Sixth-order Differential Equation Research Articles

Pull-in instability analysis of electrostatically actuated microplate with rectangular shape

As the size of the micro-electro-mechanical systems (MEMS) continues to decrease, the classical elasticity continuum theory may be inefficient to describe their mechanical behaviors. By introducing the strain gradient elasticity into the classical Kirchhoff plate theory, the size-dependent model for electrostatically actuated microplate-based MEMS is developed. The sixth-order partial differential equation (PDE), derived with the help of the principle of minimum potential energy, can be numerically solved by utilizing generalized differential quadrature (GDQ) method and pseudo arc-length algorithm. The model, with three material length scale parameters (MLSPs) included, can predict prominent size-dependent normalized pull-in voltage with the reduction of characteristic structural size, especially when the plate dimension is comparable to the MLSP (on the order of microns). This study may be helpful to characterize the mechanical properties of electrostatically actuated MEMS, or guide the design of microplate-based devices for a wide range of potential applications.

Stability and Robust Regulation of Battery-Driven Boost Converter With Simple Feedback

This paper investigates the problem of regulating the output voltage of a battery-driven boost converter, where the load is uncertain or changes within a certain range. The battery is modeled with a second order circuit with two capacitors. A state-space approach is developed for estimating the parameters of the battery. By using the state-space averaging method, the open-loop system for regulating the output voltage is described as a sixth order differential equation with a bilinear term and input constraints. A simple saturated state feedback is designed by solving some optimization problem with linear matrix inequality constraints. The optimized controller is very close to an integrator feedback. Using the newly developed Lyapunov method, we analyze the stability and regulation of the closed-loop bilinear system including the battery dynamics. Computation shows that both the optimized state feedback and the integrator feedback can achieve practically global regulation in the presence of uncertain load and uncertain battery voltage. The results are validated by experimental systems. The effect of discontinuous conduction mode on transient response is discussed via simulation and experiment.

Numerical Assessment of the Boundary Layer Effect Predicted by the Shear Flexible Beam Theory with the Sixth-Order Differential Equations

The analytical solutions of shear flexible beams with displacement boundary conditions are derived by using the new sixth-order differential equation beam theory presented by Shi and Voyiadjis (ASME J. Appl. Mech., Vol. 78, 021019, 2011), in which the boundary layer effects are included. The accuracy of the boundary layer effects predicted by the new sixth-order beam theory is evaluated by the finite element analysis in this study. The numerical results show that the new sixth-order beam theory is capable of taking account of the displacement boundary conditions of shear deformable beams and predicting good results of the boundary layer effects induced by the displacement boundaries and the continuity constraints.

STATIC DEFLECTION ANALYSIS OF FLEXURAL SIMPLY SUPPORTED SECTORIAL MICRO-PLATE USING P-VERSION FINITE-ELEMENT METHOD

In this paper, flexural Kirchhoff plate theory is utilized for static analysis of isotropic sectorial micro-plates based on a modified couple stress theory containing one material length scale parameter. The Levy method is implemented and the resulting sixth-order differential equation is solved for the unknown deflection using the p-version finite-element method. The Galerkin form of this differential equation is first reduced to its weak form and then solved using hierarchical p-version finite elements with second-order global smoothness. The computed deflection distribution of the micro-plate is compared with that of the classical theory, in which micro-effects are not present. A series of studies have revealed that when the length scale parameters are considered, deflection of a sectorial plate decreases as the length scale effect is increased; in other words, the micro-plate exhibits more rigidity.

A Sixth-Order Theory of Shear Deformable Beams With Variational Consistent Boundary Conditions

This paper presents the derivation of a new beam theory with the sixth-order differential equilibrium equations for the analysis of shear deformable beams. A sixth-order beam theory is desirable since the displacement constraints of some typical shear flexible beams clearly indicate that the boundary conditions corresponding to these constraints can be properly satisfied only by the boundary conditions associated with the sixth-order differential equilibrium equations as opposed to the fourth-order equilibrium equations in Timoshenko beam theory. The present beam theory is composed of three parts: the simple third-order kinematics of displacements reduced from the higher-order displacement field derived previously by the authors, a system of sixth-order differential equilibrium equations in terms of two generalized displacements w and ϕx of beam cross sections, and three boundary conditions at each end of shear deformable beams. A technique for the analytical solution of the new beam theory is also presented. To demonstrate the advantages and accuracy of the new sixth-order beam theory for the analysis of shear flexible beams, the proposed beam theory is applied to solve analytically three classical beam bending problems to which the fourth-order beam theory of Timoshenko has created some questions on the boundary conditions. The present solutions of these examples agree well with the elasticity solutions, and in particular they also show that the present sixth-order beam theory is capable of characterizing some boundary layer behavior near the beam ends or loading points.

