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The optimal control for vibration suppression of a plate by distributed piezoelectric actuators is considered. A performance index in the form of a weighted quadratic functional of the dynamic response of a rectangular simply supported plate will be minimized within a prescribed time duration using piezoelectric patches (voltages). The minimization of the performance index over these voltages is subject to the equation of motion governing the plate's structural vibration and a set of initial and boundary conditions. The solution method is a combination of modal space expansion and direct state parameterization. Modal space expansion will transform the optimal control of a distributed parameter system into the optimal control of a lumped parameter system. Using Legendre wavelets, the quadratic optimization problem is transformed into a mathematical programming problem, where the objective is to minimize a set of unknown coefficients to obtain the optimal trajectory and the optimal control. Numerical examples will be provided to illustrate the effectiveness and the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This article uses a hybrid approach that combines Green's function and a fixed point scheme to obtain a semi-analytical solution to the RLC electrical circuit's non-integer order differential equation. The fractional differential equation's order is determined by the two parameters α ∈ ( 1 , 2 ] and β ∈ ( 0 , 1 ] . The proposed method is unique because it can be extended to general nonlinear fractional systems. Another highly desirable feature of the proposed approach is that the construction of Green's function depends on both endpoints of the time domain, and hence the produced approximate solutions are highly accurate for fractional boundary value problems. We provide examples with figures and tables to demonstrate the effectiveness of our method in solving electrical circuit problems. TThe obtained approximate solutions are compared with exact solutions or numerical simulations to validate the applicability and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, a constructed Green's function coupled with a fixed point iteration scheme will be employed to solve nonlinear dynamical problems that arise in electroanalytical chemistry. More precisely, the method will be used to mathematically model and solve the kinetics of the amperometric enzyme. A main property that makes the proposed method superior to other iterative methods is the way it handles boundary value problems, where both endpoints are taken into consideration while other iterative methods only make account of the initial point and as a result, the approximate solution may deteriorate for values that are far away from the initial point and closer to the other endpoint. Through tests on some known amperometric enzyme kinetics, the proposed method gave more accurate results than many numerical schemes that were employed for this purpose. The method is found to be easily implemented, fast, and computationally economical and attractive. [ABSTRACT FROM AUTHOR]

The control of the uptake of growth factors in tissue engineering is mathematically modelled by a partial differential equation subject to boundary and initial conditions. The main objective is to regulate cellular processes for the growth or regeneration of a tissue within an assigned terminal time. The techniques of basis function expansion and direct state parameterization were employed to yield efficient computational methods for this problem. Using Legendre and Chebyshev wavelets, the optimal control of the lumped-parameter system was transformed into a system of algebraic equations. The computational efficiency and effectiveness of the proposed method are illustrated by numerical examples. [ABSTRACT FROM PUBLISHER]

This paper will introduce novel techniques for a fractional-order model of the human liver involving the Atangana–Baleanu, Atangana–Toufik, and the Fractal fractional method with the nonsingular kernel. These techniques give more accurate and appropriate results. Existence and uniqueness have been developed with the help of fixed-point theory results. Numerical simulations are done from the developed algorithm of the model to elaborate the effect of fractional values and justify the theoretical results. Such kind of analysis will be useful for further investigation of epidemic diseases, and also provide a better understanding of disease dynamics to overcome the effect of disease in a community. [ABSTRACT FROM AUTHOR]

Abstract: Galerkin and wavelet methods for optimal boundary control of a couple of discretely connected parallel beams are proposed. First, the problem with boundary controls is converted into a problem with distributed controls. The problem is, then, reduced by a Galerkin-based approach into determining the optimal control of a linear time-invariant lumped parameter system, which will be solved by a wavelet-based method using Legendre wavelets. The integration-operational matrix and Kronecker product are utilized to significantly simplify the optimization problem into a system of linear equations. A numerical example is presented to demonstrate the applicability and the efficiency of the proposed method. [Copyright &y& Elsevier]

Recent years have marked many significant advances in numerical treatments for stochastic differential equations. Emphasis was not only on the nature and order of convergence of numerical schemes, but also on their stability features [J. Comput. Appl. Math. 125 (2000) 171; BIT 33 (1993) 654; Comput. Math. Appl. 28 (1994) 45; SIAM J. Appl. Anal. 51 (1991) 542; SIAM J. Numer. Anal. 33 (1996) 2254]. In this article we discuss mean square stability of second-order weak numerical methods. The closed form of the second moment established by [D. Talay, Simulation and numerical analysis of stochastic differential systems, INRIA Report 1313, 1990] will be used to create a mean square stability criterion for these schemes. Numerical examples will be presented to support the theoretical analysis. [Copyright &y& Elsevier]

