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Well-posedness of a kind of nonlinear coupled system of fractional differential equations

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Science China Mathematics 2016年第6期59卷 1209-1220页

作者： ZHOU XiaoJun xu chuanju School of Mathematical Sciences Xiamen University School of Mathematics and Computer Science Guizhou Normal University

We study the boundary value problem of a coupled differential system of fractional order, and prove the existence and uniqueness of solutions to the considered problem. The underlying differential system is featured by a fractional differential operator, which is defined in the Riemann-Liouville sense, and a nonlinear term in which different solution components are coupled. The analysis is based on the reduction of the given system to an equivalent system of integral equations. By means of the nonlinear alternative of Leray-Schauder,the existence of solutions of the factional differential system is obtained. The uniqueness is established by using the Banach contraction principle.

关键词： fractional differential equations nonlinear coupled system existence and uniqueness of solutions

HIGH ORDER FINITE DIFFERENCE/SPECTRAL METHODS TO A WATER WAVE MODEL WITH NONLOCAL VISCOSITY

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Journal of Computational Mathematics 2020年第4期38卷 580-605页

作者： Mohammad Tanzil Hasan chuanju xu School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen UniversityXiamen 361005China

In this paper,efficient numerical scheme is proposed for solving the water wave model with nonlocal viscous term that describe the propagation of surface water *** using the Caputo fractional derivative definition to approximate the nonlocal fractional operator,finite difference method in time and spectral method in space are constructed for the considered *** proposed method employs known 5/2 order scheme for fractional derivative and a mixed linearization for the nonlinear *** analysis shows that the proposed numerical scheme is unconditionally stable and error estimates are provided to predict that the second order backward differentiation plus 5/2 order scheme converges with order 2 in time,and spectral accuracy in *** numerical results are provided to verify the efficiency and accuracy of our theoretical ***,the decay rate of solutions are investigated.

关键词： Water waves Nonlocal viscosity Finite difference Spectral method Conver-gence order Decay rate

Existence and Uniqueness of the Weak Solution of the Space-Time Fractional Diffusion Equation and a Spectral Method Approximation

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Communications in Computational Physics 2010年第10期8卷 1016-1051页

作者： Xianjuan Li chuanju xu BNU-HKBU United International College Zhuhai 519085China School of Mathematical Sciences Xiamen University361005 XiamenChina

In this paper,we investigate initial boundary value problems of the spacetime fractional diffusion equation and its numerical *** definitions,i.e.,Riemann-Liouville definition and Caputo one,of the fractional derivative are considered in *** both cases,we establish the well-posedness of the weak ***,based on the proposed weak formulation,we construct an efficient spectral method for numerical approximations of the weak *** main contribution of this work are threefold:First,a theoretical framework for the variational solutions of the space-time fractional diffusion equation is *** find suitable functional spaces and norms in which the space-time fractional diffusion problem can be formulated into an elliptic weak problem,and the existence and uniqueness of the weak solution are then proved by using existing theory for elliptic ***,we show that in the case of Riemann-Liouville definition,the well-posedness of the space-time fractional diffusion equation does not require any initial *** contrasts with the case of Caputo definition,in which the initial condition has to be integrated into the weak formulation in order to establish the ***,thanks to the weak formulation,we are able to construct an efficient numerical method for solving the space-time fractional diffusion problem.

关键词： Space-time fractional diffusion equation existence and uniqueness spectral methods error estimates

A Numerical Comparison of Outflow Boundary Conditions for Spectral Element Simulations of Incompressible Flows

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Communications in Computational Physics 2007年第3期2卷 477-500页

作者： chuanju xu Yumin Lin School of Mathematical Sciences Xiamen UniversityXiamen 361005China.

Outflow boundary conditions(OBCs)are investigated for calculation of incompressible flows by spectral element *** OBCs,including essentialtype,natural-type,periodic-type and advection-type,are compared by carrying out a series of numerical ***,a simplified form of the so-called Orlanski’s OBCs is proposed in the context of spectral element methods,for which a new treatment technique is *** purpose of this paper is to find stable low-reflective OBCs,suitable and flexible for use of spectral element methods in simulation of incompressible flows in complex *** computation is firstly carried out for a 2D simulation of Poiseuille-B´enard channel flow with Re=10,Ri=150 and Pr=2/*** flow serves as a useful example to demonstrate the applicability of the proposed OBCs because it exhibits a feature of vortex shedding propagating through the outflow *** a 3D flow around an obstacle is computed to show the efficiency in the case of more general *** the tested OBCs,the advection-type OBCs are proven to have better behavior as compared with the others.

关键词： Spectral element methods outflow boundary conditions incompressible flows.

A Stable Arbitrarily High Order Time-Stepping Method for Thermal Phase Change Problems

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Communications in Computational Physics 2023年第2期33卷 477-510页

作者： Weiwen Wang chuanju xu School of Mathematical Sciences and Fujian Provincial Key Laboratory of MathematicalModeling and High Performance Scientific Computing Xiamen UniversityXiamen 361005P.R.China.

