Given a polynomial with symbolic/literal coefficients,a complete discrimination system is a set of explicit expressions in terms of the coefficients,which is sufficient for determining the numbers and multiplicities o...Given a polynomial with symbolic/literal coefficients,a complete discrimination system is a set of explicit expressions in terms of the coefficients,which is sufficient for determining the numbers and multiplicities of the real and imaginary roots.Though it is of great significance,such a criterion for root-classification has never been given for polynomials with degrees greater than 4.The lack of efficient tools in this aspect extremely prevents computer implementations for Tarski’s and other methods in automated theorem proving.To remedy this defect,a generic algorithm is proposed to produce a complete discrimination system for a polynomial with any degrees.This result has extensive applications in various fields,and its efficiency was demonstrated by computer implementations.展开更多
By establishing a complete discrimination system for polynomials, the problem of complete root classification for polynomials with complex coefficients is utterly solved, furthermore, the algorithm obtained is made in...By establishing a complete discrimination system for polynomials, the problem of complete root classification for polynomials with complex coefficients is utterly solved, furthermore, the algorithm obtained is made into a general program in Maple, which enables the complete discrimination system and complete root classification of a polynomial to be automatically generated by computer, without any human intervention. Besides, by using the automatic generation of root classification, a method to determine the positive definiteness of a polynomial in one or two indeterminates is automatically presented.展开更多
A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a （1＋1）-dimensional dispersive long wave equation with general coefficients to an elementary integral fo...A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a （1＋1）-dimensional dispersive long wave equation with general coefficients to an elementary integral form and obtained its all possible exact travelling wave solutions including rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions double periodic solutions. This method can be also applied to many other similar problems.展开更多
An extended Fan's algebraic method is used for constructing exact traveling wave solution of nonlinearpartial differential equations.The key idea of this method is to introduce an auxiliary ordinary differential e...An extended Fan's algebraic method is used for constructing exact traveling wave solution of nonlinearpartial differential equations.The key idea of this method is to introduce an auxiliary ordinary differential equationwhich is regarded as an extended elliptic equation and whose degree Υ is expanded to the case of r>4.The efficiency ofthe method is demonstrated by the KdV equation and the variant Boussinesq equations.The results indicate that themethod not only offers all solutions obtained by using Fu's and Fan's methods,but also some new solutions.展开更多
The concept of weak strictly positive real regions is introduced, and its properties are discussed . By using the complete discrimination system for polynomials, complete characterization of the (weak) strictly positi...The concept of weak strictly positive real regions is introduced, and its properties are discussed . By using the complete discrimination system for polynomials, complete characterization of the (weak) strictly positive real regions for transfer functions in coefficient space is given. A new effective method for robust strictly positive real synthesis is proposed. This method results in necessary and sufficient conditions for low-order stable interval polynomials and segment polynomials, and is also efficient for high-order cases. Numerical examples are provided to illustrate the effectiveness of this method.展开更多
The complete discrimination system for polynomial method is applied to the long-short-wave interaction system to obtain the classifications of single traveling wave solutions. Compared with the solutions given by the ...The complete discrimination system for polynomial method is applied to the long-short-wave interaction system to obtain the classifications of single traveling wave solutions. Compared with the solutions given by the （G~/G）-expansion method, we gain some new solutions.展开更多
In order to get the exact traveling wave solutions to nonlinear partial differential equation, the complete discrimination system for polynomial and direct integral method are applied to the considered equation. All s...In order to get the exact traveling wave solutions to nonlinear partial differential equation, the complete discrimination system for polynomial and direct integral method are applied to the considered equation. All single traveling wave solutions to the equation can be obtained. As an example, we give the solutions to (3 + 1)-dimensional breaking soliton equation.展开更多
Trial equation method is a powerful tool for obtaining exact solutions of nonlinear differential equations. In this paper, the improved Boussinesq is reduced to an ordinary differential equation under the travelling w...Trial equation method is a powerful tool for obtaining exact solutions of nonlinear differential equations. In this paper, the improved Boussinesq is reduced to an ordinary differential equation under the travelling wave transformation. Trial equation method and the theory of complete discrimination system for polynomial are used to establish exact solutions of the improved Boussinesq equation.展开更多
In the article, the nonlinear equation is reduced to an ordinary differential equation under the travelling wave transformation. Using trial equation method, the ODE is reduced to the elementary integral form. In the ...In the article, the nonlinear equation is reduced to an ordinary differential equation under the travelling wave transformation. Using trial equation method, the ODE is reduced to the elementary integral form. In the end, complete discrimination system for polynomial is used to solve the corresponding integrals and obtain the classification of all single travelling wave solutions to the equation.展开更多
By the complete discrimination system for polynomial method, we obtained the classification of single traveling wave solutions to the generalized strong nonlinear Boussinesq equation without dissipation terms in p=1.
