Płociniczak, Łukasz Numerical method for the time-fractional porous medium equation. (English) Zbl 1409.76091 SIAM J. Numer. Anal. 57, No. 2, 638-656 (2019). The aim of this paper is to establish a numerical scheme along with the study of its convergence for the time-fractional porous medium equation with Dirichlet boundary conditions on the half line. One of the main facts to be taken care of is the nonlocal and the nonlinear behavior of the governing equation. The approach followed is the reduction of the problem into a single one-dimensional Volterra integral equation for the self-similar solution and then applying a suitable discretization. The main difficulty stems from the non-Lipschitzian behavior of the nonlinearity of the corresponding integral equation. It is proved that there exists a family of schemes that is convergent for a large subset of the parameter space using the analysis of the recurrence relation for the error. An example of a method based on the midpoint quadrature is presented to illustrate the results. Reviewer: Abdallah Bradji (Annaba) Cited in 11 Documents MSC: 76M20 Finite difference methods applied to problems in fluid mechanics 76S05 Flows in porous media; filtration; seepage 35Q35 PDEs in connection with fluid mechanics 65R20 Numerical methods for integral equations 35R11 Fractional partial differential equations 45G10 Other nonlinear integral equations Keywords:porous medium equation; nonlinear diffusion; fractional derivative; finite difference method; Volterra equation PDFBibTeX XMLCite \textit{Ł. Płociniczak}, SIAM J. Numer. Anal. 57, No. 2, 638--656 (2019; Zbl 1409.76091) Full Text: DOI arXiv References: [1] M. Abramowitz and I. A. Stegun, {\it Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables}, vol. 55, Courier Corporation, Chelmsford, MA, 1965. · Zbl 0171.38503 [2] W. F. Ames and B. Pachpatte, {\it Inequalities for differential and integral equations}, vol. 197, Academic Press, Cambridge, MA, 1997. [3] F. Atkinson and L. Peletier, {\it Similarity profiles of flows through porous media}, Arch. Rational Mech. Anal., 42 (1971), pp. 369-379. · Zbl 0249.35043 [4] A. A. Awotunde, R. A. Ghanam, S. S. Al-Homidan, and N.-e. Tatar, {\it Numerical schemes for anomalous diffusion of single-phase fluids in porous media}, Commun. Nonlinear Sci. Numer. Sim., 39 (2016), pp. 381-395. · Zbl 1459.76114 [5] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, {\it Fractional calculus: Models and Numerical Methods}, vol. 3, World Scientific, Singapore, 2012. · Zbl 1248.26011 [6] A. Bhrawy, {\it A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations}, Numer. Algorithms, 73 (2016), pp. 91-113. · Zbl 1348.65143 [7] H. Brunner, {\it Volterra Integral Equations: An Introduction to Theory and Applications}, vol. 30, Cambridge University Press, Cambridge, MA, 2017. · Zbl 1376.45002 [8] E. Buckwar, {\it Iterative Approximation of the Positive Solutions of a Class of Nonlinear Volterra-type Integral Equations}, Logos Verlag, Berlin, 1997. · Zbl 0885.65154 [9] E. Buckwar, {\it On a nonlinear Volterra integral equation}, in Volterra Equations and Applications, CRC Press, Boca Raton, FL, 2000, pp. 157-162. · Zbl 0957.45005 [10] P. Bushell, {\it On a class of Volterra and Fredholm non-linear integral equations}, in Mathematical Proceedings of the Cambridge Philosophical Society, vol. 79, Cambridge University Press, Cambridge, MA, 1976, pp. 329-335. · Zbl 0316.45003 [11] L. Caffarelli and L. Silvestre, {\it An extension problem related to the fractional Laplacian}, Commun. Partial Differential Equations, 32 (2007), pp. 1245-1260. · Zbl 1143.26002 [12] J. Crank, {\it Free and Moving Boundary Problems}, Oxford University Press, Oxford, 1987. · Zbl 0629.35001 [13] N. Cusimano, F. del Teso, L. Gerardo-Giorda, and G. Pagnini, {\it Discretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions}, SIAM