HERMITE MATRIX POLYNOMIALS AND SECOND ORDER MATRIX DIFFERENTIAL EQUATIONS

In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues' formula are given.

NEW OSCILLATION CRITERIA FOR SECOND-ORDER NONLINEAR MATRIX DIFFERENTIAL EQUATIONS

Some new oscillation criteria are established for the second-order matrix differential system （r（t）Z＇（t））＇ ＋ p（t）Z＇（t） ＋ Q（t）F（Z＇（t））G（Z（t）） = 0, t ≥ to 〉 0, are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [to,∞）, rather than on the whole half-line. The results weaken the condition of Q（t） and generalize some well-known results of Wong （1999） to nonlinear matrix differential equation.

ON DETERMINATION OF THE IMPULSIVE SOLUTIONS AND IMPULSIVE PROPERTIES TO LINEAR NON-HOMOGENEOUS MATRIX DIFFERENTIAL EQUATIONS

This note contains three main results.Firstly,a complete solution of the Linear Non-Homogeneous Matrix Differential Equations(LNHMDEs)is presented that takes into account both the non-zero initial conditions of the pseudo state and the nonzero initial conditions of the input.Secondly,in order to characterise the dynamics of the LNHMDEs correctly,some important concepts such as the state,slow state(smooth state)and fast state(impulsive state)are generalized to the LNHMDE case and the solution of the LNHMDEs is separated into the smooth(slow)response and the fast(implusive)response.As a third result,a new characterization of the impulsive free initial conditions of the LNHMDEs is given.

The Exact Solutions of Such Coupled Linear Matrix Fractional Differential Equations of Diagonal Unknown Matrices by Using Hadamard Product

In this paper, we present the general exact solutions of such coupled system of matrix fractional differential equations for diagonal unknown matrices in Caputo sense by using vector extraction operators and Hadamard product. Some illustrated examples are also given to show our new approach.

New matrix method for analyzing vibration and damping effect of sandwich circular cylindrical shell with viscoelastic core

Based on the linear theories of thin cylindrical shells and viscoelastic materials, a governing equation describing vibration of a sandwich circular cylindrical shell with a viscoelastic core under harmonic excitation is derived. The equation can be written as a matrix differential equation of the first order, and is obtained by considering the energy dissipation due to the shear deformation of the viscoelastic core layer and the interaction between all layers. A new matrix method for solving the governing equation is then presented With an extended homogeneous capacity precision integration approach. Having obtained these, vibration characteristics and damping effect of the sandwich cylindrical shell can be studied. The method differs from a recently published work as the state vector in the governing equation is composed of displacements and internal forces of the sandwich shell rather than displacements and their derivatives. So the present method can be applied to solve dynamic problems of the kind of sandwich shells with various boundary conditions and partially constrained layer damping. Numerical examples show that the proposed approach is effective and reliable compared with the existing methods.

Reaction Processes of the Titanium Matrix Composites Fabricated from Ti-Al-B2O3 System

Titanium matrix composites reinforced with a-Al2O3 and TiB2 particles were fabricated by in situ synthesis from a Ti-Al-B2O3 system. The reaction processes and microstructure were analyzed by using differential scanning calorimetry（DSC）, scanning electron microscopy（SEM） and X-ray diffraction（XRD）. The results showed that the reactions in the Ti-Al-B2O3 system can occur spontaneously and consist of three steps： 1） 15 Al ＋ 7B2O3 → 7α-Al2O3 ＋ AlB12 ＋ 2B; 2） 14 B ＋ 2Al → AlB12 ＋ AlB2 and 3） 7Ti ＋ AlB（12） ＋ AlB2 → 7TiB2 ＋ 2Al. The final reinforcements were composed of α-Al2O3 and TiB2 particles, which were uniformly distributed in the titanium matrix.

