In this article,we introduce a nonlinear Caputo-type snakebite envenoming model with memory.The well-known Caputo fractional derivative is used to generalize the previously presented integer-order model into a fractio...In this article,we introduce a nonlinear Caputo-type snakebite envenoming model with memory.The well-known Caputo fractional derivative is used to generalize the previously presented integer-order model into a fractionalorder sense.The numerical solution of the model is derived from a novel implementation of a finite-difference predictor-corrector(L1-PC)scheme with error estimation and stability analysis.The proof of the existence and positivity of the solution is given by using the fixed point theory.From the necessary simulations,we justify that the first-time implementation of the proposedmethod on an epidemicmodel shows that the scheme is fully suitable and time-efficient for solving epidemic models.This work aims to show the novel application of the given scheme as well as to check how the proposed snakebite envenoming model behaves in the presence of the Caputo fractional derivative,including memory effects.展开更多
A nonconforming H^1-Calerkin mixed finite element method is analyzed for Sobolev equations on anisotropic meshes. The error estimates are obtained without using Ritz-Volterra projection.
H1-Galerkin nonconforming mixed finite element methods are analyzed for integro-differential equation of parabolic type.By use of the typical characteristic of the elements,we obtain that the Galerkin mixed approximat...H1-Galerkin nonconforming mixed finite element methods are analyzed for integro-differential equation of parabolic type.By use of the typical characteristic of the elements,we obtain that the Galerkin mixed approximations have the same rates of convergence as in the classical mixed method,but without LBB stability condition.展开更多
In this paper,we investigate a priori and a posteriori error estimates of fully discrete H^(1)-Galerkin mixed finite element methods for parabolic optimal control prob-lems.The state variables and co-state variables a...In this paper,we investigate a priori and a posteriori error estimates of fully discrete H^(1)-Galerkin mixed finite element methods for parabolic optimal control prob-lems.The state variables and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element and linear finite element,and the control vari-able is approximated by piecewise constant functions.The time discretization of the state and co-state are based on finite difference methods.First,we derive a priori error estimates for the control variable,the state variables and the adjoint state variables.Second,by use of energy approach,we derive a posteriori error estimates for optimal control problems,assuming that only the underlying mesh is static.A numerical example is presented to verify the theoretical results on a priori error estimates.展开更多
In this paper,we present a two-grid discretization scheme for semilinear parabolic integro-differential equations by H1-Galerkin mixed finite element methods.We use the lowest order Raviart-Thomas mixed finite element...In this paper,we present a two-grid discretization scheme for semilinear parabolic integro-differential equations by H1-Galerkin mixed finite element methods.We use the lowest order Raviart-Thomas mixed finite elements and continuous linear finite element for spatial discretization,and backward Euler scheme for temporal discretization.Firstly,a priori error estimates and some superclose properties are derived.Secondly,a two-grid scheme is presented and its convergence is discussed.In the proposed two-grid scheme,the solution of the nonlinear system on a fine grid is reduced to the solution of the nonlinear system on a much coarser grid and the solution of two symmetric and positive definite linear algebraic equations on the fine grid and the resulting solution still maintains optimal accuracy.Finally,a numerical experiment is implemented to verify theoretical results of the proposed scheme.The theoretical and numerical results show that the two-grid method achieves the same convergence property as the one-grid method with the choice h=H^(2).展开更多
文摘In this article,we introduce a nonlinear Caputo-type snakebite envenoming model with memory.The well-known Caputo fractional derivative is used to generalize the previously presented integer-order model into a fractionalorder sense.The numerical solution of the model is derived from a novel implementation of a finite-difference predictor-corrector(L1-PC)scheme with error estimation and stability analysis.The proof of the existence and positivity of the solution is given by using the fixed point theory.From the necessary simulations,we justify that the first-time implementation of the proposedmethod on an epidemicmodel shows that the scheme is fully suitable and time-efficient for solving epidemic models.This work aims to show the novel application of the given scheme as well as to check how the proposed snakebite envenoming model behaves in the presence of the Caputo fractional derivative,including memory effects.
基金Supported by the National Natural Science Foundation of China(No.10671184).
文摘A nonconforming H^1-Calerkin mixed finite element method is analyzed for Sobolev equations on anisotropic meshes. The error estimates are obtained without using Ritz-Volterra projection.
基金Foundation item： the National Natural Science Foundation of China （Nos. 10671184 10371113）.
文摘H1-Galerkin nonconforming mixed finite element methods are analyzed for integro-differential equation of parabolic type.By use of the typical characteristic of the elements,we obtain that the Galerkin mixed approximations have the same rates of convergence as in the classical mixed method,but without LBB stability condition.
基金This work was supported by National Natural Science Foundation of China(11601014,11626037,11526036)China Postdoctoral Science Foundation(2016M 601359)+4 种基金Scientific and Technological Developing Scheme of Jilin Province(20160520108 JH,20170101037JC)Science and Technology Research Project of Jilin Provincial Depart-ment of Education(201646)Special Funding for Promotion of Young Teachers of Beihua University,Natural Science Foundation of Hunan Province(14JJ3135)the Youth Project of Hunan Provincial Education Department(15B096)the construct program of the key discipline in Hunan University of Science and Engineering.
文摘In this paper,we investigate a priori and a posteriori error estimates of fully discrete H^(1)-Galerkin mixed finite element methods for parabolic optimal control prob-lems.The state variables and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element and linear finite element,and the control vari-able is approximated by piecewise constant functions.The time discretization of the state and co-state are based on finite difference methods.First,we derive a priori error estimates for the control variable,the state variables and the adjoint state variables.Second,by use of energy approach,we derive a posteriori error estimates for optimal control problems,assuming that only the underlying mesh is static.A numerical example is presented to verify the theoretical results on a priori error estimates.
基金Science and Technology Research Project of Jilin Provincial Department of Education(JJKH20190634KJ)The work of C.M.Liu was supported by the National Natural Science Foundation of China(11901189)+5 种基金the Key Project of Hunan Provincial Education Department(19A191)L.P.Chen was supported by Natural Science Foundation of China(11501473)the Fundamental Research Funds of the Central Universities of China(2682016CX108)The work of Y.Yang was supported by National Natural Science Foundation of China Project(11671342,11771369,11931003)the Project of Scientific Research Fund of Hunan Provincial Science and Technology Department(2018JJ2374,2018WK4006,2019YZ3003)the Key Project of Hunan Provincial Department of Education(17A210).
文摘In this paper,we present a two-grid discretization scheme for semilinear parabolic integro-differential equations by H1-Galerkin mixed finite element methods.We use the lowest order Raviart-Thomas mixed finite elements and continuous linear finite element for spatial discretization,and backward Euler scheme for temporal discretization.Firstly,a priori error estimates and some superclose properties are derived.Secondly,a two-grid scheme is presented and its convergence is discussed.In the proposed two-grid scheme,the solution of the nonlinear system on a fine grid is reduced to the solution of the nonlinear system on a much coarser grid and the solution of two symmetric and positive definite linear algebraic equations on the fine grid and the resulting solution still maintains optimal accuracy.Finally,a numerical experiment is implemented to verify theoretical results of the proposed scheme.The theoretical and numerical results show that the two-grid method achieves the same convergence property as the one-grid method with the choice h=H^(2).
