Discrete Gronwall Inequality Research Articles

Optimal error bounds of the time-splitting sine-pseudospectral method for the biharmonic nonlinear Schrödinger equation

We propose a time-splitting sine-pseudospectral (TSSP) method for the biharmonic nonlinear Schrödinger equation (BNLS) and establish its optimal error bounds. In the proposed TSSP method, we adopt the sine-pseudospectral method for spatial discretization and the second-order Strang splitting for temporal discretization. The proposed TSSP method is explicit and structure-preserving, such as time symmetric, mass conservation and maintaining the dispersion relation of the original BNLS in the discretized level. Under the assumption that the solution of the one dimensional BNLS belongs to Hm with m≥9, we prove error bounds at O(τ2+hm) and O(τ2+hm−1) in L2 norm and H1 norm respectively, for the proposed TSSP method, with τ time step and h mesh size. For general dimensional cases with d=1,2,3, the error bounds are at O(τ2+hm) and O(τ2+hm−2) in L2 and H2 norm under the assumption that the exact solution is in Hm with m≥10. The proof is based on the bound of the Lie-commutator for the local truncation error, discrete Gronwall inequality, energy method and the H1- or H2-bound of the numerical solution which implies the L∞-bound of the numerical solution. Finally, extensive numerical results are reported to confirm our optimal error bounds and to demonstrate rich phenomena of the solutions including rapidly dispersion in space of high frequency waves and soliton collisions.

Error estimate of GL‐ADI scheme for 2D multiterm nonlinear time‐fractional subdiffusion equation

In this paper, a 2D multiterm nonlinear problem of the form is considered, where each Caputo fractional derivative is of order . We use the Grünwald–Letnikov(GL) scheme on uniform mesh to discretize the multiterm Caputo fractional derivative and finite difference scheme is used for spatial discretization, and then we construct a fully discrete GL‐ADI scheme. A discrete Gronwall inequality is introduced for getting the sharp pointwise‐in‐time error estimate on uniform mesh. Numerical examples are provided to verify the sharpness of our error estimate.

A higher-order uniform accuracy scheme for nonlinear $ \psi $-Volterra integral equations in two dimension with weakly singular kernel

<abstract><p>In this paper, we proposed a higher-order uniform accuracy scheme for nonlinear $ \psi $-Volterra integral equations in two dimension with weakly singular kernel by using the modified block-by-block method. First, we constructed a high order uniform accuracy scheme method in this paper by dividing the entire domain into some small sub-domains and approximating the integration function with biquadratic interpolation in each sub-domain. Second, we rigorously proved that the convergence order of the higher order uniform accuracy scheme was $ O(h_{s}^{3+\sigma_{1} }+h_{t}^{3+\sigma_{2} }) $ with $ 0 < \sigma_{1}, \sigma_{2} < 1 $ by using the discrete Gronwall inequality. Finally, two numerical examples were used to illustrate experimental results with different values of $ \psi $ to support the theoretical results.</p></abstract>

Quasi-uniform synchronization of fractional fuzzy discrete-time delayed neural networks via delayed feedback control design

This paper is concerned with the quasi-uniform synchronization (Q-US) issues of discrete fractional fuzzy neural networks (DFFNNs) with time delays. Firstly, we analyze the neural network systems with fuzzy time delays in discrete cases, complementing the neglected factors in the current literatures. Secondly, we use discrete fractional calculus including discrete Hölder inequality, discrete Gronwall inequality, and the discrete Lyapunov functional method to enrich the discrete system analysis. In addition, to acquire the Q-US, the designed delay feedback controller has the advantage of handling complex time delays to promote the response of the system. Moreover, the obtained results presented as direct linear matrix inequalities are efficiently verified by the simulation example.

A second-order difference scheme for the nonlinear time-fractional diffusion-wave equation with generalized memory kernel in the presence of time delay

This paper investigates a class of the time-fractional diffusion-wave equation (TFDWE), which incorporates a fractional derivative in the Caputo sense of order α+1 where 0<α<1. Initially, the original problem is transformed into a new model that incorporates the fractional Riemann–Liouville integral. Subsequently, a more generalized model is considered and investigated, which includes the generalized fractional integral with memory kernel. The paper establishes the existence and uniqueness of a numerical solution for the problem. We introduce a discretization technique for approximating the generalized fractional Riemann–Liouville integral and study its fundamental properties. Based on this technique, we propose a second-order numerical scheme for approximating the generalized model. The proof of the stability of the proposed difference scheme when considered in terms of the L2 norm is established. Furthermore, a difference scheme is devised for solving a nonlinear generalized model with time delay. The convergence and stability analysis of this particular case are established using the discrete Gronwall inequality. Numerical examples are provided to evaluate the effectiveness and reliability of the proposed methods and to support our theoretical analysis.

