The explicit solution of homogeneous linear oridinary differential equations with constant coefficients

In this paper by using tensor analysis we give the explicit expressions of the solution of the initial-value problem of homogeneous linear differential equations with constant coefficients and the n th-order homogeneous linear differential equation with constant coefficients. In fact, we give the general formula for calculating the elements of the matrix exp [At]. We also give the results when the characteristic equation has the repeated roots. The present method is simpler and better than, the other methods.     

Surface Impedance Tensors of Textured Polycrystals

A formula for the surface impedance tensors of orthorhombic aggregates of cubic crystallites is given explicitly in terms of the material constants and the texture coefficients. The surface impedance tensor is a Hermitian second-order tensor which, for a homogeneous elastic half-space, maps the displacements given at the surface to the tractions needed to sustain them. This tensor plays a fundamental role in Stroh's formalism for anisotropic elasticity. In this paper we account for the effects of crystallographic texture only up to terms linear in the texture coefficients and give an explicit formula for the terms in the surface impedance tensor up to those linear in the texture coefficients. This revised version was published online in June 2006 with corrections to the Cover Date.     

Integration of Special Linear Equations of the Second Order

Under certain conditions on the coefficients, the Chazy equation with constant coefficients reduces to a second-order linear differential equation with six singular points. Investigating this equation with the use of the Schwarz derivative, we obtain linear equations in which some of these six singular points coincide. An integration procedure for these equations is considered. Their general solutions are obtained in explicit form.     

弹性力学平面问题的位移型解答

本文证明了一个线性常系数偏微分方程的通解定理,利用这个通解定理导出了弹性力学平面问题的位移通解。     

Characteristic solutions of scalar newton equations in one space dimension

Newton equations are dynamical systems on the space of fields. The solutions of a given equation which are curves of characteristic fields for its force are planar and have constant angular momentum. Separable solutions are characteristic with angular momentum equal to zero. A Newton equation is separable if and only if its characteristic equation is homogeneous. Separable equations correspond to invariants of homogeneous ordinary differential equations, and those associated with a given homogenous equation correspond to its generalized dilation symmetries. A Newton equation is compatible with the characteristic condition if and only if its characteristic equation is linear. Such equations correspond to invariants of linear ordinary differential equations. Those associated with a given linear equation correspond to the central force problems on its solution space. Regardless of compatibility, any Newton equation with a plane of characteristic fields has non-separable characteristic solutions.     

Instability of solution for the third order linear differential equation with varied coefficient

In reference [1] asymptotic stability of dynamic system with slowly changing coefficients for all characteristic roots which have negative real part has been proved by means of Liapunov’s second method. In this paper, we give some sufficient conditions of the instability for the third order linear differential equation with varied coefficient, at least one of the characteristic roots of which has positive real part, by means of Liapunov’s second method.     

Recursive-operator method in vibration problems for rod systems

Using linear differential equations with constant coefficients describing one-dimensional dynamical processes as an example, we show that the solutions of these equations and systems are related to the solution of the corresponding numerical recursion relations and one does not have to compute the roots of the corresponding characteristic equations. The arbitrary functions occurring in the general solution of the homogeneous equations are determined by the initial and boundary conditions or are chosen from various classes of analytic functions. The solutions of the inhomogeneous equations are constructed in the form of integro-differential series acting on the right-hand side of the equation, and the coefficients of the series are determined from the same recursion relations. The convergence of formal solutions as series of a more general recursive-operator construction was proved in [1]. In the special case where the solutions of the equation can be represented in separated variables, the power series can be effectively summed, i.e., expressed in terms of elementary functions, and coincide with the known solutions. In this case, to determine the natural vibration frequencies, one obtains algebraic rather than transcendental equations, which permits exactly determining the imaginary and complex roots of these equations without using the graphic method [2, pp. 448–449]. The correctness of the obtained formulas (differentiation formulas, explicit expressions for the series coefficients, etc.) can be verified directly by appropriate substitutions; therefore, we do not prove them here.     

The viscoelastic constitutive differential operator law in terms of the relaxation and retardation spectra

The case of a linear viscoelastic medium is considered. The Carson transforms of the relaxation and retardation functions are expressed in two different ways, taking on the one hand the differential operator form of the constitutive equation, and on the other hand the generalized mechanical models. By identification we deduce general explicit expressions for the constant coefficients of the differential operator law, in terms of the discrete relaxation and retardation spectra.     

