On the completely positive kernels for nonuniform meshes
Authors:
Yuanyuan Feng and Lei Li
Journal:
Quart. Appl. Math. 82 (2024), 801-817
MSC (2020):
Primary 45D05, 65R20
DOI:
https://doi.org/10.1090/qam/1684
Published electronically:
November 29, 2023
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Abstract: The complete positivity for convolutional kernels is an important property for the positivity property and asymptotic behaviors of Volterra equations. We investigate the discrete analogue of the complete positivity properties, especially for convolutional kernels on nonuniform meshes. Through an operation which we call pseudo-convolution, we introduce the complete positivity property for discrete kernels on nonuniform meshes and establish the criterion for the complete positivity. We then apply our theory to the L1 discretization of time fractional differential equations on nonuniform meshes.
References
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References
- Wei Cai, Computational methods for electromagnetic phenomena, Cambridge University Press, Cambridge, 2013. Electrostatics in solvation, scattering, and electron transport; With a foreword by Weng Cho Chew. MR 3027264
- Ph. Clément and J. A. Nohel, Abstract linear and nonlinear Volterra equations preserving positivity, SIAM J. Math. Anal. 10 (1979), no. 2, 365–388. MR 523852, DOI 10.1137/0510035
- Ph. Clément and J. A. Nohel, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal. 12 (1981), no. 4, 514–535. MR 617711, DOI 10.1137/0512045
- Bernard D. Coleman and Walter Noll, Foundations of linear viscoelasticity, Rev. Modern Phys. 33 (1961), 239–249. MR 158605, DOI 10.1103/RevModPhys.33.239
- Eduardo Cuesta, Christian Lubich, and Cesar Palencia, Convolution quadrature time discretization of fractional diffusion-wave equations, Math. Comp. 75 (2006), no. 254, 673–696. MR 2196986, DOI 10.1090/S0025-5718-06-01788-1
- Kai Diethelm, The analysis of fractional differential equations, Lecture Notes in Mathematics, vol. 2004, Springer-Verlag, Berlin, 2010. An application-oriented exposition using differential operators of Caputo type. MR 2680847, DOI 10.1007/978-3-642-14574-2
- G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra integral and functional equations, Encyclopedia of Mathematics and its Applications, vol. 34, Cambridge University Press, Cambridge, 1990. MR 1050319, DOI 10.1017/CBO9780511662805
- Natalia Kopteva, Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions, Math. Comp. 88 (2019), no. 319, 2135–2155. MR 3957889, DOI 10.1090/mcom/3410
- S. C. Kou and X. Sunney Xie, Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule, Phys. Rev. Lett. 93 (2004), no. 18, 180603.
- Dongfang Li, Chengda Wu, and Zhimin Zhang, Linearized Galerkin FEMs for nonlinear time fractional parabolic problems with non-smooth solutions in time direction, J. Sci. Comput. 80 (2019), no. 1, 403–419. MR 3954448, DOI 10.1007/s10915-019-00943-0
- Lei Li and Jian-Guo Liu, A generalized definition of Caputo derivatives and its application to fractional ODEs, SIAM J. Math. Anal. 50 (2018), no. 3, 2867–2900. MR 3809535, DOI 10.1137/17M1160318
- Lei Li and Jian-Guo Liu, A note on deconvolution with completely monotone sequences and discrete fractional calculus, Quart. Appl. Math. 76 (2018), no. 1, 189–198. MR 3733099, DOI 10.1090/qam/1479
- Lei Li and Jian-Guo Liu, Some compactness criteria for weak solutions of time fractional PDEs, SIAM J. Math. Anal. 50 (2018), no. 4, 3963–3995. MR 3828856, DOI 10.1137/17M1145549
- Lei Li, Jian-Guo Liu, and Jianfeng Lu, Fractional stochastic differential equations satisfying fluctuation-dissipation theorem, J. Stat. Phys. 169 (2017), no. 2, 316–339. MR 3704863, DOI 10.1007/s10955-017-1866-z
- Lei Li and Dongling Wang, Complete monotonicity-preserving numerical methods for time fractional ODEs, Commun. Math. Sci. 19 (2021), no. 5, 1301–1336. MR 4283533, DOI 10.4310/CMS.2021.v19.n5.a6
- Hong-lin Liao, Dongfang Li, and Jiwei Zhang, Sharp error estimate of the nonuniform L1 formula for linear reaction-subdiffusion equations, SIAM J. Numer. Anal. 56 (2018), no. 2, 1112–1133. MR 3790081, DOI 10.1137/17M1131829
- Hong-lin Liao, William McLean, and Jiwei Zhang, A discrete Grönwall inequality with applications to numerical schemes for subdiffusion problems, SIAM J. Numer. Anal. 57 (2019), no. 1, 218–237. MR 3904430, DOI 10.1137/16M1175742
- Hong-lin Liao, Nan Liu, and Pin Lyu, Discrete gradient structure of a second-order variable-step method for nonlinear integro-differential models, SIAM J. Numer. Anal. 61 (2023), no. 5, 2157–2181. MR 4648306, DOI 10.1137/22M1520050
- H.-l. Liao, T. Tang, and T. Zhou, Positive definiteness of real quadratic forms resulting from the variable-step approximation of convolution operators, arXiv preprint arXiv:2011.13383 (2020).
