Study of parametric oscillation of an electrostatically actuated microbeam using variational iteration method

This paper deals with the study of parametric oscillation of an electrostatically actuated microbeam using variational iteration method. The paper considers a micro-beam suspended between two conductive micro-plates, subjected to a same actuation voltage. The nonlinear governing differential equation of motion about static equilibrium position using calculus of variation theory and Taylor series expansion has been linearized and implementing a Galerkin based reduced order model a Mathieu type equation has been obtained. By improving variational iteration method combining with method of strained parameters transition curves, separating stable from unstable regions have been obtained. The results of variational iteration method, perturbation and direct numerical integration methods for some cases selected from different regions (stable and unstable regions) have been compared.     

Variational procedures for a class of exterior interface problems

The problem under consideration is that of determining a function which is a solution of the Helmholtz equation in a planar region exterior to a simple closed curve and of an inhomogeneous Helmholtz equation inside the curve. Jump conditions on the function and its normal derivative across the cruve are given. The problem is first transformed into one involving the inner region only with a boundary condition which is non-local. This means that the solution at a point on the boundary is a functional of its values elsewhere. This second problem is further transformed into a variational form with all boundary conditions natural. It is shown that the variational problem has a solution. Finite dimensional approximate problems are defined and they are shown to have solutions converging to the solution of the variational problem.     

On Variational Formulations for Non-Linear, Non-Potential Operators

This paper consists of two parts. In the first part the methodfor the choice of a bilinear form rendering a non-linear operatorpotential has been developed. The method constitutes an extensionof Magri's approach, the last being valid for linear operatorsonly. The second part of the paper is concerned with the method ofadding the adjoint operator. It has been proved that a systemof two coupled operator equations consisting of a given non-linearoperator equation and the equation determined by the adjointof its Frechet derivative is described by a potential operator.The variational functional has been derived. Some particularcases have been discussed.     

Dynamical systems and variational inequalities

The variational inequality problem has been utilized to formulate and study a plethora of competitive equilibrium problems in different disciplines, ranging from oligopolistic market equilibrium problems to traffic network equilibrium problems. In this paper we consider for a given variational inequality a naturally related ordinary differential equation. The ordinary differential equations that arise are nonstandard because of discontinuities that appear in the dynamics. These discontinuities are due to the constraints associated with the feasible region of the variational inequality problem. The goals of the paper are two-fold. The first goal is to demonstrate that although non-standard, many of the important quantitative and qualitative properties of ordinary differential equations that hold under the standard conditions, such as Lipschitz continuity type conditions, apply here as well. This is important from the point of view of modeling, since it suggests (at least under some appropriate conditions) that these ordinary differential equations may serve as dynamical models. The second goal is to prove convergence for a class of numerical schemes designed to approximate solutions to a given variational inequality. This is done by exploiting the equivalence between the stationary points of the associated ordinary differential equation and the solutions of the variational inequality problem. It can be expected that the techniques described in this paper will be useful for more elaborate dynamical models, such as stochastic models, and that the connection between such dynamical models and the solutions to the variational inequalities will provide a deeper understanding of equilibrium problems.     

Applications of He’s principles to partial differential equations

This paper applies the variational iteration method (VIM) and semi-inverse variational principle to obtain solutions of linear and nonlinear partial differential equations. The nonlinear model is considered from gas dynamics, fluid dynamics and Burgers equation. The linear model is the heat transfer (diffusion) equation. Results show that variational iteration method is a powerful mathematical tool for solving linear and nonlinear partial differential equations, and therefore, can be widely applied to engineering problems.     

Power Penalty Method for a Linear Complementarity Problem Arising from American Option Valuation

In this paper, we present a power penalty function approach to the linear complementarity problem arising from pricing American options. The problem is first reformulated as a variational inequality problem; the resulting variational inequality problem is then transformed into a nonlinear parabolic partial differential equation (PDE) by adding a power penalty term. It is shown that the solution to the penalized equation converges to that of the variational inequality problem with an arbitrary order. This arbitrary-order convergence rate allows us to achieve the required accuracy of the solution with a small penalty parameter. A numerical scheme for solving the penalized nonlinear PDE is also proposed. Numerical results are given to illustrate the theoretical findings and to show the effectiveness and usefulness of the method. This work was partially supported by a research grant from the University of Western Australia and the Research Grant Council of Hong Kong, Grants PolyU BQ475 and PolyU BQ493.     

