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 < β 0 < β 1 < β 2 < ⋯ < β m < 1 . The problem is first approximated by the L 1 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 − β m in time and an exponential rate of convergence in space. We finally provide some numerical experiments to show the efficacy of the theoretical results.

Jacobi Spectral Collocation Technique for Time-Fractional Inverse Heat Equations

In this paper, we introduce a numerical solution for the time-fractional inverse heat equations. We focus on obtaining the unknown source term along with the unknown temperature function based on an additional condition given in an integral form. The proposed scheme is based on a spectral collocation approach to obtain the two independent variables. Our approach is accurate, efficient, and feasible for the model problem under consideration. The proposed Jacobi spectral collocation method yields an exponential rate of convergence with a relatively small number of degrees of freedom. Finally, a series of numerical examples are provided to demonstrate the efficiency and flexibility of the numerical scheme.

On recurrent properties of Fisher--Wright's diffusion on (0,1) with mutation

A one-dimensional Fisher–Wright diffusion process on the interval ( 0 , 1 ) {(0,1)} with mutations is considered. This is a widely known model in population genetics. The goal of this paper is an exponential recurrence of the process, which also implies an exponential rate of convergence towards the invariant measure.

Asymptotic analysis of time dependent solutions for the coagulation equation with source and efflux

This article provides mathematical proof of the existence of stationary solutions for the coagulation equation including source and efflux terms. We demonstrate the convergence of time dependent solutions to these stationary solutions and highlight the exponential rate of convergence. These properties are analyzed for affine linear coagulation kernels, non-negative source terms and positive efflux rates. Numerical examples are included to demonstrate the predicted convergence behaviour.

The Navier–Stokes Equation with Time Quasi-Periodic External Force: Existence and Stability of Quasi-Periodic Solutions

We prove the existence of small amplitude, time-quasi-periodic solutions (invariant tori) for the incompressible Navier–Stokes equation on the d-dimensional torus $$\mathbb T^d$$ T d , with a small, quasi-periodic in time external force. We also show that they are orbitally and asymptotically stable in $$H^s$$ H s (for s large enough). More precisely, for any initial datum which is close to the invariant torus, there exists a unique global in time solution which stays close to the invariant torus for all times. Moreover, the solution converges asymptotically to the invariant torus for $$t \rightarrow + \infty $$ t → + ∞ , with an exponential rate of convergence $$O( e^{- \alpha t })$$ O ( e - α t ) for any arbitrary $$\alpha \in (0, 1)$$ α ∈ ( 0 , 1 ) .

Point defects in tight binding models for insulators

We consider atomistic geometry relaxation in the context of linear tight binding models for point defects. A limiting model as Fermi-temperature is sent to zero is formulated, and an exponential rate of convergence for the nuclei configuration is established. We also formulate the thermodynamic limit model at zero Fermi-temperature, extending the results of [H. Chen, J. Lu and C. Ortner, Thermodynamic limit of crystal defects with finite temperature tight binding, Arch. Ration. Mech. Anal. 230 (2018) 701–733]. We discuss the non-trivial relationship between taking zero temperature and thermodynamic limits in the finite Fermi-temperature models.

Numerical Analysis of the Chebyshev Collocation Method for Functional Volterra Integral Equations

The collocation method based on Chebyshev basis functions, coupled Picard iterative process, is proposed to solve a functional Volterra integral equation of the second kind. Using the Banach Fixed Point Theorem, we prove theorems on the existence and uniqueness solutions in the L2-norm. We also provide the convergence and stability analysis of the proposed method, which indicates that the numerical errors in the L2-norm decay exponentially, provided that the kernel function is sufficiently smooth. Numerical results are presented and they confirm the theoretical prediction of the exponential rate of convergence.

Sinc-Galerkin method for solving the time fractional convection–diffusion equation with variable coefficients

In this paper, a new numerical algorithm for solving the time fractional convection–diffusion equation with variable coefficients is proposed. The time fractional derivative is estimated using the $L_{1}$ L 1 formula, and the spatial derivative is discretized by the sinc-Galerkin method. The convergence analysis of this method is investigated in detail. The numerical solution is $2-\alpha$ 2 − α order accuracy in time and exponential rate of convergence in space. Finally, some numerical examples are given to show the effectiveness of the numerical scheme.