Search results for: coupled differential equation

Authors: Hassen M. Ouakad

Abstract:

In this paper, the influence of van der Waals, as well as electrostatic forces on the structural behavior of MEMS and NEMS actuators, has been investigated using of a Euler-Bernoulli beam continuous model. In the proposed nonlinear model, the electrostatic fringing-fields and the mid-plane stretching (geometric nonlinearity) effects have been considered. The nonlinear integro-differential equation governing the static structural behavior of the actuator has been derived. An original Galerkin-based reduced-order model has been developed to avoid problems arising from the nonlinearities in the differential equation. The obtained reduced-order model equations have been solved numerically using the Newton-Raphson method. The basic design parameters such as the pull-in parameters (voltage and deflection at pull-in), as well as the detachment length due to the van der Waals force of some investigated micro- and nano-actuators have been calculated. The obtained numerical results have been compared with some other existing methods (finite-elements method and finite-difference method) and the comparison showed good agreement among all assumed numerical techniques.

Keywords: MEMS, NEMS, fringing-fields, mid-plane stretching, Galerkin

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Authors: Seyed Amin Vakili, Sahar Sadat Vakili, Seyed Ehsan Vakili, Nader Abdoli Yazdi

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Differential equations are of fundamental importance in engineering and applied mathematics, since many physical laws and relations appear mathematically in the form of such equations. The equilibrium state of structures consisting of one-dimensional elements can be described by an ordinary differential equation. The response of these kinds of structures under the loading, namely relationship between the displacement field and loading field, can be predicted by the solution of these differential equations and on satisfying the given boundary conditions. When the effect of change of geometry under loading is taken into account in modeling of equilibrium state, then these differential equations are partially integrable in quartered. They also exhibit instability characteristics when the structures are loaded compressively. The purpose of this paper is to represent the ability of the Modified Newmark Method in analyzing flexural-torsional instability of struts for both bifurcation and non-bifurcation structural systems. The results are shown to be very accurate with only a small number of iterations. The method is easily programmed, and has the advantages of simplicity and speeds of convergence and easily is extended to treat material and geometric nonlinearity including no prismatic members and linear and nonlinear spring restraints that would be encountered in frames. In this paper, these abilities of the method will be extended to the system of linear differential equations that govern strut flexural torsional stability.

Keywords: instability, torsion, flexural, buckling, modified newmark method stability

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Authors: Deva Kanta Phukan

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An attempt is made where an electrically conducting fluid is injected from a fixed outer cylindrical casing onto an inner moving cylindrical rod. A magnetic field is applied parallel to the axis of the cylindrical rod. The basic governing set of partial differential equations for conservation of mass and momentum are reduced to a set of non-linear ordinary differential equation by introducing similarity transformation, which are integrated numerically. A perturbation solution for the case of large magnetic parameter is derived for constant Reynolds number.

Keywords: annular axi-symmetric stagnation flow, conducting fluid, magnetic field, moving cylinder

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Authors: Shangerganesh Lingeshwaran

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In this presentation, we have performed numerical simulations for a reaction-diffusion equation with various nonlinear density-dependent diffusion operators and proliferation functions. The mathematical model represented by parabolic partial differential equation is considered to study the invasion of gliomas (the most common type of brain tumors) and to describe the growth of cancer cells and response to their treatment. The unknown quantity of the given reaction-diffusion equation is the density of cancer cells and the mathematical model based on the proliferation and migration of glioma cells. A standard Galerkin finite element method is used to perform the numerical simulations of the given model. Finally, important observations on the each of nonlinear diffusion functions and proliferation functions are presented with the help of computational results.

