The Extinction Behavior of the Solutions for a Class of Reaction-Diffusion Equations

1　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＬｅｍｍａｓＴｈｅｒｅａｒｅｍａｎｙｒｅｓｕｌｔｓａｂｏｕｔｅｘｉｓｔｅｎｃｅ (ｇｌｏｂａｌｏｒｌｏｃａｌ)ａｎｄａｓｙｍｐｔｏｔｉｃｂｅｈａｖｉｏｒｏｆｓｏｌｕｔｉｏｎｓｆｏｒｒｅａｃｔｉｏｎ＿ｄｉｆｆｕｓｉｏｎｅｑｕａｔｉｏｎｓ[1- 9].Ｂｙｔｈｅａｉｄｓｏｆｒｅｓｕｌｔｓ[2 ,3]ｏｆｅｑｕａｔｉｏｎ ｕ/ ｔ=Δｕ-λ｜ｕ｜γ- 1ｕｗｉｔｈｉｎｉｔｉａｌ＿ｂｏｕｎｄａｒｙｖａｌｕｅｓ,ｐａｐｅｒ [4 ]ｓｔｕｄｉｅｄｔｈｅｐｒｏｂｌｅｍｏｆ ｕ/ ｔ=Δｕ-λ｜ｅβｔｕ｜γ- …     

Extended self-similar scaling law of multi-scale eddy structure in wall turbulence

ＩｎｔｒｏｄｕｃｔｉｏｎＭｕｃｈｗｏｒｋｈａｓｂｅｅｎｄｅｖｏｔｅｄｉｎｔｈｅｌａｓｔｆｅｗｄｅｃａｄｅｓｔｏｔｈｅｍｅａｓｕｒｅｍｅｎｔａｎｄｍｏｄｅｌｉｎｇｏｆｔｈｅｓｃａｌｉｎｇｌａｗｏｆｓｔｒｕｃｔｕｒｅｆｕｎｃｔｉｏｎｏｆｔｕｒｂｕｌｅｎｔｆｌｏｗｓ.Ｔｈｅｓｏ＿ｃａｌｌｅｄ“ｖｅｌｏｃｉｔｙｓｔｒｕｃｔｕｒｅｆｕｎｃｔｉｏｎｏｆｏｒｄｅｒｎ”ｆｏｒｔｕｒｂｕｌｅｎｔｆｌｏｗｓｉｓｄｅｆｉｎｅｄａｓ〈ΔＶ(ｒ) ｎ〉 ,ｗｈｅｒｅΔＶ(ｒ) =Ｖ(ｘ ｒ) -Ｖ(ｘ)ｉｓｔｈｅｖｅｌｏｃｉｔｙｃｏｍｐ…     

COMPUTATION FORMULAS OF GENERALIZED INVERSE PAD APPROXIMANTS USING FOR SOLUTION OF INTEGRAL EQUATIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｇｅｎｅｒａｌｉｚｅｄｉｎｖｅｒｓｅｆｕｎｃｔｉｏｎ＿ｖａｌｕｅｄＰａｄ啨ａｐｐｒｏｘｉｍａｎｔｓｗａｓｉｎｔｒｏｄｕｃｅｄｔｏｓｏｌｖｅｉｎｔｅｇｒａｌｅｑｕａｔｉｏｎｓｉｎ [1 ] .Ｌｅｔｆ(ｘ ,λ)ｂｅａｇｉｖｅｎｐｏｗｅｒｓｅｒｉｅｓｗｉｔｈｆｕｎｃｔｉｏｎ＿ｖａｌｕｅｄｃｏｅｆｆｉｃｉｅｎｔｓ,ｉ.ｅ.,ｆ(ｘ ,λ) =ｃ0 (ｘ) ｃ1(ｘ)λ ｃ2 (ｘ)λ2 … ｃｎ(ｘ)λｎ … ,( 1 )ｗｈｅｒｅｃｊ(ｘ)ｉｓａｒｅａｌｏｒｃｏｍｐｌｅｘｆｕｎｃｔｉｏｎｗｉｔｈｒｅｇａｒｄｔｏ…     

