The present paper discusses a class of nonlinear diffusion-convection equations with source. The method that we use is the conditional symmetry method. It is shown that the equation admits certain conditional symmetri...The present paper discusses a class of nonlinear diffusion-convection equations with source. The method that we use is the conditional symmetry method. It is shown that the equation admits certain conditional symmetries for coefficient functions of the equations. As a consequence, solutions to the resulting equations are obtained.展开更多
The concept of approximate generalized conditional symmetry （AGCS） for the perturbed evolution equations is introduced, and how to derive approximate conditional invariant solutions to the perturbed equations via th...The concept of approximate generalized conditional symmetry （AGCS） for the perturbed evolution equations is introduced, and how to derive approximate conditional invariant solutions to the perturbed equations via their A GCSs is illustrated with examples.展开更多
We develop the generalized conditional symmetry （GCS） approach to solve the problem of dimensional reduction of Cauchy problems for the KdV-type equations. We characterize these equations that admit certain higheror...We develop the generalized conditional symmetry （GCS） approach to solve the problem of dimensional reduction of Cauchy problems for the KdV-type equations. We characterize these equations that admit certain higherorder GCSs and show the main reduction procedure by some examples. The obtained reductions cannot be derived within the framework of the standard Lie approach.展开更多
Lie symmetry reduction of some truly ＂variable coefficient＂ wave equations which are singled out from a class of （1 ＋ 1）-dimensional variable coefficient nonlinear wave equations with respect to one and two-dimen...Lie symmetry reduction of some truly ＂variable coefficient＂ wave equations which are singled out from a class of （1 ＋ 1）-dimensional variable coefficient nonlinear wave equations with respect to one and two-dimensional algebras is carried out. Some classes of exact solutions of the investigated equations are found by means of both the reductions and some modern techniques such as additional equivalent transformations and hidden symmetries and so on. Conditional symmetries are also discussed.展开更多
New classes of exact solutions of the quasi-linear diffusion-reaction equations are obtained by seeking for the high-order conditional Lie-Baeklund symmetries of the considered equations. The method used here extends ...New classes of exact solutions of the quasi-linear diffusion-reaction equations are obtained by seeking for the high-order conditional Lie-Baeklund symmetries of the considered equations. The method used here extends the approaches of derivative-dependent functional separation of variables and the invariant subspace. Behavior to some solutions such as blow-up and quenching is also described.展开更多
This paper is devoted to the study of functional variable separation for extended nonlinear elliptic equations. By applying the functional variable separation approach to extended nonlinear elliptic equations via the ...This paper is devoted to the study of functional variable separation for extended nonlinear elliptic equations. By applying the functional variable separation approach to extended nonlinear elliptic equations via the generalized conditional symmetry, we obtain complete classification of those equations which admit functional separable solutions （FSSs） and construct some exact FSSs to the resulting equations.展开更多
We give the generalized definitions of variable separable solutions to nonlinear evolution equations, and characterize the relation between the functional separable solution and the derivative-dependent functional sep...We give the generalized definitions of variable separable solutions to nonlinear evolution equations, and characterize the relation between the functional separable solution and the derivative-dependent functional separable solution. The new definitions can unify various kinds of variable separable solutions appearing in references. As application, we classify the generalized nonlinear diffusion equations that admit special functional separable solutions and obtain some exact solutions to the resulting equations.展开更多
This paper studies variable separation of the evolution equations via the generalized conditional symmetry. To illustrate, we classify the extended nonlinear wave equation utt = A（u, ux）uxx＋B（u, ux, ut） which adm...This paper studies variable separation of the evolution equations via the generalized conditional symmetry. To illustrate, we classify the extended nonlinear wave equation utt = A（u, ux）uxx＋B（u, ux, ut） which admits the derivative- dependent functional separable solutions （DDFSSs）. We also extend the concept of the DDFSS to cover other variable separation approaches.展开更多