Keywords: glioma invasion, nonlinear diffusion, reaction-diffusion, finite eleament method

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Authors: Maryam Khazaei Pool, Lori Lewis

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This is a summary of articles based on higher order B-splines methods and the variation of B-spline methods such as Quadratic B-spline Finite Elements Method, Exponential Cubic B-Spline Method, Septic B-spline Technique, Quintic B-spline Galerkin Method, and B-spline Galerkin Method based on the Quadratic B-spline Galerkin method (QBGM) and Cubic B-spline Galerkin method (CBGM). In this paper, we study the B-spline methods and variations of B-spline techniques to find a numerical solution to the Burgers’ equation. A set of fundamental definitions, including Burgers equation, spline functions, and B-spline functions, are provided. For each method, the main technique is discussed as well as the discretization and stability analysis. A summary of the numerical results is provided, and the efficiency of each method presented is discussed. A general conclusion is provided where we look at a comparison between the computational results of all the presented schemes. We describe the effectiveness and advantages of these methods.

Keywords: Burgers’ equation, Septic B-spline, modified cubic B-spline differential quadrature method, exponential cubic B-spline technique, B-spline Galerkin method, quintic B-spline Galerkin method

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Authors: Shashank Patidar, Sumit Kumar, Atul Srivastava, Suneet Singh

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Present work is concerned with the numerical investigation of thermal response of biological tissues during laser-based photo-thermal therapy for destroying cancerous/abnormal cells with minimal damage to the surrounding normal cells. Light propagation through the biological sample is mathematically modelled by transient radiative transfer equation. In the present work, application of the Lattice Boltzmann Method is extended to analyze transport of short-pulse radiation in a participating medium.In order to determine the two-dimensional temperature distribution inside the tissue medium, the RTE has been coupled with Penne’s bio-heat transfer equation based on Fourier’s law by several researchers in last few years.

Keywords: lattice Boltzmann method, transient radiation transfer equation, dual phase lag model

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Authors: Lev Idels, Arcady Ponosov

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Stochastic fractional differential equations have recently attracted considerable attention, as they have been used to model real-world processes, which are subject to natural memory effects and measurement uncertainties. Compared to conventional hereditary differential equations, one of the advantages of fractional differential equations is related to more realistic geometric properties of their trajectories that do not intersect in the phase space. In this report, a Peano-like existence theorem for nonlinear stochastic fractional differential equations is proven under very general hypotheses. Several specific classes of equations are checked to satisfy these hypotheses, including delay equations driven by the fractional Brownian motion, stochastic fractional neutral equations and many others.

Keywords: delay equations, operator methods, stochastic noise, weak solutions

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Authors: S. O. Oyamakin, A. U. Chukwu

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Richard's growth equation being a generalized logistic growth equation was improved upon by introducing an allometric parameter using the hyperbolic sine function. The integral solution to this was called hyperbolic Richard's growth model having transformed the solution from deterministic to a stochastic growth model. Its ability in model prediction was compared with the classical Richard's growth model an approach which mimicked the natural variability of heights/diameter increment with respect to age and therefore provides a more realistic height/diameter predictions using the coefficient of determination (R2), Mean Absolute Error (MAE) and Mean Square Error (MSE) results. The Kolmogorov-Smirnov test and Shapiro-Wilk test was also used to test the behavior of the error term for possible violations. The mean function of top height/Dbh over age using the two models under study predicted closely the observed values of top height/Dbh in the hyperbolic Richard's nonlinear growth models better than the classical Richard's growth model.

Keywords: height, Dbh, forest, Pinus caribaea, hyperbolic, Richard's, stochastic

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Authors: Nia Mair Fry, Matthew Profit, Chenfeng Li

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This work focuses on the development and implementation of a fully implicit-implicit, coupled mechanical deformation and porous flow, finite element software tool. The fully implicit software accurately predicts classical fundamental analytical solutions such as the Terzaghi consolidation problem. Furthermore, it can capture other analytical solutions less well known in the literature, such as Gibson’s sedimentation rate problem and Coussy’s problems investigating wellbore stability for poroelastic rocks. The mechanical volume strains are transferred to the porous flow governing equation in an implicit framework. This will overcome some of the many current industrial issues, which use explicit solvers for the mechanical governing equations and only implicit solvers on the porous flow side. This can potentially lead to instability and non-convergence issues in the coupled system, plus giving results with an accountable degree of error. The specification of a fully monolithic implicit-implicit coupled porous media code sees the solution of both seepage-mechanical equations in one matrix system, under a unified time-stepping scheme, which makes the problem definition much easier. When using an explicit solver, additional input such as the damping coefficient and mass scaling factor is required, which are circumvented with a fully implicit solution. Further, improved accuracy is achieved as the solution is not dependent on predictor-corrector methods for the pore fluid pressure solution, but at the potential cost of reduced stability. In testing of this fully monolithic porous media code, there is the comparison of the fully implicit coupled scheme against an existing staggered explicit-implicit coupled scheme solution across a range of geotechnical problems. These cases include 1) Biot coefficient calculation, 2) consolidation theory with Terzaghi analytical solution, 3) sedimentation theory with Gibson analytical solution, and 4) Coussy well-bore poroelastic analytical solutions.

