Derived from the Boltzmann equation,the neutron transport equation describes the motions and interactions of neutrons with nuclei in nuclear devices such as nuclear reactors.The collision or fission effect are describ...Derived from the Boltzmann equation,the neutron transport equation describes the motions and interactions of neutrons with nuclei in nuclear devices such as nuclear reactors.The collision or fission effect are described as integral terms which arrive in an integro-differential neutron transport equation(IDNT).Only for mono-material or simple geometries conditions,elegant approximation can simplify the transport equation to provide analytic solutions.To solve this integro-differential equation becomes a practical engineering challenge.Recent development of deep-learning techniques provides a new approach to solve them but for some complicated conditions,it is also time consuming.To optimize solving the integro-differential equation particularly under the deep-learning method,we propose to convert the integral terms in the integro-differential neutron transport equation into their corresponding antiderivatives,providing a set of fixed solution constraint conditions for these antiderivatives,thus yielding an exact differential neutron transport equation(EDNT).The paper elucidates the physical meaning of the antiderivatives and analyzes the continuity and computational complexity of the new transport equation form.To illustrate the significant advantage of ENDT,numerical validations have been conducted using various numerical methods on typical benchmark problems.The numerical experiments demonstrate that the EDNT is compatible with various numerical methods,including the finite difference method(FDM),finite volume method(FVM),and PINN.Compared to the IDNT,the EDNT offers significant efficiency advantages,with reductions in computational time ranging from several times to several orders of magnitude.This EDNT approach may also be applicable for other integro-differential transport theories such as radiative energy transport and has potential application in astrophysics or other fields.展开更多
In this article,we establish new and more general traveling wave solutions of space-time fractional Klein–Gordon equation with quadratic nonlinearity and the space-time fractional breaking soliton equations using the...In this article,we establish new and more general traveling wave solutions of space-time fractional Klein–Gordon equation with quadratic nonlinearity and the space-time fractional breaking soliton equations using the modified simple equation method.The proposed method is so powerful and effective to solve nonlinear space-time fractional differential equations by with modified Riemann–Liouville derivative.展开更多
By discussing the zeros of periodic.solutions we give in this paper a criterion for the existence of exactly n+1 simple 4-periodic solutions of the differential delay equation x(T)= -f(x(t-1)).
A robust SQP method, which is analogous to Facchinei’s algorithm, is introduced. The algorithm is globally convergent. It uses automatic rules for choosing penalty parameter, and can efficiently cope with the possibl...A robust SQP method, which is analogous to Facchinei’s algorithm, is introduced. The algorithm is globally convergent. It uses automatic rules for choosing penalty parameter, and can efficiently cope with the possible inconsistency of the quadratic search subproblem. In addition, the algorithm employs a differentiable approximate exact penalty function as a merit function. Unlike the merit function in Facchinei’s algorithm, which is quite complicated and is not easy to be implemented in practice, this new merit function is very simple. As a result, we can use the Facchinei’s idea to construct an algorithm which is easy to be implemented in practice.展开更多
基金supported by the National High Level Talent Special Support Program of China (Grant No. J705981200002001)the Basic Research and Stable Support Research Project of Science and Technology Industry Bureau of China (Grant No. WDZC-2023050305)+1 种基金the Sichuan Province Unveiled and Leading Common Technology Project of China and Natural Science Foundation of Sichuan Province (Grant No. 2023NSFSC0066)the National Natural Science Foundation of China (Grant No. 12175220)。
文摘Derived from the Boltzmann equation,the neutron transport equation describes the motions and interactions of neutrons with nuclei in nuclear devices such as nuclear reactors.The collision or fission effect are described as integral terms which arrive in an integro-differential neutron transport equation(IDNT).Only for mono-material or simple geometries conditions,elegant approximation can simplify the transport equation to provide analytic solutions.To solve this integro-differential equation becomes a practical engineering challenge.Recent development of deep-learning techniques provides a new approach to solve them but for some complicated conditions,it is also time consuming.To optimize solving the integro-differential equation particularly under the deep-learning method,we propose to convert the integral terms in the integro-differential neutron transport equation into their corresponding antiderivatives,providing a set of fixed solution constraint conditions for these antiderivatives,thus yielding an exact differential neutron transport equation(EDNT).The paper elucidates the physical meaning of the antiderivatives and analyzes the continuity and computational complexity of the new transport equation form.To illustrate the significant advantage of ENDT,numerical validations have been conducted using various numerical methods on typical benchmark problems.The numerical experiments demonstrate that the EDNT is compatible with various numerical methods,including the finite difference method(FDM),finite volume method(FVM),and PINN.Compared to the IDNT,the EDNT offers significant efficiency advantages,with reductions in computational time ranging from several times to several orders of magnitude.This EDNT approach may also be applicable for other integro-differential transport theories such as radiative energy transport and has potential application in astrophysics or other fields.
文摘In this article,we establish new and more general traveling wave solutions of space-time fractional Klein–Gordon equation with quadratic nonlinearity and the space-time fractional breaking soliton equations using the modified simple equation method.The proposed method is so powerful and effective to solve nonlinear space-time fractional differential equations by with modified Riemann–Liouville derivative.
基金Chinese National Foundation for Natural Sciences.
文摘By discussing the zeros of periodic.solutions we give in this paper a criterion for the existence of exactly n+1 simple 4-periodic solutions of the differential delay equation x(T)= -f(x(t-1)).
基金This research is supportedin part by the National Natural Science Foundation ofChina(Grant No. 39830070).
文摘A robust SQP method, which is analogous to Facchinei’s algorithm, is introduced. The algorithm is globally convergent. It uses automatic rules for choosing penalty parameter, and can efficiently cope with the possible inconsistency of the quadratic search subproblem. In addition, the algorithm employs a differentiable approximate exact penalty function as a merit function. Unlike the merit function in Facchinei’s algorithm, which is quite complicated and is not easy to be implemented in practice, this new merit function is very simple. As a result, we can use the Facchinei’s idea to construct an algorithm which is easy to be implemented in practice.
