Subspaces for weak mild solutions of the second order abstract differential equation

The topic on the subspaces for the polynomially or exponentially bounded weak mild solutions of the following abstract Cauchy problem d^2/（dr^2）u（t,x）=Au（t,x）;u（0,x）=x,d/（dt）u（0,x）=0,x∈X is studied, where A is a closed operator on Banach space X. The case that the problem is ill-posed is treated, and two subspaces Y（A, k） and H（A, ω） are introduced. Y（A, k） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（ t, x） such that ess sup{（1＋t）^-k｜d/（dt）〈v（t,x）,x^＊〉｜：t≥0,x^＊∈X^＊,｜x^＊‖≤1}〈＋∞. H（A, ω） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（t,x）such that ess sup{e^-ωl｜d/（dt）〈v（t,x）,x^＊）｜：t≥0,x^＊∈X^＊,‖x^＊‖≤1}〈＋∞. The following conclusions are proved that Y（A, k） and H（A, ω） are Banach spaces, and both are continuously embedded in X; the restriction operator A ｜ Y（A,k） generates a once-integrated cosine operator family { C（t） }t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖Y（A,k）≤M（1＋t）^k,arbitary t≥0; the restriction operator A ｜H（A,ω） generates a once- integrated cosine operator family {C（t）}t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖H（A,ω）≤≤Me^ωt,arbitary t≥0.

Approximate Controllability of Second-Order Neutral Stochastic Differential Equations with Infinite Delay and Poisson Jumps

The modelling of risky asset by stochastic processes with continuous paths, based on Brow- nian motions, suffers from several defects. First, the path continuity assumption does not seem reason- able in view of the possibility of sudden price variations （jumps） resulting of market crashes. A solution is to use stochastic processes with jumps, that will account for sudden variations of the asset prices. On the other hand, such jump models are generally based on the Poisson random measure. Many popular economic and financial models described by stochastic differential equations with Poisson jumps. This paper deals with the approximate controllability of a class of second-order neutral stochastic differential equations with infinite delay and Poisson jumps. By using the cosine family of operators, stochastic analysis techniques, a new set of sufficient conditions are derived for the approximate controllability of the above control system. An example is provided to illustrate the obtained theory.

Second-Order Solutions for Random Interfacial Waves Under Steady Uniform Currents

In the present research, the study of Song (2004) for random interfacial waves in two-layer fluid is extended to the case of fluids moving at different steady uniform speeds. The equations describing the random displacements of the density interface and the associated velocity potentials in two-layer fluid are solved to the second order, and the wave-wave interactions of the wave components and the interactions between the waves and currents are described. As expected, the extended solutions include those obtained by Song (2004) as one special case where the steady uniform currents of the two fluids are taken as zero, and the solutions reduce to those derived by Sharma and Dean (1979) for random surface waves if the density of the upper fluid and the current of the lower fluid are both taken as zero.

Second-order random interfacial wave solutions for two-layer fluid with a free surface

A previous study （Song. 2004. Geophys Res Lett, 31 （15）：L15302） of the second-order solutions for random interracial waves is extended in a constant depth, two-layer fluid system with a rigid lid is extended into a more general case of two-layer fluid with a top free surface. The rigid boundary condition on the upper surface is replaced by the kinematical and dynamical boundary conditions of a free surface, and the equations describing the random displacements of free surface, density-interface and the associated velocity potentials in the two-layer fluid are solved to the second order using the same expansion technology as that of Song （2004. Geophys Res Lett, 31 （15）：L15302）. The results show that the interface and the surface will oscillate synchronously, and the wave fields to the first-order both at the free surface and at the density-interface are made up of a linear superposition of many waves with different amplitudes, wave numbers and frequencies. The second-order solutions describe the second-order wave-wave interactions of the surface wave components, the interface wave components and among the surface and the interface wave components. The extended solutions also include special cases obtained by Thorpe for progressive interracial waves （Thorpe. 1968a.Trans R Soc London, 263A：563～614） and standing interracial waves （Thorpe. 1968b. J Fluid Mech, 32：489-528） for the two-layer fluid with a top free surface. Moreover, the solutions reduce to those derived for random surface waves by Sharma and Dean （1979.Ocean Engineering Rep 20） if the density of the upper layer is much smaller than that of the lower layer.