High-precision stress determination in photoelasticity

Stress separation is usually achieved by solving differential equations of equilibrium after parameter determination from isochromatics and isoclinics.The numerical error resulting from the stress determination is a main concern as it is always a function of parameters in discretization.To improve the accuracy of stress calculation,a novel meshless barycentric rational interpolation collocation method(BRICM)is proposed.The derivatives of the shear stress on the calculation path are determined by using the differential matrix which converts the differential form of the equations of equilibrium into a series of algebraic equations.The advantage of the proposed method is that the auxiliary lines,grids,and error accumulation which are commonly used in traditional shear difference methods(SDMs)are not required.Simulation and experimental results indicate that the proposed meshless method is able to provide high computational accuracy in the full-field stress determination.

THE OSCILLATION OF A CLASS NONLINEAR MATRIX DIFFERENTIAL EQUATION

In this paper, we discuss the Oscillation of Matrix Differential Equation (p(t)y'(t))' + Q(t)y(t) + F(t, y(t), y'(t)) = 0 and obtain more general consequence in comparison with the paper [1]. Under the determinated condition approved a conjecture to be correct of the paper [1].

EQUISTABILITY OF THE MATRIX DIFFERENTIAL EQUATIONS

In this paper the problem about equistability of the matrix equistability of the matrix differential equations have been discussed, and some criteria for equistability have been given.

Double Symplectic Eigenfunction Expansion Method of Free Vibration of Rectangular Thin Plates

The free vibration problem of rectangular thin plates is rewritten as a new upper triangular matrix differential system. For the associated operator matrix, we find that the two diagonal block operators are Hamiltonian. Moreover, the existence and completeness of normed symplectic orthogonal eigenfunction systems of these two block operators are demonstrated. Based on the completeness, the general solution of the free vibration of rectangular thin plates is given by double symplectie eigenfunction expansion method.

Two-dimensional interactions due to moving load in generalized thermoelastic solid with diffusion

The present paper is concerned with the investigation of disturbances in＇a homogeneous, isotropic elastic medium with generalized thermoelastic diffusion, when a moving source is acting along one of the co-ordinate axis on the boundary of the medium. Eigen value approach is applied to study the disturbance in Laplace-Fourier transform domain for a two dimensional problem. The analytical expressions for displacement components, stresses, temperature field, concentration and chemical potential are obtained in the physical domain by using a numerical technique for the inversion of Laplace transform based on Fourier expansion techniques. These expressions are calculated numerically for a copper like material and depicted graphically. As special cases, the results in generalized thermoelastic and elastic media are obtained. Effect of presence of diffusion is analyzed theoretically and numerically.

New Oscillation Criteria for Second Order Linear Matrix Differential Equations with Damping

By using the positive linear functional and the monotone subhomogeneous functional, including the general means and Riccati technique, some new oscillation criteria are established for the second order linear matrix differential system （P（t）X＇（t））＇ ＋ R（t）X＇（t） ＋ Q（t）X（t） =0, t ≥ to 〉 0 where P（t）, R（t）, Q（t） are n × n real continuous matrix functions, P（t） and R（t） are commutative. Theresults improve and generalize those given in some previous papers, which can be seen by the examples given at the end of this paper.

Oscillation Theorems of the Second Order Linear Matrix Differential System with Damping

By using the Riccati technique and the technique, new oscillation criteriaare obtained for the second order matrix differential system (P(t)Y'(t))' + r(t)P(t)Y(t) + Q(t)Y (t)= 0, t ≥ t_0. These results in the present paper generalize and improve many known conclusions.Furthermore, some results are different from the most known ones in the sense that they are based onthe information only on a sequence of subintervals of [t 0,), rather than on the whole half–line.In particular, our results complement a number of existing results and handle the case that is notcovered by the known criteria.

OSCILLATORY CRITERIA OF SECOND ORDER MATRIX DIFFERENTIAL SYSTEMS

By using a generalized Riccati transformation, we establish some new oscillation criteria which improve and generalize some known criteria in literatures.

