In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green fun...In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green function about the operators is also given.展开更多
In this article, we establish the Gauss Green type theorems for Clifford-valued functions in Clifford analysis. The boundary conditions in theorems obtained are very general by using the geometric measure theoretic me...In this article, we establish the Gauss Green type theorems for Clifford-valued functions in Clifford analysis. The boundary conditions in theorems obtained are very general by using the geometric measure theoretic method. The Cauchy-Pompeiu formula for Clifford-valued functions under the weak condition will be derived as their simple application. Furthermore, Cauchy formula for monogenic functions under the weak condition is derived directly from the Cauchy-Pompeiu formula.展开更多
It is explored that the line integral is a path independent in two or three arbitrary dimensional orthogonal curvilinear coordinate systems, which is based on the integral condition with the path independent in two or...It is explored that the line integral is a path independent in two or three arbitrary dimensional orthogonal curvilinear coordinate systems, which is based on the integral condition with the path independent in two or three dimensional rectangular coordinate systems. Firstly, according to the coordinate transformation, the condition that the line integral is the path independent in the polar coordinate system is obtained easily from the Green's theorem in two-dimensional rectangular coordinate system and the condition is extended to arbitrary two-dimension orthogonal curvilinear coordinates. Secondly, through the coordinate transformation relationship and the area projection method, the Stokes formula in three-dimensional rectangular coordinate system is promoted to the spherical coordinate system and cylindrical coordinate system, and the condition that the line integral is a path independent is obtained. Furthermore, the condition is extended to arbitrary three-dimension orthogonal curvilinear coordinates. Lastly, the conclusions are made.展开更多
A class of normal-like derivatives for functions with low regularity defined on Lipschitz domains are introduced and studied.It is shown that the new normal-like derivatives,which are called the generalized normal der...A class of normal-like derivatives for functions with low regularity defined on Lipschitz domains are introduced and studied.It is shown that the new normal-like derivatives,which are called the generalized normal derivatives,preserve the major prop- erties of the existing standard normal derivatives.The generalized normal derivatives are then applied to analyze the convergence of domain decomposition methods (DDMs) with nonmatching grids and discontinuous Galerkin (DG) methods for second-order el- liptic problems.The approximate solutions generated by these methods still possess the optimal energy-norm error estimates,even if the exact solutions to the underlying elliptic problems admit very low regularities.展开更多
In this paper,the initial boundary value problems of the nonlinear Sobolev-Galpern equation are studied.The existence,uniqueness of local solution for the problem are obtained by means of a special Green's functio...In this paper,the initial boundary value problems of the nonlinear Sobolev-Galpern equation are studied.The existence,uniqueness of local solution for the problem are obtained by means of a special Green's function and the contraction mapping principle.Finally,the blow-up of solution in finite time under some assumed conditions is proved with the aid of Jensen's inequality.展开更多
文摘In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green function about the operators is also given.
基金supported by NNSF of China(11171260)RFDP of Higher Education of China(20100141110054)
文摘In this article, we establish the Gauss Green type theorems for Clifford-valued functions in Clifford analysis. The boundary conditions in theorems obtained are very general by using the geometric measure theoretic method. The Cauchy-Pompeiu formula for Clifford-valued functions under the weak condition will be derived as their simple application. Furthermore, Cauchy formula for monogenic functions under the weak condition is derived directly from the Cauchy-Pompeiu formula.
基金Funded by the Natural Science Foundation Project of CQCSTC(No.cstc2012jj A50018)the Basic Research of Chongqing Municipal Education Commission(No.KJ120631)the Science Research Foundation Project of CQNU(No.16XYY31)
文摘It is explored that the line integral is a path independent in two or three arbitrary dimensional orthogonal curvilinear coordinate systems, which is based on the integral condition with the path independent in two or three dimensional rectangular coordinate systems. Firstly, according to the coordinate transformation, the condition that the line integral is the path independent in the polar coordinate system is obtained easily from the Green's theorem in two-dimensional rectangular coordinate system and the condition is extended to arbitrary two-dimension orthogonal curvilinear coordinates. Secondly, through the coordinate transformation relationship and the area projection method, the Stokes formula in three-dimensional rectangular coordinate system is promoted to the spherical coordinate system and cylindrical coordinate system, and the condition that the line integral is a path independent is obtained. Furthermore, the condition is extended to arbitrary three-dimension orthogonal curvilinear coordinates. Lastly, the conclusions are made.
基金supported by The Key Project of Natural Science Foundation of China G10531080National Basic Research Program of China No.2005CB321702Natural Science Foundation of China G10771178.
文摘A class of normal-like derivatives for functions with low regularity defined on Lipschitz domains are introduced and studied.It is shown that the new normal-like derivatives,which are called the generalized normal derivatives,preserve the major prop- erties of the existing standard normal derivatives.The generalized normal derivatives are then applied to analyze the convergence of domain decomposition methods (DDMs) with nonmatching grids and discontinuous Galerkin (DG) methods for second-order el- liptic problems.The approximate solutions generated by these methods still possess the optimal energy-norm error estimates,even if the exact solutions to the underlying elliptic problems admit very low regularities.
文摘In this paper,the initial boundary value problems of the nonlinear Sobolev-Galpern equation are studied.The existence,uniqueness of local solution for the problem are obtained by means of a special Green's function and the contraction mapping principle.Finally,the blow-up of solution in finite time under some assumed conditions is proved with the aid of Jensen's inequality.
