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Numerical schemes for integro-differential equations with Erdélyi-Kober fractional operator. (English) Zbl 1422.65456
The paper discusses numerical schemes for discretising the fractional differential operator of Erdélyi-Kober type which arise when self-similar solutions are considered for sub-diffusive evolution equations. The authors consider how the E-Y operator can be discretised, using a range of simple methods. Then they consider how finite difference methods can be applied to an integro-differential equation with E-K operator. They derive the truncation error and give a convergence theorem that are confirmed in an example.

65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
34A08 Fractional ordinary differential equations and fractional differential inclusions
34K37 Functional-differential equations with fractional derivatives
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[1] Bader, A-S; Kiryakova, VS, Explicit solutions of fractional integral and differential equations involving erdélyi-kober operators, Appl. Math. Comput., 95, 1-13, (1998) · Zbl 0942.45001
[2] Atkinson, KE, The numerical solution of an Abel integral equation by a product trapezoidal method, SIAM J. Numer. Anal., 11, 97-101, (1974) · Zbl 0241.65085
[3] Awotunde, AA; etal., Numerical schemes for anomalous diffusion of single-phase fluids in porous media, Commun. Nonlinear Sci. Numer. Simul., 39, 381-395, (2016)
[4] Baker, CTH, A perspective on the numerical treatment of Volterra equations, J. Comput. Appl. Math., 125, 217-249, (2000) · Zbl 0976.65121
[5] Baleanu, D., Güvenç, Z.B., Machado, J.T.: New trends in nanotechnology and fractional calculus applications. Springer (2010) · Zbl 1286.35060
[6] Baleanu, D; etal., Models and numerical methods, World Sci., 3, 10-16, (2012)
[7] Bronstein, I; etal., Transient anomalous diffusion of telomeres in the nucleus of Mammalian cells, Phys. Rev. Lett., 103, 018102, (2009)
[8] Brunner, H., Houwen, P.J.: The numerical solution of Volterra equations, vol. 3. Elsevier Science Ltd (1986) · Zbl 0611.65092
[9] Buckwar, E; Luchko, Y, Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations, J. Math. Anal. Appl., 227, 81-97, (1998) · Zbl 0932.58038
[10] Chen, C; Jiang, Y-L, Lie group analysis method for two classes of fractional partial differential equations, Commun. Nonlinear Sci. Numer. Simul., 26, 24-35, (2015)
[11] Chuvilgin, LG; Ptuskin, VS, Anomalous diffusion of cosmic rays across the magnetic field, Astron. Astrophys., 279, 278-297, (1993)
[12] Costa, FS; etal., Similarity solution to fractional nonlinear space-time diffusion-wave equation, J. Math. Phys., 56, 033507, (2015) · Zbl 06423086
[13] Demir, A; Kanca, F; Ozbilge, E, Numerical solution and distinguishability in time fractional parabolic equation, Bound. Value Probl., 2015, 1, (2015) · Zbl 1382.65291
[14] El Abd, A, A method for moisture measurement in porous media based on epithermal neutron scattering, Appl. Radiat. Isot., 105, 150-157, (2015)
[15] Erdélyi, A, On fractional integration and its application to the theory of Hankel transforms, Q. J. Math., 1, 293-303, (1940) · Zbl 0025.18602
[16] Ford, NJ; Simpson, AC, The numerical solution of fractional differential equations: speed versus accuracy, Numer. Algorithm., 26, 333-346, (2001) · Zbl 0976.65062
[17] Gazizov, R.K., Ibragimov, N.H., Lukashchuk, S.Y.: Nonlinear self-adjointness, conservation laws and exact solutions of time-fractional Kompaneets equations. Commun. Nonlinear Sci. Numer. Simul. 23 (2015) · Zbl 1351.35250
[18] Gazizov, R.K., Kasatkin, A.A., Lukashchuk, S.Y.: Symmetry properties of fractional diffusion equations. In: Physica Scripta 2009. T136, p 014016 (2009)
[19] Gorenflo, R., Luchko, Y., Mainardi, F.: Wright functions as scale-invariant solutions of the diffusion-wave equation. J. Comput. Appl. Math. 118 (2000) · Zbl 0973.35012
[20] Herrmann, R.: Towards a geometric interpretation of generalized fractional integrals—Erdélyi-Kober type integrals on R N, as an example. Fractional Calc. Appl. Anal. 17 (2014) · Zbl 1305.26019
[21] Hilfer, R.: Applications of fractional calculus in physics. World Scientific (2000) · Zbl 0998.26002
[22] Ibrahim, R.W., Momani, S.: On the existence and uniqueness of solutions of a class of fractional differential equations. J. Math. Anal. Appl. 334(1) (2007) · Zbl 1123.34302
[23] Kepten, E.: Uniform contraction-expansion description of relative centromere and telomere motion. Biophys. J. 109(7) (2015)
[24] Kiryakova, V., Al-Saqabi, B.: Explicit solutions to hyper-Bessel integral equations of second kind. Comput. Math. Appl. 37(1) (1999) · Zbl 0936.45003