The Eigentensors of an Arbitrary Second-Order Tensor and their Quality Analyses

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The sufficient and necessary conditions of unconditional stability and the delay bound of the third-order neutral delay differential equation

In this paper,the sufficient and necessary conditions of the unconditional stability.andthe delay bound of the third-order neutral delay differential equation with real constantcoefficients are given.The conditions are brief and practical algebraic criterions.Furthermore,we get the delay bound     

Energy and dissipation of inhomogeneous plane waves in thermoelasticity

Inhomogeneous small-amplitude plane waves of (complex) frequency ω are propagated through a linear dissipative material. For thermoelasticity we derive an energy-dissipation equation that contains all the quadratic dependence on the field quantities, see Eq. (10). In addition, we derive a new energy-dissipation equation (Eq. (22)) involving the total energy density which contains terms linear in the field quantities as well as the usual quadratic terms. The terms quadratic in the small quantities in the energy density, energy flux and dissipation give rise to inhomogeneous plane waves of frequency 2ω and to (attenuated) constant terms. Usually these quadratic quantities are time-averaged and only the attenuated constant terms remain. We derive a new result in thermoelasticity for these terms, see Eq. (54). The present innovation is to retain the terms of frequency 2ω, since they are comparable in magnitude to the attenuated constant terms, and a new result, see Eq. (44), is derived for a general energy-dissipation equation that connects the amplitudes of the terms of the energy density, energy flux and dissipation that have frequency 2ω. Furthermore, for dissipative waves or inhomogeneous conservative waves the (complex) group velocity is related to these amplitudes rather than to the attenuated constant terms as it is for homogeneous waves in conservative materials.     

Effective characteristics of corrugated plates

Corrugated plates are widely used in modern constructions and structures, because they, in contrast to plane plates, possess greater rigidity. In many cases, such a plate can be modeled by a homogeneous anisotropic plate with certain effective flexural and tensional rigidities. Depending on the geometry of corrugations and their location, the equivalent homogeneous plate can also have rigidities of mutual influence. These rigidities allow one to take into account the influence of bending moments on the strain in the midplane and, conversely, the influence of longitudinal strains on the plate bending [1]. The behavior of the corrugated plate under the action of a load normal to the midsurface is described by equations of the theory of flexible plates with initial deflection. These equations form a coupled system of nonlinear partial differential equations with variable coefficients [2]. The dependence of the coefficients on the coordinates is determined by the corrugation geometry. In the case of a plate with periodic corrugation, the coefficients significantly vary within one typical element and depend on the values of local variables determined in each of the typical elements. There is a connection between the local and global variables, and therefore, the functions of local coordinates are simultaneously functions of global coordinates, which are sometimes called rapidly oscillating functions [3].One of the methods for solving the equations with rapidly oscillating coefficients is the asymptotic method of small geometric parameter. The standard procedure of this method usually includes preparatory stages. At the first stage, as a rule, a rectangular periodicity cell is distinguished to be a typical element. At the second stage, the scale of global coordinates is changed so that the rectangular structure periodicity cells became square cells of size l × l. The third stage consists in passing to the dimensionless global coordinates relative to the plate characteristic dimension L. As a result, the dependence between the new local variables and the new scaled dimensionless variables is such that the factor 1/α, where α=l/L ? 1 is a small geometric parameter, appears in differentiating any function of the local coordinate with respect to the global coordinate. After this, the solution of the problem in new coordinates is sought as an asymptotic expansion in the small geometric parameter [1], [4–10].We note that, in the small geometric parameter method, the asymptotic series simultaneously have the form of expansions in the gradients of functions depending only on the global coordinates. This averaging procedure can be applied to linear and nonlinear boundary value problems for differential equations with variable periodic coefficients for which the periodicity cell can be affinely transformed into the periodicity cube. In the case of an arbitrary dependence of the coefficients on the coordinates (including periodic dependence), another averaging technique can be used in linear problems. This technique is based on the possibility of the integral representation of the solution of the original problem for the linear equation with variables coefficients in terms of the solution of the same problem for an equation of the same type but with constant coefficients [11–13]. The integral representation implies that the solution of the original problem can be represented in the form of the series in the gradients of the solution of the problem for the equation with constant coefficients [13].The aim of the present paper is to develop methods for calculating effective characteristics of corrugated plates. To this end, we first write out the equilibrium equations for a flexible anisotropic plate, which is inhomogeneous in the thickness direction and in the horizontal projection, with an initial deflection. We write these equations in matrix form, which allows one to significantly reduce the length of the expressions and to simplify further calculations. After this, we average the initial matrix equations with variable coefficients. The averaging procedure implies the statement of problems such that, after solving them, we can calculate the desired effective characteristics. By way of example, we consider the case of a corrugated plate made of a homogeneous isotropic material whose corrugations are hexagonal in the horizontal projection. In this case, we obtain approximate expressions for the components of the effective tensors of flexural rigidity and longitudinal compliance and expressions for the effective plate thickness.     

Marangoni convection in a cylinder of finite size

This paper considers a liquid in a finite-size cylinder in which Marangoni instability occurs. The upper boundary of the liquid is free and deformable. The problem of the occurrence of convection in a cylindrical container is solved using the method of separation of variables. A homogeneous differential equation of the sixth order with constant coefficients and complex boundary conditions is obtained. An analytical expression for critical Marangoni numbers is derived for the case of monotonic perturbations. The case is considered where the liquid in the cylinder is weightless.     