Keywords: coupled, implicit, monolithic, porous media

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Authors: Hafez Fouad

Abstract:

Telemedicine applications have very low voltage which required High Precision Rectifier Design with high Sensitivity to operate at minimum input Voltage. In this work, we targeted 0.2V input voltage using 65 nm CMOS rectifier for Energy Harvesting Telemedicine application. The proposed rectifier which designed at 2.4GHz using two-stage structure found to perform in a better case where minimum operation voltage is lower than previous published paper and the rectifier can work at a wide range of low input voltage amplitude. The Performance Summary of Full-wave fully gate cross-coupled rectifiers (FWFR) CMOS Rectifier at F = 2.4 GHz: The minimum and maximum output voltages generated using an input voltage amplitude of 2 V are 490.9 mV and 1.997 V, maximum VCE = 99.85 % and maximum PCE = 46.86 %. The Performance Summary of Differential drive CMOS rectifier with external bootstrapping circuit rectifier at F = 2.4 GHz: The minimum and maximum output voltages generated using an input voltage amplitude of 2V are 265.5 mV (0.265V) and 1.467 V respectively, maximum VCE = 93.9 % and maximum PCE= 15.8 %.

Keywords: energy harvesting, embedded system, IoT telemedicine system, threshold voltage minimization, differential drive cmos rectifier, full-wave fully gate cross-coupled rectifiers CMOS rectifier

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Authors: Kunpeng Wang, Hongchun, Wu, Liangzhi Cao, Chuanqi Zhao

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The distributions of homogeneous neutron flux within a node were expanded into a set of analytic basis functions which satisfy the diffusion equation at any point in a triangular-z node for each energy group, and nodes were coupled with each other with both the zero- and first-order partial neutron current moments across all the interfaces of the triangular prism at the same time. Based this method, a code TABFEN has been developed and applied to solve the neutron diffusion equation in a complicated geometry. In addition, after a series of numerical derivation, one can get the neutron adjoint diffusion equations in matrix form which is the same with the neutron diffusion equation; therefore, it can be solved by TABFEN, and the low-high scan strategy is adopted to improve the efficiency. Four benchmark problems are tested by this method to verify its feasibility, the results show good agreement with the references which demonstrates the efficiency and feasibility of this method.

Keywords: analytic basis function expansion method, arbitrary triangular-z node, adjoint neutron flux, complicated geometry

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Authors: Tadas Telksnys, Zenonas Navickas, Minvydas Ragulskis

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Necessary and sufficient conditions for the existence of second order solitary solutions to the Hodgkin-Huxley equation are derived in this paper. The generalized multiplicative operator of differentiation helps not only to construct closed-form solitary solutions but also automatically generates conditions of their existence in the space of the equation's parameters and initial conditions. It is demonstrated that bright, kink-type solitons and solitary solutions with singularities can exist in the Hodgkin-Huxley equation.

Keywords: Hodgkin-Huxley equation, solitary solution, existence condition, operator method

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Authors: A. Amar

Abstract:

The angular distributions of the nuclear systems under consideration have been analyzed in the framework of the optical model (OM), where the real part was taken in the crystal model form. A crystal model (CM) has been applied to deuteron elastically scattered by ⁶,⁷Li and ⁹Be. A crystal model (CM) + distorted-wave Born approximation (DWBA) + dynamic polarization potential (DPP) potential has been applied to deuteron elastically scattered by ⁶,⁷Li and 9Be. Also, a crystal model has been applied to ⁶Li elastically scattered by ¹⁶O and ²⁸Sn in addition to the ⁷Li+⁷Li system and the ¹²C(alpha,⁸Be) ⁸Be reaction. The continuum-discretized coupled-channels (CDCC) method has been applied to the ⁷Li+⁷Li system and agreement between the crystal model and the continuum-discretized coupled-channels (CDCC) method has been observed. In general, the models succeeded in reproducing the differential cross sections at the full angular range and for all the energies under consideration.

