A SPACE-TIME SPECTRAL METHOD FOR THE TIME FRACTIONAL DIFFUSION EQUATION

National NSF of China [10531080]; Ministry of Education of China; 973 High Performance Scientific Computation Research Program [2005CB321703]In this paper, we consider the numerical solution of the time fractional diffusion equation. Essentially, the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term. In the former case, the first-order time derivative is replaced by a fractional derivative, making the problem global in time. We propose a spectral method in both temporal and spatial discretizations for this equation. The convergence of the method is proven by providing a priori error estimate. Numerical tests are carried out to confirm the theoretical results. Thanks to the spectral accuracy in both space and time of the proposed method, the storage requirement due to the "global time dependence" can be considerably relaxed, and therefore calculation of the long-time solution becomes possible

A pseudo-spectral method based on reproducing kernel for solving the time-fractional diffusion-wave equation

In this paper, we focus on the development and study of the finite difference/pseudo-spectral method to obtain an approximate solution for the time-fractional diffusion-wave equation in a reproducing kernel Hilbert space. Moreover, we make use of the theory of reproducing kernels to establish certain reproducing kernel functions in the aforementioned reproducing kernel Hilbert space. Furthermore, we give an approximation to the time-fractional derivative term by applying the finite difference scheme by our proposed method. Over and above, we present an appropriate technique to derive the numerical solution of the given equation by utilizing a pseudo-spectral method based on the reproducing kernel. Then, we provide two numerical examples to support the accuracy and efficiency of our proposed method. Finally, we apply numerical experiments to calculate the quality of our approximation by employing discrete error norms. © 2022, The Author(s)

New developments in Functional and Fractional Differential Equations and in Lie Symmetry

Delay, difference, functional, fractional, and partial differential equations have many applications in science and engineering. In this Special Issue, 29 experts co-authored 10 papers dealing with these subjects. A summary of the main points of these papers follows:Several oscillation conditions for a first-order linear differential equation with non-monotone delay are established in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, whereas a sharp oscillation criterion using the notion of slowly varying functions is established in A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay. The approximation of a linear autonomous differential equation with a small delay is considered in Approximation of a Linear Autonomous Differential Equation with Small Delay; the model of infection diseases by Marchuk is studied in Around the Model of Infection Disease: The Cauchy Matrix and Its Properties. Exact solutions to fractional-order Fokker–Planck equations are presented in New Exact Solutions and Conservation Laws to the Fractional-Order Fokker–Planck Equations, and a spectral collocation approach to solving a class of time-fractional stochastic heat equations driven by Brownian motion is constructed in A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise. A finite difference approximation method for a space fractional convection-diffusion model with variable coefficients is proposed in Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients; existence results for a nonlinear fractional difference equation with delay and impulses are established in On Nonlinear Fractional Difference Equation with Delay and Impulses. A complete Noether symmetry analysis of a generalized coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry is provided in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, and new soliton solutions of a fractional Jaulent soliton Miodek system via symmetry analysis are presented in New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis

From phenomenological modelling of anomalous diffusion through continuous-time random walks and fractional calculus to correlation analysis of complex systems