Conservation laws of a nonlinear (1+1) wave equation

In this paper, further study of the conservation laws of the nonlinear (1+1) wave equation involving two arbitrary functions of the dependent variable is performed. This equation is not derivable from a variational principle. By writing the equation, admitting a partial Lagrangian, in the partial Euler–Lagrange   form, partial Noether operators associated with the partial Lagrangian are obtained for all possible cases of the functions. These partial Noether operators do not form a Lie algebra in general. Partial Noether operators aid via a formula in the construction of the conservation laws of the equation. We obtain new conservation laws for the equation which have not been presented in the earlier literature.     

Variation and series approach to the Thomas-Fermi equation

The Thomas-Fermi equation describing the screening of the Coulomb potential inside heavy neutral atoms is reconsidered. An accurate representation for its numerical solution was obtained by means of the variational principle. The proposed new solution has more precise asymptotic behaviour at large distances from the origin and allows us to obtain the exact value of the initial slope. The obtained new variational solution can also be developed in power series similar to the Baker’s ones but more precise even than some series solutions that have been recently obtained within the homotopy analysis method and a modified variational method.     

Numerical simulation of the modified regularized long wave equation by He's variational iteration method

The modified regularized long wave (MRLW) equation, with some initial conditions, is solved numerically by variational iteration method. This method is useful for obtaining numerical solutions with high degree of accuracy. The variational iteration solution for the MRLW equation converges to its exact solution. Moreover, the conservation laws properties of the MRLW equation are also studied. Finally, interaction of two and three solitary waves is shown. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

On singularity of a nonlinear variational sine-Gordon equation

We study a sine-Gordon-type of nonlinear variational wave equation whose wave speed is a sinusoidal function of the wave amplitude. This equation arises naturally from long waves on a dipole chain in the continuum limit, which provides a crude model for some polymers. Using characteristic methods, we describe a blow-up result for the one-dimensional nonlinear variational sine-Gordon equation, which shows that smooth solutions breakdown in finite time.     

A variational approach to transonic potential flow problems

This paper continues considerations of transonic potential flow problems by variational methods. A functional which is associated with a boundary value problem for the (full) potential equation and which possesses a real physical meaning is minimized over a class of admissible functions. These functions have to satisfy a non‐linear local entropy condition and a certain boundness constraint. Though this class is not a compact set of the underlying Hilbert space and though the functional need not be convex, the existence of a solution to the established variational problem can be proved by direct methods of the calculus of variations. Furthermore, some properties of minimizers concerning uniqueness, relation to the potential equation, and behaviour on supersonic regions are derived. Copyright © 2000 John Wiley & Sons, Ltd.     

弹性力学中的立兹法和屈列弗兹法的一般推导

本文采用一般的数学表示形式推导了线弹性力学中的立兹法和屈列弗兹法,证明了立兹法给出相应泛函极值的上限,屈列弗兹法则给出其下限.同时发现,特征值问题(例如自振频率问题)泛函变分法中的屈列弗兹法同求特征值的放松边界条件下限法是一致的.当然,此处的推导结果,也适用于一类泛函的变分法中,这类泛函的欧拉方程是线性正定的.     

A penalty approximation method for a semilinear parabolic double obstacle problem

In this work, we present a novel power penalty method for the approximation of a global solution to a double obstacle complementarity problem involving a semilinear parabolic differential operator and a bounded feasible solution set. We first rewrite the double obstacle complementarity problem as a double obstacle variational inequality problem. Then, we construct a semilinear parabolic partial differential equation (penalized equation) for approximating the variational inequality problem. We prove that the solution to the penalized equation converges to that of the variational inequality problem and obtain a convergence rate that is corresponding to the power used in the formulation of the penalized equation. Numerical results are presented to demonstrate the theoretical findings.     

