Existence and Asymptotic Behavior of Solutions of the Cauchy Problem x (x′)γ = f(t,x, x′), x(0) = 0

We consider a singular Cauchy problem and prove the existence of continuously differentiable solutions with required asymptotic properties.     

THE EXISTENCE OF LIMIT CYCLES FOR THE SYSTEM X=Q(x,y),Y=P(x)

ＴＨＥＥＸＩＳＴＥＮＣＥＯＦＬＩＭＩＴＣＹＣＬＥＳＦＯＲＴＨＥＳＹＳＴＥＭX＝Ｑ（ｘ，ｙ），Y＝Ｐ（ｘ）ＸｕＲｏｎｇ－ｌｉａｎｇ（徐荣良）；ＺｈｏｕＧｕｏ－ｃａｉ（周国才）；ＳｕｎＺｈａｏ（孙昭）（ＴａｉｙｕａｎＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏ...     

The existence of limit cycles for system\dot x=Q(x,y), Y=P(x)

In [1], by a transformation on the Lienard equation system

Bäcklund transformations for the equation ∂2u/∂x1∂x1 + ∂2u/∂x2∂x2−f (u)

Bäcklund transformations for the equation ² u/x¹x ¹+ ²u/x ² x ²=f (u)here f is an arbitrary function) is studied in this paper, using the procedure of Wahlquist and Estabrook (WEP). We conclude that the condition d ² f/du ²=f is sufficient for the existence of Bäcklund transformations for the equation of our interest. A special case of our results leads to the conclusion of Leibbrandt ^[1,2].     

Neue numerische Verfahren zur Berechnung des charakteristischen Exponenten der verallgemeinerten Mathieuschen Differentialgleichung (1+2 γcos2x)y′′(x)+(λ−2h 2 cos2x)y(x)=0

Ohne Zusammenfassung Vorgelegt von J. Meixner     

A simple,closed approximate solution for the diffusion equationc t=(c m cx) x

So far, only formal power series are known for this equation (except fundamental solution). In order to obtain closed expressions, especially, for the penetration parameter, we derive systematically an approximate solution (for Dirichlet boundary conditions). Although being very simple, it yields quite satisfactory results form1. This is demonstrated by numerical calculations.

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Eine einfache, geschlossene Näherungslösung für die Diffusionsgleichungc _t=(c ^m c_x) _x

Zusammenfassung Bisher sind nur formale Potenzreihen-Lösungen für diese Gleichung bekannt. Um geschlossene Ausdrücke insbesondere für den Eindringparameter zu erhalten, leiten wir systematisch eine Näherungslösung ab (für Dirichletsche Randbedingungen). Trotz ihrer Einfachheit liefert sie ganz zufriedenstellende Ergebnisse fürm1, was durch numerische Rechnungen gezeigt wird.

     

Barnett-Lothe tensors and their associated tensors for monoclinic materials with the symmetry plane at x 3=0

The three Barnett-Lothe tensors S, H, L and the three associated tensors S(), H(), L() appear frequently in the real form solutions to two-dimensional anisotropic elasticity problems. Explicit expressions of the components of these tensors are derived and presented for monoclinic materials whose plane of material symmetry is at x ₃=0. We use the algebraic formalism for these tensors but the results are derived not by the straight-forward substitution of the complex matrices A and B into the formulae. Instead, we find the product –AB ^-1, whose real and imaginary parts are SL ^-1 and L ^-1, respectively. The tensors S, H, L are then determined from SL ^-1 and L ^-1. For S(), H(), L() we again avoid the direct substitution by employing an alternate approach. The new approaches require minimal algebra and, at the same time, provide simple and concise expressions for the components of these tensors. Although the new approaches can be extended, in principle, to monoclinic materials whose plane of symmetry is not at x ₃=0 and to materials of general anisotropy, the explicit expressions for these materials are too complicated. More studies are needed for these materials.     