[25] Kiryakova, V.S.: Generalized fractional calculus and applications. CRC Press (1993) · Zbl 0882.26003
[26] Kiryakova, V.S., Al-Saqabi, B.N.: Transmutation method for solving Erdélyi-Kober fractional differintegral equations. J. Math. Anal. Appl. 1997(1) · Zbl 0879.45005
[27] Kober, H.: On fractional integrals and derivatives. Q. J. Math. 11 (1940) · Zbl 0025.18502
[28] Küntz, M., Lavallée, P.: Experimental evidence and theoretical analysis of anomalous diffusion during water infiltration in porous building materials. J. Phys. D. Appl. Phys. 34(16) (2001)
[29] Li, C., Yi, Q., Chen, A.: Finite difference methods with non-uniform meshes for nonlinear fractional differential equations. J. Comput. Phys. 316 (2016) · Zbl 1349.65246
[30] Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47(3) (2009) · Zbl 1193.35243
[31] Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225(2) (2007) · Zbl 1126.65121
[32] Linz, P.: Analytical and numerical methods for Volterra equations, vol. 7. Siam (1985) · Zbl 0566.65094
[33] Luchko, Y.F., Srivastava, H.M.: The exact solution of certain differential equations of fractional order by using operational calculus. Comput. Math. Appl. 29 (8) (1995) · Zbl 0824.44011
[34] Luchko, Y.: Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation. Fractional Calc. Appl. Anal. 15(1) (2012) · Zbl 1276.26018
[35] Lyness, J., Ninham, B.W.: Numerical quadrature and asymptotic expansions. Math. Comput. 21(98) (1967) · Zbl 0178.18402
[36] Mainardi, F., Luchko, Y., Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation. In: arXiv preprint arXiv:0702419 (2007) · Zbl 06413561
[37] Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1) (2000) · Zbl 0984.82032
[38] Miller, K.S., Ross, B.: An introduction to the fractional calculus and fractional differential equations (1993) · Zbl 0789.26002
[39] Pablo, A., et al.: A fractional porous medium equation. Adv. Math. 226(2) (2011) · Zbl 1208.26016
[40] Pachepsky, Y., Timlin, D., Rawls, W.: Generalized Richards’ equation to simulate water transport in unsaturated soils. J. Hydrol. 272(1) (2003) · JFM 66.0522.01
[41] Pagnini, G.: Erdélyi-Kober fractional diffusion. Fractional Calc. Appl. Anal. 15(1) (2012) · Zbl 1276.26021
[42] Płociniczak, Ł.: Analytical studies of a time-fractional porous medium equation. Derivation, approximation and applications. Commun. Nonlinear Sci. Numer. Simul. 24(1), 169-183 (2015) · Zbl 0976.65121
[43] Płociniczak, Ł, Approximation of the erdélyi-kober operator with application to the time-fractional porous medium equation, SIAM J. Appl. Math., 74, 1219-1237, (2014) · Zbl 1309.26010
[44] Płociniczak, Ł, Diffusivity identification in a nonlinear time-fractional diffusion equation, Fractional Calc. Appl. Anal., 19, 843-866, (2016) · Zbl 1344.35165
[45] Płociniczak, Ł; Okrasińska, H, Approximate self-similar solutions to a nonlinear diffusion equation with time-fractional derivative, Physica D: Nonlinear Phenomena, 261, 85-91, (2013) · Zbl 1286.35060
[46] Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A.: Advances in fractional calculus, vol. 4, p 9. Springer (2007) · Zbl 1116.00014
[47] Sahadevan, R; Bakkyaraj, T, Invariant analysis of time fractional generalized Burgers and Korteweg-de Vries equations, J. Math. Anal. Appl., 393, 341-347, (2012) · Zbl 1245.35142
[48] Sahadevan, R; Bakkyaraj, T, Invariant subspace method and exact solutions of certain nonlinear time fractional partial differential equations, Fractional Calc. Appl. Anal., 18, 146-162, (2015) · Zbl 06413561
[49] Santos-León, JC, Asymptotic expansions for trapezoidal type product integration rules, J. Comput. Appl. Math., 91, 219-230, (1998) · Zbl 0930.65012
[50] Sneddon, I.N.: The use in mathematical physics of Erdelyi-Kober operators and of some of their generalizations. In: Fractional Calculus and its applications, pp 37-79. Springer (1975) · Zbl 1245.35142
[51] Sun, HG; etal., A fractal Richards equation to capture the non-Boltzmann scaling of water transport in unsaturated media, Adv. Water Resour., 52, 292-295, (2013)
[52] Wang, JR; Dong, XW; Zhou, Y, Analysis of nonlinear integral equations with erdélyi-kober fractional operator, Commun. Nonlinear Sci. Numer. Simul., 17, 3129-3139, (2012) · Zbl 1298.45011
[53] Weiss, M; Hashimoto, H; Nilsson, T, Anomalous protein diffusion in living cells as seen by fluorescence correlation spectroscopy, Biophys. J., 84, 4043-4052, (2003)
[54] Weiss, R, Product integration for the generalized Abel equation, Math. Comput., 26, 177-190, (1972) · Zbl 0257.45015
[55] Zhokh, AA; Trypolskyi, AI; Strizhak, PE, An investigation of anomalous time-fractional diffusion of isopropyl alcohol in mesoporous silica, Int. J. Heat Mass Trans., 104, 493-502, (2017)
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