Some remarks on the dynamics of cell membranes

Motivated by the desire to model the entry of 1,25D into a cell by receptor mediated endocytosis, we have formulated the problem as the dynamics of a bilayer membrane. We have discussed setting the problem as a variational problem using the Helfrich modeling of the bilayer in terms of the free energy. Using a Lagrangian formulation we arrive at the Euler–Lagrange equations for the system. The model we have used depends on the amount of reagent in the neighborhood of the upper membrane. The problem thereby reduces to a moving boundary problem, which is dependent on a diffusion equation for a region changing with time. In order to solve this problem we seek the correct Neumann function for this altered. This is accomplished by deriving a Hadamard variational formula for the diffusion equation. We also offer an iterative procedure for solving this non-linear problem.     

Analysis of the time fractional nonlinear diffusion equation from diffusion process

Under investigation in this paper is a time fractional nonlinear diffusion equation which can be utilized to express various diffusion processes. The symmetry of this considered equation has been obtained via fractional Lie group approach with the sense of Riemann-Liouville (R-L) fractional derivative. Based on the symmetry, this equation can be changed into an ordinary differential equation of fractional order. Moreover, some new invariant solutions of this considered equation are found. Lastly, utilising the Noether theorem and the general form of Noether type theorem, the conservation laws are yielded to the time fractional nonlinear diffusion equation, respectively. Our discovery that there are no conservation laws under the general form of Noether type theorem case. This result tells us the symmetry of this equation is not variational symmetry of the considered functional. These rich results can give us more information to interpret this equation.     

Extended general variational inequalities

In this work, we introduce and consider a new class of general variational inequalities involving three nonlinear operators, which is called the extended general variational inequalities. Noor [M. Aslam Noor, Projection iterative methods for extended general variational inequalities, J. Appl. Math. Comput. (2008) (in press)] has shown that the minimum of nonconvex functions can be characterized via these variational inequalities. Using a projection technique, we establish the equivalence between the extended general variational inequalities and the general nonlinear projection equation. This equivalent formulation is used to discuss the existence of a solution of the extended general variational inequalities. Several special cases are also discussed.     

A variational method applied to a linear hyperbolic initial boundary value problem

This paper deals with the application of a variational method to a boundary value problem of the wave equation. Starting with an initial boundary value problem (which is given) introduction of a boundary condition at the final time leads to a boundary value problem with one of the initial conditions redundant. This redundant initial condition is used by the trial function of the direct method (of the Ritz type) which is employed to stationarize the variational principle.     

A uniqueness theorem for a free boundary problem

In this paper, we prove a uniqueness theorem for a free boundary problem which is given in the form of a variational inequality. This free boundary problem arises as the limit of an equation that serves as a basic model in population biology. Apart from the interest in the problem itself, the techniques used in this paper, which are based on the regularity theory of variational inequalities and of harmonic functions, are of independent interest, and may have other applications.

     

磁电弹性体修正后的H-R混合变分原理和状态向量方程

以三维弹性体的Hellinger-Reissner(H-R)混合变分原理为基础,建立了三维磁电弹性体修正后的H-R混合变分原理,通过变分运算得到了磁电弹性板的状态向量方程,并应用该原理导出了平面内离散元素的状态向量方程,为半解析法在磁电弹性板问题上的应用奠定了理论基础．最后指出:纯弹性体、单一压电体或单一压磁体修正后的H-R混合变分原理都是目前原理的特例．     

非周期Dirac 方程的稳态解

本文研究Dirac方程-iΣαkku＋aβu＋M（x）u=g（x,｜u｜）u的解,其中M（x）是位势函数,g（x,｜u｜）u在无穷远处关于u是超线性的.本文用变分法来研究这一问题.借助于与此方程的＂极限方程＂相关的某个辅助系统,构造了变分泛函ΦM的环绕水平,使得建立在ΦM环绕结构上的极小极大值CM满足0〈CM〈C,这里C是＂极限方程＂的最小能量.从而可以证明（C）c条件对所有c〈C成立,因此得到了方程的最小能量解.