Necessary and Sufficient Conditions for the Invertibility of the Nonlinear Difference Operator \left( {\mathcal{D}x} \right)\left( t \right) = x\left( {t + 1}\right) - f\left( {x\left( t \right)} \right) in the Space of Functions Bounded and Continuous on the Axis

We find necessary and sufficient conditions for the nonlinear difference operator $\left( {\mathcal{D}x} \right)\left( t \right) = x\left( {t + 1} \right) - f\left( {x\left( t \right)} \right)$ $t \in \mathbb{R}$ , where $f:\mathbb{R} \to \mathbb{R}$ is a continuous function, to have the inverse in the space of functions bounded and continuous on $\mathbb{R}$ .     

含x^5项强非线性系统的共振解和亚谐解

本文应用强非线性系统频闪法,对公式进行适当变换,求出了方程x+ax+βx~3+γx~5=ε(-μx+δcosΩt)的共振解及亚谐解存在的条件及其解析表达式,并与数值计算结果作了比较.     

Direct numerical simulation of fluid flow in a 5x5 square rod bundle

Characterization of parallel flow through rod bundles is of key importance in assessing the performance and safety of several engineering systems, including a majority of nuclear reactor concepts. Inhomogeneities in the bundle cross-section can present complex flow phenomena, including varying local conditions of turbulence. With the ever-increasing capabilities of high-performance computing, Direct Numerical Simulation (DNS) of turbulent flows is becoming more feasible. Through resolving all scales of turbulence, DNS can serve as a “numerical experiment,” and can provide substantial insight into flow physics, but at considerable computational cost. Thus to date, the DNS in open literature for rod bundle flows is relatively scarce, and largely limited to unit-cell domains. Since wall effects are important in rod bundle flows, a multiple-pin DNS study can expand understanding of rod bundle flows while providing valuable reference data for evaluating reduced-resolution techniques. In this work, DNS of a 5x5 square bare rod bundle representative of typical light water reactor fuel dimensions was performed using the spectral element code Nek5000. Turbulent microscales based on an advanced Reynolds-Averaged Navier–Stokes model were used to establish the required DNS resolution. Velocity and Reynolds stress fields are analyzed in detail, and invariant analysis is used for further investigation into flow physics. The results show stark changes in the structure of turbulence in the edge gaps, suggesting the presence of gap vortices in these regions. In addition, turbulent kinetic energy budgets are presented to more fully illustrate the various turbulent processes. These data can prove useful for rigorous evaluation of lower-fidelity turbulence modeling approaches.     

Flamelet Based NO x -Radiation Integrated Modelling of Turbulent Non-premixed Flame using Reynolds-stress Closure

A methodology of extending laminar flamelet model in its adiabatic form to a non-adiabatic form which can account for radiative heat loss as well as its effect on NO _x pollutant has been developed. Coupling of radiation submodel with flamelet model is based on the enthalpy defect concept. Pollutant NO _x has been calculated from solution of its transport equation containing source term which is derived from flamelet calculations. Flamelet calculations adopted GRI 2.11 reaction mechanism which accounts for detailed carbon and NO _x chemistry. Depending on consideration of variation in scalar dissipation within flamelet calculations, the non-adiabatic form has been further divided into non-adiabatic model with single (NADS) and multiple scalar dissipation rates (NADM). Bluff-body stabilized CH₄/H₂ flame has been chosen as the test case to assess the capability of non-adiabatic models. Turbulence closure has been achieved with a Reynolds stress transport model. Calculations have also been carried out with a modified k-ε model for evaluation of relative performance of the two turbulence closures. Performance of non-adiabatic flamelet models in regard to the overall structure of the flame is reasonably good and the agreement is similar to that of the adiabatic flamelet model thereby indicating weakly radiating nature of the flame. However, the NADM model results in minor but encouraging improvement in NO mass fraction predictions by reducing the extent of overprediction observed with the adiabatic model. In contrast, the NADS model results in overprediction over and above the adiabatic predictions thereby showing that, it is imperative to consider variation in scalar dissipation rate in flamelet calculations to capture the effect of radiation on NO. The results also show that employing the modified k-ε model instead of the Reynolds stress transport model for turbulence closure in NADM calculations results in considerable overprediction in centerline NO mass fractions.     

