Numerical method for the estimation of the fractional parameters in the fractional mobile/immobile advection–diffusion model

References

• M.A. Abdelkawy, M.A. Zaky, A.H. Bhrawy, and D. Baleanu, Numerical simulation of time variable fractional order mobile–immobile advection–dispersion model, Rom. Rep. Phys. 67(3) (2015), pp. 1–19.
   Web of Science ®Google Scholar
• D. Baleanu, K. Diethelm, E. Scalas, and J.J. Trujillo, Fractional Calculus, World Scientific, New Jersey, 2012.
   Google Scholar
• J.-L. Battaglia, O. Cois, L. Puigsegur, and A. Oustaloup, Solving an inverse heat conduction problem using a non-integer identified model, Int. J. Heat Mass Transf. 44 (2001), pp. 2671–2680. doi: 10.1016/S0017-9310(00)00310-0
   Web of Science ®Google Scholar
• C.M. Chen, F. Liu, and K. Burrage, Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation, Appl. Math. Comput. 198(2) (2008), pp. 754–769.
   Web of Science ®Google Scholar
• S. Chen, F. Liu, X. Jiang, I. Turner, and K. Burrage, Fast finite difference approximation for identifying parameters in a two-dimensional space-fractional nonlocal model with variable diffusivity coefficients, SIAM J. Numer. Anal. 54(2) (2016), pp. 606–624. doi: 10.1137/15M1019301
   Web of Science ®Google Scholar
• J. Cheng, J. Nakagawa, M. Yamamoto, and T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Problems 25 (2009), p. 115002. doi: 10.1088/0266-5611/25/11/115002
   Web of Science ®Google Scholar
• X.-L. Ding and J.J. Nieto, Analytical solutions for coupling fractional partial differential equations with Dirichlet boundary conditions, Commun. Nonlinear Sci. Numer. Simul. 52 (2017), pp. 165–176. doi: 10.1016/j.cnsns.2017.04.020
   Web of Science ®Google Scholar
• H.R. Ghazizadeh, A. Azimi, and M. Maerefat, An inverse problem to estimate relaxation parameter and order of fractionality in fractional single-phase-lag heat equation, Int. J. Heat Mass Transf. 55 (2012), pp. 2095–2101. doi: 10.1016/j.ijheatmasstransfer.2011.12.012
   Web of Science ®Google Scholar
• R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
   Google Scholar
• P.-T. Hsu and Y.-H. Chu, An inverse non-Fourier heat conduction problem approach for estimating the boundary condition in electronic device, Appl. Math. Model. 28 (2004), pp. 639–652. doi: 10.1016/j.apm.2003.10.010
   Web of Science ®Google Scholar
• C.P. Li, Z. Zhao, and Y.Q. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl. 62(3) (2011), pp. 855–875. doi: 10.1016/j.camwa.2011.02.045
   Web of Science ®Google Scholar
• F. Liu and K. Burrage, Novel techniques in parameter estimation for fractional dynamical models arising from biological systems, Comput. Math. Appl. 62(3) (2011), pp. 822–833. doi: 10.1016/j.camwa.2011.03.002
   Web of Science ®Google Scholar
• F. Liu, K. Burrage, and N.A. Hamilton, Some novel techniques of parameter estimation for dynamical models in biological systems, IMA J. Appl. Math. 78(2) (2011), pp. 235–260. doi: 10.1093/imamat/hxr046
   Web of Science ®Google Scholar
• F. Liu, P. Zhuang, and K. Burrage, Numerical methods and analysis for a class of fractional advection dispersion models, Comput. Math. Appl. 64(10) (2012), pp. 2990–3007. doi: 10.1016/j.camwa.2012.01.020
   Web of Science ®Google Scholar
• F. Liu, P. Zhuang, and Q. Liu, Numerical Methods of Fractional Partial Differential Equations and Applications, Science Press, China, 2015 (in Chinese).
   Google Scholar
• H. Ma and Y. Yang, Jacobi spectral collocation method for the time variable-order fractional mobile–immobile advection–dispersion solute transport model, East Asian J. Appl. Math. 6(03) (2016), pp. 337–352. doi: 10.4208/eajam.141115.060616a
   Web of Science ®Google Scholar
• M.M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection–dispersion flow equations, J. Comput. Appl. Math. 172(1) (2004), pp. 65–77. doi: 10.1016/j.cam.2004.01.033
   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(1) (2000), pp. 1–77. doi: 10.1016/S0370-1573(00)00070-3
   Web of Science ®Google Scholar
• A. Mohebbi and M. Dehghan, High-order compact solution of the one-dimensional heat and advection–diffusion equations, Appl. Math. Model. 34 (2010), pp. 3071–3084. doi: 10.1016/j.apm.2010.01.013
   Web of Science ®Google Scholar
• D.A. Murio, Time fractional IHCP with Caputo fractional derivatives, Comput. Math. Appl. 56 (2008), pp. 2371–2381. doi: 10.1016/j.camwa.2008.05.015
   Web of Science ®Google Scholar
• K. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
   Google Scholar
• M.N. Özisik, Inverse Heat Transfer: Fundamentals and Applications, Taylor & Francis, New York, 2000.
