Active vibration control of piezoelectric plates

The control of thermally induced vibrations of a rectangular plate is investigated. An optimization problem is formulated to determine the control voltage needed to perform vibration suppression with least control effort. By eigenfunction expansion, the optimal control problem will be converted from a distributed to a lumped parameter system. By utilizing the variational theory, an explicit optimal control criterion will be derived

Green’s Function Iterative Method for Solving a Class of Boundary Value Problems Arising in Heat Transfer

In this study, a new algorithm based on Green’s function fixed point iterations is developed and implemented to solve a class of nonlinear boundary value problems that arise in heat transfer. The method will be employed to determine the efficiency of convective straight fins with temperature dependent thermal conductivity. The main idea of this method is to find an appropriate Green’s function that will be incorporated into a linear integral operator. By applying the fixed point theorem, an iterative formula for successive approximations will be obtained. Effectiveness and accuracy were noted when approximate results obtained by the proposed method reveal significant agreement with existed exact solutions and/or approximate solutions obtained by other renowned methods

Variational Iteration Method for Nonlinear Singular Two-Point Boundary Value Problems Arising in Human Physiology

The variational iteration method is applied to solve a class of nonlinear singular boundary value problems that arise in physiology. The process of the method, which produces solutions in terms of convergent series, is explained. The Lagrange multipliers needed to construct the correctional functional are found in terms of the exponential integral and Whittaker functions. The method easily overcomes the obstacle of singularities. Examples will be presented to test the method and compare it to other existing methods in order to confirm fast convergence and significant accuracy

A Galerkin-Parameterization Method for the Optimal Control of Smart Microbeams

A proposed computational method is applied to damp out the excess vibrations in smart microbeams, where the control action is implemented using piezoceramic actuators. From a mathematical point of view, we wish to determine the optimal boundary actuators that minimize a given energy-based performance measure. The minimization of the performance measure over the actuators is subjected to the full motion of the structural vibrations of the micro-beams. A direct state-control parametrization approach is proposed where the shifted Legendre polynomials are employed to solve the optimization problem. Legendre operational matrix and the properties of Kronecker product are utilized to find the approximated optimal trajectory and optimal control law of the lumped parameter systems with respect to the quadratic cost function by solving linear algebraic equations. Numerical examples are provided to demonstrate the applicability and efficiency of the proposed approach

An Efficient Semi-Analytical Solution of a One-Dimensional Curvature Equation that Describes the Human Corneal Shape

In this paper, a numerical approach is proposed to find a semi analytical solution for a prescribed anisotropic mean curvature equation modeling the human corneal shape. The method is based on an integral operator that is constructed in terms of Green’s function coupled with the implementation of Picard’s or Mann’s fixed point iteration schemes. Using the contraction principle, it will be shown that the method is convergent for both fixed point iteration schemes. Numerical examples will be presented to demonstrate the applicability, efficiency, and high accuracy of the proposed method

Spline-based numerical treatments of Bratu-type equations,

Communicated by Ayman Badawi MSC2010 CLASSIFICATION: 65D07; 65N25; 65N35 Keywords: Nonlinear elliptic eigenvalue problem; Bratu-type equations; cubic spline collocation method; adaptive spline collocation; order of convergence. Abstract Three different spline-based approaches for solving Bratu and Bratu-type equations are presented. The classical cubic spline collocation method, an adaptive spline collocation on nonuniform partitions, and an optimal collocation method are derived for solving Bratu-type equations. Numerical examples are presented to verify the efficiency and accuracy of these methods when compared to other numerical schemes. The fourth order of convergence for the optimal method is verified. Introduction A nonlinear elliptic eigenvalue problem has the form Equation (2.1) arises in many fields of science and engineering such as radiative heat transfer, combustion theory, and nanotechnology Bratu equation, which is a special case of equation (2.1), is a boundary value problem in one-dimensional planar coordinates that has the form u + λe For λ > 0, the exact solution of equatio

Solution of a Complex Nonlinear Fractional Biochemical Reaction Model

This paper discusses a complex nonlinear fractional model of enzyme inhibitor reaction where reaction memory is taken into account. Analytical expressions of the concentrations of enzyme, substrate, inhibitor, product, and other complex intermediate species are derived using Laplace decomposition and differential transformation methods. Since different rate constants, large initial concentrations, and large time domains are unavoidable in biochemical reactions, different dynamics will result; hence, the convergence of the approximate concentrations may be lost. In this case, the proposed analytical methods will be coupled with Padé approximation. The validity and accuracy of the derived analytical solutions will be established by direct comparison with numerical simulations

Solution of a Complex Nonlinear Fractional Biochemical Reaction Model

This paper discusses a complex nonlinear fractional model of enzyme inhibitor reaction where reaction memory is taken into account. Analytical expressions of the concentrations of enzyme, substrate, inhibitor, product, and other complex intermediate species are derived using Laplace decomposition and differential transformation methods. Since different rate constants, large initial concentrations, and large time domains are unavoidable in biochemical reactions, different dynamics will result; hence, the convergence of the approximate concentrations may be lost. In this case, the proposed analytical methods will be coupled with Padé approximation. The validity and accuracy of the derived analytical solutions will be established by direct comparison with numerical simulations

A Mathematical Model of Risk Factors in HIV/AIDS Transmission Dynamics: Observational Study of Female Sexual Network in India

In this paper, a mathematical model for the transmission dynamics of HIV/AIDS epidemic with emphasis on the role of female sex workers is considered. The model is a system of nine nonlinear differential equations that represent nine different groups of an HIV population. A modified approach of the homotopy perturbation method is used to derive an approximate analytical expression for each of the nine different groups that form HIV population. The analytical results are shown to be consistent with the numerical results obtained by the highly accurate fourth-order Runge-Kutta method. The analytical solution will simplify studying the effect of each parameter on the governing equation and identifying the dynamics of HIV prevalence. Thus, effective prevention strategies can be adopted

Adaptation of the Novel Cubic B-Spline Algorithm for Dealing with Conformable Systems of Differential Boundary Value Problems concerning Two Points and Two Fractional Parameters

Recently, conformable calculus has appeared in many abstract uses in mathematics and several practical applications in engineering and science. In addition, many methods and numerical algorithms have been adapted to it. In this paper, we will demonstrate, use, and construct the cubic B-spline algorithm to deal with conformable systems of differential boundary value problems concerning two points and two fractional parameters in both regular and singular types. Here, several linear and nonlinear examples will be presented, and a model for the Lane-Emden will be one of the applications presented. Indeed, we will show the complete construction of the used spline through the conformable derivative along with the convergence theory, and the error orders together with other results that we will present in detail in the form of tables and graphs using Mathematica software. Through the results we obtained, it became clear to us that the spline approach is effective and fast, and it requires little compulsive and mathematical burden in solving the problems presented. At the end of the article, we presented a summary that contains the most important findings, what we calculated, and some future suggestions