We exploit higher-order conditional symmetry to reduce initial-value problems for evolution equations toCauchy problems for systems of ordinary differential equations (ODEs).We classify a class of fourth-order evoluti...We exploit higher-order conditional symmetry to reduce initial-value problems for evolution equations toCauchy problems for systems of ordinary differential equations (ODEs).We classify a class of fourth-order evolutionequations which admit certain higher-order generalized conditional symmetries (GCSs) and give some examples to showthe main reduction procedure.These reductions cannot be derived within the framework of the standard Lie approach,which hints that the technique presented here is something essential for the dimensional reduction of evolu tion equations.展开更多
The generalized conditional symmetry approach is applied to study the variable separation of the extended wave equations. Complete classification of those equations admitting functional separable solutions is obtained...The generalized conditional symmetry approach is applied to study the variable separation of the extended wave equations. Complete classification of those equations admitting functional separable solutions is obtained and exact separable solutions to some of the resulting equations are constructed.展开更多
Invariant subspace method is exploited to obtain exact solutions of the two- component b-family system. It is shown that the two-component b-family system admits the generalized functional separable solutions. Further...Invariant subspace method is exploited to obtain exact solutions of the two- component b-family system. It is shown that the two-component b-family system admits the generalized functional separable solutions. Furthermore, blow up and behavior of those exact solutions are also investigated.展开更多
We present basic theory of variable separation for (1 + 1)-dimensional nonlinear evolution equations withmixed partial derivatives.As an application,we classify equations u_(xt)=A(u,u_x)u_(xxx)+B(u,u_x) that admits de...We present basic theory of variable separation for (1 + 1)-dimensional nonlinear evolution equations withmixed partial derivatives.As an application,we classify equations u_(xt)=A(u,u_x)u_(xxx)+B(u,u_x) that admits derivative-dependent functional separable solutions (DDFSSs) and illustrate how to construct those DDFSSs with some examples.展开更多
The generalized conditional symmetry is developed to study the variable separation for equations of type uxt = A（u,ux）uxx ＋ B（u, ux）. Complete classification of those equations which admit derivative-dependent fu...The generalized conditional symmetry is developed to study the variable separation for equations of type uxt = A（u,ux）uxx ＋ B（u, ux）. Complete classification of those equations which admit derivative-dependent functional separable solutions is obtained and some of their exact separable solutions are constructed.展开更多
The functionally generalized variable separation of the generalized nonlinear diffusion equations ut = A（u, Ux）Uxx ＋ B（u, ux） is studied by using the conditional Lie-Blicklund symmetry method. The variant forms o...The functionally generalized variable separation of the generalized nonlinear diffusion equations ut = A（u, Ux）Uxx ＋ B（u, ux） is studied by using the conditional Lie-Blicklund symmetry method. The variant forms of the considered equations, which admit the corresponding conditional Lie--Biicklund symmetries, are characterized. To construct functionally gener- alized separable solutions, several concrete examples defined on the exponential and trigonometric invariant subspaces are provided.展开更多
By using the approximate derivative-dependent functional variable separation approach, we study the quasi-linear diffusion equations with a weak source ut = （A（u）Ux）x ＋ eB（u, Ux）. A complete classification of t...By using the approximate derivative-dependent functional variable separation approach, we study the quasi-linear diffusion equations with a weak source ut = （A（u）Ux）x ＋ eB（u, Ux）. A complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is listed. As a consequence, some approxi- mate solutions to the resulting perturbed equations are constructed via examples.展开更多
The functionally generalized variable separation solutions of a general KdV-type equations u_t=u_(xxx) +A(u, u_x)u_(xx) + B(u, u_x) are investigated by developing the conditional Lie-Backlund symmetry method. A comple...The functionally generalized variable separation solutions of a general KdV-type equations u_t=u_(xxx) +A(u, u_x)u_(xx) + B(u, u_x) are investigated by developing the conditional Lie-Backlund symmetry method. A complete classification of the considered equations, which admit multi-dimensional invariant subspaces governed by higher-order conditional Lie-B¨acklund symmetries, is presented. As a result, several concrete examples are provided to construct functionally generalized variable separation solutions of some resulting equations.展开更多
