Two types of new algorithms for finding explicit analytical solutions of nonlinear differential equations

ＩｎｔｒｏｄｕｃｔｉｏｎＦｏｒｓｅｖｅｒａｌｈｕｎｄｒｅｄｙｅａｒｓ,ｉｔｈａｓｂｅｅｎｂｏｔｈｉｍｐｏｒｔａｎｔａｎｄｄｉｆｆｉｃｕｌｔｐｒｏｊｅｃｔｔｏｃｏｎｓｔｒｕｃｔｔｈｅａｎａｌｙｔｉｃａｌｓｏｌｕｔｉｏｎｓｆｏｒｎｏｎｌｉｎｅａｒｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓ.Ｍａｎｙｍａｔｈｅｍａｔｉｃｉａｎｓａｎｄｔｈｅｏｒｅｔｉｃａｌｐｈｙｓｉｃｉｓｔｓｈａｖｅｄｏｎｅａｌｏｔｏｆｗｏｒｋ[1～ 5 ].Ｈｏｗｅｖｅｒ,ｔｈｅｒｅｅｘｉｓｔｌｏｔｓｏｆｉｍｐｏｒｔａｎｔａｎｄｂｅｉｎｇｏｆｐ…     

A class of nonlinear boundary value problems for the second-order<Emphasis Type="Italic">E</Emphasis><Subscript>2</Subscript> class elliptic systems in general form

A class of nonlinear boundary value problems (BVP) for the second-order E₂ class elliptic systems in general form is discussed. By introducing a kind of transformation, this kind of BVP is reduced to a class of generalized nonlinear Riemann-Hilbert BVP. And then some singular integral operators are introduced to establish the equivalent nonlinear singular integral equations. The solvability is proved under some suitable hypotheses by means of the properties of singular integral operators and the function theoretic methods. Foundation items: the National Natural Science Foundation of China (19671056); Shanghai Municipal Natural Scientific Foundation (99ZA14030, 01ZA14023); Jiangxi Provincial Natural Scientific Foundation (981102, 0211014) Biographies: LI Ming-zhong (1935−); XU Ding-hua (1967−)     

1:2 Internal Resonance of Coupled Dynamic System with Quadratic and Cubic Nonlinearities

The1:2 internal resonance of coupled dynamic system with quadratic and cubic nonlinearities is studied. The normal forms of this system in1:2 internal resonance were derived by using the direct method of normal form. In the normal forms, quadratic and cubic nonlinearities were remained. Based on a new convenient transformation technique, the4-dimension bifurcation equations were reduced to3-dimension. A bifurcation equation with one-dimension was obtained. Then the bifurcation behaviors of a universal unfolding were studied by using the singularity theory. The method of this paper can be applied to analyze the bifurcation behavior in strong internal resonance on4-dimension center manifolds. Paper from Chen Yu-shu, Member of Editorial Commuttee, AMM Foundation item: the National Natural Science Foundation of China (1990510); the National Key Basic Research Special Fund (G1998020316); the Doctoral Point Fund of Education Committee of China (D09901) Biography: Chen Yu-shu (1931-)     

Poincaré-cartan integral invariants of Birkhoffian systems

Based on modern differential geometry, the symplectic structure of a Birkhoffian system which is an extension of conservative and nonconservative systems is analyzed. An integral invariant of Poincaré-Cartan’s type is constructed for Birkhoffian systems. Finally, one-dimensional damped vibration is taken as an illustrative example and an integral invariant of Poincaré’s type is found.     

On the formal solution of initial value problem of Navier-Stokes equation

Ｉｎｒｅｆｅｒｅｎｃｅ [1 ] ,ｗｅｄｉｓｃｕｓｓｅｄｔｈｅｉｎｓｔａｂｉｌｉｔｙｏｆＮａｖｉｅｒ＿ＳｔｏｋｅｓｅｑｕａｔｉｏｎｏｎＲ3×Ｒ ,ａｎｄ ,ｉｎｔｈｅｉｌｌｕｓｔｒａｔｉｏｎｏｆｉｔ,ｗｅｐｒｅｓｅｎｔｅｄｓｏｍｅｅｘａｍｐｌｅｓｉｎｔｈｅｆｏｒｍｏｆｅｘａｃｔｓｏｌｕｔｉｏｎ .Ｔｈｅｂａｓｉｓｏｆｔｈｅｃｏｎｓｔｒｕｃｔｉｏｎｏｆｔｈｅｍｉｓｔｈｅ“ｅｑｕａｔｉｏｎｓｅｃｏｎｄａｉｒｅ”ｏｆｔｈｅｅｑｕａｔｉｏｎ .Ｉｎｆａｃｔ,ｔｈｅｓｅｅｘａｃｔｓｏｌｕｔｉｏｎｓａｒｅｓｏｍｅｌｉｍｉｔ…     