Semi-discrete breather in a helicoidal DNA double chain-model

The nonlinear dynamics of DNA molecular chain is studied for longitudinal and transversal motions through a new discrete helicoidal zigzag model with four degrees of freedom. We take into account the Stokes and hydrodynamical viscous forces. In the semi-discrete approximation, we show that the coupled nonlinear partial differential equations for the longitudinal and transversal out-of-phase motions can be reduced to the nonlinear Schrödinger equation with complex coefficients, allowing analytical breather soliton solution. We found analytically as well as numerically that increasing the damping constant reduces the amplitude and increases the width of the soliton. When the zigzag angle decreases, the height of the soliton increases, but its width remains constant. The linear stability analysis of the system is performed. The growth rate of the instability and the instability regions are discussed as the functions of damping constant, zigzag angle and system parameters.     

分布轴向力作用下隔水管横向自由振动分析

隔水管固有频率的精确计算对保证隔水管的安全使用和防止共振的发生有着极为重要的意义.在分析中,考虑了分布轴向力和顶张力的共同作用,建立了隔水管横向振动力学模型;基于牛顿定律和纵横弯曲梁理论,对微单元受力分析,得到隔水管横向自由振动的四阶偏微分方程;利用分离变量法将四阶偏微分方程简化为四阶变系数常微分方程;采用积分法求解四阶变系数常微分方程,得到隔水管横向自由振动固有频率的解析解.结果表明:(1)分布轴向力作用下隔水管横向自由振动的固有频率和振型,与将分布轴向力简化为集中力作用下隔水管的固有频率和振型有很大差别;(2)顶张力一定时,随着分布轴向力减小,隔水管固有频率增大;分布轴向力一定时,随着顶张力增大,隔水管固有频率增大;(3)采用积分法求解隔水管横向振动特性时,计算精度高,为隔水管的优化设计提供了可靠的理论依据.     

Instability of solution for the fourth order linear differential equation with varied coefficient

In this paper,we give some sufficient conditions of the instability for the fourth order linear differential equation with varied coefficient,at least one of the characteristic roots of which has positive real part,by means of Liapunov’s second method.     

On paradoxes of the formal approach to determining the eigenvalues of one-dimensional boundary value problems

The conventional approach to determining the eigenvalues of a one-dimensional boundary value problem consists in writing out the solution of the differential equation in general form containing indeterminate coefficients and constructing a system of homogeneous linear algebraic equations for these coefficients on the basis of the expressions for the boundary conditions. The eigenvalue is determined from the condition that the determinant of the system thus constructed is zero. In the classical problems (of string, rod, etc. vibrations), this method, as a rule, does not cause any difficulties, although several examples in which the zero value of the frequency satisfying the characteristic equation thus constructed is not an eigenfrequency were constructed and investigated, for example, in [1, p. 220]. We show that in some cases more complicated than the classical ones a similar situation can lead to paradoxical conclusions and erroneous results.     

A damped simple pendulum of constant amplitude

A simple pendulum acted on by gravity and subjected to a resistance proportional to the velocity of the bob is considered. If the length of the string and the mass of the bob are held constant, the amplitude of the bob decreases gradually because of the damping. We want to keep the maximum swing of the bob constant for all time; this we achieve by varying the length of the string, the mass of the bob or both. The key to the solution of our problem is a second-order nonlinear differential equation having arbitrary nonlinearity and an arbitrary coefficient function, for which we give the exact integral. We also give an application of this differential equation to a boundary-value problem for a nonlinear generalization of a hypergeometric equation.     

Exponential Dichotomies,the Fredholm Alternative,and Transverse Homoclinic Orbits in Partial Functional Differential Equations

In this paper we investigate the transversality of homoclinic orbits in partial functional differential equations. We first discuss the exponential dichotomies for linear operator equations. Then we show that the Fredholm Alternative holds if the homogeneous equation has exponential dichotomies on R. Transversality of homoclinic orbits for periodically perturbed partial functional differential equations is studied using the Liapunov-Schmidt method and the Melnikov integral. Ams Subject Classifications: 35R10; 58F14.     

The ray method for the deflection of a floating flexible platform in short waves

In this paper, we explain how the ‘ray method’ can be used to describe the deflection, due to short waves, of a very large floating platform in finite or infinite water depth. The elastic properties of the platform are isotropic, but may be distributed inhomogeneously. In the first section, we give a derivation of the equation for the phase and amplitude functions. Then an integro-differential equation for the determination of the deflection is used to find the initial condition for amplitude along the characteristics. For the homogeneous two-dimensional platform in water of finite depth, an exact solution in the form of a superposition of modes can be obtained. This simplified problem serves as a ‘canonical’ problem for problems with the same structure locally. In the last section, we give some result for a semi-infinite platform with varying elasticity coefficient, the mass distribution being taken constant.