International Journal of Differential Equations

Spectral Collocation Method for Fractional Differential/Integral Equations with Generalized Fractional Operator

Qinwu Xu and Zhoushun Zheng

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Abstract

Generalized fractional operators are generalization of the Riemann-Liouville and Caputo fractional derivatives, which include Erdélyi-Kober and Hadamard operators as their special cases. Due to the complicated form of the kernel and weight function in the convolution, it is even harder to design high order numerical methods for differential equations with generalized fractional operators. In this paper, we first derive analytical formulas for α-th(α>0) order fractional derivative of Jacobi polynomials. Spectral approximation method is proposed for generalized fractional operators through a variable transform technique. Then, operational matrices for generalized fractional operators are derived and spectral collocation methods are proposed for differential and integral equations with different fractional operators. At last, the method is applied to generalized fractional ordinary differential equation and Hadamard-type integral equations, and exponential convergence of the method is confirmed. Further, based on the proposed method, a kind of generalized grey Brownian motion is simulated and properties of the model are analyzed.

Article information

Source
Int. J. Differ. Equ., Volume 2019 (2019), Article ID 3734617, 14 pages.

Dates
Received: 30 September 2018
Accepted: 13 November 2018
First available in Project Euclid: 26 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1551150374

Digital Object Identifier
doi:10.1155/2019/3734617

Mathematical Reviews number (MathSciNet)
MR3898784

Citation

Xu, Qinwu; Zheng, Zhoushun. Spectral Collocation Method for Fractional Differential/Integral Equations with Generalized Fractional Operator. Int. J. Differ. Equ. 2019 (2019), Article ID 3734617, 14 pages. doi:10.1155/2019/3734617. https://projecteuclid.org/euclid.ijde/1551150374