On the existence of solutions of boundary value problems of duffing type systems

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ [1 ] ,ｕｎｄｅｒａｇｒｏｕｐｏｆｖｅｒｙｅｘｔｅｎｓｉｖｅｃｏｎｄｉｔｉｏｎｓ,ＳＨＥＮＺｕ＿ｈｅｓｔｕｄｉｅｄｔｈｅｅｘｉｓｔｅｎｃｅｏｆａｕｎｉｑｕｅ 2π＿ｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｏｆｔｈｅｓｙｓｔｅｍｏｆｏｒｄｉｎａｒｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｕ″(ｔ) Ｇ(ｕ(ｔ) ) =ｐ(ｔ) ,(1 )ｗｈｅｒｅＧ :Ｒｎ →Ｒｈａｓａｃｏｎｔｉｎｕｏｕｓｓｅｃｏｎｄｐａｒｔｉａｌｄｅｒｉｖａｔｉｖｅｓ,ａｎｄｐ:Ｒ→Ｒｎｉｓｃｏｎｔｉｎｕｏｕｓａｎｄ2π＿ｐｅｒ…     

Precise integration method for LQG optimal measurement feedback control problem

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｌｉｎｅａｒｓｙｓｔｅｍｏｆｔｈｅｍｅａｓｕｒｅｍｅｎｔｆｅｅｄｂａｃｋｃｏｎｔｒｏｌ ｘ=Ａｘ Ｂｗ Ｂ2 ｕ , ( 1 )ｙ =Ｃｘ ｖ ,( 2 )ｗｈｅｒｅｘｉｓｔｈｅｎ＿ｄｉｍｅｎｓｉｏｎａｌｓｔａｔｅｖｅｃｔｏｒ,ｙｉｓａｑ＿ｖｅｃｔｏｒｏｆｍｅａｓｕｒｅｍｅｎｔｓ,ｕｉｓａｎｍ＿ｖｅｃｔｏｒｏｆｃｏｎｔｒｏｌｉｎｐｕｔｓ,ｗａｎｄｖａｒｅｌ＿ｖｅｃｔｏｒ,ｑ＿ｖｅｃｔｏｒｏｆｗｈｉｔｅ＿ｎｏｉｓｅｐｒｏｃｅｓｓｗｉｔｈｋｎｏｗｎｓｔａｔｉｓｔｉｃａｌｐｒｏｐｅ…     

DYNAMICS IN NEWTONIAN-RIEMANNIAN SPACE-TIME (Ⅱ)

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅｐａｐｅｒ (Ⅰ ) (ｓｅｅ:ＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓａｎｄＭｅｃｈａｎｉｃｓ (ＥｎｇｌｉｓｈＥｄｉｔｉｏｎ) ,2 0 0 0 ,2 1 (7) :82 5-83 5) ,ｗｅｈａｖｅｅｓｔａｂｌｉｓｈｅｄｔｈｅｄｙｎａｍｉｃｓｉｎＮｅｗｔｏｎｉａｎ＿Ｒｉｅｍａｎｎｉａｎｓｐａｃｅ＿ｔｉｍｅ,ａｎｄｇｉｖｅｉｔｓａｐｐｌｉｃａｔｉｏｎｔｏＨｙｄｒｏｍｅｃｈａｎｉｃｓ.ＴｈｅｒｅｉｓＧａｌｉｌｅｏ’ｓｐｒｉｎｃｉｐｌｅｏｆｒｅｌａｔｉｖｉｔｙｉｎＣｌａｓｓｉｃａｌＮｅｗｔｏｎ’ｓｍｅｃｈａ…     

THE EXACT SOLITARY WAVE SOLUTIONS FOR THE KLEIN- GORDON-SCHRDINGER EQUATIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅｐｒｅｓｅｎｔｐａｐｅｒｗｅｃｏｎｓｉｄｅｒｔｈｅｃｏｕｐｌｅｄＫｌｅｉｎ＿Ｇｏｒｄｏｎ＿Ｓｃｈｒ ｄｉｎｇｅｒ (ＫＧＳ)ｅｑｕａｔｉｏｎｓａｓｆｏｌｌｏｗｓ:ｉψｔ+ 12 Δψ=-φψ ,φｔｔ-Δφ+ｍ2 φ=｜ψ｜2 .( 1 )( 2 )( 1 )ａｎｄ ( 2 )ｄｅｓｃｒｉｂｅａｃｌａｓｓｉｃａｌｍｏｄｅｌｏｆｉｎｔｅｒａｃｔｉｏｎｏｆｎｕｃｌｅｏｎｆｉｅｌｄｗｉｔｈｍｅｓｏｎｆｉｅｌｄ[1],ｗｈｅｒｅψｉｓａｃｏｍｐｌｅｘｓｃａｌａｒｎｕｃｌｅｏｎｆｉｅｌｄ ,φ＿ａｒｅａｌｍｅｓｏｎｆｉ…     

