A semi-analytic collocation method for space fractional parabolic PDE

References

• T. Aboelenen and H.M. El-Hawary, A high-order nodal discontinuous Galerkin method for a linearized fractional Cahn–Hilliard equation, Comput. Math. Appl. 73 (2017), pp. 1197–1217. doi: 10.1016/j.camwa.2016.07.018
   Web of Science ®Google Scholar
• T. Atanackovic, S. Pilipovic, B. Stankovic, and D. Zorica, Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes, Wiley-ISTE, London, 2014.
   Google Scholar
• A.K. Bahşı and S. Yalçinbaş, Numerical solutions and error estimations for the space fractional diffusion equation with variable coefficients via Fibonacci collocation method, SpringerPlus 5 (2016), p. 1375. doi: 10.1186/s40064-016-2853-6
   PubMed Web of Science ®Google Scholar
• A.H. Bhrawy and D. Baleanu, A spectral Legendre–Gauss–Lobatto collocation method for a space-fractional advection diffusion equations with variable coefficients, Rep. Math. Phys. 72 (2013), pp. 219–233. doi: 10.1016/S0034-4877(14)60015-X
   Web of Science ®Google Scholar
• P. Borwein, T. Erdéyi, and J. Zhang, Mütz systems and orthogonal Mütz-Legendre polynomials, Trans. Amer. Math. Soc. 342 (1994), pp. 523–542.
   Web of Science ®Google Scholar
• A.V. Chechkin, R. Gorenflo, and I.M. Sokolov, Fractional diffusion in inhomogeneous media, J. Phys. A 38 (2005), pp. 679–684. doi: 10.1088/0305-4470/38/42/L03
   Google Scholar
• M. Chen and W. Deng, Fourth order difference approximations for space Rriemann–Liouville derivatives based on weighted and shifted Lubich difference operators, Commun. Comput. Phys. 16(2) (2014), pp. 516–540. doi: 10.4208/cicp.120713.280214a
   Web of Science ®Google Scholar
• M. Chen and W. Deng, A second-order numerical method for two-dimensional two-sided space fractional convection diffusion equation, Appl. Math. Model. 38(13) (2014), pp. 3244–3259. doi: 10.1016/j.apm.2013.11.043
   Web of Science ®Google Scholar
• K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer, Berlin, 2010.
   Google Scholar
• K. Diethelm, N.J. Ford, A.D. Freed, and Y. Luchko, Algorithms for the fractional calculus: A selection of numerical methods, Comput. Methods Appl. Mech. Eng. 194 (2005), pp. 743–773. doi: 10.1016/j.cma.2004.06.006
   Web of Science ®Google Scholar
• S. Duoa, L. Ju, and Y. Zhang, A fast algorithm for solving the space–time fractional diffusion equation, Comp. Math. Appl. https://doi.org/10.1016/j.camwa.2017.04.008.
   Web of Science ®Google Scholar
• Z. Hao, K. Fan, W. Cao, and Z. Sun, A finite difference scheme for semilinear space-fractional diffusion equations with time delay, Appl. Math. Comput. 275 (2016), pp. 238–254.
   Web of Science ®Google Scholar
• M.M. Khader, On the numerical solutions for the fractional diffusion equation, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), pp. 2535–2542. doi: 10.1016/j.cnsns.2010.09.007
   Web of Science ®Google Scholar
• A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Vol. 204, Elsevier, Amsterdam, 2006.
   Google Scholar
• H.G. Lee, A second-order operator splitting Fourier spectral method for fractional-in-space reaction–diffusion equations, J. Comp. Appl. Math. 333 (2018), pp. 395–403. doi: https://doi.org/10.1016/j.cam.2017.09.007.
   Web of Science ®Google Scholar
• Y. Liu, Y. Yan, and M. Khan, Discontinuous Galerkin time stepping method for solving linear space fractional partial differential equations, Appl. Numer. Math. 115 (2017), pp. 200–213. doi: 10.1016/j.apnum.2017.01.009
   Web of Science ®Google Scholar
• R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339 (2000), pp. 1–77. doi: 10.1016/S0370-1573(00)00070-3
   Web of Science ®Google Scholar
• P. Mokhtary, F. Ghoreishi, and H.M. Srivastava, The Müntz–Legendre Tau method for fractional differential equations, Appl. Math. Model. 40 (2016), pp. 671–684. doi: 10.1016/j.apm.2015.06.014
   Web of Science ®Google Scholar
• A. Pedas and E. Tamme, Piecewise polynomial collocation for linear boundary value problems of fractional differential equations, J. Comput. Appl. Math. 236 (2012), pp. 3349–3359. doi: 10.1016/j.cam.2012.03.002
   Web of Science ®Google Scholar
• I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999.
