This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schr?dinger(NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grü...This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schr?dinger(NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grünwald-Letnikov difference(WSGD) operator for the spatial fractional Laplacian. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. By introducing the differentiation matrices, the semi-discrete fractional nonlinear Schr?dinger(FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations(ODEs) in matrix formulations. Two kinds of time discretization methods are proposed for the semi-discrete formulation. One is based on the Crank-Nicolson(CN) method which can be proved to preserve the fully discrete mass and energy conservation. The other one is the compact implicit integration factor(c IIF) method which demands much less computational effort. It can be shown that the cIIF scheme can approximate CN scheme with the error O(τ~2). Finally numerical results are presented to demonstrate the method’s conservation, accuracy, efficiency and the capability of capturing blow-up.展开更多
This paper studies an Ebola epidemic model with an exponential nonlinear incidence function that considers the efficacy and the behaviour change.The current model also incorporates a new density-dependent treatment th...This paper studies an Ebola epidemic model with an exponential nonlinear incidence function that considers the efficacy and the behaviour change.The current model also incorporates a new density-dependent treatment that catches the impact of the disease transmission on the treatment.Firstly,we provide a theoretical study of the nonlinear differential equations model obtained.More precisely,we derive the effective reproduction number and,under suitable conditions,prove the stability of equilibria.Afterwards,we show that the model exhibits the phenomenon of backward-bifurcation whenever the bifurcation parameter and the reproduction number are less than one.We find that the bi-stability and backward-bifurcation are not automatically connected in epidemic models.In fact,when a backward-bifurcation occurs,the disease-free equilibrium may be globally stable.Numerically,we use well-known standard tools to fit the model to the data reported for the 2018–2020 Kivu Ebola outbreak,and perform the sensitivity analysis.To control Ebola epidemics,our findings recommend a combination of a rapid behaviour change and the implementation of a proper treatment strategy with a high level of efficacy.Secondly,we propose and analyze a fractional-order Ebola epidemic model,which is an extension of the first model studied.We use the Caputo operator and construct the Grünwald-Letnikov nonstandard finite difference scheme,and show its advantages.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.61573008 and 61703290)Laboratory of Computational Physics(Grant No.6142A0502020717)National Science Foundation of USA(Grant No.DMS-1620108)
文摘This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schr?dinger(NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grünwald-Letnikov difference(WSGD) operator for the spatial fractional Laplacian. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. By introducing the differentiation matrices, the semi-discrete fractional nonlinear Schr?dinger(FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations(ODEs) in matrix formulations. Two kinds of time discretization methods are proposed for the semi-discrete formulation. One is based on the Crank-Nicolson(CN) method which can be proved to preserve the fully discrete mass and energy conservation. The other one is the compact implicit integration factor(c IIF) method which demands much less computational effort. It can be shown that the cIIF scheme can approximate CN scheme with the error O(τ~2). Finally numerical results are presented to demonstrate the method’s conservation, accuracy, efficiency and the capability of capturing blow-up.
基金C.Tadmon acknowledges good working conditions at the institute of Mathematics,University of Mainz,where this paper has been finalised during a research stay supported by the Alexander von Humboldt Foundation.
文摘This paper studies an Ebola epidemic model with an exponential nonlinear incidence function that considers the efficacy and the behaviour change.The current model also incorporates a new density-dependent treatment that catches the impact of the disease transmission on the treatment.Firstly,we provide a theoretical study of the nonlinear differential equations model obtained.More precisely,we derive the effective reproduction number and,under suitable conditions,prove the stability of equilibria.Afterwards,we show that the model exhibits the phenomenon of backward-bifurcation whenever the bifurcation parameter and the reproduction number are less than one.We find that the bi-stability and backward-bifurcation are not automatically connected in epidemic models.In fact,when a backward-bifurcation occurs,the disease-free equilibrium may be globally stable.Numerically,we use well-known standard tools to fit the model to the data reported for the 2018–2020 Kivu Ebola outbreak,and perform the sensitivity analysis.To control Ebola epidemics,our findings recommend a combination of a rapid behaviour change and the implementation of a proper treatment strategy with a high level of efficacy.Secondly,we propose and analyze a fractional-order Ebola epidemic model,which is an extension of the first model studied.We use the Caputo operator and construct the Grünwald-Letnikov nonstandard finite difference scheme,and show its advantages.