J. Numer. Anal., 56 (2018), pp. 1243-1272. · Zbl 1468.65138 [14] E. N. de Azevedo, P. L. de Sousa, R. E. de Souza, M. Engelsberg, M. d. N. d. N. Miranda, and M. A. Silva, {\it Concentration-dependent diffusivity and anomalous diffusion: A magnetic resonance imaging study of water ingress in porous zeolite}, Phys. Rev. E, 73 (2006), 011204. [15] A. de Pablo, F. Quiros, A. Rodriguez, and J. L. Vazquez, {\it A fractional porous medium equation}, Adv. Math., 226 (2011), pp. 1378-1409. · Zbl 1208.26016 [16] L. Dedić, M. Matić, and J. Pečarić, {\it On Euler midpoint formulae}, ANZIAM J., 46 (2005), pp. 417-438. · Zbl 1098.26012 [17] D. del Castillo-Negrete, B. Carreras, and V. Lynch, {\it Nondiffusive transport in plasma turbulence: A fractional diffusion approach}, Phys. Rev. Lett., 94 (2005), 065003. [18] F. del Teso, {\it Finite difference method for a fractional porous medium equation}, Calcolo, 51 (2014), pp. 615-638. · Zbl 1310.76115 [19] K. Diethelm and N. J. Ford, {\it Analysis of fractional differential equations}, J. Math. Anal. Appl., 265 (2002), pp. 229-248. · Zbl 1014.34003 [20] A. E.-G. El Abd and J. J. Milczarek, {\it Neutron radiography study of water absorption in porous building materials: Anomalous diffusion analysis}, J. Phys. D Appl. Phys., 37 (2004), 2305. [21] V. J. Ervin, N. Heuer, and J. P. Roop, {\it Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation}, SIAM J. Numer. Anal., 45 (2007), pp. 572-591. · Zbl 1141.65089 [22] F. J. Gaspar and C. Rodrigo, {\it Multigrid waveform relaxation for the time-fractional heat equation}, SIAM J. Sci. Comput., 39 (2017), pp. A1201-A1224. · Zbl 1371.65103 [23] G. Gripenberg, {\it Unique solutions of some Volterra integral equations}, Math. Scand., 48 (1981), pp. 59-67. · Zbl 0463.45002 [24] Y. Huang and A. Oberman, {\it Numerical methods for the fractional laplacian: A finite difference-quadrature approach}, SIAM J. Numer. Anal., 52 (2014), pp. 3056-3084. · Zbl 1316.65071 [25] V. S. Kiryakova, {\it Generalized Fractional Calculus and Applications}, CRC Press, Boca Raton, FL, 1993. · Zbl 0882.26003 [26] V. S. Kiryakova and B. N. Al-Saqabi, {\it Transmutation method for solving Erdélyi-Kober fractional differintegral equations}, J. Math. Anal. Appl., 211 (1997), pp. 347-364. · Zbl 0879.45005 [27] J. Klafter, S. Lim, and R. Metzler, {\it Fractional dynamics: Recent advances}, World Scientific, Singapore, 2012. · Zbl 1238.93005 [28] R. Klages, G. Radons, and I. M. Sokolov, {\it Anomalous transport: Foundations and applications}, John Wiley & Sons, New York, 2008. [29] M. Küntz and P. Lavallée, {\it Experimental evidence and theoretical analysis of anomalous diffusion during water infiltration in porous building materials}, J. Phys. D Appl. Phys., 34 (2001), 2547. [30] M. Kwaśnicki, {\it Ten equivalent definitions of the fractional laplace operator}, Fract. Calc. Appl. Anal., 20 (2017), pp. 7-51. · Zbl 1375.47038 [31] M. Levandowsky, B. White, and F. Schuster, {\it Random movements of soil amebas}, Acta Protozoologica, 36 (1997), pp. 237-248. [32] X. Li and C. Xu, {\it A space-time spectral method for the time fractional diffusion equation}, SIAM J. Numer. Anal., 47 (2009), pp. 2108-2131.ļearpage · Zbl 1193.35243 [33] P. Linz, {\it Analytical and Numerical Methods for Volterra Equations}, vol. 7, SIAM, Philadelphia, 1985. · Zbl 0566.65094 [34] R. Metzler and J. Klafter, {\it The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics}, J. Phys. A: Math. Gen., 37 (2004), R161. · Zbl 1075.82018 [35] B. P. Moghaddam and J. A. T. Machado, {\it 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), pp. 1262-1269. · Zbl 1412.65084 [36] W. Okrasiński, {\it On a nonlinear ordinary differential equation}, Ann. Polon. Math., 3(49), 1989, pp. 237-245. · Zbl 0685.34038 [37] Y. Pachepsky, D. Timlin, and W. Rawls, {\it Generalized