Fractional diffusion model generalised by the distributed-order operator involving variable diffusion coefficients

The diffusion process plays a crucial role in various fields, such as fluid dynamics, microorganisms, heat conduction and food processing. Since molecular diffusion usually takes place in complex materials and disordered media, there still exist many challenges in describing the diffusion process in the real world. Fractional calculus is a powerful tool for modelling complex physical processes due to its non-local property. This research generalises a fractional diffusion model by using the distributed-order operator in time and the Riesz fractional derivative in space. Moreover, variable diffusion coefficients are introduced to better capture the diffusion complexity. The fractional diffusion model is discretised by the finite element method in space. The approximation of the distributed-order operator is implemented by Simpson’s rule and the L2-1σ formula. A numerical example is provided to verify the effectiveness of the proposed numerical methods. This generalised fractional diffusion model may offer more insights into characterising diffusion behaviours in complex and disordered media. References A. A. Alikhanov. A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280 (2015), pp. 424–438. doi: 10.1016/j.jcp.2014.09.031 R. L. Burden, J. D. Faires, and A. M. Burden. Numerical Analysis. Cengage Learning, 2015. url: https://au.cengage.com/c/isbn/9781305253667/ W. Ding, S. Patnaik, S. Sidhardh, and F. Semperlotti. Applications of distributed-order fractional operators: A review. Entropy 23.1 (2021), p. 110. doi: 10.3390/e23010110 G. Gao, A. A. Alikhanov, and Z. Sun. The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations. J. Sci. Comput. 73.1 (2017), pp. 93–121. doi: 10.1007/s10915-017-0407-x P. Gouze, Y. Melean, T. Le Borgne, M. Dentz, and J. Carrera. Non-Fickian dispersion in porous media explained by heterogeneous microscale matrix diffusion. Water Resour. Res. 44.11 (2016), pp. 2276–2283. doi: 10.1029/2007WR006690 S. E. Maier, Y. Sun, and R. V. Mulkern. Diffusion imaging of brain tumors. NMR Biomed. 23.7 (2010), pp. 849–864. doi: 10.1002/nbm.1544 M. M. Meerschaert. Fractional calculus, anomalous diffusion, and probability. Fractional Dynamics: Recent Advances. Ed. by J. Klafter, S. C. Lim, and R. Metzler. World Sci., 2011, pp. 265–284. doi: 10.1142/9789814340595_0011 I. Podlubny. Fractional differential equations. New York: Academic Press, 1999. url: https://shop.elsevier.com/books/fractional-differential-equations/podlubny/978-0-12-558840-9 S. Qin, F. Liu, and I. W. Turner. A 2D multi-term time and space fractional Bloch-Torrey model based on bilinear rectangular finite elements. Commun. Nonlin. Sci. Numer. Sim. 56 (2018), pp. 270–286. doi: 10.1016/j.cnsns.2017.08.014 J. N. Reddy. An introduction to the finite element method. Vol. 1221. McGraw-Hill New York, 2004. url: https://www.accessengineeringlibrary.com/content/book/9781259861901 J. P. Roop. Variational solution of the fractional advection dispersion equation. PhD thesis. Clemson University, 2004. url: https://tigerprints.clemson.edu/arv_dissertations/1466/ J. A. Tenreiro Machado, M. F. Silva, R. S. Barbosa, I. S. Jesus, C. M. Reis, M. G. Marcos, and A. F. Galhano. Some applications of fractional calculus in engineering. Math. Prob. Eng. 2010, 639801 (2010). doi: 10.1155/2010/639801 T. Xu, F. Liu, S. Lü, and V. V. Anh. Finite difference/finite element method for two-dimensional time–space fractional Bloch–Torrey equations with variable coefficients on irregular convex domains. In: Comput. Math. App. 80.12 (2020), pp. 3173–3192. doi: 10.1016/j.camwa.2020.11.007 O. C. Zienkiewicz, R. L. Taylor, P. Nithiarasu, and J. Zhu. The finite element method. Vol. 3. McGraw-Hill London, 197