Applying Differential Transform Method to Nonlinear Partial Differential Equations: A Modified Approach

This paper proposes another use of the Differential transform method (DTM) in obtaining approximate solutions to nonlinear partial differential equations (PDEs). The idea here is that a PDE can be converted to an ordinary differential equation (ODE) upon using a wave variable, then applying the DTM to the resulting ODE. Three equations, namely, Benjamin-Bona-Mahony (BBM), Cahn-Hilliard equation and Gardner equation are considered in this study. The proposed method reduces the size of the numerical computations and use less rules than the usual DTM method used for multi-dimensional PDEs. The results show that this new approach gives very accurate solutions

Hybrid deterministic stochastic systems with microscopic look-ahead dynamics

We study the impact of stochastic mechanisms on a coupled hybrid system consisting of a general advection-diffusion-reaction partial differential equation and a spatially distributed stochastic lattice noise model. The stochastic dynamics include both spin-flip and spin-exchange type interparticle interactions. Furthermore, we consider a new, asymmetric, single exclusion pro- cess, studied elsewhere in the context of traffic flow modeling, with an one-sided interaction potential which imposes advective trends on the stochastic dynamics. This look-ahead stochastic mechanism is responsible for rich nonlinear behavior in solutions. Our approach relies heavily on first deriving approximate differential mesoscopic equations. These approximations become exact either in the long range, Kac interaction partial differential equation case, or, given sufficient time separation con- ditions, between the partial differential equation and the stochastic model giving rise to a stochastic averaging partial differential equation. Although these approximations can in some cases be crude, they can still give a first indication, via linearized stability analysis, of the interesting regimes for the stochastic model. Motivated by this linearized stability analysis we choose particular regimes where interacting nonlinear stochastic waves are responsible for phenomena such as random switching, convective instability, and metastability, all driven by stochasticity. Numerical kinetic Monte Carlo simulations of the coarse grained hybrid system are implemented to assist in producing solutions and understanding their behavior

A novel rational harmonic balance approach for periodic solutions of conservative nonlinear oscillators

An analytical approximate procedure for a class of conservative single degree-of-freedom nonlinear oscillators with odd non-linearity is proposed. This technique is based on the generalized harmonic balance method in which analytical approximate solutions have rational forms. Unlike the classical harmonic balance techniques, in this new procedure the approximate solution and the restoring force are expanded in Fourier series prior to substituting them in the nonlinear differential equation. This approach gives us not only a truly periodic solution but also the frequency of the motion as a function of the amplitude of oscillation. Four nonlinear oscillators are presented to illustrate the usefulness and effectiveness of the proposed technique. The most significant features of this method are its simplicity and its excellent accuracy for the whole range of oscillation amplitude values and the results reveal that this technique is very effective and convenient for solving a class of conservative nonlinear oscillatory systems

NONLINEAR INSTABILITIES IN MEM/NEM ELECTROSTATIC SWITCHES

The aim of this thesis is to develop suitable mathematical models for the purpose of investigating nonlinear instabilities in Micro-Electro-Mechanical (MEM) and Nano- Electro-Mechanical (NEM) electrostatic switches. The proposed models capture the influence of electric field fringing, intermolecular forces, surface stress and surface elasticity. Based on Euler-Bernoulli assumptions, a surface elasticity model and the generalized Young-Laplace equation, effects of surface stress and surface elasticity are incorporated in the models, while the intermolecular force effects are modelled using quantum mechanics. The derived governing equation representing static pull-in behaviour of switches is inherently nonlinear due to the driving electrostatic force and intermolecular forces which become dominant at nanoscale. Since no exact solutions are available for the resulting nonlinear differential equation, an approach based on homotopy perturbation method (HPM) is proposed to construct approximate analytical solutions, as well as to characterize the instability behaviour. Numerical solutions obtained via finite difference method (FDM) are employed for validating the analytical results. HPM in conjunction with Adomian decomposition method (ADM) has been employed for approximate analytical predictions. To this end, the solutions for the fourth-order two- point boundary value problem (BVP) representing MEM/NEM electrostatic switches are constructed in terms of a convergent series. The pull-in parameters, including pull-in voltage, detachment length and low-voltage actuation windows, are investigated in detail using the above methods and also via a lumped parameter model. HPM analytical solutions are found to be more accurate and reliable compared to those predicted via the lumped parameter model. HPM solutions also tend to overestimate the static deflection, and underestimate pull-in voltage and detachment length compared to the FDM numerical solutions. However, its relative differences to the FDM numerical solutions are within an acceptable range for design purposes. HPM is concluded to work well for the static pull-in in parameter determination, and is preferred since it is straightforward to implement and could save computation efforts while not losing accuracy. Predictions via HPM and FDM also revealed that the influence of surface effects on the pull-in instability of MEM/NEM switches is significant and the exclusion of surface effects in the analysis may result in an erroneous estimation of the pull-in parameters. Further, the concept of Casimir actuated switches is proposed for the purpose of ensuring the physical realization of a new class of the switchable devices using pure Casmir force actuation. To this end, a new idea of Casimir-force actuation window has been introduced for the purpose of ensuring designs that yield functional Casimir actuated switches. The present study is envisaged to be beneficial for the design and applications of MEM/NEM electrostatic as well as Casimir actuated switches. The methodology presented in this thesis may be also used for the analysis of actuation systems, which may involve other types of nonlinear actuation forces