The fractional derivatives in the sense of Caputo and the homotopy analysis method are used to construct an approximate solution for the nonlinear space-time fractional derivatives Klein-Gordon equation. The numerical...The fractional derivatives in the sense of Caputo and the homotopy analysis method are used to construct an approximate solution for the nonlinear space-time fractional derivatives Klein-Gordon equation. The numerical results show that the approaches are easy to implement and accurate when applied to the nonlinear space-time fractional derivatives Klein- Gordon equation. This method introduces a promising tool for solving many space-time fractional partial differential equations. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.展开更多
In this paper,we present the existence and uniqueness of fixed points and common fixed points for Reich and Chatterjea pairs of self-maps in complete metric spaces.Furthermore,we study fixed point theorems for Reich a...In this paper,we present the existence and uniqueness of fixed points and common fixed points for Reich and Chatterjea pairs of self-maps in complete metric spaces.Furthermore,we study fixed point theorems for Reich and Chatterjea nonexpansive mappings in a Banach space using the Krasnoselskii-Ishikawa iteration method associated withSλand consider some applications of our results to prove the existence of solutions for nonlinear integral and nonlinear fractional differential equations.We also establish certain interesting examples to illustrate the usability of our results.展开更多
In this paper, we get many new analytical solutions of the space-time nonlinear fractional modified KDV-Zakharov Kuznetsov (mKDV-ZK) equation by means of a new approach namely method of undetermined coefficients based...In this paper, we get many new analytical solutions of the space-time nonlinear fractional modified KDV-Zakharov Kuznetsov (mKDV-ZK) equation by means of a new approach namely method of undetermined coefficients based on a fractional complex transform. These solutions have physics meanings in natural sciences. This method can be used to other nonlinear fractional differential equations.展开更多
Based on the Crank-Nicolson and the weighted and shifted Grunwald operators,we present an implicit difference scheme for the Riesz space fractional reaction-dispersion equations and also analyze the stability and the ...Based on the Crank-Nicolson and the weighted and shifted Grunwald operators,we present an implicit difference scheme for the Riesz space fractional reaction-dispersion equations and also analyze the stability and the convergence of this implicit difference scheme.However,after estimating the condition number of the coefficient matrix of the discretized scheme,we find that this coefficient matrix is ill-conditioned when the spatial mesh-size is sufficiently small.To overcome this deficiency,we further develop an effective banded M-matrix splitting preconditioner for the coefficient matrix.Some properties of this preconditioner together with its preconditioning effect are discussed.Finally,Numerical examples are employed to test the robustness and the effectiveness of the proposed preconditioner.展开更多
Motivated by the widely used ans¨atz method and starting from the modified Riemann–Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional pa...Motivated by the widely used ans¨atz method and starting from the modified Riemann–Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper.展开更多
In this paper,we have considered the multi-dimensional space fractional diffusion equations with variable coefficients.The fractional operators(derivative/integral)are used based on the Caputo definition.This study pr...In this paper,we have considered the multi-dimensional space fractional diffusion equations with variable coefficients.The fractional operators(derivative/integral)are used based on the Caputo definition.This study provides an analytical approach to determine the analytical solution of the considered problems with the help of the two-step Adomian decomposition method(TSADM).Moreover,new results have been obtained for the existence and uniqueness of a solution by using the Banach contraction principle and a fixed point theorem.We have extended the dimension of the space fractional diffusion equations with variable coefficients into multi-dimensions.Finally,the generalized problems with two different types of the forcing term have been included demonstrating the applicability and high efficiency of the TSADM in comparison to other existing numerical methods.The diffusion coefficients do not require to satisfy any certain conditions/restrictions for using the TSADM.There are no restrictions imposed on the problems for diffusion coefficients,and a similar procedures of the TSADM has followed to the obtained analytical solution for the multi-dimensional space fractional diffusion equations with variable coefficients.展开更多
In this paper,we consider the numerical solutions of the semilinear Riesz space-fractional diffusion equations(RSFDEs)with time delay,which constitute an important class of differential equations of practical signific...In this paper,we consider the numerical solutions of the semilinear Riesz space-fractional diffusion equations(RSFDEs)with time delay,which constitute an important class of differential equations of practical significance.We develop a novel implicit alternating direction method that can effectively and efficiently tackle the RSFDEs in both two and three dimensions.The numerical method is proved to be uniquely solvable,stable and convergent with second order accuracy in both space and time.Numerical results are presented to verify the accuracy and efficiency of the proposed numerical scheme.展开更多
文摘The fractional derivatives in the sense of Caputo and the homotopy analysis method are used to construct an approximate solution for the nonlinear space-time fractional derivatives Klein-Gordon equation. The numerical results show that the approaches are easy to implement and accurate when applied to the nonlinear space-time fractional derivatives Klein- Gordon equation. This method introduces a promising tool for solving many space-time fractional partial differential equations. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.