In this article,we have studied a nonlinear time–space fractional longitudinal wave equation in the context of the conformable fractional derivative.Through the soliton ansatz method and a direct integration approach...In this article,we have studied a nonlinear time–space fractional longitudinal wave equation in the context of the conformable fractional derivative.Through the soliton ansatz method and a direct integration approach with the symmetry condition,new soliton and solitary wave solutions are derived.Furthermore,the existing conditions of these obtained solutions are also given in this text.These new results add to the existing literature.We believe that they can provide a new window into the understanding of this model.展开更多
We investigate a generalized form of the Boussinesq equation, relevant for nerve pulse propagation in biological membranes. The generalized conditional symmetry （GCS） method is applied in order to obtain the conditi...We investigate a generalized form of the Boussinesq equation, relevant for nerve pulse propagation in biological membranes. The generalized conditional symmetry （GCS） method is applied in order to obtain the conditions that enable the equation to admit a special class of second-order GCSs. For the case of quadratic nonlinearities, we outline a new class of invariant solutions.展开更多
As an extension to the derivative-dependent functional variable separation approach, the approximate derivative-dependent functional variable separation approach is proposed, and it is applied to study the generalized...As an extension to the derivative-dependent functional variable separation approach, the approximate derivative-dependent functional variable separation approach is proposed, and it is applied to study the generalized diffusion equations with perturbation. Complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is obtained. As a result, the corresponding approximate derivative-dependent functional separable solutions to some resulting perturbed equations are derived by way of examples.展开更多
Under special conditions on data set and underlying distribution, the limit of finite sample breakdown point of Tukey＇s halfspace median （1） has been obtained in the literature. In this paper, we establish the resu...Under special conditions on data set and underlying distribution, the limit of finite sample breakdown point of Tukey＇s halfspace median （1） has been obtained in the literature. In this paper, we establish the result under weaker assumptions imposed on underlying distribution （weak smoothness） and on data set （not necessary in general position）. The refined representation of Tukey＇s sample depth regions for data set not necessary in general position is also obtained, as a by-product of our derivation.展开更多
基金The project supported by National Natural Science Foundation of China under Grant No. 10371098 and the Program for New Century Excellent Talents in Universities under Grant No. NCET-04-0968.
文摘The present paper discusses a class of nonlinear diffusion-convection equations with source. The method that we use is the conditional symmetry method. It is shown that the equation admits certain conditional symmetries for coefficient functions of the equations. As a consequence, solutions to the resulting equations are obtained.
基金The project supported by National Natural Science Foundation of China under Grant No. 10447007, the China Postdoctoral Science Foundation, and the Natural Science Foundation of Shanxi Province under Grant No. 2005A13
文摘The concept of approximate generalized conditional symmetry （AGCS） for the perturbed evolution equations is introduced, and how to derive approximate conditional invariant solutions to the perturbed equations via their A GCSs is illustrated with examples.
基金supported by National Natural Science Foundation of China under Grant Nos.10447007 and 10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.2005A13
文摘We develop the generalized conditional symmetry （GCS） approach to solve the problem of dimensional reduction of Cauchy problems for the KdV-type equations. We characterize these equations that admit certain higherorder GCSs and show the main reduction procedure by some examples. The obtained reductions cannot be derived within the framework of the standard Lie approach.
基金Supported by the National Key Basic Research Project of China under Grant No.2010CB126600the National Natural Science Foundation of China under Grant No.60873070+2 种基金Shanghai Leading Academic Discipline Project No.B114the Postdoctoral Science Foundation of China under Grant No.20090450067Shanghai Postdoctoral Science Foundation under Grant No.09R21410600
文摘Lie symmetry reduction of some truly ＂variable coefficient＂ wave equations which are singled out from a class of （1 ＋ 1）-dimensional variable coefficient nonlinear wave equations with respect to one and two-dimensional algebras is carried out. Some classes of exact solutions of the investigated equations are found by means of both the reductions and some modern techniques such as additional equivalent transformations and hidden symmetries and so on. Conditional symmetries are also discussed.