Similarity reductions for 2+1-dimensional variable coefficient generalized kadomtsev-petviashvili equation

ＩｎｔｒｏｄｕｃｔｉｏｎＤｕｒｉｎｇｔｈｅｓｔｕｄｙｏｆｓｈａｌｌｏｗｗａｔｅｒｗａｖｅｓ,ｐｅｏｐｌｅｏｂｔａｉｎｅｄｍａｎｙｗｅｌｌ＿ｋｎｏｗｎｃｏｍｐｌｅｔｅｌｙｉｎｔｅｇｒａｂｌｅｍｏｄｅｌｓ,ｓｕｃｈａｓＫｄＶｅｑｕａｔｉｏｎ,ｍＫｄＶｅｑｕａｔｉｏｎ,Ｂｏｕｓｓｉｎｅｓｑｅｑｕａｔｉｏｎ,Ｗｈｉｔｈａｍ＿Ｂｒｏｅｒ＿ＫａｕｐｅｑｕａｔｉｏｎａｎｄＫｈｏｋｈｌｏｖ＿Ｚａｂｏｌｏｔｓｋａｙａｅｑｕａｔｉｏｎｅｔｃ.[1～8].Ｆｏｒ2 1＿ｄｉｍｅｎｓｉｏｎａｌｖａｒｉａｂｌｅｃｏｅｆｆｉｃｉｅｎｔｇ…     

Ishikawa iterative process for constructing solutions ofm-accretive operator equations

The convergence of theIshikawa iteration sequences with errors for constructing solutions of m-accretive operator equations is characterized. Moreover, the error estimates of approximate solutions for locally Lipschitzian and m-accretive operator equations are established. Foundation items: the National Natural Science Foundation of China (19801023); the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China Biography: ZENG Liu-chuan (1965-), Professor     

TheC k instability of navier-Stokes equation appending polynomials of unknown functions

Applying the theory of stratification, the solution space structure about a class of deformed Navier-Stokes equation isdetermined. It is proved that such kind of equation has no C^k (k≥2) stable solution by the fact that the strate transversale is a null set. Foundation item: the National Natural Science Foundation of China (19971054), Shanghai Natural Science Foundation (99ZA14034) Biography: HE You-hua (1960-)     

Block H-matrices and spectrum of block matrices

The block H-matrices are studied by the concept of G-functions, several concepts of block matrices are introduced. Equivalent characters of block H-matrices are obtained. Spectrum localizations characterized by G-functions for block matrices are got. Foundation item: Sichuan Youth Science & Technology Foundation of China (Jsa1081) Biography: HUANG Ting-zhu (1964-)     

Stability and strict positive realness of convex polytopes of interval polynomials

For an uncertain system described by convex combination of interval polynomials, its Hurwitz-stability can be guaranteed by certain subset composed of vertices and edges. Furthermore, the testing set does not increase when the order of given system increases. Foundation item: the National Natural Science Foundation of China (69925307) Biography: WANG Zhi-zhen (1974-)     

On the stability boundary of hamiltonian systems

The criterion for the points in the parameter space being on the stability boundary of linear Hamiltonian system depending on arbitrary numbers of parameters was given, through the sensitivity analysis of eigenvalues and eigenvectors. The results show that multiple eigenvalues with Jordan chain take a very important role in the stability of Hamiltonian systems. Foundation item: the National Natural Science Foundation of China (10072012); the National Natural Science Foundation of Russia Biography: QI Zhao-hui (1964-)     

T -stress in piezoelectric solid

The non-singular and bounded terms for stresses near the crack tip were investigated. The crack problem in a transversely isotropic piezoelectric solid for the plane problem was dealt with. The principle of superpsition and the Plemelj formulation were introduced. The non-singular terms are given by solving Rieman-Hilbert problem. It is shown that the non-singular terms are influenced by the elastic and electric constants. Contributed by WANG Biao Foundation items: the Nature Science Foundation of Shandong of China (Q99F15); the Post Doctoral Science Foundation of Heilongjiang Provice of China Biography: MA Hao (1967≈), Professor, Doctor     

MIXED FINITE ELEMENT METHODS FOR THE SHALLOWWATER EQUATIONS INCLUDING CURRENT AND SILTSEDIMENTATION （Ⅱ）——THE DISCRETE-TIMECASE ALONG CHARACTERISTICS

The mixed finite element(MFE) methods for a shallow water equation systemconsisting of water dynamics equations, silt transport equation, and the equation of bottomtopography change were derived. A fully discrete MFE scheme for the discrete-time alongcharacteristics is presented and error estimates are established. The existence andconvergence of MFE solution of the discrete current velocity, elevation of the bottomtopography, thickness of fluid column, and mass rate of sediment is demonstrated.     

