Asymptotic Solutions of Boundary Value Problems for Third-Order Ordinary Differential Equations with Turning Points

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｓｉｎｇｕｌａｒｐｅｒｔｕｒｂａｔｉｏｎｓｏｆｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓｆｏｒｏｒｄｉｎａｒｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｗｉｔｈｎｏｔｕｒｎｉｎｇｐｏｉｎｔｈａｄｂｅｅｎｓｔｕｄｉｅｄｖｅｒｙｅａｒｌｙ ,ｂｕｔｔｈｅｓｔｕｄｙｏｆｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓｗｉｔｈｔｕｒｎｉｎｇｐｏｉｎｔｐｒｏｃｅｅｄｅｄｒａｔｈｅｒｓｌｏｗｌｙｏｗｉｎｇｔｏｔｈｅｓｉｎｇｕｌａｒｐｏｉｎｔｏｆｔｈｅｄｅｇｅｎｅｒａｔｅｄｅｑｕａｔｉｏ…     

Positive Solutions of Boundary Value Problems for Second-Order Singular Nonlinear Differential Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ,ｗｅｓｈａｌｌｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｓｉｎｇｕｌａｒｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓ (ＢＶＰ)ｕ″ ｇ(ｔ)ｆ(ｕ) =0 ,　　 0 <ｔ<1 ,αｕ(0 ) -βｕ′(0 ) =0 ,　　γｕ(1 ) δｕ′(1 ) =0 ,(1 )ｗｈｅｒｅα ,β,γ ,δ≥ 0 ,ρ:=βγ αγ αδ>0 ,ｆ∈Ｃ([0 ,∞ ) ,[0 ,∞ ) ) ,ｇｍａｙｂｅｓｉｎｇｕｌａｒａｔｔ=0ａｎｄ/ｏｒｔ=1 .Ｔｈｉｓｐｒｏｂｌｅｍａｒｉｓｅｓｎａｔｕｒａｌｌｙｉｎｔｈｅｓｔｕｄｙｏｆｒａｄｉａｌｌｙｓｙｍｍｅｔ…     

ON THE ASYMPTOTIC SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR A CLASS OF SYSTEMS OF NONLINEAR DIFFERENTIAL EQUATIONS (Ⅰ)

ＩｎｔｒｏｄｕｃｔｉｏｎＯｗｉｎｇｔｏｔｈｅｆａｃｔｔｈａｔｔｈｅｓｏｌｕｔｉｏｎｓｏｆｔｈｅＤｉｒｉｃｈｌｅｔｐｒｏｂｌｅｍｓｆｏｒａｃｌａｓｓｏｆｓｙｓｔｅｍｓｏｆｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｏｆｔｈｅｆｏｒｍｕ″=ｖ ,εｖ″ ｆ(ｘ ,ｕ ,ｕ′)ｖ′ -ｇ(ｘ ,ｕ ,ｕ′)ｖ=0　　 ( 0 <ｘ <1,0 <ε 1) ,ｕ( 0 ;ε) =0 ,ｕ( 1;ε) =0 ,ｖ( 0 ;ε) =α ,ｖ( 1;ε) =β,( 1)ｈａｖｅｔｈｅｑｕａｌｉｔａｔｉｖｅｐｒｏｐｅｒｔｉｅｓｏｆｓｏｌｕｔｉｏｎｓｏｆａｍｏｒｅｃｏｍｐｌｉｃａｔｅｄｓｙｓｔｅ…     

