An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations

In this paper, we consider the initial boundary value problem of the two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations. An alternating direction implicit (ADI) spectral method is developed based on Legendre spectral approximation in space and finite difference discretization in time. Numerical stability and convergence of the schemes are proved, the optimal error is $O(N^{-r}+\tau^2)$, where $N, \tau, r$ are the polynomial degree, time step size and the regularity of the exact solution, respectively. We also consider the non-smooth solution case by adding some correction terms. Numerical experiments are presented to confirm our theoretical analysis. These techniques can be used to model diffusion and transport of viscoelastic non-Newtonian fluids

Numerical Simulation for Solute Transport in Fractal Porous Media

A modified Fokker-Planck equation with continuous source for solute transport in fractal porous media is considered. The dispersion term of the governing equation uses a fractional-order derivative and the diffusion coefficient can be time and scale dependent. In this paper, numerical solution of the modified Fokker-Planck equation is proposed. The effects of different fractional orders and fractional power functions of time and distance are numerically investigated. The results show that motions with a heavy tailed marginal distribution can be modelled by equations that use fractional-order derivatives and/or time and scale dependent dispersivity

Novel numerical analysis of multi-term time fractional viscoelastic non-Newtonian fluid models for simulating unsteady MHD Couette flow of a generalized Oldroyd-B fluid

In recent years, non-Newtonian fluids have received much attention due to their numerous applications, such as plastic manufacture and extrusion of polymer fluids. They are more complex than Newtonian fluids because the relationship between shear stress and shear rate is nonlinear. One particular subclass of non-Newtonian fluids is the generalized Oldroyd-B fluid, which is modelled using terms involving multi-term time fractional diffusion and reaction. In this paper, we consider the application of the finite difference method for this class of novel multi-term time fractional viscoelastic non-Newtonian fluid models. An important contribution of the work is that the new model not only has a multi-term time derivative, of which the fractional order indices range from 0 to 2, but also possesses a special time fractional operator on the spatial derivative that is challenging to approximate. There appears to be no literature reported on the numerical solution of this type of equation. We derive two new different finite difference schemes to approximate the model. Then we establish the stability and convergence analysis of these schemes based on the discrete $H^1$ norm and prove that their accuracy is of $O(\tau+h^2)$ and $O(\tau^{\min\{3-\gamma_s,2-\alpha_q,2-\beta\}}+h^2)$, respectively. Finally, we verify our methods using two numerical examples and apply the schemes to simulate an unsteady magnetohydrodynamic (MHD) Couette flow of a generalized Oldroyd-B fluid model. Our methods are effective and can be extended to solve other non-Newtonian fluid models such as the generalized Maxwell fluid model, the generalized second grade fluid model and the generalized Burgers fluid model.Comment: 19 pages, 8 figures, 3 table

Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach

An unstructured mesh control volume method for two-dimensional space fractional diffusion equations with variable coefficients on convex domains

In this paper, we propose a novel unstructured mesh control volume method to deal with the space fractional derivative on arbitrarily shaped convex domains, which to the best of our knowledge is a new contribution to the literature. Firstly, we present the finite volume scheme for the two-dimensional space fractional diffusion equation with variable coefficients and provide the full implementation details for the case where the background interpolation mesh is based on triangular elements. Secondly, we explore the property of the stiffness matrix generated by the integral of space fractional derivative. We find that the stiffness matrix is sparse and not regular. Therefore, we choose a suitable sparse storage format for the stiffness matrix and develop a fast iterative method to solve the linear system, which is more efficient than using the Gaussian elimination method. Finally, we present several examples to verify our method, in which we make a comparison of our method with the finite element method for solving a Riesz space fractional diffusion equation on a circular domain. The numerical results demonstrate that our method can reduce CPU time significantly while retaining the same accuracy and approximation property as the finite element method. The numerical results also illustrate that our method is effective and reliable and can be applied to problems on arbitrarily shaped convex domains.Comment: 18 pages, 5 figures, 9 table