文摘In this paper,we present the existence and uniqueness of fixed points and common fixed points for Reich and Chatterjea pairs of self-maps in complete metric spaces.Furthermore,we study fixed point theorems for Reich and Chatterjea nonexpansive mappings in a Banach space using the Krasnoselskii-Ishikawa iteration method associated withSλand consider some applications of our results to prove the existence of solutions for nonlinear integral and nonlinear fractional differential equations.We also establish certain interesting examples to illustrate the usability of our results.
文摘In this paper, we get many new analytical solutions of the space-time nonlinear fractional modified KDV-Zakharov Kuznetsov (mKDV-ZK) equation by means of a new approach namely method of undetermined coefficients based on a fractional complex transform. These solutions have physics meanings in natural sciences. This method can be used to other nonlinear fractional differential equations.
基金supported by the National Natural Science Foundation of China(Grant No.12161030)by the Hainan Provincial Natural Science Foundation of China(Grant No.121RC537).
文摘Based on the Crank-Nicolson and the weighted and shifted Grunwald operators,we present an implicit difference scheme for the Riesz space fractional reaction-dispersion equations and also analyze the stability and the convergence of this implicit difference scheme.However,after estimating the condition number of the coefficient matrix of the discretized scheme,we find that this coefficient matrix is ill-conditioned when the spatial mesh-size is sufficiently small.To overcome this deficiency,we further develop an effective banded M-matrix splitting preconditioner for the coefficient matrix.Some properties of this preconditioner together with its preconditioning effect are discussed.Finally,Numerical examples are employed to test the robustness and the effectiveness of the proposed preconditioner.
基金Supported by National Natural Science Foundation of China under Grant Nos.11071278,111471004the Fundamental Research Funds for the Central Universities of GK201302026 and GK201102007
文摘Motivated by the widely used ans¨atz method and starting from the modified Riemann–Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper.
文摘In this paper,we have considered the multi-dimensional space fractional diffusion equations with variable coefficients.The fractional operators(derivative/integral)are used based on the Caputo definition.This study provides an analytical approach to determine the analytical solution of the considered problems with the help of the two-step Adomian decomposition method(TSADM).Moreover,new results have been obtained for the existence and uniqueness of a solution by using the Banach contraction principle and a fixed point theorem.We have extended the dimension of the space fractional diffusion equations with variable coefficients into multi-dimensions.Finally,the generalized problems with two different types of the forcing term have been included demonstrating the applicability and high efficiency of the TSADM in comparison to other existing numerical methods.The diffusion coefficients do not require to satisfy any certain conditions/restrictions for using the TSADM.There are no restrictions imposed on the problems for diffusion coefficients,and a similar procedures of the TSADM has followed to the obtained analytical solution for the multi-dimensional space fractional diffusion equations with variable coefficients.
基金supported by National Natural Science Foundation(NSF)of China(Grant No.11501238)NSF of Guangdong Province(Grant No.2016A030313119)and NSF of Huizhou University(Grant No.hzu201806)+2 种基金supported by the startup fund from City University of Hong Kong and the Hong Kong RGC General Research Fund(projects Nos.12301420,12302919 and 12301218)supported by the NSF of China No.11971221the Shenzhen Sci-Tech Fund No.JCYJ20190809150413261,JCYJ20180307151603959,and JCYJ20170818153840322,and Guangdong Provincial Key Laboratory of Computational Science and Material Design(No.2019B030301001).
文摘In this paper,we consider the numerical solutions of the semilinear Riesz space-fractional diffusion equations(RSFDEs)with time delay,which constitute an important class of differential equations of practical significance.We develop a novel implicit alternating direction method that can effectively and efficiently tackle the RSFDEs in both two and three dimensions.The numerical method is proved to be uniquely solvable,stable and convergent with second order accuracy in both space and time.Numerical results are presented to verify the accuracy and efficiency of the proposed numerical scheme.
基金Supported by the Natural Science Foundation of Fujian Province(2008J0204)Fujian Provincial Department of Education Category the Projects(JA09242)Wuyi University Special Research Fund for Young Teachers(xq201022)