An R (x)-orthonormal theory for the vibration performance of a non-smooth symmetric composite beam with complex interface

A composite beam is symmetric if both the material property and support are symmetric with respect to the middle point. In order to study the free vibration performance of the symmetric composite beams with different complex nonsmooth/discontinuous interfaces, we develop an R(x)-orthonormal theory, where R(x) is an integrable flexural rigidity function. The R(x)-orthonormal bases in the linear space of boundary functions are constructed, of which the second-order derivatives of the boundary functions are asked to be orthonormal with respect to the weight function R(x). When the vibration modes of the symmetric composite beam are expressed in terms of the R(x)-orthonormal bases we can derive an eigenvalue problem endowed with a special structure of the coefficient matrix A :=[aij ],aij= 0 if i + j is odd. Based on the special structure we can prove two new theorems, which indicate that the characteristic equation of A can be decomposed into the product of the characteristic equations of two sub-matrices with dimensions half lower. Hence, we can sequentially solve the natural frequencies in closed-form owing to the specialty of A. We use this powerful new theory to analyze the free vibration performance and the vibration modes of symmetric composite beams with three different interfaces.     

Elastodynamics and Elastostatics by a Unified Method of Potentials for x 3-Convex Domains

A new general solution in terms of two scalar potential functions for classical elastodynamics of x ₃-convex domains is presented. Through the establishment and usage of a set of basic mathematical lemmas, a demonstration of its connection to Kovalevshi–Iacovache–Somigliana elastodynamic solution, and thus its completeness, is realized with the aid of the theory of repeated wave equations and Boggio’s theorem. With the time dependence of the potentials suppressed, the new decomposition can, unlike Lamé’s, degenerate to a complete solution for elastostatic problems.      

The global structural stability and application of the toroidal differntial equations\left\{ {\begin{array}{*{20}c} {\dot x = a_1 [\sin (x - y) - \sin x] - \Delta \omega } \\ {\dot y = a_2 [\sin (y - x) - \sin y] - \Delta \omega } \\ \end{array} } \right.

In this paper, the global structural stability of the toroidal differential equationshas been obtained, and applied to the cross-coupled phase - locked loop, where Δω≥0     

Classification of the structure of positive radial solutions to\Delta u + K(|x|)u^p = 0inR n

Positive radial solutions to the semilinear elliptic equation \(\Delta u + K(|x|)u^p = 0\) inR ⁿ are studied, wherep > 1,n > 2 andK ≧ 0. It is shown that, under a general condition onK(r) andp, the structure of positive radial solutions becomes one of three types. We give sharp criteria to classify the type of the structure, and apply the result to the conformal scalar curvature equation.     

On an analogue of the Euler-Cauchy polygon method for the numerical solution of ux y = f(x,y, u,ux, uy)

This paper develops, with an eye on the numerical applications, an analogue of the classical Euler-Cauchy polygon method (which is used in the solution of the ordinary differential equation dy/dx=f(x, y), y(x ₀)=y ₀) for the solution of the following characteristic boundary value problem for a hyperbolic partial differential equation u _xy =f(x, y, u, u _x , y _y ), u(x, y ₀)=(x), u(x ₀, y)=(y), where (x ₀)=(y ₀). The method presented here, which may be roughly described as a process of bilinear interpolation, has the advantage over previously proposed methods that only the tabulated values of the given functions (x) and (y) are required for its numerical application. Particular attention is devoted to the proof that a certain sequence of approximating functions, constructed in a specified way, actually converges to a solution of the boundary value problem under consideration. Known existence theorems are thus proved by a process which can actually be employed in numerical computation.

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