Keywords: optical model (OM), crystal model (CM), distorted-wave born approximation (DWBA), dynamic polarization potential (DPP), the continuum-discretized coupled-channels (CDCC) method, and deuteron elastically scattered by ⁶, ⁷Li and ⁹Be

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Authors: Alexander Blokhin, Ekaterina Kruglova, Boris Semisalov

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Mathematical model based on the mesoscopic theory of polymer dynamics is developed for numerical simulation of the flows of polymeric liquid between two coaxial cylinders. This model is a system of nonlinear partial differential equations written in the cylindrical coordinate system and coupled with the heat conduction equation including a specific dissipation term. The stationary flows similar to classical Poiseuille ones are considered, and the resolving equations for the velocity of flow and for the temperature are obtained. For solving them, a fast pseudospectral method is designed based on Chebyshev approximations, that enables one to simulate the flows through the channels with extremely small relative values of the radius of inner cylinder. The numerical analysis of the dependance of flow on this radius and on the values of dissipation constant is done.

Keywords: dynamics of polymeric liquid, heat dissipation, singularly perturbed problem, pseudospectral method, Chebyshev polynomials, stabilization technique

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Authors: A. I. Ma’ali, R. B. Adeniyi, A. Y. Badeggi, U. Mohammed

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An error estimation of the integrated formulation of the Lanczos tau method for some class of ordinary differential equations was reported. This paper is concern with the generalization of tau approximants and their corresponding error estimates for some class of ordinary differential equations (ODEs) characterized by m + s =3 (i.e for m =1, s=2; m=2, s=1; and m=3, s=0) where m and s are the order of differential equations and number of overdetermination, respectively. The general result obtained were validated with some numerical examples.

Keywords: approximant, error estimate, tau method, overdetermination

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Authors: Nicolò Vaiana, Giorgio Serino

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The nonlinear time history analysis of seismically base-isolated structures can require a significant computational effort when the behavior of each seismic isolator is predicted by adopting the widely used differential equation Bouc-Wen model. In this paper, a nonlinear exponential model, able to simulate the response of seismic isolation bearings within a relatively large displacements range, is described and adopted in order to reduce the numerical computations and speed up the nonlinear dynamic analysis. Compared to the Bouc-Wen model, the proposed one does not require the numerical solution of a nonlinear differential equation for each time step of the analysis. The seismic response of a 3d base-isolated structure with a lead rubber bearing system subjected to harmonic earthquake excitation is simulated by modeling each isolator using the proposed analytical model. The comparison of the numerical results and computational time with those obtained by modeling the lead rubber bearings using the Bouc-Wen model demonstrates the good accuracy of the proposed model and its capability to reduce significantly the computational effort of the analysis.

Keywords: base isolation, computational efficiency, nonlinear exponential model, nonlinear time history analysis

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Authors: Lina Wu, Ye Li

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An equivalent relation between a harmonic form and a closed co-closed form is established on a complete non-compact manifold. This equivalence has been generalized for a differential k-form ω from Lq spaces to non-Lq spaces when q=2 in the context of p-balanced growth where p=2. Especially for a simple differential k-form on a complete non-compact manifold, the equivalent relation has been verified with the extended scope of q for from finite q-energy in Lq spaces to infinite q-energy in non-Lq spaces when with 2-balanced growth. Generalized Hadamard Theorem, Cauchy-Schwarz Inequality, and Calculus skills including Integration by Parts as well as Convergent Series have been applied as estimation techniques to evaluate growth rates for a differential form. In particular, energy growth rates as indicated by an appropriate power range in a selected test function lead to a balance between a harmonic differential form and a closed co-closed differential form. Research ideas and computational methods in this paper could provide an innovative way in the study of broadening Lq spaces to non-Lq spaces with a wide variety of infinite energy growth for a differential form.