- Hong-lin Liao and Zhimin Zhang, Analysis of adaptive BDF2 scheme for diffusion equations, Math. Comp. 90 (2021), no. 329, 1207–1226. MR 4232222, DOI 10.1090/mcom/3585
- Yumin Lin and Chuanju Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225 (2007), no. 2, 1533–1552. MR 2349193, DOI 10.1016/j.jcp.2007.02.001
- R. J. Loy and R. S. Anderssen, Interconversion relationships for completely monotone functions, SIAM J. Math. Anal. 46 (2014), no. 3, 2008–2032. MR 3218823, DOI 10.1137/120891988
- Pin Lyu and Seakweng Vong, A symmetric fractional-order reduction method for direct nonuniform approximations of semilinear diffusion-wave equations, J. Sci. Comput. 93 (2022), no. 1, Paper No. 34, 25. MR 4483525, DOI 10.1007/s10915-022-02000-9
- W. McLean, V. Thomée, and L. B. Wahlbin, Discretization with variable time steps of an evolution equation with a positive-type memory term, J. Comput. Appl. Math. 69 (1996), no. 1, 49–69. MR 1391611, DOI 10.1016/0377-0427(95)00025-9
- Richard K. Miller and Alan Feldstein, Smoothness of solutions of Volterra integral equations with weakly singular kernels, SIAM J. Math. Anal. 2 (1971), 242–258. MR 287258, DOI 10.1137/0502022
- R. K. Miller, On Volterra integral equations with nonnegative integrable resolvents, J. Math. Anal. Appl. 22 (1968), 319–340. MR 227707, DOI 10.1016/0022-247X(68)90176-5
- Gianpietro Del Piero and Luca Deseri, On the concepts of state and free energy in linear viscoelasticity, Arch. Rational Mech. Anal. 138 (1997), no. 1, 1–35. MR 1463802, DOI 10.1007/s002050050035
- R. J. Plemmons, $M$-matrix characterizations. I. Nonsingular $M$-matrices, Linear Algebra Appl. 18 (1977), no. 2, 175–188. MR 444681, DOI 10.1016/0024-3795(77)90073-8
- René L. Schilling, Renming Song, and Zoran Vondraček, Bernstein functions, 2nd ed., De Gruyter Studies in Mathematics, vol. 37, Walter de Gruyter & Co., Berlin, 2012. Theory and applications. MR 2978140, DOI 10.1515/9783110269338
- O. Stenzel, The physics of thin film optical spectra, Springer, 2005.
- Martin Stynes, Eugene O’Riordan, and José Luis Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal. 55 (2017), no. 2, 1057–1079. MR 3639581, DOI 10.1137/16M1082329
- Tao Tang, Haijun Yu, and Tao Zhou, On energy dissipation theory and numerical stability for time-fractional phase-field equations, SIAM J. Sci. Comput. 41 (2019), no. 6, A3757–A3778. MR 4036095, DOI 10.1137/18M1203560
- Dennis G. Weis, Asymptotic behavior of some nonlinear Volterra integral equations, J. Math. Anal. Appl. 49 (1975), 59–87. MR 367596, DOI 10.1016/0022-247X(75)90162-6
- David Vernon Widder, The Laplace transform, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, NJ, 1941. MR 5923
- Q. Zhan, M. Zhuang, Z. Zhou, J.-G. Liu, and Q. H. Liu, Complete-Q model for poro-viscoelastic media in subsurface sensing: Large-scale simulation with an adaptive DG algorithm, IEEE Trans. Geosci. Remote Sensing 57 (2019), no. 7, 4591–4599.
- R. Zwanzig, Nonlinear generalized Langevin equations, J. Stat. Phys. 9 (1973), no. 3, 215–220.
- Robert Zwanzig, Nonequilibrium statistical mechanics, Oxford University Press, New York, 2001. MR 2012558
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Additional Information
Yuanyuan Feng
Affiliation:
School of Mathematical Sciences, Key Laboratory of MEA (Ministry of Education) and Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, People’s Republic of China
MR Author ID:
1265454
Email:
yyfeng@math.ecnu.edu.cn
Lei Li
Affiliation:
School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
Email:
leili2010@sjtu.edu.cn
Received by editor(s):
September 8, 2023
Received by editor(s) in revised form:
November 5, 2023
Published electronically:
November 29, 2023
Additional Notes:
This work was financially supported by the National Key R&D Program of China, Project Number 2021YFA1002800 and 2020YFA0712000. The work of the first author was partially sponsored by NSFC 12301283, Shanghai Sailing program 23YF1410300 and Science and Technology Commission of Shanghai Municipality (No. 22DZ2229014). The work of the second author was partially supported by NSFC 12371400 and 12031013, Shanghai Science and Technology Commission (Grant No. 21JC1403700, 20JC144100), the Strategic Priority Research Program of Chinese Academy of Sciences, Grant No. XDA25010403.
The second author is the corresponding author.
Article copyright:
© Copyright 2023
Brown University
