This article uses the reflecting function of Mironenko to study some complicated differential equations which are equivalent to the Abel equation. The results are applied to discuss the behavior of solutions of these ...This article uses the reflecting function of Mironenko to study some complicated differential equations which are equivalent to the Abel equation. The results are applied to discuss the behavior of solutions of these complicated differential equations.展开更多
This paper is devoted to the conditions of the existence of CC-center for the generalized Abel equations.Using some new original methods,we obtain extended results of the main theorems in the paper by Llibre and Valls...This paper is devoted to the conditions of the existence of CC-center for the generalized Abel equations.Using some new original methods,we obtain extended results of the main theorems in the paper by Llibre and Valls(2020)and the one by Zhou(2020),respectively.The proofs in this paper are much simpler than the previous ones.展开更多
In this paper, the solution, more general than [1], of a weak singular integral equation integral(0)(pi)integral(-infinity)(infinity) p(s,psi)d sk(psi)d psi=F(r,theta), (r,theta)epsilon (Q) over bar=Q+partial derivati...In this paper, the solution, more general than [1], of a weak singular integral equation integral(0)(pi)integral(-infinity)(infinity) p(s,psi)d sk(psi)d psi=F(r,theta), (r,theta)epsilon (Q) over bar=Q+partial derivative Q subject to constraint p(s,psi)=0, for (s,psi)=(r,theta)is not an element of Q={r,theta)/F(r,theta)>c*} is found p=2/pi[root w g'(0)+integral(0)(w) root w-u g '(u)du] where k and F are given continuous functions; (s,psi) is a local polar coordinating with origin at M(r,theta); (r,theta) is the global polar coordinating with origin at O(0,0) F(r,theta)=c* (const.) is the boundary contour partial derivative Q of the considered range Q; g(w)=F(r,theta)/[pi k(psi(0))]; g'=dg/dw; w=N-r(2)sin(2)(theta+psi(0)); psi(0) and N are mean values. The solution shown in type (2.19) of [1] is a special case of the above solution and only suits F(r,theta)=w. The solution of a rigid cone contact with elastic half space, more simple and clear than Love's (1939), is given as an example of application.展开更多
Using the boundary integral equation method, the problem of an external circular crack in a three-dimensional infinite elastic body under asymmetric loadings is investigated. The two-dimensional singular boundary inte...Using the boundary integral equation method, the problem of an external circular crack in a three-dimensional infinite elastic body under asymmetric loadings is investigated. The two-dimensional singular boundary integral equations of the problem were reduced to a system of Abel integral equations by means of Fourier series and hypergeometric functions. The exact solutions of stress intensity factors ore obtained for the problem of an external circular crack under asymmetric loadings, which are even more universal than the results obtained by the use of Hankel transform method. The results demonstrate that the boundary integral equation method has great potential as a new analytic method.展开更多
The approach of word problems is adopted to study the centers of nonautonomous cubic equations.We answer various questions that arise from this investigation. In particular.we demonstrate that a recent condition for t...The approach of word problems is adopted to study the centers of nonautonomous cubic equations.We answer various questions that arise from this investigation. In particular.we demonstrate that a recent condition for the existence of a center is not necessary.展开更多
In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of un...In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of unknown functions at an arbitrary point and integration method, such that the given system of singular integral equations is converted into a system of linear equations with respect to unknown functions and their derivatives. The required solutions are obtained by solving the resulting linear system. The proposed method gives a very satisfactory solution,which can be performed by any symbolic mathematical packages such as Maple, Mathematica, etc. Our proposed approach provides a significant advantage that the m-th order approximate solutions are equal to exact solutions if the exact solutions are polynomial functions of degree less than or equal to m. We present an error analysis for the proposed method to emphasize its reliability. Six numerical examples are provided to show the accuracy and the efficiency of the suggested scheme for which the exact solutions are known in advance.展开更多
In this paper we study the decision problem of the center or focus for a class of planar polynomial fields which can be changed into Abel equation. Taking a Poincaré return map h(x) , we can calculate any orde...In this paper we study the decision problem of the center or focus for a class of planar polynomial fields which can be changed into Abel equation. Taking a Poincaré return map h(x) , we can calculate any order derivative of h(x) at x=0 and obtain the focus value of each order. The new method in this paper avoids the recurence operation and reduces the work in calculating the focus value.展开更多
This paper deals with the existence of Darboux first integrals for the planar polynomial differential systems x=x-y+P n+1(x,y)+xF2n(x,y),y=x+y+Q n+1(x,y)+yF2n(x,y),where P i(x,y),Q i(x,y)and F i(x,y)are homogeneous po...This paper deals with the existence of Darboux first integrals for the planar polynomial differential systems x=x-y+P n+1(x,y)+xF2n(x,y),y=x+y+Q n+1(x,y)+yF2n(x,y),where P i(x,y),Q i(x,y)and F i(x,y)are homogeneous polynomials of degree i.Within this class,we identify some new Darboux integrable systems having either a focus or a center at the origin.For such Darboux integrable systems having degrees 5and 9 we give the explicit expressions of their algebraic limit cycles.For the systems having degrees 3,5,7 and 9and restricted to a certain subclass we present necessary and sufficient conditions for being Darboux integrable.展开更多
文摘This article uses the reflecting function of Mironenko to study some complicated differential equations which are equivalent to the Abel equation. The results are applied to discuss the behavior of solutions of these complicated differential equations.