Keywords: closed forms, co-closed forms, harmonic forms, L^q spaces, p-balanced growth, simple differential k-forms

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Authors: Mustafa Bayram Gücen, Coşkun Yakar

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In this study, we have investigated the strict stability of fuzzy differential systems and we compare the classical notion of strict stability criteria of ordinary differential equations and the notion of strict stability of fuzzy differential systems. In addition that, we present definitions of stability and strict stability of fuzzy differential equations and also we have some theorems and comparison results. Strict Stability is a different stability definition and this stability type can give us an information about the rate of decay of the solutions. Lyapunov’s second method is a standard technique used in the study of the qualitative behavior of fuzzy differential systems along with a comparison result that allows the prediction of behavior of a fuzzy differential system when the behavior of the null solution of a fuzzy comparison system is known. This method is a usefull for investigating strict stability of fuzzy systems. First of all, we present definitions and necessary background material. Secondly, we discuss and compare the differences between the classical notion of stability and the recent notion of strict stability. And then, we have a comparison result in which the stability properties of the null solution of the comparison system imply the corresponding stability properties of the fuzzy differential system. Consequently, we give the strict stability results and a comparison theorem. We have used Lyapunov second method and we have proved a comparison result with scalar differential equations.

Keywords: fuzzy systems, fuzzy differential equations, fuzzy stability, strict stability

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Authors: Nara Guimarães, Marcelo Aquino Martorano, Douglas Gouvêa

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An investigation into Cahn-Hilliard equation was carried out through numerical simulation to identify a possible phase separation for one and two dimensional domains. It was observed that this equation can reproduce important mass fluxes necessary for phase separation within the miscibility gap and for coalescence of particles.

Keywords: Cahn-Hilliard equation, miscibility gap, phase separation, dimensional domains

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Authors: Ishit Sheth, Chandrasekhar Jinendran, Chinmaya Ranjan Sahu

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A fourteen degree of freedom (DOF) ride and handling control mathematical model is developed for a car using generalized boltzmann hamel equation which will create a basis for design of ride and handling controller. Mathematical model developed yield equations of motion for non-holonomic constrained systems in quasi-coordinates. The governing differential equation developed integrates ride and handling control of car. Model-based systems engineering approach is implemented for simulation using matlab/simulink, vehicle’s response in different DOF is examined and later validated using commercial software (ADAMS). This manuscript involves detailed derivation of full car vehicle model which provides response in longitudinal, lateral and yaw motion to demonstrate the advantages of the developed model over the existing dynamic model. The dynamic behaviour of the developed ride and handling model is simulated for different road conditions.

Keywords: Full Vehicle Model, MBSE, Non Holonomic Constraints, Boltzmann Hamel Equation

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Authors: Manasa Kalla, Narasimha Raju Chebrolu, Ashok Chatterjee

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We have studied the quantum transport properties of a single molecular transistor in the presence of an external magnetic field using the Keldysh Green function technique. We also used the Anderson-Holstein-Caldeira-Leggett Model to describe the single molecular transistor that consists of a molecular quantum dot (QD) coupled to two metallic leads and placed on a substrate that acts as a heat bath. The phonons are eliminated by the Lang-Firsov transformation and the effective Hamiltonian is used to study the effect of an external magnetic field on the spectral density function, Tunneling Current, Differential Conductance and Spin polarization. A peak in the spectral function corresponds to a possible excitation. In the presence of a magnetic field, the spin-up and spin-down states are degenerate and this degeneracy is lifted by the magnetic field leading to the splitting of the central peak of the spectral function. The tunneling current decreases with increasing magnetic field. We have observed that even the differential conductance peak in the zero magnetic field curve is split in the presence electron-phonon interaction. As the magnetic field is increased, each peak splits into two peaks. And each peak indicates the existence of an energy level. Thus the number of energy levels for transport in the bias window increases with the magnetic field. In the presence of the electron-phonon interaction, Differential Conductance in general gets reduced and decreases faster with the magnetic field. As magnetic field strength increases, the spin polarization of the current is increasing. Our results show that a strongly interacting QD coupled to metallic leads in the presence of external magnetic field parallel to the plane of QD acts as a spin filter at zero temperature.