Thermal phase change problems are widespread in mathematics,nature,and *** are particularly useful in simulating the phenomena of melting and solidification in materials *** this paper we propose a novel class of arbitrarily high-order and unconditionally energy stable schemes for a thermal phase changemodel,which is the coupling of a heat transfer equation and a phase field *** unconditional energy stability and consistency error estimates are rigorously proved for the proposed schemes.A detailed implementation demonstrates that the proposed method requires only the solution of a system of linear elliptic equations at each time step,with an efficient scheme of sufficient accuracy to calculate the solution at the first *** is observed from the comparison with the classical explicit Runge-Kutta method that the new schemes allow to use larger time *** time step size strategies can be applied to further benefit from this unconditional *** experiments are presented to verify the theoretical claims and to illustrate the accuracy and effectiveness of our method.

关键词： Thermal phase change problem gradient flows unconditional energy stability auxiliary variable Runge-Kutta methods phase field

Spectral Optimization Methods for the Time Fractional Diffusion Inverse Problem

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Numerical Mathematics(Theory,Methods and Applications) 2013年第3期6卷 499-519页

作者： Xingyang Ye chuanju xu School of Mathematical Sciences Xiamen UniversityXiamen 361005FujianChina School of Science Jimei UniversityXiamen 361021FujianChina

An inverse problem of reconstructing the initial condition for a time fractional diffusion equation is *** the basis of the optimal control framework,the uniqueness and first order necessary optimality condition of the minimizer for the objective functional are established,and a time-space spectral method is proposed to numerically solve the resulting minimization *** contribution of the paper is threefold:1)a priori error estimate for the spectral approximation is derived;2)a conjugate gradient optimization algorithm is designed to efficiently solve the inverse problem;3)some numerical experiments are carried out to show that the proposed method is capable to find out the optimal initial condition,and that the convergence rate of the method is exponential if the optimal initial condition is smooth.

关键词： Time fractional diffusion equation inverse problem spectral method error estimate conjugate gradient method.

Analysis of a system of autonomous fractional differential equations

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International Journal of Biomathematics 2017年第7期10卷 57-74页

作者： Xiaojun Zhou chuanju xu School of Mathematical Sciences Guizhou Normal University Guiyang 550001 P. R. China School of Mathematical Sciences Xiamen University Xiamen 361005 P. R. China

In this work, we study a system of autonomous fractional differential equations. The differential operator is taken in the Caputo sense. Using the monotone iterative technique combined with the method of upper and lower solutions, we investigate the existence and uniqueness of solutions for coupled system which are nonlinear fractional differential equations, moreover, we obtain the dependence of the solution on the initial values. In addition, we give an important example that is a two-patch subdiffusive predator-prey metapopulation model, investigate the solvability and give the numerical results with this model. The numerical simulation indicates that the results of the suhdiffusive model approximate to the two-patch predator-prey metapopulation model with the order a approach to 1.

关键词： Caputo fractional derivative autonomous fractional differential equations upper and lower solutions two-patch subdiffusive predator-prey metapopulation model.

A High-Order Scheme for Fractional Ordinary Differential Equations with the Caputo-Fabrizio Derivative

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Communications on Applied Mathematics and Computation 2020年第2期2卷 179-199页

作者： Junying Cao Ziqiang Wang chuanju xu School of Data Science and Information Engineering Guizhou Minzu UniversityGuiyang 550025China School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen UniversityXiamen 361005China

In this paper, we consider numerical solutions of fractional ordinary diferential equations with the Caputo-Fabrizio derivative, and construct and analyze a high-order time-stepping scheme for this equation. The proposed method makes use of quadratic interpolation function in sub-intervals, which allows to produce fourth-order convergence. A rigorous stability and convergence analysis of the proposed scheme is given. A series of numerical examples are presented to validate the theoretical claims. Traditionally a scheme having fourth-order convergence could only be obtained by using block-by-block technique. The advantage of our scheme is that the solution can be obtained step by step, which is cheaper than a block-by-block-based approach.

关键词： Caputo-Fabrizio derivative Fractional diferential equations High-order numerical scheme

A Unstructured Nodal Spectral-Element Method for the Navier-Stokes Equations

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Communications in Computational Physics 2012年第6期12卷 315-336页

作者： Lizhen Chen Jie Shen chuanju xu School of Mathematical Sciences Xiamen University361005 XiamenChina Department of Mathematics Purdue UniversityWest LafayetteIN47907USA

An unstructured nodal spectral-elementmethod for theNavier-Stokes equations is developed in this *** method is based on a triangular and tetrahedral rational approximation and an easy-to-implement nodal basis which fully enjoys the tensorial product *** allows arbitrary triangular and tetrahedralmesh,affording greater flexibility in handling complex domains while maintaining all essential features of the usual spectral-element *** details of the implementation and some numerical examples are provided to validate the efficiency and flexibility of the proposed method.

关键词： Navier-Stokes equations unstructured mesh triangular spectral element method