Study on Exact Analytical Solutions for Two Systems of Nonlinear Evolution Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＤｕｒｉｎｇｔｈｅｃｏｕｒｓｅｏｆｓｔｕｄｙｉｎｇｔｈｅｗａｔｅｒｗａｖｅ,ｍａｎｙｃｏｍｐｌｅｔｅｌｙｉｎｔｅｇｒａｂｌｅｍｏｄｅｌｓｗｅｒｅｏｂｔａｉｎｅｄ ,ｓｕｃｈａｓＫｄＶｅｑｕａｔｉｏｎ ,ｍＫｄＶｅｑｕａｔｉｏｎ ,(2 1 )＿ｄｉｍｅｎｓｉｏｎａｌＫＰｅｑｕａｔｉｏｎ ,ｃｏｕｐｌｅｄＫｄＶｅｑｕａｔｉｏｎｓ,ｖａｒｉａｎｔＢｏｕｓｓｉｎｅｓｑｅｑｕａｔｉｏｎｓ ,ＷＫＢｅｑｕａｔｉｏｎｓｅｔｃ .[1- 13 ].Ｉｎｏｒｄｅｒｔｏｆｉｎｄｅｘｐｌｉｔｉｃｅｘａｃｔｓｏｌｕｔｉｏ…     

Lyapunov Stability and Adaptive Regulation of a Class of Nonlinear Nonautonomous Second-Order Differential Equations

This paper presents an adaptive regulation scheme for a class ofordinary nonlinear nonautonomous second-order differential equationswhich includes as particular cases a number of particular differentialequations which occur in applications. The unforced reference model isproposed to be a stable differential parametrization within the generalclass dealt with. Therefore, some sufficient Lyapunov's stabilityconditions for such a class are previously investigated which can beused, in particular, to set an appropriate reference model. Theresulting closed-loop adaptive scheme is proved to be stable and itinvolves a parameter estimation scheme of least-squares type which isproved to possess all suitable properties in terms of estimatesboundedness and asymptotic convergence of the estimates to finite limitsas well as time-integrability of the squared adaptation error.     

Boundary Integral Equations in Dynamic Problems for Elastic Plates

Initial-boundary value problems with Dirichlet and Neumann conditions arising in the theory of bending of plates with transverse shear deformation are reduced to time-dependent boundary integral equations by means of layer potentials. The solvability of these equations is then investigated in Sobolev-type spaces. This revised version was published online in June 2006 with corrections to the Cover Date.     

Singular Perturbation of Boundary Value Problem for Quasilinear Third-Order Ordinary Differential Equations Involving Two Small Parameters

ＩｎｔｒｏｄｕｃｔｉｏｎＡｂｏｕｔｓｉｎｇｕｌａｒｐｅｒｔｕｒｂａｔｉｏｎｏｆｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｆｏｒｓｅｃｏｎｄ＿ｏｒｄｅｒｏｒｄｉｎａｒｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｗｏｒｋｓｏｆａｌａｒｇｅｎｕｍｂｅｒｗｅｒｅｃｏｎｓｉｄｅｒｅｄｏｎｌｙｆｏｒｔｈｅｃａｓｅｏｆｉｎｖｏｌｖｉｎｇｏｎｅｓｍａｌｌｐａｒａｍｅｔｅｒ[1- 4].Ｏｎｌｙａｆｅｗｗｏｒｋｓｗｅｒｅｃｏｎｓｉｄｅｒｅｄｆｏｒｔｈｅｃａｓｅｏｆｉｎｖｏｌｖｉｎｇｔｗｏｓｍａｌｌｐａｒａｍｅｔｅｒｓ[5 - 8]…     

Uniform Convergence Series to Solve Nonlinear Partial Differential Equations: Application to Beam Dynamics

Extended trigonometric series of uniform convergence are proposed as a method to solve the nonlinear dynamic problemsgoverned by partial differential equations. In particular, the method isapplied to the solution of a uniform beam supported at its ends withnonlinear rotational springs and subjected to dynamic loads. The beam isassumed to be both material and geometrically linear and the end springs are of the Duffing type. The action may be a continuous load q = q(x, t) within a certain range and/or concentrated dynamic moments at the boundaries. The adopted solution satisfies the differential equation, the initial conditions, andthe nonlinear boundary conditions. It has been previously demonstrated that, due to the uniform convergence of the series, the method yieldsarbitrary precision results. An illustration example shows theefficiency of the method.     