Finite Time Stability Results for Neural Networks Described by Variable-Order Fractional Difference Equations

Variable-order fractional discrete calculus is a new and unexplored part of calculus that provides extraordinary capabilities for simulating multidisciplinary processes. Recognizing this incredible potential, the scientific community has been researching variable-order fractional discrete calculus applications to the modeling of engineering and physical systems. This research makes a contribution to the topic by describing and establishing the first generalized discrete fractional variable order Gronwall inequality that we employ to examine the finite time stability of nonlinear Nabla fractional variable-order discrete neural networks. This is followed by a specific version of a generalized variable-order fractional discrete Gronwall inequality described using discrete Mittag–Leffler functions. A specific version of a generalized variable-order fractional discrete Gronwall inequality represented using discrete Mittag–Leffler functions is shown. As an application, utilizing the contracting mapping principle and inequality approaches, sufficient conditions are developed to assure the existence, uniqueness, and finite-time stability of the equilibrium point of the suggested neural networks. Numerical examples, as well as simulations, are provided to show how the key findings can be applied.

A Stable and Convergent Hybridized Discontinuous Galerkin Method for Time-Fractional Telegraph Equations

In this paper, we extend the application of the hybridized discontinuous Galerkin (HDG) method to solve time-fractional telegraph equations. In fact, we use an HDG method for space discretization and L1 and L2 finite difference schemes using non-uniform meshes for time discretization. Thanks to a special kind of discrete Gronwall inequality, we prove that the HDG method is unconditionally stable and it is convergent with the optimal spatial order of convergence. Two numerical experiments are tested to confirm the theoretical results.

On discrete tempered fractional calculus and its application

Discrete tempered fractional calculus, as a fresh generalization of discrete fractional calculus, takes both the advantages of discrete fractional calculus and its tempered fractional counterpart, which has great potential value in the interpretation of special physical phenomena. However, its essential characteristics are far from clarifying. Meanwhile, the primary goal of this study is to enrich the vital characteristics of discrete tempered fractional calculus in the presence of required domains for diverse scenarios. First, the reciprocal properties of the tempered fractional summation and difference operators, as well as the discrete version of Taylor’s formula, are established. The formulae of the discrete Laplace transform for discrete tempered fractional calculus are also determined with the aid of discrete convolution. Then, the well-posedness of tempered fractional difference equation is investigated, along with the exploration on the corresponding Volterra summation equation, and the existence, uniqueness and continuous dependence as well. In the aspect of application, the criterion of Ulam–Hyers stability of tempered fractional difference equation on finite time interval is obtained in light of a novel discrete Gronwall inequality. Besides, all examples are showed to verify the effectiveness of the main results.

L1/LDG Method for Caputo-Hadamard Time Fractional Diffusion Equation.

In this paper, a class of discrete Gronwall inequalities is proposed. It is efficiently applied to analyzing the constructed L1/local discontinuous Galerkin (LDG) finite element methods which are used for numerically solving the Caputo-Hadamard time fractional diffusion equation. The derived numerical methods are shown to be -robust using the newly established Gronwall inequalities, that is, it remains valid when . Numerical experiments are given to demonstrate the theoretical statements.

Closed‐loop consensus control of partial difference multi‐agent systems via variable gain iterative learning strategy

AbstractIn the research on consensus control of multi‐agent systems (MASs), it is the key to ensure that the state quantities (speed, displacement, etc.) of all agents tend to be consensus. Under this premise, how to accelerate the convergence rate of the consensus error is also an important issue. Aiming at the accelerated consensus problem of MASs composed of partial difference equations, based on the network topology and the output form of each follower, a distributed closed‐loop accelerated iterative learning control (ILC) protocol with variable gain was proposed, which was designed to improve the speed of consensus error. With the help of basic mathematical tools such as discrete Gronwall inequality and contraction mapping method, the sufficient conditions for the convergence of the consensus error are derived and analyzed. Finally, numerical simulations show that the proposed acceleration control protocol is more effective than the traditional open‐loop and closed‐loop ILC protocols.