Nonrigid spherical real analytic hypersurfaces in ℂ2

A Levi nondegenerate real analytic hypersurface M of ℂ2 represented in local coordinates (z, w) ∈ ℂ2 by a complex defining equation of the form , which satisfies an appropriate reality condition, is spherical if and only if its complex graphing function Θ satisfies an explicitly written sixth-order polynomial complex partial differential equation. In the rigid case (known before), this system simplifies considerably, but in the general nonrigid case, its combinatorial complexity shows well why the two fundamental curvature tensors constructed by Cartan [Sur la géométrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes, II, Ann. Scuola Norm. Sup. Pisa 1 (1932), pp. 333–354] in his classification of hypersurfaces have, since then, never been reached in parametric representation.

RESULTS FOR SIXTH ORDER POSITIVELY HOMOGENEOUS EQUATIONS

We consider positively homogeneous the sixth order differential equations of the type x (6) = h(t, x), where hpossesses the property that h(t, cx) = ch(t, x) for c ≥ 0. This class includes the linear equations x (6) = p(t)x and piece‐wise linear ones x (6) = k 2 x+ - k 1 x− . We consider conjugate points and angles associated with extremal solutions and prove some comparison results.

Advection/diffusion of large scale magnetic field in accretion disks

Abstract. Activity of the nuclei of galaxies and stellar mass systems involving disk accretion to black holes is thought to be due to (1) a small-scale turbulent magnetic field in the disk (due to the magneto-rotational instability or MRI) which gives a large viscosity enhancing accretion, and (2) a large-scale magnetic field which gives rise to matter outflows and/or electromagnetic jets from the disk which also enhances accretion. An important problem with this picture is that the enhanced viscosity is accompanied by an enhanced magnetic diffusivity which acts to prevent the build up of a significant large-scale field. Recent work has pointed out that the disk's surface layers are non-turbulent and thus highly conducting (or non-diffusive) because the MRI is suppressed high in the disk where the magnetic and radiation pressures are larger than the thermal pressure. Here, we calculate the vertical (z) profiles of the stationary accretion flows (with radial and azimuthal components), and the profiles of the large-scale, magnetic field taking into account the turbulent viscosity and diffusivity due to the MRI and the fact that the turbulence vanishes at the surface of the disk. We derive a sixth-order differential equation for the radial flow velocity vr(z) which depends mainly on the midplane thermal to magnetic pressure ratio β>1 and the Prandtl number of the turbulence P=viscosity/diffusivity. Boundary conditions at the disk surface take into account a possible magnetic wind or jet and allow for a surface current in the highly conducting surface layer. The stationary solutions we find indicate that a weak (β>1) large-scale field does not diffuse away as suggested by earlier work.