Richards’ equation to simulate water transport in unsaturated soils}, J. Hydrology, 272 (2003), pp. 3-13. [38] Ł. Płociniczak, {\it Approximation of the Erdélyi-Kober operator with application to the time-fractional porous medium equation}, SIAM J. Appl. Math., 74 (2014), pp. 1219-1237. · Zbl 1309.26010 [39] Ł. Płociniczak, {\it Analytical studies of a time-fractional porous medium equation. derivation, approximation and applications}, Commun. Nonlinear Sci. Numer. Sim., 24 (2015), pp. 169-183. · Zbl 1440.76148 [40] Ł. Płociniczak and H. Okrasińska, {\it Approximate self-similar solutions to a nonlinear diffusion equation with time-fractional derivative}, Phys. D Nonlinear Phenomena, 261 (2013), pp. 85-91. · Zbl 1286.35060 [41] Ł. Płociniczak and H. Okrasińska-Płociniczak, {\it Numerical method for Volterra equation with a power-type nonlinearity}, Appl. Math. Comput., 337 (2018), pp. 452-460. · Zbl 1286.35060 [42] Ł. Płociniczak and M. Świtała, {\it Existence and uniqueness results for a time-fractional nonlinear diffusion equation}, J. Math. Anal. Appl., 462(2) (2018), pp. 1425-1434. · Zbl 1386.35453 [43] N. Ramos, J. Delgado, and V. De Freitas, {\it Anomalous diffusion during water absorption in porous building materials-experimental evidence}, in Defect and Diffusion Forum, vol. 273, Trans Tech Publ, 2008, pp. 156-161. [44] S. Schaufler, W. Schleich, and V. Yakovlev, {\it Scaling and asymptotic laws in subrecoil laser cooling}, Europhys. Lett., 39 (1997), 383. [45] I. N. Sneddon, {\it The use in mathematical physics of Erdélyi-Kober operators and of some of their generalizations}, in Fractional Calculus and its Applications, Springer, New York, 1975, pp. 37-79. · Zbl 0325.26005 [46] D. Stan, F. del Teso, and J. L. Vázquez, {\it Finite and infinite speed of propagation for porous medium equations with fractional pressure}, C. R. Math., 352 (2014), pp. 123-128. · Zbl 1284.35352 [47] H. Sun, M. M. Meerschaert, Y. Zhang, J. Zhu, and W. Chen, {\it A fractal Richards’ equation to capture the non-Boltzmann scaling of water transport in unsaturated media}, Adv. Water Res., 52 (2013), pp. 292-295. [48] T. Sungkaworn, M.-L. Jobin, K. Burnecki, A. Weron, M. J. Lohse, and D. Calebiro, {\it Single-molecule imaging reveals receptor-g protein interactions at cell surface hot spots}, Nature, 550 (2017), p. 543. [49] C. Tadjeran, M. M. Meerschaert, and H.-P. Scheffler, {\it A second-order accurate numerical approximation for the fractional diffusion equation}, J. Comput. Phys., 213 (2006), pp. 205-213. · Zbl 1089.65089 [50] J. L. Vázquez, {\it Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators}, Discrete Continuous Dynam. Syst., 7 (2014), pp. 857-885. · Zbl 1290.26010 [51] J. L. Vázquez, {\it The mathematical theories of diffusion: Nonlinear and fractional diffusion}, in Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions, Springer, New York, 2017, pp. 205-278. · Zbl 1492.35151 [52] S. B. Yuste and L. Acedo, {\it An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations}, SIAM J. Numer. Anal., 42 (2005), pp. 1862-1874. · Zbl 1119.65379 [53] F. Zeng, F. Liu, C. Li, K. Burrage, I. Turner, and V. Anh, {\it A Crank-Nicolson adi spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation}, SIAM J. Numer. Anal., 52 (2014), pp. 2599-2622. · Zbl 1382.65349 [54] A. A. Zhokh, A. A. Trypolskyi, and P. E. Strizhak, {\it Application of the time-fractional diffusion equation to methyl alcohol mass transfer in silica}, in Theory and Applications of Non-integer Order Systems, Springer, New York, 2017, pp. 501-510. · Zbl 1448.76169 [55] P. Zhuang, F. Liu, V. Anh, and I. Turner, {\it Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term}, SIAM J. Numer. Anal., 47 (2009), pp. 1760-1781. · Zbl 1204.26013 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. 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