Convergence analysis of a fast second‐order time‐stepping numerical method for two‐dimensional nonlinear time–space fractional Schrödinger equation

AbstractIn this article, we consider the two‐dimensional nonlinear time–space fractional Schrödinger equation with space described by the fractional Laplacian. A second‐order fractional backward difference formula in the temporal direction while Fourier spectral method in the spatial direction is proposed to solve the model numerically. In the numerical implementation, a fast method is applied based on a globally uniform approximation of the trapezoidal rule for the integral on the real line to decrease the memory requirement and computational cost. By using the generalized discrete Gronwall inequality developed by Dixon and McKee and the temporal–spatial error splitting argument, the convergence of the fast time‐stepping numerical method is also proved in a simple manner without imposing the Courant‐Friedrichs‐Lewy (CFL) condition. Finally, some numerical results are provided to support the theoretical analysis.

Error estimate of the fast L1 method for time-fractional subdiffusion equations

This paper analyzes the convergence of the fast L1 method for the semi-linear time-factional subdiffusion equations. The tool we use is the generalized discrete Gronwall inequality. We show that the difference between the solution of the L1 method and the solution of the fast L1 method can be arbitrarily small and is independent of the sizes of the time and/or space grids. Our proof is very simple and is helpful to be used to simplify the convergence analysis of the fast methods for time-fractional evolution equations.

A Mass-Preserving Two-Step Lagrange–Galerkin Scheme for Convection-Diffusion Problems

A mass-preserving two-step Lagrange–Galerkin scheme of second order in time for convection-diffusion problems is presented, and convergence with optimal error estimates is proved in the framework of L^2-theory. The introduced scheme maintains the advantages of the Lagrange–Galerkin method, i.e., CFL-free robustness for convection-dominated problems and a symmetric and positive coefficient matrix resulting from the discretization. In addition, the scheme conserves the mass on the discrete level if the involved integrals are computed exactly. Unconditional stability and error estimates of second order in time are proved by employing two new key lemmas on the truncation error of the material derivative in conservative form and on a discrete Gronwall inequality for multistep methods. The mass-preserving property is achieved by the Jacobian multiplication technique introduced by Rui and Tabata in 2010, and the accuracy of second order in time is obtained based on the idea of the multistep Galerkin method along characteristics originally introduced by Ewing and Russel in 1981. For the first time step, the mass-preserving scheme of first order in time by Rui and Tabata in 2010 is employed, which is efficient and does not cause any loss of convergence order in the ell ^infty (L^2)- and ell ^2(H^1_0)-norms. For the time increment Delta t, the mesh size h and a conforming finite element space of polynomial degree k in {mathbb {N}}, the convergence order is of O(Delta t^2 + h^k) in the ell ^infty (L^2)cap ell ^2(H^1_0)-norm and of O(Delta t^2 + h^{k+1}) in the ell ^infty (L^2)-norm if the duality argument can be employed. Error estimates of O(Delta t^{3/2}+h^k) in discrete versions of the L^infty (H^1_0)- and H^1(L^2)-norm are additionally proved. Numerical results confirm the theoretical convergence orders in one, two and three dimensions.

Splitting extrapolation algorithms for solving linear delay Volterra integral equations with a spatial variable

This paper presents a numerical algorithm for solving linear delay Volterra integral equations with a spatial variable. The theory on compact operators is utilized to prove the existence and uniqueness of solution for the original equation, by combining the trapezoidal quadrature formula with interpolation to obtain an approximate equation. Then, we validate the existence of the approximate solution by analyzing the convergence of iterative sequence. The uniqueness of the approximate solution is proved by applying a two-dimensional discrete Gronwall inequality. Next, the convergence of the approximate solution and the asymptotic expansion of errors are obtained. Moreover, a higher accuracy solution is obtained by means of splitting extrapolation algorithms. Some numerical experiments are provided to demonstrate the efficiency of the proposed method.

Convergence analysis of the time-stepping numerical methods for time-fractional nonlinear subdiffusion equations

In 1986, Dixon and McKee (Z Angew Math Mech 66:535–544, 1986) developed a discrete fractional Gronwall inequality, which can be seen as a generalization of the classical discrete Gronwall inequality. However, this generalized discrete Gronwall inequality and its variant (Al-Maskari and Karaa in SIAM J Numer Anal 57:1524–1544, 2019) have not been widely applied in the numerical analysis of the time-stepping methods for the time-fractional evolution equations. The main purpose of this paper is to show how to apply the generalized discrete Gronwall inequality to prove the convergence of a class of time-stepping numerical methods for time-fractional nonlinear subdiffusion equations, including the popular fractional backward difference type methods of order one and two, and the fractional Crank-Nicolson type methods. We obtain the optimal \(L^2\) error estimate in space discretization for multi-dimensional problems. The convergence of the fast time-stepping numerical methods is also proved in a simple manner. The present work unifies the convergence analysis of several existing time-stepping schemes. Numerical examples are provided to verify the effectiveness of the present method.