One-Dimensional Consolidation of Layered and Visco-Elastic Soils under Arbitrary Loading

ＩｎｔｒｏｄｕｃｔｉｏｎＡｆｔｅｒＴｅｒｚａｇｈｉ,ａｌｏｔｏｆｓｃｈｏｌａｒｓｈａｖｅｄｏｎｅｏｕｔｓｔａｎｄｉｎｇｒｅｓｅａｒｃｈｏｎｔｈｅｐｒｏｂｌｅｍｓｏｆｏｎｅ＿ｄｉｍｅｎｓｉｏｎａｌｃｏｎｓｏｌｉｄａｔｉｏｎ .Ｗｉｌｓｏｎ ( 1974) [1] ａｎｄＢａｌｉｇｈ( 1978) [2 ] ｅｔａｌ.ｈａｖｅｓｔｕｄｉｅｄｔｈｅｃｏｎｓｏｌｉｄａｔｉｏｎｕｎｄｅｒｔｈｅｒｅｃｔａｎｇｕｌａｒｌｏａｄｉｎｇ .Ｉｎ 1988,ＷＵｅｔａｌ.[3] ｗｏｒｋｅｄｏｕｔｔｈｅｇｅｎｅｒａｌｓｏｌｕｔｉｏｎｓｏｆｏｎｅ＿ｄｉｍｅｎ…     

IMD BASED NONLINEAR GALERKIN METHOD

ＩｎｔｒｏｄｕｃｔｉｏｎＴｏｍａｋｅｔｈｅｌｏｎｇｔｅｒｍｓｉｍｕｌａｔｉｏｎｏｆｔｈｅＮａｖｉｅｒ＿Ｓｔｏｋｅｓｅｑｕａｔｉｏｎｓｐｏｓｓｉｂｌｅｄｕｒｉｎｇｔｈｅｌａｓｔｄｅｃａｄｅ,ｔｈｅｓｏ＿ｃａｌｌｅｄｎｏｎｌｉｎｅａｒＧａｌｅｒｋｉｎｍｅｔｈｏｄ (ＮＧＭ) ,ｗｈｉｃｈｉｓｂａｓｅｄｕｐｏｎｔｈｅｃｏｎｃｅｐｔｓｏｆｔｈｅＩｎｅｒｔｉａｌＭａｎｉｆｏｌｄ (ＩＭ) [1]ａｎｄｔｈｅＡｐｐｒｏｘｉｍａｔｅＩｎｅｒｔｉａｌＭａｎｉｆｏｌｄ (ＡＩＭ) [2 ],ｗａｓｃｏｎｓｔｒｕｃｔｅｄ(ｓｅｅＲｅｆｓ.…     

Periodic boundary value problems for first-order integro-differential equations of mixed type

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｐｅｒｉｏｄｉｃｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍ(ＰＢＶＰ,ｆｏｒｓｈｏｒｔ)ｆｏｒｆｉｒｓｔ＿ｏｒｄｅｒｉｎｔｅｇｒｏ＿ｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｏｆｍｉｘｅｄｔｙｐｅｕ′=ｆ(ｔ,ｕ,Ｔ1ｕ,Ｔ2ｕ)　　(ａ.ｅ.ｔ∈Ｉ),(1)ｕ(0)=ｕ(2π),(2)ｗｈｅｒｅＩ=[0,2π],ｆｓａｔｉｓｆｉｅｓＣａｒａｔｈｅｏｄｏｒｙ’ｓｃｏｎｄｉｔｉｏｎｓ,Ｔ1ｉｓａＶｏｌｔｅｒｒａｉｎｔｅｇｒａｌｏｐｅｒａｔｏｒ,Ｔ2ｉｓａＦｒｅｄｈｏｌｍｉｎｔｅ…     

Equivalence between the Local Boundary Integral Equation and the Mean Value Theorem in the Theory of Elasticity

The boundary element method based on a boundary integral equation has been very successful in computational mechanics. Atluri et al. [4] recently developed a new meshless method using the local boundary integral equations. It eliminates the tedious step of mesh generation and thus greatly simplifies the numerical computation process. This paper shows the equivalence between the local boundary integral equation and the mean value theorem in the theory of elasticity. In addition, it gives new proofs for the mean value theorem of elasticity and its converse based on the concept of a companion solution. This revised version was published online in June 2006 with corrections to the Cover Date.     