基金supported by the National Natural Science Foundation of China under Grant No. 10671156the Program for New Century Excellent Talents in Universities under Grant No. NCET-04-0968
文摘New classes of exact solutions of the quasi-linear diffusion-reaction equations are obtained by seeking for the high-order conditional Lie-Baeklund symmetries of the considered equations. The method used here extends the approaches of derivative-dependent functional separation of variables and the invariant subspace. Behavior to some solutions such as blow-up and quenching is also described.
基金The project supported by National Natural Science Foundation of China under Grant No. 10447007 and the Natural Science Foundation of Shaanxi Province of China under Grant No. 2005A13
文摘This paper is devoted to the study of functional variable separation for extended nonlinear elliptic equations. By applying the functional variable separation approach to extended nonlinear elliptic equations via the generalized conditional symmetry, we obtain complete classification of those equations which admit functional separable solutions （FSSs） and construct some exact FSSs to the resulting equations.
基金The project supported by National Natural Science Foundation of China under Grant Nos.10447007 and 10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.2005A13
文摘We give the generalized definitions of variable separable solutions to nonlinear evolution equations, and characterize the relation between the functional separable solution and the derivative-dependent functional separable solution. The new definitions can unify various kinds of variable separable solutions appearing in references. As application, we classify the generalized nonlinear diffusion equations that admit special functional separable solutions and obtain some exact solutions to the resulting equations.
基金Project supported by the National Natural Science Foundation of China （Grant Nos 10371098, 10447007 and 10475055）, the Natural Science Foundation of Shaanxi Province of China （Grant No 2005A13）.
文摘This paper studies variable separation of the evolution equations via the generalized conditional symmetry. To illustrate, we classify the extended nonlinear wave equation utt = A（u, ux）uxx＋B（u, ux, ut） which admits the derivative- dependent functional separable solutions （DDFSSs）. We also extend the concept of the DDFSS to cover other variable separation approaches.
基金National Natural Science Foundation of China under Grant Nos.10447007 and 10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.2005A13
文摘We exploit higher-order conditional symmetry to reduce initial-value problems for evolution equations toCauchy problems for systems of ordinary differential equations (ODEs).We classify a class of fourth-order evolutionequations which admit certain higher-order generalized conditional symmetries (GCSs) and give some examples to showthe main reduction procedure.These reductions cannot be derived within the framework of the standard Lie approach,which hints that the technique presented here is something essential for the dimensional reduction of evolu tion equations.
文摘The generalized conditional symmetry approach is applied to study the variable separation of the extended wave equations. Complete classification of those equations admitting functional separable solutions is obtained and exact separable solutions to some of the resulting equations are constructed.
基金supported by NSFC(11471260)the Foundation of Shannxi Education Committee(12JK0850)
文摘Invariant subspace method is exploited to obtain exact solutions of the two- component b-family system. It is shown that the two-component b-family system admits the generalized functional separable solutions. Furthermore, blow up and behavior of those exact solutions are also investigated.
基金National Natural Science Foundation of China under Grant Nos.10447007 and 10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.2005A13
文摘We present basic theory of variable separation for (1 + 1)-dimensional nonlinear evolution equations withmixed partial derivatives.As an application,we classify equations u_(xt)=A(u,u_x)u_(xxx)+B(u,u_x) that admits derivative-dependent functional separable solutions (DDFSSs) and illustrate how to construct those DDFSSs with some examples.