This document contains more than one topic, but they are all connected in ei- ther physical analogy, analytic/numerical resemblance or because one is a building block of another. The topics are anomalous diffusion, modelling of stylised facts based on an empirical random walker diffusion model and null-hypothesis tests in time series data-analysis reusing the same diffusion model. Inbetween these topics are interrupted by an introduction of new methods for fast production of random numbers and matrices of certain types. This interruption constitutes the entire chapter on random numbers that is purely algorithmic and was inspired by the need of fast random numbers of special types. The sequence of chapters is chrono- logically meaningful in the sense that fast random numbers are needed in the first topic dealing with continuous-time random walks (CTRWs) and their connection to fractional diffusion. The contents of the last four chapters were indeed produced in this sequence, but with some temporal overlap. While the fast Monte Carlo solution of the time and space fractional diffusion equation is a nice application that sped-up hugely with our new method we were also interested in CTRWs as a model for certain stylised facts. Without knowing economists [80] reinvented what physicists had subconsciously used for decades already. It is the so called stylised fact for which another word can be empirical truth. A simple example: The diffusion equation gives a probability at a certain time to find a certain diffusive particle in some position or indicates concentration of a dye. It is debatable if probability is physical reality. Most importantly, it does not describe the physical system completely. Instead, the equation describes only a certain expectation value of interest, where it does not matter if it is of grains, prices or people which diffuse away. Reality is coded and “averaged” in the diffusion constant. Interpreting a CTRW as an abstract microscopic particle motion model it can solve the time and space fractional diffusion equation. This type of diffusion equation mimics some types of anomalous diffusion, a name usually given to effects that cannot be explained by classic stochastic models. In particular not by the classic diffusion equation. It was recognised only recently, ca. in the mid 1990s, that the random walk model used here is the abstract particle based counterpart for the macroscopic time- and space-fractional diffusion equation, just like the “classic” random walk with regular jumps ±∆x solves the classic diffusion equation. Both equations can be solved in a Monte Carlo fashion with many realisations of walks. Interpreting the CTRW as a time series model it can serve as a possible null- hypothesis scenario in applications with measurements that behave similarly. It may be necessary to simulate many null-hypothesis realisations of the system to give a (probabilistic) answer to what the “outcome” is under the assumption that the particles, stocks, etc. are not correlated. Another topic is (random) correlation matrices. These are partly built on the previously introduced continuous-time random walks and are important in null- hypothesis testing, data analysis and filtering. The main ob jects encountered in dealing with these matrices are eigenvalues and eigenvectors. The latter are car- ried over to the following topic of mode analysis and application in clustering. The presented properties of correlation matrices of correlated measurements seem to be wasted in contemporary methods of clustering with (dis-)similarity measures from time series. Most applications of spectral clustering ignores information and is not able to distinguish between certain cases. The suggested procedure is sup- posed to identify and separate out clusters by using additional information coded in the eigenvectors. In addition, random matrix theory can also serve to analyse microarray data for the extraction of functional genetic groups and it also suggests an error model. Finally, the last topic on synchronisation analysis of electroen- cephalogram (EEG) data resurrects the eigenvalues and eigenvectors as well as the mode analysis, but this time of matrices made of synchronisation coefficients of neurological activity

An Efficient Hybrid Numerical Scheme for Nonlinear Multiterm Caputo Time and Riesz Space Fractional-Order Diffusion Equations with Delay

In this paper, we construct and analyze a linearized finite difference/Galerkin-Legendre spectral scheme for the nonlinear multiterm Caputo time fractional-order reaction-diffusion equation with time delay and Riesz space fractional derivatives. The temporal fractional orders in the considered model are taken as 0<β0<β1<β2<⋯<βm<1. The problem is first approximated by the L1 difference method on the temporal direction, and then, the Galerkin-Legendre spectral method is applied on the spatial discretization. Armed by an appropriate form of discrete fractional Grönwall inequalities, the stability and convergence of the fully discrete scheme are investigated by discrete energy estimates. We show that the proposed method is stable and has a convergent order of 2-βm in time and an exponential rate of convergence in space. We finally provide some numerical experiments to show the efficacy of the theoretical results. © 2021 A. K. Omran et al.A. K. Omran is funded by a scholarship under the joint executive program between the Arab Republic of Egypt and Russian Federation. M. A. Zaky wishes to acknowledge the support of the Nazarbayev University Program (091019CRP2120). M. A. Zaky wishes also to acknowledge the partial support of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant “Dynamical Analysis and Synchronization of Complex Neural Networks with Its Applications”)

Fourier spectral methods for fractional-in-space reaction-diffusion equations

Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is computationally demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reactiondiffusion equations. The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is show-cased by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models,together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator

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

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