Keywords: Anderson-Holstein model, Caldeira-Leggett model, spin-polarization, quantum dots

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Authors: Beata Jackowska-Zduniak

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We formulate and analyze a mathematical model describing dynamics of the hypothalamus-pituitary-thyroid homoeostatic mechanism in endocrine system. We introduce to this system two types of couplings and delay. In our model, feedback controls the secretion of thyroid hormones and delay reflects time lags required for transportation of the hormones. The influence of delayed feedback on the stability behaviour of the system is discussed. Analytical results are illustrated by numerical examples of the model dynamics. This system of equations describes normal activity of the thyroid and also a couple of types of malfunctions (e.g. hyperthyroidism).

Keywords: mathematical modeling, ordinary differential equations, endocrine system, delay differential equation

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Authors: Fang Liu, JunShuai Xue, JiaJia Yao, XueYan Yang, ZuMao Li, GuanLin Wu, HePeng Zhang, ZhiPeng Sun

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III-Nitride resonant tunneling diode (RTD) is one of the most promising candidates for multiple-valued logic (MVL) elements. Here, we report a monolithic integration of GaN resonant tunneling diodes to realize multiple negative differential resistance (NDR) regions for MVL application. GaN RTDs, composed of a 2 nm quantum well embedded in two 1 nm quantum barriers, are grown by plasma-assisted molecular beam epitaxy on free-standing c-plane GaN substrates. Negative differential resistance characteristic with a peak current density of 178 kA/cm² in conjunction with a peak-to-valley current ratio (PVCR) of 2.07 is observed. Statistical properties exhibit high consistency showing a peak current density standard deviation of almost 1%, laying the foundation for the monolithic integration. After complete electrical isolation, two diodes of the designed same area are connected in series. By solving the Poisson equation and Schrodinger equation in one dimension, the energy band structure is calculated to explain the transport mechanism of the differential negative resistance phenomenon. Resonant tunneling events in a sequence of the series-connected RTD pair (SCRTD) form multiple NDR regions with nearly equal peak current, obtaining three stable operating states corresponding to ternary logic. A frequency multiplier circuit achieved using this integration is demonstrated, attesting to the robustness of this multiple peaks feature. This article presents a monolithic integration of SCRTD with multiple NDR regions driven by the resonant tunneling mechanism, which can be applied to a multiple-valued logic field, promising a fast operation speed and a great reduction of circuit complexity and demonstrating a new solution for nitride devices to break through the limitations of binary logic.

Keywords: GaN resonant tunneling diode, multiple-valued logic system, frequency multiplier, negative differential resistance, peak-to-valley current ratio

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Authors: Binyam Teferi

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In this paper, a theoretical analysis of blood flow in the presence of thermal radiation and chemical reaction under the influence of time dependent magnetic field intensity has been studied. The unsteady non linear partial differential equations of blood flow considers time dependent stretching velocity, the energy equation also accounts time dependent temperature of vessel wall, and concentration equation includes time dependent blood concentration. The governing non linear partial differential equations of motion, energy, and concentration are converted into ordinary differential equations using similarity transformations solved numerically by applying ode45. MATLAB code is used to analyze theoretical facts. The effect of physical parameters viz., permeability parameter, unsteadiness parameter, Prandtl number, Hartmann number, thermal radiation parameter, chemical reaction parameter, and Schmidt number on flow variables viz., velocity of blood flow in the vessel, temperature and concentration of blood has been analyzed and discussed graphically. From the simulation study, the following important results are obtained: velocity of blood flow increases with both increment of permeability and unsteadiness parameter. Temperature of the blood increases in vessel wall as Prandtl number and Hartmann number increases. Concentration of the blood decreases as time dependent chemical reaction parameter and Schmidt number increases.