Numerical solution of two-dimensional weakly singular Volterra integral equations with non-smooth solutions

Various numerical methods have been proposed for the solution of weakly singular Volterra integral equations but, for the most part, authors have dealt with linear or one-dimensional weakly singular Volterra integral equations, or have assumed that these equations have smooth solutions. The main purpose of this paper is to propose and analyse a numerical method for the solution of two-dimensional nonlinear weakly singular Volterra integral equations of the second kind. In general the solutions of these equations exhibit singularities in their derivatives at t=0 even if the forcing functions are smooth. To overcome these difficulties a simple smoothing change of variables is proposed. By applying this transformation an equation is obtained which, while still being weakly singular, can have a solution as smooth as is required. We then solve this transformed integral equation using Navot’s quadrature rule for computing integrals with an end point singularity. A new extension of a discrete Gronwall inequality allows us to prove convergence and obtain an error estimate. The theoretical results are then verified by numerical examples.

Finite Difference Solution to a Nonlinear Time-Fractional Logistic Model

We set forth a time-fractional logistic model and give an implicit finite difference scheme for solving of the model. The L^2 stability and convergence of the scheme are proved with the aids of discrete Gronwall inequality, and numerical examples are presented to support the theoretical analysis.

A fast and high-order numerical method for nonlinear fractional-order differential equations with non-singular kernel

Efficient and fast explicit methods are proposed to solve nonlinear Caputo-Fabrizio fractional differential equations, where Caputo-Fabrizio operator is a new proposed fractional derivative with a smooth kernel. The proposed methods produce the second-order for linear interpolation and the third-order accuracy for quadratic interpolation, respectively. The convergence analysis is proved by using discrete Gronwall's inequality. Furthermore, applying the recurrence relation of the memory term, it reduces CPU time executed the proposed methods. The proposed fast algorithm requires approximately O(N) arithmetic operations while O(N2) is required in case of the regular predictor-corrector schemes, where N is the total number of the time step. The following numerical examples demonstrate the accuracy of the proposed methods as well as the efficiency: nonlinear fractional differential equations, time-fraction sub-diffusion, and time-fractional advection-diffusion equation. Numerical experiments also verify the theoretical convergence rates.

An improved error analysis for a second-order numerical scheme for the Cahn–Hilliard equation

In this paper we present an error analysis for a second order accurate numerical scheme for the 2-D and 3-D Cahn–Hilliard (CH) equation, with an improved convergence constant. The unique solvability, unconditional energy stability, and a uniform-in-time H2 stability of this numerical scheme have already been established. However, a standard error estimate gives a convergence constant in an order of exp(CTε−m0), with m0 a positive integer and the interface width parameter ε being small, which comes from the application of discrete Gronwall inequality. To overcome this well-known difficulty, we apply a spectrum estimate for the linearized Cahn–Hilliard operator (Alikakos and Fusco, 1993; Chen, 1994; Feng and Prohl, 2004), perform a detailed numerical analysis, and get an improved estimate, in which the convergence constant depends on 1ε only in a polynomial order, instead of the exponential order.

α-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation

An initial-boundary value problem of the form ${D}_{t}^{\alpha } u+{\varDelta }^{2}u-c{\varDelta } u =f$ is considered, where ${D}_{t}^{\alpha }$ is a Caputo temporal derivative of order α ∈ (0,1) and c is a nonnegative constant. The spatial domain ${\varOmega } \subset \mathbb {R}^{d}$ for some d ∈{1,2,3}, with Ω bounded and convex. The boundary conditions are u = Δu = 0 on ∂Ω. A priori bounds on the solution are established, given sufficient regularity and compatibility of the data; typical solutions have a weak singularity at the initial time t = 0. The problem is rewritten as a system of two second-order differential equations, then discretised using standard finite elements in space together with the L1 discretisation of ${D}_{t}^{\alpha }$ on a graded temporal mesh. The numerical method computes approximations ${u_{h}^{n}}$ and ${{p}_{h}^{n}}$ of u(⋅,tn) and Δu(⋅,tn) at each time level tn. The stability of the method (i.e. a priori bounds on $\|{{u}_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{p_{h}^{n}}\|_{L^{2}({\varOmega })}$ ) is established by means of a new discrete Gronwall inequality that is α-robust, i.e. remains valid as α → 1−. Error bounds on $\|u(\cdot , t_{n}) - {u_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{\varDelta } u(\cdot , t_{n}) - {{p}_{h}^{n}}\|_{L^{2}({\varOmega })}$ are then derived; these bounds are of optimal order in the spatial and temporal mesh parameters for each fixed value of α, and they are α-robust if one considers α → 1−.