Grey 모형을 이용한 다목적댐의 유입 홍수량과 하류 하천 홍수량 실시간 예측

본 연구에서는 다목적댐의 효율적인 홍수관리와 조기 홍수 경보시스템의 정확성을 향상시키기 위하여 두 가지 모형이 제안되었다. 두 모형은 상류 유입 홍수량과 하류 하천의 홍수량을 실시간으로 예측할 수 있는 능력을 각각 가지고 있다. 이들 모형은 남강댐 상류와 하류 홍수량의 실측치와 모의치를 비교하여 보정 및 검정되었으며, 실제 상황에서 모형의 홍수량 예측 능력이 평가되었다. 상류 유입량 예측 모형은 Grey 시스템 이론에 근거하였으며, 모형의 예측능력을 고려하여 6차 모형을 선정하였다. 서로 다른 자료 세트를 사용하여 보정된 모형들을 사용하여 예측한 홍수량과 실측자료를 비교하여 가장 적정한 모형이 선정되었으며, 검정 결과를 검토한 결과 선정된 모형이 양호한 예측결과를 제시하는 것으로 나타났다. 댐 하류 하천 홍수량 예측 모형은 Grey 모형과 수정 Muskingum 홍수 추적 모형을 병합하여 구성되었으며, 보정 및 검정을 통해서 모형의 예측 능력이 평가되었다. 제안된 모형들을 실시간 홍수량 예측에 적용한 결과, 비교적 양호한 예측결과를 나타냈다. 또한, 모형의 정확도를 향상시키기 위해서는 유출 단계를 고려한 모형의 보정 및 적용이 필요하다는 것이 밝혀졌다. To efficiently carry out the flood management of a multipurpose dam, two flood forecasting models are developed, each of which has the capabilities of forecasting upstream inflows and flood discharges downstream of a dam, respectively. The models are calibrated, validated, and evaluated by comparison of the observed and the runoff forecasts upstream and downstream of Namgang Dam. The upstream inflow forecasting model is based on the Grey system theory and employs the sixth order differential equation. By comparing the inflows forecasted by the models calibrated using different data sets with the observed in validation, the most appropriate model is determined. To forecast flood discharges downstream of a dam, a Grey model is integrated with a modified Muskingum flow routing model. A comparison of the observed and the forecasted values in validation reveals that the model can provide good forecasts for the dam's flood management. The applications of the two models to forecasting floods in real situations show that they provide reasonable results. In addition, it is revealed that to enhance the prediction accuracy, the models are necessary to be calibrated and applied considering runoff stages; the rising, peak, and falling stages.

Spectral methods in linear stability. Applications to thermal convection with variable gravity field

The onset of convection in a horizontal layer of fluid heated from below in the presence of a gravity field varying across the layer is numerically investigated. The eigenvalue problem governing the linear stability of the mechanical equilibria of the fluid, in the case of free boundaries, is a sixth order differential equation with Dirichlet and hinged boundary conditions. It is transformed into a system of second order differential equations supplied only with Dirichlet boundary conditions. Then it is solved using two distinct classes of spectral methods namely, weighted residuals (Galerkin type) methods and a collocation (pseudospectral) method, both based on Chebyshev polynomials. The methods provide a fairly accurate approximation of the lower part of the spectrum without any scale resolution restriction. The Viola's eigenvalue problem is considered as a benchmark one. A conjecture is stated for the first eigenvalue of this problem.

Tensionless contact of a finite beam resting on Reissner foundation

A more generalized model of a beam resting on a tensionless Reissner foundation is presented. Compared with the Winkler foundation model, the Reissner foundation model is a much improved one. In the Winkler foundation model, there is no shear stress inside the foundation layer and the foundation is assumed to consist of closely spaced, independent springs. The presence of shear stress inside Reissner foundation makes the springs no longer independent and the foundation to deform as a whole. Mathematically, the governing equation of a beam on Reissner foundation is sixth order differential equation compared with fourth order of Winkler one. Because of this order change of the governing equation, new boundary conditions are needed and related discussion is presented. The presence of the shear stress inside the tensionless Reissner foundation together with the unknown feature of contact area/length makes the problem much more difficult than that of Winkler foundation. In the model presented here, the effects of beam dimension, gap distance, loading asymmetry and foundation shear stress on the contact length are all incorporated and studied. As the beam length increases, the results of a finite beam with zero gap distance converge asymptotically to those obtained by the previous model for an infinitely long beam.

On a class of 4D Kähler bases and AdS5supersymmetric black holes

We construct a class of toric Kahler manifolds, M_4, of real dimension four, a subset of which corresponds to the Kahler bases of all known 5D asymptotically AdS_5 supersymmetric black-holes. In a certain limit, these Kahler spaces take the form of cones over Sasaki spaces, which, in turn, are fibrations over toric manifolds of real dimension two. The metric on M_4 is completely determined by a single function H(x), which is the conformal factor of the two dimensional space. We study the solutions of minimal five dimensional gauged supergravity having this class of Kahler spaces as base and show that in order to generate a five dimensional solution H(x) must obey a simple sixth order differential equation. We discuss the solutions in detail, which include all known asymptotically AdS_5 black holes as well as other spacetimes with non-compact horizons. Moreover we find an infinite number of supersymmetric deformations of these spacetimes with less spatial isometries than the base space. These deformations vanish at the horizon, but become relevant asymptotically.