POSITIVE SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR SECOND-ORDER SINGULAR NONLINEAR DIFFERENTIAL EQUATIONS

New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the conditions 0 less than or equal to f(0)(+) < M-1, m(1) < f(infinity)(-)less than or equal to infinity or 0 less than or equal to f(infinity)(+)< M-1, m(1) < f (-)(0)less than or equal to infinity where f(0)(+) = lim(u -->0)f(u)/u, f(infinity)(-)= lim(u --> infinity)f(u)/u, f(0)(-)= lim(u -->0)f(u)/u, f(infinity)(+) = lim(u --> infinity)f(u)/u, g may be singular at t = 0 and/or t = 1. The proof uses a fixed point theorem in cone theory.

Second-order random wave solutions for interfacial internal waves in N-layer density-stratified fluid

This paper studies the random internal wave equations describing the density interface displacements and the velocity potentials of N-layer stratified fluid contained between two rigid walls at the top and bottom. The density interface displacements and the velocity potentials were solved to the second-order by an expansion approach used by Longuet-Higgins （1963） and Dean （1979） in the study of random surface waves and by Song （2004） in the study of second- order random wave solutions for internal waves in a two-layer fluid. The obtained results indicate that the first-order solutions are a linear superposition of many wave components with different amplitudes, wave numbers and frequencies, and that the amplitudes of first-order wave components with the same wave numbers and frequencies between the adjacent density interfaces are modulated by each other. They also show that the second-order solutions consist of two parts： the first one is the first-order solutions, and the second one is the solutions of the second-order asymptotic equations, which describe the second-order nonlinear modification and the second-order wave-wave interactions not only among the wave components on same density interfaces but also among the wave components between the adjacent density interfaces. Both the first-order and second-order solutions depend on the density and depth of each layer. It is also deduced that the results of the present work include those derived by Song （2004） for second-order random wave solutions for internal waves in a two-layer fluid as a particular case.

Exact solutions for unsteady unidirectional flows of a generalized second-order fluid through a rectangular conduit

The exact solutions are obtained for unsteady unidirectional flows of a generalized second-order fluid through a rectangular conduit. The fractional calculus in the constitutive relationship of a non-Newtonian fluid is introduced. We construct the solutions by means of Fourier transform and the discrete Laplace transform of the sequential derivatives and the double finite Fourier transform. The solutions for Newtonian fluid between two infinite parallel plates appear as limiting cases of our solutions.

ON GLOBAL MEROMORPHIC SOLUTIONS OF SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS WITH MEROMORPHIC COEFFICIENTS

The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory of such equations.

A Semi-Analytical Method for Solutions of a Certain Class of Second Order Ordinary Differential Equations

This paper presents the theory and applications of a new computational technique referred to as Differential Transform Method (DTM) for solving second order linear ordinary differential equations, for both homogeneous and nonhomogeneous cases. For the robustness and efficiency of the method, four examples are considered. The results indicate that the DTM is reliable and accurate when compared to the exact solutions of the solved problems.

Explicit Iterative Methods of Second Order and Approximate Inverse Preconditioners for Solving Complex Computational Problems

Explicit Exact and Approximate Inverse Preconditioners for solving complex linear systems are introduced. A class of general iterative methods of second order is presented and the selection of iterative parameters is discussed. The second order iterative methods behave quite similar to first order methods and the development of efficient preconditioners for solving the original linear system is a decisive factor for making the second order iterative methods superior to the first order iterative methods. Adaptive preconditioned Conjugate Gradient methods using explicit approximate preconditioners for solving efficiently large sparse systems of algebraic equations are also presented. The generalized Approximate Inverse Matrix techniques can be efficiently used in conjunction with explicit iterative schemes leading to effective composite semi-direct solution methods for solving large linear systems of algebraic equations.

MAXIMAL SUBSPACES FOR SOLUTIONS OF THE SECOND ORDER ABSTRACT CAUCHY PROBLEM

For a continuous, increasing function ω ： R^＋ →R^＋／{0} of finite exponential type, this paper introduces the set Z（A, ω） of all x in a Banach space X for which the second order abstract differential equation （2） has a mild solution such that [ω（t）]^-1u（t,x） is uniformly continues on R^＋, and show that Z（A, ω） is a maximal Banach subspace continuously embedded in X, where A ∈ B（X） is closed. Moreover, A[z（A,ω） generates an O（ω（t）） strongly continuous cosine operator function family.

MULTIPLICITY OF PERIODIC SOLUTIONS FOR SECOND ORDER HAMILTONIAN SYSTEMS WITH MIXED NONLINEARITIES

The multiplicity of periodic solutions for a class of second order Hamiltonian system with superquadratic plus subquadratic nonlinearity is studied in this paper.Obtained via the Symmetric Mountain Pass Lemma,two results about infinitely many periodic solutions of the systems extend some previously known results.