   Google Scholar
• I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
   Google Scholar
• R. Schumer, D.A. Benson, M.M. Meerschaert, and B. Baeumer, Fractal mobile/immobile solute transport, Water Resour. Res. 39(10) (2003), p. 1296. doi: 10.1029/2003WR002141
   Web of Science ®Google Scholar
• W.F. Spotz and G.F. Carey, Extension of high-order compact schemes to time-dependent problems, Numer. Methods Partial Differential Equations 17(6) (2001), p. 657–672. doi: 10.1002/num.1032
   Web of Science ®Google Scholar
• Z.-Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56(2) (2006), pp. 193–209. doi: 10.1016/j.apnum.2005.03.003
   Web of Science ®Google Scholar
• H. Sun, W. Chen, C. Li, and Y. Chen, Finite difference schemes for variable-order time fractional diffusion equation, Int. J. Bifurc. Chaos 22(4) (2012), p. 1250085. doi: 10.1142/S021812741250085X
   Web of Science ®Google Scholar
• H. Wei, W. Chen, H. Sun, and X. Li, A coupled method for inverse source problem of spatial fractional anomalous diffusion equations, Inverse Probl. Sci. Eng. 18(7) (2010), pp. 945–956. doi: 10.1080/17415977.2010.492515
   Web of Science ®Google Scholar
• H. Wei, W. Chen, and H. Sun, Homotopy method for parameter determination of solute transport with fractional advection–dispersion equation, CMES Comput. Model. Eng. Sci. 100(2) (2014), pp. 85–103.
   Web of Science ®Google Scholar
• S. Wei, W. Chen, and Y.C. Hon, Characterizing time dependent anomalous diffusion process: A survey on fractional derivative and nonlinear models, Phys. A 462 (2016), pp. 1244–1251. doi: 10.1016/j.physa.2016.06.145
   Web of Science ®Google Scholar
• B. Yu and X.Y. Jiang, Numerical identification of the fractional derivatives in the two-dimensional fractional Cable equation, J. Sci. Comput. 68(1) (2016), pp. 252–272. doi: 10.1007/s10915-015-0136-y
   Web of Science ®Google Scholar
• B. Yu, X.Y. Jiang, and H.Y. Xu, A novel compact numerical method for solving the two-dimensional non-linear fractional reaction–subdiffusion equation, Numer. Algorithms 68 (2015), pp. 923–950. doi: 10.1007/s11075-014-9877-1
   Web of Science ®Google Scholar
• B. Yu, X.Y. Jiang, and H. Qi, An inverse problem to estimate an unknown order of a Riemann–Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid, Acta Mech. Sin. 31 (2015), pp. 153–161. doi: 10.1007/s10409-015-0408-7
   Web of Science ®Google Scholar
• B. Yu, X.Y. Jiang, and C. Wang, Numerical algorithms to estimate relaxation parameters and Caputo fractional derivative for a fractional thermal wave model in spherical composite medium, Appl. Math. Comput. 274 (2016), pp. 106–118.
   Web of Science ®Google Scholar
• Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation, Inverse Probl. 27 (2011), p. 035010.
   Web of Science ®Google Scholar
• Y. Zhang, D.A. Benson, and D.M. Reeves, Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications, Adv. Water. Resour. 32(4) (2009), pp. 561–581. doi: 10.1016/j.advwatres.2009.01.008
   Web of Science ®Google Scholar
• H. Zhang, F. Liu, M.S. Phanikumar, and M.M. Meerschaert, A novel numerical method for the time variable fractional order mobile–immobile advection–dispersion model, Comput. Math. Appl. 66(5) (2013), pp. 693–701. doi: 10.1016/j.camwa.2013.01.031
   Web of Science ®Google Scholar
• P. Zhuang, F. Liu, V. Anh, and I. Turner, Stability and convergence of an implicit numerical method for the non-linear fractional reaction–subdiffusion process, IMA J. Appl. Math. 74(2009), pp. 645–667. doi: 10.1093/imamat/hxp015
   Web of Science ®Google Scholar