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References

  • B. Baeumer, D. A. Benson, M. M. Meerschaert, and S. W. Wheat-craft, “Subordinated advection-dispersion equation for contaminant transport,” Water Resources Research, vol. 37, no. 6, pp. 1543–1550, 2001.
  • E. Barkai, R. Metzler, and J. Klafter, “From continuous time random walks to the fractional Fokker-Planck equation,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 61, no. 1, pp. 132–138, 2000.MR1736459
  • A. Blumen, G. Zumofen, and J. Klafter, “Transport aspects in anomalous diffusion: Lévy walks,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 40, no. 7, pp. 3964–3973, 1989.
  • J. P. Bouchaud and A. Georges, “Anomalous diffusion in dis-ordered media: statistical mechanisms, models and physical applications,” Physics Reports, vol. 195, no. 4-5, pp. 127–293, 1990.MR1081295
  • M. Raberto, E. Scalas, and F. Mainardi, “Waiting-times and returns in high-frequency financial data: an empirical study,” Physica A: Statistical Mechanics and its Applications, vol. 314, no. 1–4, pp. 749–755, 2002.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, New York, NY, USA, Elsevier, 2006.MR2218073
  • S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Inte-grals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.MR1347689
  • V. Kiryakova, “A brief story about the operators of the generalized fractional calculus,” Fractional Calculus & Applied Analysis. An International Journal for Theory and Applications, vol. 11, no. 2, pp. 203–220, 2008.MR2401328
  • S. L. Kalla, “On operators of fractional integration, I,” Mathematicae Notae, vol. 22, pp. 89–93, 1970/71.MR0333084
  • S. L. Kalla, “On operators of fractional integration, II,” Mathematicae Notae, vol. 25, pp. 29–35, 1976.MR0473730
  • V. S. Kiryakova, Generalized Fractional Calculus and Applications, Long-man & J. Wiley, Harlow, New York, NY, USA, 1994.MR1265940
  • A. Mura and F. Mainardi, “A class of self-similar stochastic processes with stationary increments to model anomalous diffusion in physics,” Integral Transforms and Special Functions, vol. 20, no. 3-4, pp. 185–198, 2009.MR2501791
  • O. P. Agrawal, “Some generalized fractional calculus operators and their applications in integral equations,” Fractional Calculus and Applied Analysis An International Journal for Theory and Applications, vol. 15, no. 4, pp. 700–711, 2012.MR2974327
  • G. Pagnini, “Erdélyi-Kober fractional diffusion,” Fractional Calculus and Applied Analysis An International Journal for Theory and Applications, vol. 15, no. 1, pp. 117–127, 2012.MR2872114
  • A. Mura and G. Pagnini, “Characterizations and simulations of a class of stochastic processes to model anomalous diffusion,” Journal of Physics A: Mathematical and General, vol. 41, no. 28, 285003, 22 pages, 2008.MR2430462
  • E. K. Lenzi, L. R. Evangelista, M. K. Lenzi, H. V. Ribeiro, and E. C. de Oliveira, “Solutions for a non-Markovian diffusion equa-tion,” Physics Letters A, vol. 374, no. 41, pp. 4193–4198, 2010.MR2684866
  • B. Al-Saqabi and V. S. Kiryakova, “Explicit solutions of fractional integral and differential equations involving Erdéryi-Kober operators,” Applied Mathematics and Computation, vol. 95, no. 1, pp. 1–13, 1998.MR1630272
  • L. A. Hanna and Y. F. Luchko, “Operational calculus for the Caputo-type fractional Erdélyi-Kober derivative and its appli-cations,” Integral Transforms and Special Functions, vol. 25, no. 5, pp. 359–373, 2014.MR3172049
  • Y. Xu, Z. He, and Q. Xu, “Numerical solutions of fractional advection-diffusion equations with a kind of new generalized fractional derivative,” International Journal of Computer Mathematics, vol. 91, no. 3, pp. 588–600, 2014.MR3217590
  • Y. Xu, Z. He, and O. P. Agrawal, “Numerical and analytical solu-tions of new generalized fractional diffusion equation,” Computers & Mathematics with Applications, vol. 66, no. 10, pp. 2019–2029, 2013.
  • Y. Xu and O. P. Agrawal, “Numerical solutions and analysis of diffusion for new generalized fractional Burgers equation,” Fractional Calculus and Applied Analysis An International Journal for Theory and Applications, vol. 16, no. 3, pp. 709–736, 2013.MR3071210
  • X. Li and C. Xu, “A space-time spectral method for the time fractional diffusion equation,” SIAM Journal on Numerical Analysis, vol. 47, no. 3, pp. 2108–2131, 2009.MR2519596
  • Q. Xu and J. S. Hesthaven, “Stable multi-domain spectral penalty methods for fractional partial differential equations,” Journal of Computational Physics, vol. 257, pp. 241–258, 2014.MR3129533
  • R. Mittal and S. Pandit, “Quasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynamical systems,” Engineering Computations, vol. 35, no. 5, pp. 1907–1931, 2018.
  • C. Li, F. Zeng, and F. Liu, “Spectral approximations to the frac-tional integral and derivative,” Fractional Calculus and Applied Analysis, vol. 15, no. 3, pp. 383–406, 2012.MR2944106
  • M. Zheng, F. Liu, I. Turner, and V. Anh, “A novel high order space-time spectral method for the time fractional Fokker-Planck equation,” SIAM Journal on Scientific Computing, vol. 37, no. 2, pp. A701–A724, 2015.MR3319851
  • F. Zeng, F. Liu, C. Li, K. Burrage, I. Turner, and V. Anh, “A Crank-Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation,” SIAM Journal on Numerical Analysis, vol. 52, no. 6, pp. 2599–2622, 2014.MR3274369
  • E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, “Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations,” Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, vol. 35, no. 12, pp. 5662–5672, 2011.MR2820942
  • M. Zayernouri and G. E. Karniadakis, “Exponentially accurate spectral and spectral element methods for fractional ODEs,” Journal of Computational Physics, vol. 257, pp. 460–480, 2014.MR3129544
  • X. Zhao and Z. Zhang, “Superconvergence points of fractional spectral interpolation,” SIAM Journal on Scientific Computing, vol. 38, no. 1, pp. A598–A613, 2016.MR3463700
  • F. Chen, Q. Xu, and J. S. Hesthaven, “A multi-domain spectral method for time-fractional differential equations,” Journal of Computational Physics, vol. 293, pp. 157–172, 2015.MR3342464
  • Z. Mao and J. Shen, “Efficient spectral-Galerkin methods for fractional partial differential equations with variable coefficients,” Journal of Computational Physics, vol. 307, pp. 243–261, 2016.MR3448207
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.MR1658022
  • J. Hadamard, “Essai sur l'étude des fonctions données par leur développement de Taylor,” Journal de Mathématiques Pures et Appliquées, vol. 4, pp. 101–186, 1892.
  • A. Erdélyi and H. Kober, “Some remarks on Hankel transforms,” Quarterly Journal of Mathematics, vol. 11, pp. 212–221, 1940.MR0003270
  • I. Dimovski, “Operational calculus for a class of differential operators,” Comptes Rendus De L Academie Bulgare Des Sciences, vol. 19, pp. 1111–1114, 1966.MR0205001
  • S. B. Yakubovich and Y. F. Luchko, The hypergeometric approach to integral transforms and convolutions, vol. 287 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht-Boston-London, 1994.MR1304259
  • Y. Luchko, “Operational rules for a mixed operator of the Erdélyi-Kober type,” Fractional Calculus and Applied Analysis, vol. 7, no. 3, pp. 339–364, 2004.MR2252570
  • G. Szegö, Orthogonal polynomials, American Mathematical Society, Providence, 1992.MR0106295
  • M. Zayernouri and G. E. Karniadakis, “Fractional Sturm-Liou-ville eigen-problems: theory and numerical approximation,” Journal of Computational Physics, vol. 252, pp. 495–517, 2013.MR3101519
  • S. Chen, J. Shen, and L.-L. Wang, “Generalized Jacobi functions and their applications to fractional differential equations,” Mathematics of Computation, vol. 85, no. 300, pp. 1603–1638, 2016.MR3471102
  • A. A. Kilbas, “Hadamard-type integral equations and fractional calculus operators,” in Singular integral operators, factorization and applications, vol. 142 of Oper. Theory Adv. Appl., pp. 175–188, Birkhäuser, Basel, 2003.MR2167367 \endinput