MAXIMAL ELEMENTS FOR GB-MAJORIZED MAPPINGS IN PRODUCT G-CONVEX SPACES AND APPLICATIONS (Ⅰ)

A new family of set-valued mappings from a topological space into generalized convex spaces was introduced and studied. By using the continuous partition of unity theorem and Brouwer fixed point theorem, several existence theorems of maximal elements for the family of set-valued mappings were proved under noncompact setting of product generalized convex spaces. These theorems improve, unify and generalize many important results in recent literature. Contributed by DING Xie-ping Foundation items: the National Natural Science Foundation of China (19871059); the Natural Science Foundation of Education Department of Sichuan Province of China ([2000]25) Biography: DING Xie-ping (1938∼)     

An automatic constraint violation stabilization method for differential/ algebraic equations of motion in multibody system dynamics

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｄｙｎａｍｉｃｅｑｕａｔｉｏｎｓｏｆｍｏｔｉｏｎｏｆｍｕｌｔｉｂｏｄｙｓｙｓｔｅｍｓｗｉｔｈｃｏｎｓｔｒａｉｎｔｓａｒｅｔｈｅｆｏｌｌｏｗｉｎｇｄｉｆｆｅｒｅｎｔｉａｌ／ａｌｇｅｂｒａｉｃｅｑｕａｔｉｏｎｓ,ｉ．ｅ．,Ｅ...     

New Explicit and Exact Travelling Wave Solutions for a Class of Nonlinear Evolution Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＷｉｔｈｔｈｅｒａｐｉｄｄｅｖｅｌｏｐｍｅｎｔｏｆｓｃｉｅｎｃｅａｎｄｔｅｃｈｎｏｌｏｇｙ ,ｔｈｅｓｔｕｄｙｋｅｒｎｅｌｏｆｍｏｄｅｒｎｓｃｉｅｎｃｅｉｓｃｈａｎｇｅｄｆｒｏｍｌｉｎｅａｒｔｏｎｏｎｌｉｎｅａｒｓｔｅｐｂｙｓｔｅｐ .Ｍａｎｙｎｏｎｌｉｎｅａｒｓｃｉｅｎｃｅｐｒｏｂｌｅｍｓｃａｎｓｉｍｐｌｙａｎｄｅｘａｃｔｌｙｂｅｄｅｓｃｒｉｂｅｄｂｙｕｓｉｎｇｔｈｅｍａｔｈｅｍａｔｉｃａｌｍｏｄｅｌｏｆｎｏｎｌｉｎｅａｒｅｑｕａｔｉｏｎ .Ｕｐｔｏｎｏｗ ,ｍａｎｙｉｍｐｏｒ…     

CONTROLLABILITY OF A CLASS OF HYBRID DYNAMIC SYSTEMS (Ⅱ)—SINGLE TIME-DELAY CASE

The controllability for switched linear systems with time-delay in controls is first investigated. The whole work contains three parts. This is the second part. The definition and determination of controllability of switched linear systems with single time-delay in control functions is mainly investigated. The sufficient and necessary conditions for the oneperiodic, multiple-periodic controllability of periodic-type systems and controllability of periodic systems are presented, respectively. Contributed by YE Qing-kai Foundation items: the National Natural Science Foundation of China (69925307, 60274001); the National Key Basic Research and Development Program (2002CB312200); the Postdoctoral Program Foundation of China Biography: XIE Guang-ming (1972 ∼), Doctor (E-mail: xiegming@mech.pku.edu.cn)     

MAXIMAL ELEMENTS FOR G-B-MAJORIZED MAPPINGS IN PRODUCT G-CONVEX SPACES AND APPLICATIONS (Ⅱ)

By applying existence theorems of maximal elements for a family of G_B-majorized mappings in a product space of G-convex spaces, some coincidence theorem, Fan-Browder type fixed point theorem and some existence theorems of solutions for a system of minimax inequalities are proved under noncompact setting of G-convex spaces. These theorems improve and generalize many important known results in literature. Contributed by Duping Xie-ping Foundation items: the National Natural Science Foundation of China (19871059); the Natural Science Foundation of Education Department of Sichuan Province ([2000]25) Biography: Duping Xie-ping (1938∼), Professor (E-mail: dingxip@sichu.edu.cn)     

ON THE CONVERGENCE PROBLEMS OF ISHIKAWA AND MANN ITERATIVE PROCESSES WITH ERROR FOR Φ-PSEUDO CONTRACTIVE TYPE MAPPINGS

The purpose of this paper is to introduce the concept of Φ-pseudo contractive type mapping and to study the convergence problem of Ishikawa and Mann iterative processes with error for this kind of mappings. The results presented in this paper improve and extend many authors’recent results.