nonlinear Sci. Numer. Simul., 92:105475, 2021. https://doi.org/10.1016/j.cnsns.2020.105475
Z. Moniri, B.P. Moghaddam and M.Z. Roudbaraki. An efficient and robust numerical solver for impulsive control of fractional chaotic systems. Journal of Function Spaces, 2023:9077924, 2023. https://doi.org/10.1155/2023/9077924
D.A. Murio. The Mollification Method and the Numerical Solution of Ill-Posed Problems. John Wiley & Sons Inc., New York, 1993. https://doi.org/10.1002/9781118033210
S. Qiu, Z. Wang and A. Xie. Multivariate numerical derivative by solving an inverse heat source problem. Inverse Problems in Science and Engineering, 26(8):1178–1197, 2018. https://doi.org/10.1080/17415977.2017.1386187
R. Qu. A new approach to numerical differentiation and integration. Math. Comput. Modelling, 24(10):55–68, 1996. https://doi.org/10.1016/S08957177(96)00164-1
A. Ramm and A. Smirnova. On stable numerical differentiation. Mathematics of Computation, 70(235):1131–1153, 2001. https://doi.org/10.1090/S0025-571801-01307-2
P.G. Surkov. Approximate calculation of the Caputo-type fractional derivative from inaccurate data. dynamical approach. Fract. Calc. Appl. Anal., 24(3):895– 922, 2021. https://doi.org/10.1515/fca-2021-0038
Y. Wang and J. Liu. On the edge detection of an image by numerical differentiations for gray function. Math. Method. Appl. Sci., 41(6):2466–2479, 2018. https://doi.org/10.1002/mma.4752
Y.B. Wang, X.Z. Jia and J. Cheng. A numerical differentiation method and its application to reconstruction of discontinuity. Inverse Problems, 18(6):1461–1476, 2002. https://doi.org/10.1088/0266-5611/18/6/301
Z. Wang, S. Qiu, Z. Ye and B. Hu. Posteriori selection strategies of regularization parameters for Lanczos’ generalized derivatives. Appl. Math. Lett., 111:106645, 2021. https://doi.org/10.1016/j.aml.2020.106645
Z. Wang, H. Wang and S. Qiu. A new method for numerical differentiation based on direct and inverse problems of partial differential equations. Appl. Math. Lett., 43:61–67, 2015. https://doi.org//10.1016/j.aml.2014.11.016
Z. Wang and R. Wen. Numerical differentiation for high orders by an integration method. J. Comput. Appl. Math., 234(3):941–948, 2010. https://doi.org/10.1016/j.cam.2010.01.056
Z. Wang and R. Wen. Lanczos’ method for numerical differentiation of the first and second order. Numerical Mathematics A Journal of Chinese Universities, 34(2):160–178, 2012.
T. Wei and Y.C. Hon. Numerical differentiation by radial basis functions approximation. Adv. Comput. Math., 27:247–272, 2007. https://doi.org/10.1007/s10444-005-9001-0
T. Wei and M. Li. High order numerical derivatives for one-dimensional scattered noisy data. Appl. Math. Comput., 175(2):1744–1759, 2006. https://doi.org/10.1016/j.amc.2005.09.018
Q. Yang, D. Chen, T. Zhao and Y. Chen. Fractional calculus in image processing: a review. Fract. Calc. Appl. Anal., 19(5):1222–1249, 2016. https://doi.org/10.1515/fca-2016-0063
W. Zhang, C. Wu, Z. Ruan and S. Qiu. A Jacobi spectral method for calculating fractional derivative based on mollification regularization. Asymptotic Analysis, 136(1):61–77, 2024. https://doi.org/10.3233/ASY-231869
Z. Zhao and L. You. A numerical differentiation method based on Legendre expansion with super order Tikhonov regularization. Appl. Math. Comput., 393:125811, 2021. https://doi.org/10.1016/j.amc.2020.125811