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We consider a class of numerical approximations to the Caputo fractional derivative. Our assumptions permit the use of nonuniform time steps, such as is appropriate for accurately resolving the behavior of a solution whose temporal derivatives are singular at t = 0. The main result is a type of fractional Grönwall inequality and we illustrate its use by outlining some stability and convergence estimates of schemes for fractional reaction-subdiffusion problems. This approach extends earlier work that used the familiar LI approximation to the Caputo fractional derivative, and will facilitate the analysis of higher order and linearized fast schemes.

Stability and convergence of the L1 formula on nonuniform time grids are studied for solving linear react ion-sub diffusion equations with the Caputo derivative. A discrete fractional Gronwall inequality is developed for the nonuniform L1 formula by introducing a discrete convolution kernel of Riemann-Liouville fractional integral. To simplify the consistency analysis of the nonuniform L1 formula, we bound the local truncation error in a discrete convolution form and consider a global convolution error involving the discrete Riemann-Liouville integral kernel. With the help of discrete fractional Gronwall inequality and global consistency error analysis, a sharp error estimate reflecting the regularity of solution is obtained for a simple L1 scheme. Numerical examples are provided to verify the sharpness of the error analysis.

We address spin transport in the easy-axis Heisenberg spin chain subject to different integrability-breaking perturbations. We find subdiffusive spin transport characterized by dynamical exponent z = 4 up to a timescale parametrically long in the anisotropy. In the limit of infinite anisotropy, transport is subdiffusive at all times; for finite anisotropy, one eventually recovers diffusion at late times but with a diffusion constant independent of the strength of the perturbation and solely fixed by the value of the anisotropy. We provide numerical evidence for these findings, and we show how they can be understood in terms of the dynamical screening of the relevant quasiparticle excitations and effective dynamical constraints. Our results show that the diffusion constant of near-integrable diffusive spin chains is generically not perturbative in the integrability-breaking strength.

Opinions in human societies are measured by political polls on time scales of months to years. Such opinion polls do not resolve the effects of individual interactions but constitute a stochastic process. Voter models with zealots (individuals who do not change their opinions) can describe the mean-field dynamics in systems where no consensus is reached. We show that for large populations, the voter model with zealots is equivalent to the noisy voter model and it has a single characteristic time scale associated with the number of zealots in the population. We discuss which parameters are observable in real data by analysing time series of approval ratings of several political leaders that match the statistical behaviour of the voter model using the technique of the time-averaged mean squared displacement. The characteristic time scale of political opinions in societies is around 12 months, so it cannot be resolved by analysing election data, for which the resolution is several years. The effective population size in all fitted data sets is much smaller than the real population size, which indicates positive correlations of successive voter model steps. We also discuss the heterogeneity of voters as a cause of subdiffusion on long time scales, i.e. slow changes in the society.

We consider two inverse problems for a generalized subdiffusion equation that use the final overdetermination condition. Firstly, we study a problem of reconstruction of a specific space-dependent component in a source term. We prove existence, uniqueness and stability of the solution to this problem. Based on these results, we consider an inverse problem of identification of a space-dependent coefficient of a linear reaction term. We prove the uniqueness and local existence and stability of the solution to this problem.

In this article, we consider two inverse problems with a generalized fractional derivative. The first problem, IP1, is to reconstruct the function u based on its value and the value of its fractional derivative in the neighborhood of the final time. We prove the uniqueness of the solution to this problem. Afterwards, we investigate the IP2, which is to reconstruct a source term in an equation that generalizes fractional diffusion and wave equations, given measurements in a neighborhood of final time. The source to be determined depends on time and all space variables. The uniqueness is proved based on the results for IP1. Finally, we derive the explicit solution formulas to the IP1 and IP2 for some particular cases of the generalized fractional derivative.

The aim of this paper is to develop and analyze high-order time stepping schemes for approximately solving semilinear subdiffusion equations. We apply the convolution quadrature generated by k-step backward differentiation formula (BDFk) to discretize the time-fractional derivative with order α ∊(0,1) and modify the starting steps in order to achieve optimal convergence rate. This method has already been well-studied for the linear fractional evolution equations in Jin, Li, and Zhou [SIAM J. Sci. Comput, 39 (2017), pp. A3129-A3152], while the numerical analysis for the nonlinear problem is still missing in the literature. By splitting the nonlinear potential term into an irregular linear part and a smoother nonlinear part and using the generating function technique, we prove that the convergence order of the corrected BDFfc scheme is O(τmin(k, 1+2α-e)) without imposing further assumption on the regularity of the solution. Numerical examples are provided to support our theoretical results.

In this paper, we study the orthogonal Gauss collocation method (OGCM) with an arbitrary polynomial degree for the numerical solution of a two-dimensional (2D) fourth-order subdiffusion model. This numerical method involves solving a coupled system of partial differential equations by using OGCM in space together with the L1 scheme in time on a graded mesh. The approximations w h n and v h n of w ( · , t n ) and Δ w ( · , t n ) are constructed. The stability of w h n and v h n are proved, and the a priori bounds of ‖ w h n ‖ and ‖ v h n ‖ are established, remaining α -robust as α → 1 - . Then, the error ‖ w ( · , t n ) - w h n ‖ and ‖ Δ w ( · , t n ) - v h n ‖ are estimated with α -robust at each time level. In addition, superconvergence results of the first-order and second-order derivative approximations are proved. These new error bounds are desirable and natural, as that they are optimal in both temporal and spatial mesh parameters for each fixed α . Finally some numerical results are provided to support our theoretical findings.

Many-body localization for a system of bosons trapped in a one-dimensional lattice is discussed. Two models that may be realized for cold atoms in optical lattices are considered. The model with a random on-site potential is compared with previously introduced random interactions model. While the origin and character of the disorder in both systems is different they show interesting similar properties. In particular, many-body localization appears for a sufficiently large disorder as verified by a time evolution of initial density wave states as well as using statistical properties of energy levels for small system sizes. Starting with different initial states, we observe that the localization properties are energy-dependent which reveals an inverted many-body localization edge in both systems (that finding is also verified by statistical analysis of energy spectrum). Moreover, we consider computationally challenging regime of transition between many body localized and extended phases where we observe a characteristic algebraic decay of density correlations which may be attributed to subdiffusion (and Griffiths-like regions) in the studied systems. Ergodicity breaking in the disordered Bose-Hubbard models is compared with the slowing-down of the time evolution of the clean system at large interactions.