基金Supported by NSFC(Grant Nos.12171491 and 12071006)。
文摘This paper is devoted to the conditions of the existence of CC-center for the generalized Abel equations.Using some new original methods,we obtain extended results of the main theorems in the paper by Llibre and Valls(2020)and the one by Zhou(2020),respectively.The proofs in this paper are much simpler than the previous ones.
文摘In this paper, the solution, more general than [1], of a weak singular integral equation integral(0)(pi)integral(-infinity)(infinity) p(s,psi)d sk(psi)d psi=F(r,theta), (r,theta)epsilon (Q) over bar=Q+partial derivative Q subject to constraint p(s,psi)=0, for (s,psi)=(r,theta)is not an element of Q={r,theta)/F(r,theta)>c*} is found p=2/pi[root w g'(0)+integral(0)(w) root w-u g '(u)du] where k and F are given continuous functions; (s,psi) is a local polar coordinating with origin at M(r,theta); (r,theta) is the global polar coordinating with origin at O(0,0) F(r,theta)=c* (const.) is the boundary contour partial derivative Q of the considered range Q; g(w)=F(r,theta)/[pi k(psi(0))]; g'=dg/dw; w=N-r(2)sin(2)(theta+psi(0)); psi(0) and N are mean values. The solution shown in type (2.19) of [1] is a special case of the above solution and only suits F(r,theta)=w. The solution of a rigid cone contact with elastic half space, more simple and clear than Love's (1939), is given as an example of application.
基金国家自然科学基金,West Foundation of Ministry Education of China
文摘Using the boundary integral equation method, the problem of an external circular crack in a three-dimensional infinite elastic body under asymmetric loadings is investigated. The two-dimensional singular boundary integral equations of the problem were reduced to a system of Abel integral equations by means of Fourier series and hypergeometric functions. The exact solutions of stress intensity factors ore obtained for the problem of an external circular crack under asymmetric loadings, which are even more universal than the results obtained by the use of Hankel transform method. The results demonstrate that the boundary integral equation method has great potential as a new analytic method.
文摘The approach of word problems is adopted to study the centers of nonautonomous cubic equations.We answer various questions that arise from this investigation. In particular.we demonstrate that a recent condition for the existence of a center is not necessary.
文摘In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of unknown functions at an arbitrary point and integration method, such that the given system of singular integral equations is converted into a system of linear equations with respect to unknown functions and their derivatives. The required solutions are obtained by solving the resulting linear system. The proposed method gives a very satisfactory solution,which can be performed by any symbolic mathematical packages such as Maple, Mathematica, etc. Our proposed approach provides a significant advantage that the m-th order approximate solutions are equal to exact solutions if the exact solutions are polynomial functions of degree less than or equal to m. We present an error analysis for the proposed method to emphasize its reliability. Six numerical examples are provided to show the accuracy and the efficiency of the suggested scheme for which the exact solutions are known in advance.
文摘In this paper we study the decision problem of the center or focus for a class of planar polynomial fields which can be changed into Abel equation. Taking a Poincaré return map h(x) , we can calculate any order derivative of h(x) at x=0 and obtain the focus value of each order. The new method in this paper avoids the recurence operation and reduces the work in calculating the focus value.
基金supported by National Natural Science Foundation of China (Grant No. 11271252)Ministerio de Economiay Competitidad of Spain (Grant No. MTM2008-03437)+2 种基金 Agència de Gestió d’Ajuts Universitaris i de Recerca of Catalonia (Grant No. 2009SGR410)ICREA Academia,Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110073110054)a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme (Grant Nos. FP7-PEOPLE-2012-IRSES-316338 and 318999)
文摘This paper deals with the existence of Darboux first integrals for the planar polynomial differential systems x=x-y+P n+1(x,y)+xF2n(x,y),y=x+y+Q n+1(x,y)+yF2n(x,y),where P i(x,y),Q i(x,y)and F i(x,y)are homogeneous polynomials of degree i.Within this class,we identify some new Darboux integrable systems having either a focus or a center at the origin.For such Darboux integrable systems having degrees 5and 9 we give the explicit expressions of their algebraic limit cycles.For the systems having degrees 3,5,7 and 9and restricted to a certain subclass we present necessary and sufficient conditions for being Darboux integrable.