ASYMPTOTIC THEORY OF INITIAL VALUE PROBLEMS FOR NONLINEAR PERTURBED KLEIN-GORDON EQUATIONS

Introduction InthispaperasymptotictheoryofthefollowinginitialvalueproblemforanonlinearKlein Gordonequationisconsidered.tt-Δ =εF(t,x,,ε),t>0,x∈R2,(0,x,ε)=0(x,ε),t(0,x,ε)=1(x,ε),x∈R2,(1)where(t,x)isarealvaluedunknownfunction,Δ=2i     

Loss of uniqueness of numerical solutions of the borehole problem modelled with enhanced media

It is well known that, initial boundary value problems involving constitutive equations modeling the degradation of the strength of materials are not well posed, which renders computations questionable. To overcome this issue it is necessary to enhance the models by incorporating some internal length. It has been shown that such an enhancement restores the objectivity of the computation as spurious mesh dependency is avoided. However, at least for simple problems (e.g. one dimensional ones), it has been proven that uniqueness of the underlying mathematical problem itself is not restored. Moreover numerical modeling of element tests yields several solutions. This paper demonstrates that several numerical solutions can be obtained also for less simple problems, namely the borehole problems. Even when a defect is introduced in the computed problems, different numerical solutions are found. Contrary to the one dimensional problem there is no proof that this loss of uniqueness comes from the underlying mathematical problem. It is our opinion that this is an inherent property of initial boundary value problems where, broadly speaking, strong degradation of the mechanical properties is modeled. In any case, it is necessary to be aware of this issue. For problems involving constitutive equation modeling strength degradation, it is important to try to find other solutions than the one obtained by using routinely a numerical code. The failure patterns of the different solutions found are however similar to experimental observations. This possible loss of uniqueness can then be seen as a counterpart of the difficulties encountered when attempting to reproduce experiments. This is crucial when dealing with geomaterials.     

求解非线性动力系统周期解的改进打靶法

针对有周期解的动力系统边值问题可以转化为初值问题这一特点，改进了周期解的打靶 法数值求解. 在计算边界条件代数方程关于待定初值参数导数的过程中利用前一次 Runge-Kutta方法计算得到的节点函数值并通过再次利用Runge-Kutta方法获得了该导数值. 用此方法求解了Duffing方程及非线性转子---轴承系统的周期解,用Floquet理论判断了 周期解的稳定性,与普通打靶法作了比较,验证了方法的有效性.     

On the initial-boundary value problem for the Boltzmann equation

A theorem on the traces of the solutions of initial-boundary value problems for the Boltzmann equation is proved. This result makes it possible to extend a recent theorem of existence proved by Hamdache to more realistic situations.     

A refined theory of elastic thick plates for extensional deformation

Based on elasticity theory, various two-dimensional (2D) equations and solutions for extensional deformation have been deduced systematically and directly from the three-dimensional (3D) theory of thick rectangular plates by using the Papkovich–Neuber solution and the Lur’e method without ad hoc assumptions. These equations and solutions can be used to construct a refined theory of thick plates for extensional deformation. It is shown that the displacements and stresses of the plate can be represented by the displacements and transverse normal strain of the midplane. In the case of homogeneous boundary conditions, the exact solutions for the plate are derived, and the exact equations consist of three governing differential equations: the biharmonic equation, the shear equation, and the transcendental equation. With the present theory a solution of these can satisfy all the fundamental equations of 3D elasticity. Moreover, the refined theory of thick plate for bending deformation constructed by Cheng is improved, and some physical or mathematical explanations and proof are provided to support our justification. It is important to note that the refined theory is consistent with the decomposition theorem by Gregory. In the case of nonhomogeneous boundary conditions, the approximate governing differential equations and solutions for the plate are accurate up to the second-order terms with respect to plate thickness. The correctness of the stress assumptions in the classic plane-stress problems is revised. In an example it is shown that the exact or accurate solutions may be obtained by applying the refined theory deduced herein.     