Positive solutions of second-order differential systems with nonhomogeneous boundary conditions

This paper is to investigate the positive solutions of the systems of second-order ordinary differential equations with nonhomogeneous multi-point boundary conditions. By the lower and upper solutions method, Schauder fixed point theorem and fixed point index theory, under certain conditions, it is proved that there exist appropriate regions of parameters in which the problem has at least two, at least one or no positive solution.

Existence and Multiplicity of Periodic solutions for the Non-autonomous Second-order Hamiltonian Systems

In this paper,we study the existence and multiplicity of periodic solutions of the non-autonomous second-order Hamiltonian systems■where T> 0.Under suitable assumptions on F,some new existence and multiplicity theorems are obtained by using the least action principle and minimax methods in critical point theory.

Picard Boundary Value Problems of Second Order p-Laplacian Differential Equations

Suffcient conditions for the existence of at least one solution of two-point boundary value problems for second order nonlinear differential equations [φ(x(t))] + kx(t) + g(t,x(t)) = p(t),t ∈(0,π) x(0) = x(π) = 0 are established,where [φ(x)] =(|x |p-2x) with p > 1.Our result is new even when [φ(x)] = x in above problem,i.e.p = 2.Examples are presented to illustrate the effciency of the theorem in this paper.

Eigenstructure assignment in a class of second-order dynamic systems

Eigenstructure assignment using the proportional-plus-derivative feedback controller in a class of secondorder dynamic system is investigated. Simple, general, complete parametric expressions for both the closed-loop eigenvector matrix and the feedback gains are established based on two simple Smith form reductions. The approach utilizes directly the original system data and involves manipulations only on n-dimensional matrices. Furthermore, it reveals all the degrees of freedom which can be further utilized to achieve additional system specifications. An example shows the effect of the proposed approach.

Horizontal dynamic response of a large-diameter pipe pile considering the second-order effect of axial force

The second-order effect of axial force on horizontal vibrating characteristics of a large-diameter pipe pile is theoretically investigated.Governing equations of the pile-soil system are established based on elastodynamics.Threedimensional wave equations of soil are decoupled through differential transformation and variable separation.Consequently,expressions of soil displacements and horizontal resistances can be obtained.An analytical solution of the pile is derived based on continuity conditions between the pile and soil,subsequently from which expressions of the complex impedances are deduced.Analyses are carried out to examine the second-order effect of axial force on the horizontal vibrating behavior of the pipe pile.Some conclusions can be summarized as follows： stiffness and damping factors are decreased with the application of axial force on the pile head; distributions of the pile horizontal displacement and rotation angle are regenerated due to the second-order effect of the applied axial force; and redistributions of the bending moment and shearing force occur due to the second-order effect of the applied axial force.

Two parametric approaches for eigenstructure assignment in second-order linear systems

This paper considers eigenstructure assignment in second-order linear systems via proportional plus derivative feedback. It is shown that the problem is closely related to a type of so-called second-order Sylvester matrix equations. Through establishing two general parametric solutions to this type of matrix equations, two complete parametric methods for the proposed eigenstructure assignment problem are presented. Both methods give simple complete parametric expressions for the feedback gains and the closed-loop eigenvector matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable; the second one utilizes the right factorization of the system, and allows the closed-loop eigenvalues to be set undetermined and sought via certain optimization procedures. An example shows the effectiveness of the proposed approaches. Keywords Second-order linear systems - Eigenstructure assignment - Proportional plus derivative feedback - Parametric solution - Singular value decompoition - Right factorization This work was supported in part by the Chinese Outstanding Youth Foundation (No.69504002).

SOLVING SECOND ORDER DIFFERENTIAL EQUATIONS IN QUANTUM MECHANICS BYORDER REDUCTION

Solving the famous Hermite, Legendre, Laguerre and Chebyshev equations requires different techniques of unique character for each equation. By reducing these differential equations of second order to a common solvable differential equation of first order, a simple common solution is provided to cover all the existing standard solutions of these named equations. It is easier than the method of generating functions and more powerful than the Probenius method of power series.

Oscillation Theorems for a Class of Nonlinear Second Order Differential Equations with Damping

The oscillatory behavior of solutions of a class of second order nonlinear differential equations with damping is studied and some new sufficient conditions are obtained by using the refined integral averaging technique. Some well known results in the literature are extended. Moreover, two examples are given to illustrate the theoretical analysis.