   Google Scholar
• M.U. Rehman and R.A. Khan, Numerical solutions to initial and boundary value problems for linear fractional partial differential equations, Appl. Math. Model. 37 (2013), pp. 5233–5244. doi: 10.1016/j.apm.2012.10.045
   Web of Science ®Google Scholar
• S.Y. Reutskiy, The backward substitution method for multipoint problems with linear Volterra–Fredholm integro-differential equations of the neutral type, J. Comp. Appl. Math. 296 (2016), pp. 724–738. doi: 10.1016/j.cam.2015.10.013
   Web of Science ®Google Scholar
• S.Y. Reutskiy, A new semi-analytical collocation method for solving multi-term fractional partial differential equations with time variable coefficients, Appl. Math. Model. 45 (2017), pp. 238–254. doi: 10.1016/j.apm.2016.12.029
   Web of Science ®Google Scholar
• A. Saadatmandi and M. Dehghan, A tau approach for solution of the space fractional diffusion equation, Comput. Math. Appl. 62 (2011), pp. 1135–1142. doi: 10.1016/j.camwa.2011.04.014
   Web of Science ®Google Scholar
• X. Si, C. Wang, Y. Shen, and L. Zheng, Numerical method to initial-boundary value problems for fractional partial differential equations with time-space variable coefficients, Appl. Math. Model. 40 (2016), pp. 4397–4411. doi: 10.1016/j.apm.2015.11.039
   Web of Science ®Google Scholar
• E. Sousa, A second order explicit finite difference method for the fractional advection diffusion equation, Comp. Math. Appl. 64 (2013), pp. 3141–3152. doi: 10.1016/j.camwa.2012.03.002
   Web of Science ®Google Scholar
• H.G. Sun, W. Chen, C. Li, and Y.Q. Chen, Fractional differential models for anomalous diffusion, Physica A 389 (2010), pp. 2719–2724. doi: 10.1016/j.physa.2010.02.030
   Web of Science ®Google Scholar
• N.H. Sweilam, A.M. Nagy, and A.A. El-Sayed, Second kind shifted Chebyshev polynomials for solving space fractional order diffusion equation, Chaos Solitons Fract. 73 (2015), pp. 141–147. doi: 10.1016/j.chaos.2015.01.010
   Web of Science ®Google Scholar
• C. Tadjeran, M.M. Meerschaert, and H.P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys. 213 (2006), pp. 205–213. doi: 10.1016/j.jcp.2005.08.008
   Web of Science ®Google Scholar
• Y. Takeuchi, Y. Yoshimoto, and R. Suda, Second order accuracy finite difference methods for space-fractional partial differential equations, J. Comput. Appl. Math. 320 (2017), pp. 101–119. doi: 10.1016/j.cam.2017.01.013
   Web of Science ®Google Scholar
• W. Tian, H. Zhou, and W. Deng, A class of second order difference approximations for solving space fractional diffusion equations, Math. Comp. 84(294) (2015), pp. 1703–1727. doi: 10.1090/S0025-5718-2015-02917-2
   Web of Science ®Google Scholar
• Z. Weng, S. Zhai, and X. Feng, A Fourier spectral method for fractional-in-space Cahn–Hilliard equation, Appl. Math. Model. 42 (2017), pp. 462–477. doi: 10.1016/j.apm.2016.10.035
   Web of Science ®Google Scholar
• H. Xu, X. Jianga, and B. Yu, Numerical analysis of the space fractional Navier–Stokes equations, Appl. Math. Lett. 69 (2017), pp. 94–100. doi: 10.1016/j.aml.2017.02.006
   Web of Science ®Google Scholar
• H. Zhang, X. Jiang, and X. Yang, A time-space spectral method for the time-space fractional Fokker–Planck equation and its inverse problem, Appl. Math. Comput. 320 (2018), pp. 495–511.
   Web of Science ®Google Scholar