By using the complete discrimination system for the polynomial method, the classification of single traveling wave solutions to the generalized Kadomtsev-Petviashvili equation without dissipation terms in p=2?is obtai...By using the complete discrimination system for the polynomial method, the classification of single traveling wave solutions to the generalized Kadomtsev-Petviashvili equation without dissipation terms in p=2?is obtained.展开更多
By the complete discrimination system for polynomials, we classify exact traveling wave solutions to the Zhiber-Shabat equation, and compute some new traveling wave solutions.
基金Project supported by the National Natural Science Foundation of China.
文摘Given a polynomial with symbolic/literal coefficients,a complete discrimination system is a set of explicit expressions in terms of the coefficients,which is sufficient for determining the numbers and multiplicities of the real and imaginary roots.Though it is of great significance,such a criterion for root-classification has never been given for polynomials with degrees greater than 4.The lack of efficient tools in this aspect extremely prevents computer implementations for Tarski’s and other methods in automated theorem proving.To remedy this defect,a generic algorithm is proposed to produce a complete discrimination system for a polynomial with any degrees.This result has extensive applications in various fields,and its efficiency was demonstrated by computer implementations.
基金Project supported by the "Scale" Plan and the "863" Plan of China.
文摘By establishing a complete discrimination system for polynomials, the problem of complete root classification for polynomials with complex coefficients is utterly solved, furthermore, the algorithm obtained is made into a general program in Maple, which enables the complete discrimination system and complete root classification of a polynomial to be automatically generated by computer, without any human intervention. Besides, by using the automatic generation of root classification, a method to determine the positive definiteness of a polynomial in one or two indeterminates is automatically presented.
文摘A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a （1＋1）-dimensional dispersive long wave equation with general coefficients to an elementary integral form and obtained its all possible exact travelling wave solutions including rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions double periodic solutions. This method can be also applied to many other similar problems.
基金National Natural Science Foundation of China under Grant No.10672053
文摘An extended Fan's algebraic method is used for constructing exact traveling wave solution of nonlinearpartial differential equations.The key idea of this method is to introduce an auxiliary ordinary differential equationwhich is regarded as an extended elliptic equation and whose degree Υ is expanded to the case of r>4.The efficiency ofthe method is demonstrated by the KdV equation and the variant Boussinesq equations.The results indicate that themethod not only offers all solutions obtained by using Fu's and Fan's methods,but also some new solutions.
文摘The concept of weak strictly positive real regions is introduced, and its properties are discussed . By using the complete discrimination system for polynomials, complete characterization of the (weak) strictly positive real regions for transfer functions in coefficient space is given. A new effective method for robust strictly positive real synthesis is proposed. This method results in necessary and sufficient conditions for low-order stable interval polynomials and segment polynomials, and is also efficient for high-order cases. Numerical examples are provided to illustrate the effectiveness of this method.
基金Project supported by the Scientific Research Fund of Education Department of Heilongjiang Province of China (Grant No.12531475)
文摘The complete discrimination system for polynomial method is applied to the long-short-wave interaction system to obtain the classifications of single traveling wave solutions. Compared with the solutions given by the （G~/G）-expansion method, we gain some new solutions.
文摘In order to get the exact traveling wave solutions to nonlinear partial differential equation, the complete discrimination system for polynomial and direct integral method are applied to the considered equation. All single traveling wave solutions to the equation can be obtained. As an example, we give the solutions to (3 + 1)-dimensional breaking soliton equation.
文摘Trial equation method is a powerful tool for obtaining exact solutions of nonlinear differential equations. In this paper, the improved Boussinesq is reduced to an ordinary differential equation under the travelling wave transformation. Trial equation method and the theory of complete discrimination system for polynomial are used to establish exact solutions of the improved Boussinesq equation.
文摘In the article, the nonlinear equation is reduced to an ordinary differential equation under the travelling wave transformation. Using trial equation method, the ODE is reduced to the elementary integral form. In the end, complete discrimination system for polynomial is used to solve the corresponding integrals and obtain the classification of all single travelling wave solutions to the equation.
文摘By the complete discrimination system for polynomial method, we obtained the classification of single traveling wave solutions to the generalized strong nonlinear Boussinesq equation without dissipation terms in p=1.
文摘By using the complete discrimination system for the polynomial method, the classification of single traveling wave solutions to the generalized Kadomtsev-Petviashvili equation without dissipation terms in p=2?is obtained.
文摘By the complete discrimination system for polynomials, we classify exact traveling wave solutions to the Zhiber-Shabat equation, and compute some new traveling wave solutions.