基于S-R和分解定理的三维几何非线性无网格法

S-R(strain-rotation)和分解定理克服了经典有限变形理论的一些缺点, 使其可以为几何非线性数值分析提供可靠的理论基础. 对于大变形问题, 由于无网格法(element-free method)避免了对单元网格的依赖, 从而从根本上避免了有限单元法(finite element method, FEM)的单元畸变问题, 保证了求解精度. 因此, 将无网格法和S-R和分解定理结合起来势必能建立一套更加合理可靠的几何非线性数值计算方法. 目前基于S-R 定理的无网格数值方法研究较少并且只能用于二维平面问题的求解, 但实际上绝大多数问题都必须以三维模型来进行处理, 因此建立适用于三维情况的S-R无网格法是非常有必要的. 本文给出了适用于三维情况的S-R 无网格法: 采用由更新拖带坐标法和势能率原理推导出来的增量变分方程, 利用基于全局弱式的无网格Galerkin 法(EFG)得到了用于求解三维空间问题的离散格式. 利用MATLAB编制三维S-R 无网格法程序, 对受均布载荷的三维悬臂梁和四边简支矩形板结构的非线性弯曲问题进行了计算. 最后将所得的数值结果与已有文献进行了比较, 验证了本文的三维S-R无网格数值算法的合理性、有效性和准确性. 本文的三维S-R无网格数值算法可以作为一种可靠的三维几何非线性数值分析方法.      

Boundary element calculation of shallow water waves

A boundary element method is proposed for studying periodic shallow water problems. The numerical model is based on the shallow water equation. The key feature of this method is that the boundary integral equations are derived using the weighted residual method and the fundamental solutions for shallow water wave problems are obtained by solving the simultaneous singular equations. The accuracy of this method is studied for the wave reflection problem in a rectangular tank. As a result of this test, it has been shown that the number of element divisions and the distribution of nodes are significant to the accuracy. For numerical examples of external problems, the wave diffraction problems due to single cylindrical, double cylindrical and plate obstructions are analysed and compared with the exact and other numerical solutions. Relatively accurate solutions are obtained.     

Existence of solutions for periodic boundary value problems for second-order integro-differential equations

ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔ(Ｅ,｜·｜)ｂｅａｒｅａｌＢａｎａｃｈｓｐａｃｅｗｉｔｈａｐａｒｔｉａｌｏｒｄｅｒｉｎｔｒｏｄｕｃｅｄｂｙａｒｅｇｕｌａｒｃｏｎｅＫｏｆＥ.Ｉｎｔｈｉｓｐａｐｅｒ,ｔｈｅｅｘｉｓｔｅｎｃｅｏｆｓｏｌｕｔｉｏｎｓｏｆｔｈｅｆｏｌｌｏｗｉｎｇｐｅｒｉｏｄｉｃｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓ(ＰＢＶＰ)ｗｉｌｌｂｅｉｎｖｅｓｔｉｇａｔｅｄ:　　(Ⅰ)ｕ″=ｆ(ｔ,ｕ,Ｔｕ)　ａ.ｅ.ｔ∈Ｊ,ｕ(0)=ｕ(ａ),ｕ′(0)=ｕ′(ａ),ｗｈｅｒｅｆ∈Ｃ(Ｊ×Ｅ×Ｅ,Ｅ),Ｊ=[0,ａ](ａ>0),ａｎｄ(Ｔ…     

On constructing a Green’s function for a semi-infinite beam with boundary damping

The main aim of this paper is to contribute to the construction of Green’s functions for initial boundary value problems for fourth order partial differential equations. In this paper, we consider a transversely vibrating homogeneous semi-infinite beam with classical boundary conditions such as pinned, sliding, clamped or with a non-classical boundary conditions such as dampers. This problem is of important interest in the context of the foundation of exact solutions for semi-infinite beams with boundary damping. The Green’s functions are explicitly given by using the method of Laplace transforms. The analytical results are validated by references and numerical methods. It is shown how the general solution for a semi-infinite beam equation with boundary damping can be constructed by the Green’s function method, and how damping properties can be obtained.