基金The project supported by National Natural Science Foundation of China under Grant Nos. 10371098, 10447007, aria 10475055, the Natural Science Foundation of Shaanxi Province of China under Grant No. 2005A13, and the Special Research Project of Educational Department of Shaanxi Province under Grant No. 03JK060
文摘The generalized conditional symmetry is developed to study the variable separation for equations of type uxt = A（u,ux）uxx ＋ B（u, ux）. Complete classification of those equations which admit derivative-dependent functional separable solutions is obtained and some of their exact separable solutions are constructed.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11371293,11401458,and 11501438)the National Natural Science Foundation of China,Tian Yuan Special Foundation(Grant No.11426169)the Natural Science Basic Research Plan in Shaanxi Province of China(Gran No.2015JQ1014)
文摘The functionally generalized variable separation of the generalized nonlinear diffusion equations ut = A（u, Ux）Uxx ＋ B（u, ux） is studied by using the conditional Lie-Blicklund symmetry method. The variant forms of the considered equations, which admit the corresponding conditional Lie--Biicklund symmetries, are characterized. To construct functionally gener- alized separable solutions, several concrete examples defined on the exponential and trigonometric invariant subspaces are provided.
基金Project supported by the National Natural Science Foundation of China(Grant No.10671156)the Natural Science Foundation of Shaanxi Province of China(Grant No.SJ08A05)
文摘By using the approximate derivative-dependent functional variable separation approach, we study the quasi-linear diffusion equations with a weak source ut = （A（u）Ux）x ＋ eB（u, Ux）. A complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is listed. As a consequence, some approxi- mate solutions to the resulting perturbed equations are constructed via examples.
基金Supported by the National Science Foundation of China under Grant Nos.11371293,11501419the Mathematical Discipline Foundation of Shaanxi Province of China under Grant No.14TSXK02+1 种基金the Natural Science Foundation of Weinan Normal University under Grant No.16ZRRC05 and 15YKS005Natural Science Foundation of Hebei Province of China under Grant No.A2018207030
文摘The functionally generalized variable separation solutions of a general KdV-type equations u_t=u_(xxx) +A(u, u_x)u_(xx) + B(u, u_x) are investigated by developing the conditional Lie-Backlund symmetry method. A complete classification of the considered equations, which admit multi-dimensional invariant subspaces governed by higher-order conditional Lie-B¨acklund symmetries, is presented. As a result, several concrete examples are provided to construct functionally generalized variable separation solutions of some resulting equations.
基金supported by the Yue-Qi Scholar of the China University of Mining and Technology(No.102504180004)。
文摘In this article,we have studied a nonlinear time–space fractional longitudinal wave equation in the context of the conformable fractional derivative.Through the soliton ansatz method and a direct integration approach with the symmetry condition,new soliton and solitary wave solutions are derived.Furthermore,the existing conditions of these obtained solutions are also given in this text.These new results add to the existing literature.We believe that they can provide a new window into the understanding of this model.
文摘We investigate a generalized form of the Boussinesq equation, relevant for nerve pulse propagation in biological membranes. The generalized conditional symmetry （GCS） method is applied in order to obtain the conditions that enable the equation to admit a special class of second-order GCSs. For the case of quadratic nonlinearities, we outline a new class of invariant solutions.
基金Supported by the National Natural Science Foundation of China under Grant No.10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.SJ08A05
文摘As an extension to the derivative-dependent functional variable separation approach, the approximate derivative-dependent functional variable separation approach is proposed, and it is applied to study the generalized diffusion equations with perturbation. Complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is obtained. As a result, the corresponding approximate derivative-dependent functional separable solutions to some resulting perturbed equations are derived by way of examples.
基金Supported by NSF of China(Grant Nos.11601197,11461029 and 61563018)Ministry of Education Humanity Social Science Research Project of China(Grant No.15JYC910002)+2 种基金China Postdoctoral Science Foundation Funded Project(Grant Nos.2016M600511 and 2017T100475)NSF of Jiangxi Province(Grant Nos.20171ACB21030,20161BAB201024 and 20161ACB20009)the Key Science Fund Project of Jiangxi Provincial Education Department(Grant Nos.GJJ150439,KJLD13033 and KJLD14034)
文摘Under special conditions on data set and underlying distribution, the limit of finite sample breakdown point of Tukey＇s halfspace median （1） has been obtained in the literature. In this paper, we establish the result under weaker assumptions imposed on underlying distribution （weak smoothness） and on data set （not necessary in general position）. The refined representation of Tukey＇s sample depth regions for data set not necessary in general position is also obtained, as a by-product of our derivation.
