A numerical algorithm is developed for the approximation of the solution to certain boundary value problems involving the third-order ordinary differential equation associated with draining and coating flows. The auth...A numerical algorithm is developed for the approximation of the solution to certain boundary value problems involving the third-order ordinary differential equation associated with draining and coating flows. The authors show that the approximate so- lutions obtained by the numerical algorithm developed by using uonpolynomial quintic spline functions ave better than those produced by other spline and domain decomposition methods. The algorithm is tested on two problems associated with draining and coating flows to demonstrate the practical usefulness of the approach.展开更多
In this paper, the problem of limit cycles for a class of nonpolynomial planar vector felds is investigated. First, based on Liapunov method theory, we obtain some sufcient conditions for determining the origin as the...In this paper, the problem of limit cycles for a class of nonpolynomial planar vector felds is investigated. First, based on Liapunov method theory, we obtain some sufcient conditions for determining the origin as the critical point of such nonpolynomial planar vector felds to be the focus or center. Then, using Dulac criterion, we establish some sufcient conditions for the nonexistence of limit cycles of this nonpolynomial planar vector felds. And then, according to Hopf bifurcation theory, we analyze some sufcient conditions for bifurcating limit cycles from the origin. Finally, by transforming the nonpolynomial planar vector felds into the generalized Li′enard planar vector felds, we discuss the existence, uniqueness and stability of limit cycles for the former and latter planar vector felds. Some examples are also given to illustrate the efectiveness of our theoretical results.展开更多
In this paper, the boundary value problems for nonlinear third order differential equations are treated. A generic approach based on nonpolynomial quintic spline is developed to solve such boundary value problem. We s...In this paper, the boundary value problems for nonlinear third order differential equations are treated. A generic approach based on nonpolynomial quintic spline is developed to solve such boundary value problem. We show that the approximate solutions of such problems obtained by the numerical algorithm developed using nonpolynomial quintic spline functions are better than those produced by other numerical methods. The algorithm is tested on a problem to demonstrate the practical usefulness of the approach.展开更多
The numerical computation of partial differential equations(PDEs)is highly important across numerous scientific and engineering disciplines.The accuracy and convergence of integration-based methods depend primarily on...The numerical computation of partial differential equations(PDEs)is highly important across numerous scientific and engineering disciplines.The accuracy and convergence of integration-based methods depend primarily on the ability to perform analytical integration over complex domains.Owing to the inherent challenges posed by the complexities of irregular integration domains and general integrands,this paper introduces an innovative analytical method for nonpolynomial integration over complex domains for the first time.This method is initially applied within the framework of the numerical manifold method(NMM)to address the inevitable trigonometric and exponential polynomial integrations encountered in the analysis of the Laplace equation problem.First,a comprehensive overview of the fundamentals of the NMM and the simplex integration(SI)method is provided in this paper.Subsequently,the NMM framework for solving the Laplace equation is elaborated upon,with a focus on deriving closed-form formulas for trigonometric and exponential polynomial integration.Finally,a series of rigorous numerical experiments is conducted,where the proposed method demonstrates improved accuracy and efficiency.In conclusion,this study innovatively enhances the NMM by introducing the SI method for nonpolynomial functions over complex domains,which is a promising approach for increasing accuracy and convergence across various integration-based methods.This groundbreaking achievement has not yet been reported in the publicly available literature.展开更多
文摘A numerical algorithm is developed for the approximation of the solution to certain boundary value problems involving the third-order ordinary differential equation associated with draining and coating flows. The authors show that the approximate so- lutions obtained by the numerical algorithm developed by using uonpolynomial quintic spline functions ave better than those produced by other spline and domain decomposition methods. The algorithm is tested on two problems associated with draining and coating flows to demonstrate the practical usefulness of the approach.
基金Supported by the Natural Science Foundation of Anhui Education Committee(KJ2007A003)the"211 Project"for Academic Innovative Teams of Anhui University(KJTD002B)+3 种基金the Doctoral Scientifc Research Project for Anhui Medical University(XJ201022)the Key Project for Hefei Normal University(2010kj04zd)the Provincial Excellent Young Talents Foundation for Colleges and Universities of Anhui Province(2011SQRL126)the Academic Innovative Scientifc Research Project of Postgraduates for Anhui University(yfc100020,yfc100028)
文摘In this paper, the problem of limit cycles for a class of nonpolynomial planar vector felds is investigated. First, based on Liapunov method theory, we obtain some sufcient conditions for determining the origin as the critical point of such nonpolynomial planar vector felds to be the focus or center. Then, using Dulac criterion, we establish some sufcient conditions for the nonexistence of limit cycles of this nonpolynomial planar vector felds. And then, according to Hopf bifurcation theory, we analyze some sufcient conditions for bifurcating limit cycles from the origin. Finally, by transforming the nonpolynomial planar vector felds into the generalized Li′enard planar vector felds, we discuss the existence, uniqueness and stability of limit cycles for the former and latter planar vector felds. Some examples are also given to illustrate the efectiveness of our theoretical results.
文摘In this paper, the boundary value problems for nonlinear third order differential equations are treated. A generic approach based on nonpolynomial quintic spline is developed to solve such boundary value problem. We show that the approximate solutions of such problems obtained by the numerical algorithm developed using nonpolynomial quintic spline functions are better than those produced by other numerical methods. The algorithm is tested on a problem to demonstrate the practical usefulness of the approach.
基金supported by the Natural Science Foundation of Shanghai(Grant No.21ZR1468500)the Fundamental Research Funds for the Central Universities(Grant No.22120240299)。
文摘The numerical computation of partial differential equations(PDEs)is highly important across numerous scientific and engineering disciplines.The accuracy and convergence of integration-based methods depend primarily on the ability to perform analytical integration over complex domains.Owing to the inherent challenges posed by the complexities of irregular integration domains and general integrands,this paper introduces an innovative analytical method for nonpolynomial integration over complex domains for the first time.This method is initially applied within the framework of the numerical manifold method(NMM)to address the inevitable trigonometric and exponential polynomial integrations encountered in the analysis of the Laplace equation problem.First,a comprehensive overview of the fundamentals of the NMM and the simplex integration(SI)method is provided in this paper.Subsequently,the NMM framework for solving the Laplace equation is elaborated upon,with a focus on deriving closed-form formulas for trigonometric and exponential polynomial integration.Finally,a series of rigorous numerical experiments is conducted,where the proposed method demonstrates improved accuracy and efficiency.In conclusion,this study innovatively enhances the NMM by introducing the SI method for nonpolynomial functions over complex domains,which is a promising approach for increasing accuracy and convergence across various integration-based methods.This groundbreaking achievement has not yet been reported in the publicly available literature.
