Buckling Electrothermal NEMS Actuators: Analytic Design for Very Slender Beams

Analytic approximations are presented for the response of buckling-mode electrothermal actuators with very slender beams with a width-to-length ratio of W/L≤0.001 of the type found in nanoelectromechanical systems (NEMS). The results are found as closed-form solutions to the Euler beam bending theory rather than by an iterative numerical solution or a time-consuming finite element analysis. Expressions for transverse deflections and stiffness are presented for actuators with the common raised cosine and chevron pre-buckled shapes. The approximations are valid when the effects of bending dominate over those of axial compression. A few higher-order approximations are also presented for less slender beams with 0.001≤W/L≤0.01.

Optimal Replenishment Strategy for Inventory Mechanism with a Known Price Increase and Backordering in Finite Horizon

For the finite horizon inventory mechanism with a known price increase and backordering, based on minimizing the inventory cost, we establish two mixed integer optimization models. By buyer’s cost analysis, we present the closed-form solutions to the models, and by comparing the minimum cost of the two strategies, we provide an optimal ordering policy to the buyer. Numerical examples are presented to illustrate the validity of the model, and sensitivity analysis on major parameters is also made to show some insights to the inventory model.

On exact solutions of fractional differential-difference equations with Ψ-Riemann–Liouville derivative

The aim of this work is to modify the invariant subspace method (ISM) in order to obtain closed form solutions of fractional differential-difference equations with Ψ-Riemann–Liouville (Ψ-RL) fractional derivative for first time. We have investigated the cases of two-dimensional and the three-dimensional invariant subspaces (ISs) in the suggested scheme. Using the modified ISM, new exact generalized solutions for the general fractional mKdV Lattice equation and the fractional Volterra lattice system are obtained. Compared with similar solution techniques in literature, the presented solution scheme is highly efficient and is capable to find new general exact solutions which cannot be attained by other methods.

A Study of (2+1)-Dimensional Konopelchenko-Dubrovsky (KD) System: Closed-Form Solutions, Solitary Waves, Bifurcation Analysis and Quasi-Periodic Solution

Nonlinear evolution equations (NLEEs) are extensively used to establish the elementary propositions of natural circumstances. In this work, we study the Konopelchenko-Dubrovsky (KD) equation which depicts non-linear waves in mathematical physics with weak dispersion. The considered model is investigated using the combination of generalized exponential rational function (GERF) method and dynamical system method. The GERF method is utilized to generate closedform invariant solutions to the (2+1)-dimensional KD model in terms of trigonometric, hyperbolic, and exponential forms with the assistance of symbolic computations. Moreover, three-dimensional graphics are displayed to depict the behavior of obtained solitary wave solutions. The model is observed to have single and multiple soliton profiles, kink-wave profiles, and periodic oscillating nonlinear waves. These generated solutions have never been published in the literature. All the newly generated soliton solutions are checked by putting them back into the associated system with the soft computation via Wolfram Mathematica. Moreover, the system is converted into a planer dynamical system using a certain transformation and the analysis of bifurcation is examined. Furthermore, the quasi-periodic solution is investigated numerically for the perturbed system by inserting definite periodic forces into the considered model. With regard to the parameter of the perturbed model, two-dimensional and three-dimensional phase portraits are plotted.

Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations

In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential equation into an integer order ordinary differential equation. Using the novel (G’/G2)-expansion method, several exact solutions with extra free parameters are found in the form of hyperbolic, trigonometric, and rational function solutions. When parameters are given appropriate values along with distinct values of fractional order α travelling wave solutions such as singular periodic waves, singular kink wave soliton solutions are formed which are forms of soliton solutions. Also, the solutions obtained by the proposed method depend on the value of the arbitrary parameters H. Previous results are re-derived when parameters are given special values. Furthermore, for numerical presentations in the form of 3D and 2D graphics, the commercial software Mathematica 10 is incorporated. The method is accurately depicted, and it provides extra general exact solutions.

Pricing and hedging bond options and sinking-fund bonds under the CIR model

<abstract><p>This article derives simple closed-form solutions for computing Greeks of zero-coupon and coupon-bearing bond options under the CIR interest rate model, which are shown to be accurate, easy to implement, and computationally highly efficient. These novel analytical solutions allow us to extend the literature in two other directions. First, the static hedging portfolio approach is used for pricing and hedging American-style plain-vanilla zero-coupon bond options under the CIR model. Second, we derive analytically the comparative static properties of sinking-fund bonds under the same interest rate modeling setup.</p></abstract>

Exact finite beam element for open thin walled doubly symmetric members under torsional and warping moments

Starting with total potential energy variational principle, the governing equilibrium coupled equations for the torsional-warping static analysis of open thin-walled beams under various torsional and warping moments are derived. The formulation captures shear deformation effects due to warping. The exact closed form solutions for torsional rotation and warping deformation functions are then developed for the coupled system of two equations. The exact solutions are subsequently used to develop a family of shape functions which exactly satisfy the homogeneous form of the governing coupled equations. A super-convergent finite beam element is then formulated based on the exact shape functions. Key features of the beam element developed include its ability to (a) eliminate spatial discretization arising in commonly used finite elements, and (e) eliminate the need for time discretization. The results based on the present finite element solution are found to be in excellent agreement with those based on exact solution and ABAQUS finite beam element solution at a small fraction of the computational and modelling cost involved.

A Square Equal-Area Map Projection with Low Angular Distortion, Minimal Cusps, and Closed-Form Solutions

A novel square equal-area map projection is proposed. The projection combines closed-form forward and inverse solutions with relatively low angular distortion and minimal cusps, a combination of properties not manifested by any previously published square equal-area projection. Thus, the new projection has lower angular distortion than any previously published square equal-area projection with a closed-form solution. Utilizing a quincuncial arrangement, the new projection places the north pole at the center of the square and divides the south pole between its four corners; the projection can be seamlessly tiled. The existence of closed-form solutions makes the projection suitable for real-time visualization applications, both in cartography and in other areas, such as for the display of panoramic images.

Impact-Angle and Terminal-Maneuvering-Acceleration Constrained Guidance against Maneuvering Target

A new, highly constrained guidance law is proposed against a maneuvering target while satisfying both impact angle and terminal acceleration constraints. Here, the impact angle constraint is addressed by solving an optimal guidance problem in which the target’s maneuvering acceleration is time-varying. To deal with the terminal acceleration constraint, the closed-form solutions of the new guidance are needed. Thus, a novel engagement system based on the guidance considering the target maneuvers is put forward by choosing two angles associated with the relative velocity vector and line of sight (LOS) as the state variables, and then the system is linearized using small angle assumptions, which yields a special linear time-varying (LTV) system that can be solved analytically by the spectral-decomposition-based method. For the general case where the closing speed, which is the speed of approach of the missile and target, is allowed to change with time arbitrarily, the solutions obtained are semi-analytical. In particular, when the closing speed changes linearly with time, the completely closed-form solutions are derived successfully. By analyzing the generalized solutions, the stability domain of the guidance coefficients is obtained, in which the maneuvering acceleration of the missile can converge to zero finally. Here, the key to investigating the stability domain is to find the limits of some complicated integral terms of the generalized solutions by skillfully using the squeeze theorem. The advantages of the proposed guidance are demonstrated by conducting trajectory simulations.