TRANSFER MATRIX METHOD FOR ANALYZING VIBRATION AND DAMPING CHARACTERISTICS OF ROTATIONAL SHELL WITH PASSIVE CONSTRAINED LAYER DAMPING TREATMENT

The first order differential matrix equations of the host shell and constrained layer for a sandwich rotational shell are derived based on the thin shell theory.Employing the layer wise principle and first order shear deformation theory, only considering the shearing deformation of the viscoelastic layer, the integrated first order differential matrix equation of a passive constrained layer damping rotational shell is established by combining with the normal equilibrium equation of the viscoelastic layer.A highly precise transfer matrix method is developed by extended homogeneous capacity precision integration technology.The numerical results show that present method is accurate and effective.

A universal solution to one-dimensional oscillatory integrals

How to calculate the highly oscillatory integrals is the bottleneck that restraints the research of light wave and electromagnetic wave＇s propagation and scattering. Levin method is a classical quadrature method for this type of integrals. Unfortunately it is susceptible to the system of linear equations＇ ill-conditioned behavior. We bring forward a universal quadrature method in this paper, which adopts Chebyshev differential matrix to solve the ordinary differential equation （ODE）. This method can not only obtain the indefinite integral＇ function values directly, but also make the system of linear equations well-conditioned for general oscillatory integrals. Furthermore, even if the system of linear equations in our method is ill-conditioned, TSVD method can be adopted to solve them properly and eventually obtain accurate integral results, thus making a breakthrough in Levin method＇s susceptivity to the system of linear equations＇ ill-conditioned behavior.

Input-output finite-time stability of time-varying linear singular systems

This paper studies the input-output finite-time stabilization problem for time-varying linear singular sys- tems. The output and the input refer to the controlled output and the disturbance input, respectively. Two classes of dis- turbance inputs are considered, which belong to L-two and L-infinity. Sufficient conditions are firstly provided which guarantee the input-output finite-time stability. Based on this, state feedback controllers are designed such that the resultant closed-loop systems are input-output finite-time stable. The conditions are presented in terms of differential linear matrix inequalities. Finally, an example is presented to show the validity of the proposed results.

Shape Memory Effect,Thermal Expansion and Damping Property of Friction Stir Processed NiTip/Al Composite

NiTi particles reinforced aluminum （NiTip/Al） composite was prepared via friction stir processing, elimi- nating interfacial reaction and/or elemental diffusion. The NiTip in the composite maintained the intrinsic characteristic of a reversible thermoelastic phase transformation even after heat-treatment. The shape memory characteristic of the NiTip decreased the coefficient of thermal expansion of the Al matrix, and an apparent two-way shape memory effect was observed in the composite. The composite owned a good combination of adjustable damping and thermal physical properties.

AN IMPLEMENTATION OF THE COLLOCATION METHOD FOR INITIAL VALUE PROBLEMS

An implementation of the standard collocation method based on polynomial interpolation is presented in a matrix framework in this paper.The underlying differentiation matrix can be partitioned to yield a superconvergent implicit multistep-like method to solve the initial value problem numerically.The first-and second-order versions of this method are L-stable.

A Hybrid Mathematical Model of Tumor-Induced Angiogenesis with Blood Perfusion

Angiogenesis, the growth of new blood vessel from existing ones, is a pivotal stage in cancer development,and is an important target for cancer therapy. We develop a hybrid mathematical model to understand the mechanisms behind tumor-induced angiogenesis. This model describes uptake of Tumor Angiogenic Factor（TAF）at extracellular level, uses partial differential equation to describe the evolution of endothelial cell density including TAF induced proliferation, chemotaxis to TAF, and haptotaxis to extracellular matrix. In addition we also consider the phenomenon of blood perfusion in the micro-vessels. The model produces sprout formation with realistic morphological and dynamical features, including the so-called brush border effect, the dendritic branching and fusing of the capillary sprouts forming a vessel network. The model also demonstrates the effects of individual mechanisms in tumor angiogenesis： Chemotaxis to TAF is the key driving mechanisms for the extension of sprout cell; endothelial proliferation is not absolutely necessary for sprout extension; haptotaxis to Extra Cellular Matrix（ECM） gradient provides additional guidance to sprout extension, suggesting potential targets for anti-angiogenic therapies.