×

Fractional filter method for recovering the historical distribution for diffusion equations with coupling operator of local and nonlocal type. (English) Zbl 1486.65153

Summary: The purpose of this paper is to investigate the problem of recovering the historical distribution for diffusion equations in which the diffusion operators are described by the coupling of local and nonlocal type. The problem essentially arises in many real-world circumstances including the biological population dynamic where a population competes for the resources and diffuses by a combination of the Brownian and Lévy processes. We first design a typical example to illustrate the ill-posed nature of the problem. A fractional filter method is then proposed to achieve reliable approximations of the problem. The stability and convergence of the proposed method are gingerly analyzed. Four numerical examples, with the support from the finite difference method and the fast Fourier transform, are implemented to validate the theoretical results including the ill-posedness and the effect of regularization. The numerical results agree with the theoretical analysis.

MSC:

65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
65N20 Numerical methods for ill-posed problems for boundary value problems involving PDEs
65T50 Numerical methods for discrete and fast Fourier transforms
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
35R30 Inverse problems for PDEs
35R25 Ill-posed problems for PDEs
47J06 Nonlinear ill-posed problems
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abatangelo, N., Cozzi, M.: An elliptic boundary value problem with fractional nonlinearity. arXiv:2005.09515 (2020) · Zbl 1479.35443
[2] Abatangelo, N., Valdinoci, E.: Getting acquainted with the fractional Laplacian. In: Contemporary research in elliptic PDEs and related topics, volume 33 of Springer INdAM Ser., pp. 1-105. Springer, Cham (2019) · Zbl 1432.35216
[3] Andreu-Vaillo, F, Mazón, J.M., Rossi, J.D., Toledo-Melero, J.J.: Nonlocal diffusion problems Number, vol. 165. American Mathematical Soc. (2010) · Zbl 1214.45002
[4] Barles, G.; Chasseigne, E.; Ciomaga, A.; Imbert, C., Lipschitz regularity of solutions for mixed integro-differential equations, J. Diff. Equ., 252, 11, 6012-6060 (2012) · Zbl 1298.35033 · doi:10.1016/j.jde.2012.02.013
[5] Barles, G.; Chasseigne, E.; Ciomaga, A.; Imbert, C., Large time behavior of periodic viscosity solutions for uniformly parabolic integro-differential equations, Calc. Var Partial Diff. Equ., 50, 1-2, 283-304 (2014) · Zbl 1293.35037 · doi:10.1007/s00526-013-0636-2
[6] Biagi, S., Dipierro, S., Valdinoci, E., Vecchi, E.: A Faber-Krahn inequality for mixed local and nonlocal operators. arXiv:2104.00830 (2021) · Zbl 1473.35622
[7] Biagi, S., Vecchi, E., Dipierro, S., Valdinoci, E.: Semilinear elliptic equations involving mixed local and nonlocal operators. Proc. R. Soc. Edinburgh Sect. A: Math:1-31 (2020) · Zbl 1473.35622
[8] Blazevski, D., del Castillo-Negrete, D.: Local and nonlocal anisotropic transport in reversed shear magnetic fields: Shearless cantori and nondiffusive transport. Phys. Rev. E, 87(6):063106 (2013)
[9] Bucur, C., Valdinoci, E.: Nonlocal diffusion and applications, vol. 20. Springer (2016) · Zbl 1377.35002
[10] dosSantos, B.C., Oliva, S.M., Rossi, J.D.: A local/nonlocal diffusion model. Appl. Anal., 1-34 (2021)
[11] Cabré, X.; Serra, J., An extension problem for sums of fractional Laplacians and 1-D symmetry of phase transitions, Nonlinear Anal., 137, 246-265 (2016) · Zbl 1386.35430 · doi:10.1016/j.na.2015.12.014
[12] Caffarelli, L., Valdinoci, E.: A priori bounds for solutions of a nonlocal evolution PDE. In: Analysis and numerics of partial differential equations, volume 4 of Springer INdAM Ser., pp. 141-163. Springer, Milan (2013) · Zbl 1273.35067
[13] Cesbron, L.: On the derivation of non-local diffusion equations in confined spaces. PhD thesis, University of Cambridge (2017)
[14] del Castillo-Negrete, D, Chacon, L: Parallel heat transport in integrable and chaotic magnetic fields. Phys. Plasmas 19(5), 056112 (2012)
[15] del Castillo-Negrete, D., Chacon, L.: Local and nonlocal parallel heat transport in general magnetic fields. Phys. Rev. Lett. 106(19), 195004 (2011)