On the Trajectory Method for the Reconstruction of Differential Equations from Time Series

This work investigates the trajectory method [1] for thereconstruction of ordinary differential equations (ODEs) from timeseries. The potentials of the method are analyzed for dynamical systemsdescribed by second- and third-order ODEs, focusing in particular on therole of the parameters of the method and on the influence of the qualityof the time series in terms of noise, length and sampling frequency.Typical models are investigated, such as the van der Pol, the linearmechanical, the Duffing and the Rössler equations, resulting in arobust and versatile method which is capable of allowing interestingapplications to experimental cases. The method is then applied to themeasured time series of a nonlinear mechanical oscillator, a typicalvelocity oscillation of the bursting phenomenon in near-wall turbulenceand the averaged annual evolution of rainfall, temperature andstreamflow over a hydrological basin.     

Asymptotic Solution for a Kind of Boundary Layer Problem

This paper studies the partial differential equation with a small coefficient in the highest-order item. This kind of equation is also named as boundary layer problem. The Burgers equation and modified Burgers equation are analyzed in this approach. First, these equations are transferred into the strong nonlinear ones, and then the corresponding strong nonlinear equations are solved based on the perturbation method. The results from the asymptotic method are comparable with those obtained from numerical computation. An erratum to this article is available at .     

The Asymptotic Behavior of Solution for the Singularly Perturbed Initial Boundary Value Problems of the Reaction Diffusion Equations in a Part of Domain

Ｔｈｅａｕｔｈｏｒｓｓｔｕｄｉｅｄａｃｌａｓｓｏｆｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓｉｎ [1 ] -[7] .Ｎｏｗｗｅｒａｉｓｅａｃｌａｓｓｏｆｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓｏｎａｐａｒｔｏｆｄｏｍａｉｎ .Ｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｉｎｉｔｉａｌｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｆｏｒｔｈｅｒｅａｃｔｉｏｎｄｉｆｆｕｓｉｏｎｅｑｕａｔｉｏｎｓ　　ｕｔ-λε(ｘ) ( μ(ｕ)ｕｘ) ｘ Ｋｘ(ｕ) ｆ(ｘ ,ｔ,ｕ) =0 ,(ｔ,ｘ) ∈ ( 0 ,Ｔ)× ( ( 0 ,α) ∪ (…     

On the Asymptotic Behavior of Solutions of a System of Nonlinear Differential Equations

We investigate the asymptotic behavior of a system of nonlinear differential equations of a special form at infinity. We also propose a method for the reduction of more general systems of nonlinear differential equations to this form, which enables one to study their asymptotic properties.     

Plane Elastic Boundary Value Problem Posed by Displacement and Force Orientations on a Closed Contour

A boundary value problem (BVP) of the plane elasticity posed in terms of the orientations of forces and displacements is considered. The main aim of the present paper is to investigate the solvability of BVPs of this kind. Firstly, analysis of two cases is performed: the case of a circle with special orientations of force and displacement vectors on the circumference and the case of an arbitrary contour with coaxial orientations of these vectors. The solutions obtained indicate that the problem can have a certain number of solutions or be unsolvable. Then the BVP is reduced to a boundary integral equation and its solvability is investigated for the general case of a smooth simple-connected closed contour. As a result, the number of linearly independent solutions is determined. This number only depends upon the angle between the force and displacement vectors. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the problem of preventing blowing-up and quenching for semilinear heat equation

In this paper,the global existence of solutions to the IVP=Δu+g(t)f(u) (t>0),u|_(t=0)=u_0(x)and the (?)PVPu_t=Δu-g(t,x)f(u)(t>0,x∈Ω),u|_(t-0)=u|_(?)(?)is investigated. As has been done in [6]the (?)duction of factor g(t) or g(t.x) innonlinear term is to prevent(?) occurrance of blowing-up or quenching of solutions.It isshown in this paper that most of the restrictions on f,g and u_0 in the theorems of[6] maybe cancelled or relaxed,that the smallness of g is required only for t large,and thatunder certain conditions controlling initial state can avoid blowing-up.     