Anomalous diffusion in rotating Casson fluid through a porous medium

This paper investigates the space-fractional anomalous diffusion in unsteady Casson fluid through a porous medium, based on an uncoupled continuous time random walk. The influences of binary chemical reaction and activation energy between two horizontal rotating parallel plates are taken into account. The governing equations of motion are reduced to a set of nonlinear differential equations by time derivatives discretization and generalized transformation, which are solved by bvp4c and implicit finite difference method (IFDM). Stability and convergence of IFDM are proved and some numerical comparisons to the previous study are presented with excellent agreement. The effects of involved physical parameters such as fractional derivative parameter, rotation parameter and time parameter are presented and analyzed through graphs. Results indicate that the increase of fractional derivative parameter triggers concentration increase near the lower plate, while it causes a reduction near the upper plate. It is worth mentioning that the decrease of heat transfer rate on the plate is observed with the higher time parameter.</p

Effects of fractional mass transfer and chemical reaction on MHD flow in a heterogeneous porous medium

This paper presents a study on space fractional anomalous convective-diffusion and chemical reaction in the magneto-hydrodynamic fluid over an unsteady stretching sheet. The fractional diffusion model is derived from decoupled continuous time random walks in a heterogeneous porous medium. A novel transformation which features time finite difference is introduced to reduce the governing equations into ordinary differential ones in each time level. Numerical solutions are established by an implicit finite difference scheme. The stability and convergence of the method are analyzed. Results show that increasing fractional derivative parameter enhances concentration near the surface while an opposite phenomenon occurs far away from the wall. There is a reduction of mass transfer rate on the sheet with an increase in the fractional derivative parameter. Moreover, the numerical solutions are compared with exact solutions and good agreement has been observed.</p