[16] Dell’Oro, F., Pata, V.: Second order linear evolution equations with general dissipation. Appl. Math. Optim., 1-41 (2019)
[17] Dipierro, S., Lippi, EP, Valdinoci, E: Linear theory for a mixed operator with N,eumann conditions. arXiv:2006.03850 (2020)
[18] Dipierro, S., Lippi, E.P., Valdinoci, E.: (Non) local logistic equations with Neumann conditions. arXiv:2101.02315 (2021)
[19] Dipierro, S.; Valdinoci, E., Description of an ecological niche for a mixed local/nonlocal dispersal: an evolution equation and a new Neumann condition arising from the superposition of Brownian and Lévy processes, Physica A: Stat. Mech. Appl., 575, 126052 (2021) · Zbl 1528.60037 · doi:10.1016/j.physa.2021.126052
[20] Dipierro, S.; Valdinoci, E.; Vespri, V., Decay estimates for evolutionary equations with fractional time-diffusion, J. Evol Equ., 19, 2, 435-462 (2019) · Zbl 1461.35049 · doi:10.1007/s00028-019-00482-z
[21] Fury, MA; Hughes, RJ, Regularization for a class of ill-posed evolution problems in Banach space, Semigroup Forum, 85, 2, 191-212 (2012) · Zbl 1267.47014 · doi:10.1007/s00233-011-9353-3
[22] Hào, DN; Duc, NV, Stability results for the heat equation backward in time, J. Math. Anal Appl., 353, 2, 627-641 (2009) · Zbl 1170.35097 · doi:10.1016/j.jmaa.2008.12.018
[23] Henry, B.I., Langlands, T.A.M., Straka, P.: An introduction to fractional diffusion. In: Complex Physical, Biophysical and Econophysical Systems, pp. 37-89. World Scientific (2010) · Zbl 1221.60047
[24] Hildebrand, M., Skødt, H., Showalter, K.: Spatial symmetry breaking in the Belousov-Zhabotinsky reaction with light-induced remote communication. Phys. Rev. Lett. 87(8), 088303 (2001)
[25] Jakobsen, ER; Karlsen, KH, Continuous dependence estimates for viscosity solutions of integro-PDEs, J. Differ. Equ., 212, 2, 278-318 (2005) · Zbl 1082.45008 · doi:10.1016/j.jde.2004.06.021
[26] Jakobsen, ER; Karlsen, KH, A “maximum principle for semicontinuous functions” applicable to integro-partial differential equations, NoDEA Nonlinear Differ. Equ. Appl., 13, 2, 137-165 (2006) · Zbl 1105.45006 · doi:10.1007/s00030-005-0031-6
[27] Khanh, TQ; Hoa, NV, On the axisymmetric backward heat equation with non-zero right hand side: regularization and error estimates, J. Comput. Appl. Math., 335, 156-167 (2018) · Zbl 1524.35755 · doi:10.1016/j.cam.2017.11.036
[28] Vo, AK; Truong, MTN; Duy, NHM; Tuan, NH, The Cauchy problem of coupled elliptic sine-Gordon equations with noise: analysis of a general kernel-based regularization and reliable tools of computing, Comput. Math. Appl., 73, 1, 141-162 (2017) · Zbl 1368.65217 · doi:10.1016/j.camwa.2016.11.001
[29] Le, T.M., Pham, Q.H., Luu, P.H.: On an asymmetric backward heat problem with the space and time-dependent heat source on a disk. J. Inverse Ill-Posed Probl. 27(1), 103-115 (2019) · Zbl 1411.35132
[30] Liu, Q.; Liu, F.; Turner, I.; Anh, V., Approximation of the Lévy-Feller advection-dispersion process by random walk and finite difference method, J. Comput. Phys., 222, 1, 57-70 (2007) · Zbl 1112.65006 · doi:10.1016/j.jcp.2006.06.005
[31] Minh, TL; Khieu, TT; Khanh, TQ; Vo, H-H, On a space fractional backward diffusion problem and its approximation of local solution, J. Comput. Appl. Math., 346, 440-455 (2019) · Zbl 1404.65142 · doi:10.1016/j.cam.2018.07.016
[32] Nam, PT; Trong, DD; Tuan, NH, The truncation method for a two-dimensional nonhomogeneous backward heat problem, Appl. Math. Comput., 216, 12, 3423-3432 (2010) · Zbl 1197.65131
[33] Nicola, E.M., Bär, M., Engel, H.: Wave instability induced by nonlocal spatial coupling in a model of the light-sensitive belousov-zhabotinsky reaction. Phys. Rev. E 73(6), 066225 (2006)
[34] Zheng, G-H; Zhang, Q-G, Recovering the initial distribution for space-fractional diffusion equation by a logarithmic regularization method, Appl. Math. Lett., 61, 143-148 (2016) · Zbl 1347.65152 · doi:10.1016/j.aml.2016.06.002
[35] Zheng, G-H; Zhang, Q-G, Determining the initial distribution in space-fractional diffusion by a negative exponential regularization method, Inverse Probl. Sci Eng., 25, 7, 965-977 (2017) · Zbl 1369.65137 · doi:10.1080/17415977.2016.1209750
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.