Computer computation of the method of multiple scales—dirichlet problem for a class of system of nonlinear differential equations

The method of boundary layer with multiple scales and computer algebra were applied to study the asymptotic behavior of solution of boundary value problems for a class of system of nonlinear differential equations. The asymptotic expansions of solution were constructed. The remainders were estimated. And an example was analysed. It provides a new foreground for the application of the method of boundary layer with multiple scales. Contributed by Jiang Fu-ru, Original Member of Editorial Committe, AMM Biography: Xie La-bing (1976∼); Jiang Fu-ru(1927∼)     

Periodic Solutions for Some Systems of Delay Differential Equations

In this paper, we give sufficient conditions for the existence of periodic orbits of some systems of delay differential equations with a unique delay having 3, 4 or n equations. Moreover, we provide examples of delay systems satisfying the different sets of sufficient conditions.     

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…     

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

ＩｎｔｒｏｄｕｃｔｉｏｎＷｉｔｈｔｈｅｒａｐｉｄｄｅｖｅｌｏｐｍｅｎｔｏｆｓｃｉｅｎｃｅａｎｄｔｅｃｈｎｏｌｏｇｙ ,ｔｈｅｓｔｕｄｙｋｅｒｎｅｌｏｆｍｏｄｅｒｎｓｃｉｅｎｃｅｉｓｃｈａｎｇｅｄｆｒｏｍｌｉｎｅａｒｔｏｎｏｎｌｉｎｅａｒｓｔｅｐｂｙｓｔｅｐ .Ｍａｎｙｎｏｎｌｉｎｅａｒｓｃｉｅｎｃｅｐｒｏｂｌｅｍｓｃａｎｓｉｍｐｌｙａｎｄｅｘａｃｔｌｙｂｅｄｅｓｃｒｉｂｅｄｂｙｕｓｉｎｇｔｈｅｍａｔｈｅｍａｔｉｃａｌｍｏｄｅｌｏｆｎｏｎｌｉｎｅａｒｅｑｕａｔｉｏｎ .Ｕｐｔｏｎｏｗ ,ｍａｎｙｉｍｐｏｒ…     

Study of Explicit Analytic Solutions for the Nonlinear Coupled Scalar Field Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＷｉｔｈｔｈｅｒａｐｉｄｄｅｖｅｌｏｐｍｅｎｔｏｆｎｏｎｌｉｎｅａｒｓｃｉｅｎｃｅ,Ｍａｎｙｐｈｅｎｏｍｅｎａｉｎｐｈｙｓｉｃｓ,ｍｅｃｈａｎｉｃｓ,ｃｈｅｍｉｓｔｒｙａｎｄｂｉｏｌｏｇｙｅｔｃ.ｃａｎｂｅｄｅｓｃｒｉｂｅｄｓｉｍｐｌｙａｎｄｅｘａｃｔｌｙｂｙｔｈｅｍａｔｈｅｍａｔｉｃａｌｍｏｄｅｌ＿ｎｏｎｌｉｎｅａｒｅｑｕａｔｉｏｎｓ[1- 7].Ｏｎｔｈｅｃｏｎｔｒａｒｙ ,ｉｎｏｒｄｅｒｔｏｓｔｕｄｙｔｈｅｓｅｐｈｅｎｏｍｅｎａｑｕａｎｔｉｔａｔｉｖｅｌｙ .Ｉｔｉｓｖｅｒｙｉｍ…     

Energy Approach to the Formulation of Boundary Problems of Nonlinear Thermomechanics for Elastic Systems

An energy approach is proposed to derive the physical constitutive equations of nonlinear thermomechanics for inertial elastic systems. A potential of local inertial thermodynamic state and a potential of thermoelastic energy dissipation are introduced. The variational formulation of nonlinear boundary problems of thermoelasticity is implemented on the basis of the Hamiltonian energy functional. Sufficient conditions for the convexity of the functional are formulated __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 9, pp. 52–59, September 2005.