A numerical method for the fractional Fitzhugh&ndash;Nagumo monodomain model

A fractional FitzHugh&ndash;Nagumo monodomain model with zero Dirichlet boundary conditions is presented, generalising the standard monodomain model that describes the propagation of the electrical potential in heterogeneous cardiac tissue. The model consists of a coupled fractional Riesz space nonlinear reaction-diffusion model and a system of ordinary differential equations, describing the ionic fluxes as a function of the membrane potential. We solve this model by decoupling the space-fractional partial differential equation and the system of ordinary differential equations at each time step. Thus, this means treating the fractional Riesz space nonlinear reaction-diffusion model as if the nonlinear source term is only locally Lipschitz. The fractional Riesz space nonlinear reaction-diffusion model is solved using an implicit numerical method with the shifted Grunwald&ndash;Letnikov approximation, and the stability and convergence are discussed in detail in the context of the local Lipschitz property. Some numerical examples are given to show the consistency of our computational approach. References B. Baeumer, M. Kovaly, and M. M. Meerschaert, Fractional reproduction-dispersal equations and heavy tail dispersal kernels, Bulletin of Mathematical Biology 69:2281&ndash;2297, 2007. doi:10.1007/s11538-007-9220-2 B. Baeumer, M. Kovaly, and M. M. Meerschaert, Numerical solutions for fractional reaction-diffusion equations, Computers and Mathematics with Applications 55:2212&ndash;2226, 2008. doi:10.1016/j.camwa.2007.11.012 N. Badie and N. Bursac, Novel micropatterned cardiac cell cultures with realistic ventricular microstructure, Biophys J 96:3873&ndash;3885, 2009. doi:10.1016/j.bpj.2009.02.019 A. Bueno-Orovio, D. Kay, K. Burrage, Fourier spectral methods for fractional-in-space reaction-diffusion equations, Technical report, University of Oxford, 2013. A. Bueno-Orovioy, D. Kay, V. Grau, B. Rodriguez and K. Burrage, Fractional dffusion models of electrical propagation in cardiac tissue: electrotonic effects and the modulated dispersion of repolarization, Technical report, University of Oxford, 2013. K. F. Decker, J. Heijman, J. R. Silva, T. J. Hund and Y. Rudy, Properties and ionic mechanisms of action potential adaptation, restitution, and accommodation in canine epicardium, Am. J. Physiol Heart Circ. Physiol. 296:H1017&ndash;H1026, 2009. doi:10.1152/ajpheart.01216.2008 J. S. Frank and G. A. Langer, The myocardial interstitium: its structure and its role in ionic exchange, J Cell Biol 60:586&ndash;601, 1974. doi:10.1083/jcb.60.3.586 A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (Lond), 117:500&ndash;544, 1952. http://jp.physoc.org/content/117/4/500.html R. FitzHugh, Impulses and Physiological States in Theoretical Models of Nerve Membrane, Biophys. J., 1:445&ndash;466, 1961. doi:10.1016/S0006-3495(61)86902-6 D. Kay, I. W. Turner, N. Cusimano and K. Burrage, Reflections from a boundary: reflecting boundary conditions for space-fractional partial differential equations on bounded domains, Technical report, University of Oxford, 2013. . F. Liu, V. Anh and I. Turner, Numerical solution of space fractional Fokker-Planck equation. J. Comp. and Appl. Math., 166:209&ndash;219, 2004. doi:10.1016/j.cam.2003.09.028 F. Liu, P. Zhuang, V. Anh and I. Turner and K. Burrage, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comp., 191:12&ndash;20, 2007. doi:10.1016/j.amc.2006.08.162 R. Magin, O. Abdullah, D. Baleanu and X. Zhou, Anomalous diffusion expressed through fractional order differential operators in the Bloch&ndash;Torrey equation, Journal of Magnetic Resonance 190:255&ndash;270, 2008. doi:10.1016/j.jmr.2007.11.007 M. M. Meerschaert, J. Mortensenb and S. W. Wheatcraft, Fractional vector calculus for fractional advection-dispersion, Physica A, 367:181&ndash;190, 2006. doi:10.1016/j.physa.2005.11.015 L. C. McSpadden, R. D. Kirkton and N. Bursac, Electrotonic loading of anisotropic cardiac monolayers by unexcitable cells depends on connexin type and expression level, Am. J. Physiol. Cell Physiol. 297:C339&ndash;C351, 2009. doi:10.1152/ajpcell.00024.2009 J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 50:2061&ndash;2070, 1962. doi:10.1109/JRPROC.1962.288235 S. F. Roberts, J. G. Stinstra and C. S. Henriquez, Effect of nonuniform interstitial space properties on impulse propagation: a discrete multidomain model, Biophys J 95:3724&ndash;3737, 2008. doi:10.1529/biophysj.108.137349 J. Sundnes, G. T. Lines, X. Cai, B. F. Nielsen, K. A. Mardal and A. Tveitio, Computing the electrical activity in the heart, Springer-Verlag, 2006. G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, Clarendon Press, Oxford, 1985. F. J. Valdes-Parada, J. A. Ochoa-Tapia and J. Alvarez-Ramirez, Effective medium equations for fractional Fick law in porous media, Physica A, 373:339&ndash;353, 2007. doi:10.1016/j.physa.2006.06.007 Q. Yang, F. Liu and I. Turner, Stability and convergence of an effective numerical method for the time-space fractional Fokker-Planck equation with a nonlinear source term, International Journal of Differential Equations, 2010:464321, 2010, doi:10.1155/2010/464321 W. Ying, A multilevel adaptive approach for computational cardiology, PhD thesis, Duke University, 2005. Q. Yu, F. Liu, I. Turner and K. Burrage, A computationally effective alternating direction method for the space and time fractional Bloch-Torrey equation in 3-D, Appl. Math. Comp., 219:4082&ndash;4095, 2012. doi:10.1016/j.amc.2012.10.056 Q. Yu, F. Liu, I. Turner and K. Burrage, Stability and convergence of an implicit numerical method for the space and time fractional Bloch-Torrey equation, the special issue of Fractional Calculus and Its Applications in-Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 371:20120150, 2013. doi:10.1098/rsta.2012.0150 Q. Yu, F. Liu, I. Turner and K. Burrage, Numerical simulation of the fractional Bloch equations, J. Comp. Appl. Math., 255:635&ndash;651, 2014. doi:10.1016/j.cam.2013.06.027 P. Zhuang, F. Liu, V. Anh and I. Turner, Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Num. Anal., 47:1760&ndash;1781, 2009. doi:10.1137/08073059