Exact dynamic stiffness matrix of a three-dimensional shear beam with doubly asymmetric cross-section

The dynamic member stiffness matrix of a three-dimensional shear beam with doubly asymmetric cross-section is derived exactly from the governing, sixth-order differential equation of motion. Such a formulation accounts for the uniform distribution of mass in the member and necessitates the solution of a transcendental eigenvalue problem. This is achieved using the Wittrick–Williams algorithm, where the necessary parameters are developed using a generalised procedure. An example is given to clarify the theory, together with a small parametric study that indicates when lateral–torsional coupling may safely be ignored. The work also holds considerable potential in its application to the approximate analysis of asymmetric, multi-storey, three-dimensional frame structures.

Sinc-Galerkin method for solving nonlinear boundary-value problems

The sinc-Galerkin method is used to approximate solutions of nonlinear problems involving nonlinear second-, fourth-, and sixth-order differential equations with homogeneous and nonhomogeneous boundary conditions. The scheme is tested on four nonlinear problems. The results demonstrate the reliability and efficiency of the algorithm developed.

Existence and nonexistence of nontrivial solutions of semilinear sixth-order ordinary differential equations

The existence of nontrivial periodic solutions of semilinear sixth-order differential equations arising in phase transition models is discussed via variational methods.

An instability result for certain system of sixth order differential equations

The main purpose of this paper is to give sufficient conditions for the instability of the trivial solution X=0 of Eq. (1.1).

A note on the numerical solution of high-order differential equations

Numerical solution of high-order differential equations with multi-boundary conditions is discussed in this paper. Motivated by the discrete singular convolution algorithm, the use of fictitious points as additional unknowns is proposed in the implementation of locally supported Lagrange polynomials. The proposed method can be regarded as a local adaptive differential quadrature method. Two examples, an eigenvalue problem and a boundary-value problem, which are governed by a sixth-order differential equation and an eighth-order differential equation, respectively, are employed to illustrate the proposed method.

Free boson formulation of boundary states inW3minimal models and the critical Potts model

We develop a Coulomb gas formalism for boundary conformal field theory having a $W$ symmetry and illustrate its operation using the three state Potts model. We find that there are free-field representations for six $W$ conserving boundary states, which yield the fixed and mixed physical boundary conditions, and two $W$ violating boundary states which yield the free and new boundary conditions. Other $W$ violating boundary states can be constructed but they decouple from the rest of the theory. Thus we have a complete free-field realization of the known boundary states of the three state Potts model. We then use the formalism to calculate boundary correlation functions in various cases. We find that the conformal blocks arising when the two point function of $\phi_{2,3}$ is calculated in the presence of free and new boundary conditions are indeed the last two solutions of the sixth order differential equation generated by the singular vector.

The quantum cosmological wavefunction at very early times for a quadratic gravity theory

The quantum cosmological wavefunction for a quadratic gravity theory derived from the heterotic string effective action is obtained near the inflationary epoch and during the initial Planck era. Neglecting derivatives with respect to the scalar field, the wavefunction would satisfy a third-order differential equation near the inflationary epoch which has a solution that is singular in the scale factor limit a(t) → 0. When scalar field derivatives are included, a sixth-order differential equation is obtained for the wavefunction and the solution by Mellin transform is regular in the a → 0 limit. It follows that inclusion of the scalar field in the quadratic gravity action is necessary for consistency of the quantum cosmology of the theory at very early times.

Soliton solutions of two bidirectional sixth-order partial differential equations belonging to the KP hierarchy

In this letter, we analyse two bidirectional sixth-order partial differential equations, which are reductions in (1 + 1) dimensions of equations belonging to the KP hierarchy. They have fourth-order and fifth-order Lax pairs, respectively. We derive their Backlund transformations and, from the nonlinear superposition formula, we can build their soliton solutions like a Grammian. The interesting dynamics of these solitons is that they may describe not only the overtaking collision but also the head-on collision of solitary waves of different type and shape.