Keywords: stretching velocity, similarity transformations, time dependent magnetic field intensity, thermal radiation, chemical reaction

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Authors: S. Mousavian, F. Mousavian, V. Nikkhah Rashidabad

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Cubic equations of state like Redlich–Kwong (RK) EOS have been proved to be very reliable tools in the prediction of phase behavior. Despite their good performance in compositional calculations, they usually suffer from weaknesses in the predictions of saturated liquid density. In this research, RK equation was modified. The result of this study shows that modified equation has good agreement with experimental data.

Keywords: equation of state, modification, ammonia, genetic algorithm

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Authors: Muna Alghabshi, Edmana Krishnan

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A nonlinear Schrodinger equation has been considered for solving by mapping methods in terms of Jacobi elliptic functions (JEFs). The equation under consideration has a linear evolution term, linear and nonlinear dispersion terms, the Kerr law nonlinearity term and three terms representing the contribution of meta materials. This equation which has applications in optical fibers is found to have soliton solutions, shock wave solutions, and singular wave solutions when the modulus of the JEFs approach 1 which is the infinite period limit. The equation with special values of the parameters has also been solved using the tanh method.

Keywords: Jacobi elliptic function, mapping methods, nonlinear Schrodinger Equation, tanh method

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Authors: Ali Raheli, Rahim Habibifar, Behzad Mohammadi-Alasti, Mahdi Abbasgholipour

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This paper presents vibration behavior of a FGM micro-beam and its pull-in instability under a nonlinear electrostatic pressure. An exponential function has been applied to show the continuous gradation of the properties along thickness. Nonlinear integro-differential-electro-mechanical equation based on Euler–Bernoulli beam theory has been derived. The governing equation in the static analysis has been solved using Step-by-Step Linearization Method and Finite Difference Method. Fixed points or equilibrium positions and singular points have been shown in the state control space. In order to find the response to a step DC voltage, the nonlinear equation of motion has been solved using Galerkin-based reduced-order model and time histories and phase portrait for different applied voltages have been shown. The effects of electrostatic pressure on stability of FGM micro-beams having various amounts of the ceramic constituent have been investigated.

Keywords: FGM, MEMS, nonlinear vibration, electrical, dynamic pull-in voltage

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Authors: Benedict Barnes, Anthony Y. Aidoo

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A Divergence Regularization Method (DRM) is used to regularize the ill-posed Helmholtz equation where the boundary deflection is inhomogeneous in a Hilbert space H. The DRM incorporates a positive integer scaler which homogenizes the inhomogeneous boundary deflection in Cauchy problem of the Helmholtz equation. This ensures the existence, as well as, uniqueness of solution for the equation. The DRM restores all the three conditions of well-posedness in the sense of Hadamard.

Keywords: divergence regularization method, Helmholtz equation, ill-posed inhomogeneous Cauchy boundary conditions

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Authors: A. M. Sagir

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Discrete linear multistep block method of uniform order for the solution of first order Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs) is presented in this paper. The approach of interpolation and collocation approximation are adopted in the derivation of the method which is then applied to first order ordinary differential equations with associated initial conditions. The continuous hybrid formulations enable us to differentiate and evaluate at some grids and off – grid points to obtain four discrete schemes, which were used in block form for parallel or sequential solutions of the problems. Furthermore, a stability analysis and efficiency of the block method are tested on ordinary differential equations, and the results obtained compared favorably with the exact solution.

Keywords: block method, first order ordinary differential equations, hybrid, self-starting

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Authors: Raoudha Chaabane, Faouzi Askri, Sassi Ben Nasrallah

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A new numerical algorithm is developed to solve coupled convection-radiation heat transfer in a two dimensional enclosure. Radiative heat transfer in participating medium has been carried out using the control volume finite element method (CVFEM). The radiative transfer equations (RTE) are formulated for absorbing, emitting and scattering medium. The density, velocity and temperature fields are calculated using the two double population lattice Boltzmann equation (LBE). In order to test the efficiency of the developed method the Rayleigh Benard convection with and without radiative heat transfer is analyzed. The obtained results are validated against available works in literature and the proposed method is found to be efficient, accurate and numerically stable.

Keywords: participating media, LBM, CVFEM- radiation coupled with convection

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