In recent years, many numerical methods for solving stochastic differential equations have been developed. Some of these methods converge in the weak sense and some others converge in the mean square sense. One of the important features of numerical methods is their stability behavior. In this paper, we focus our attention on stability in expectation (e. stability) of numerical methods of second-order accuracy in the weak sense. The region of e. stability for these methods will be discussed. The possibility of enlarging regions of e. stability will be described. Some numerical examples will be discussed to support the theoretical study. [ABSTRACT FROM AUTHOR]

The primary goal of this article is to present novel analytical solutions for the coupled nonlinear equation found in the mediated electron transfer process at polymer-modified conducting ultramicroelectrodes. Taylor’s series method is utilized to obtain approximate analytical solutions for the reaction-diffusion equations, allowing for the determination of the substrate and mediator concentrations and the current response concerning the substrate concentration at the electrode’s surface. The impact of different factors on concentration and current is also explored. The derived analytical results are in strong agreement with numerical results and with other analytical outcomes from the literature. [ABSTRACT FROM AUTHOR]

We consider different distribution functions as possible fits for the light curves (time profiles) of Terrestrial Gamma-ray Flashes (TGFs): the piecewise Gaussian and the piecewise exponential, which correspond to the assumption of exponential growth and decay of the electrons that emit the radiation, and the inverse Gaussian and Ornstein-Uhlenbeck functions, which correspond to the assumption of an exit-time process. We compute maximum likelihood estimations (MLEs) for each of these four functions for a 100 TGFs recorded by the RHESSI and AGILE satellites, and compared those values with MLEs of the lognormal distribution function, which has been very widely used. The analysis of time profiles of these TGFs shows that all five distributions fit the data equally well. The results obtained in this work, combined with the results obtained for the analysis of BATSE and FERMI TGFs (Abukhaled et al. in J. Geophys. Res. Space Phys. 119:5918–5930, 2014) show that the TGF data as recorded thus far by various satellite gamma-ray detectors, are not powerful enough to discriminate between those various mathematical functions, which, if one function were preferred, would identify some physical features of the phenomenon. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper considers the problem of controlling the solution of an initial boundary-value problem for a wave equation with time-dependent sound speed. The control problem is to determine the optimal sound speed function which damps the vibration of the system by minimizing a given energy performance measure. The minimization of the energy performance measure over sound speed is subjected to the equation of motion of the system with imposed initial and boundary conditions. Using the modal space technique, the optimal control of distributed parameter systems is simplified into the optimal control of bilinear time-invariant lumped-parameter systems. A wavelet-based method for evaluating the modal optimal control and trajectory of the bilinear system is proposed. The method employs finite CAS wavelets to approximate modal control and state variables. Numerical examples are presented to demonstrate the effectiveness of the method in reducing the energy of the system. [ABSTRACT FROM AUTHOR]

Abstract: The optimal control of transverse vibration of two Euler–Bernoulli beams coupled in parallel by discrete springs is considered. An index of performance is formulated which consists of a modified energy functional of two coupled structures at a specified time and penalty functions involving the point control forces. The minimization of the performance index over these forces is subject to the equation of motion governing the structural vibrations, the imposed initial condition as well as the boundary conditions. By use of the modal space technique, the optimal control of distributed parameter systems is simplified into the optimal control of a linear time-invariant lumped-parameter systems. A computationally attractive method based on Legendre wavelets in time domain for solving the optimal control of the lumped parameter systems for any finite interval is proposed. Legendre wavelet integral operational matrix and the properties of a Kronecker product are used to find the approximated optimal trajectory and optimal law of the linear systems with respect to a quadratic cost function by only solving a linear system of algebraic equations. This method provides a straightforward and convenient approach for digital computation. A numerical example is provided to demonstrate the applicability and effectiveness of the proposed method. [Copyright &y& Elsevier]

A mathematical model of electrostatic interaction with reaction-generated pH change on the kinetics of immobilized enzyme is discussed. The model involves the coupled system of non-linear reaction–diffusion equations of substrate and hydrogen ion. The non-linear term in this model is related to the Michaelis–Menten reaction of the substrate and non-Michaelis–Menten kinetics of hydrogen ion. The approximate analytical expression of concentration of substrate and hydrogen ion has been derived by solving the non-linear reactions using Taylor's series method. Reaction rate and effectiveness factor are also reported. A comparison between the analytical approximation and numerical solution is also presented. The effects of external mass transfer coefficient and the electrostatic potential on the overall reaction rate were also discussed. [ABSTRACT FROM AUTHOR]

An efficient analytical method is applied to solve a system of convection-diffusion equations in the pseudo-first-order EC-catalytic mechanism at a rotating disk electrode. A simple closed form analytical expression for the concentration of oxidized and reduced catalyst species in the electrochemical reaction is obtained and analyzed for various values of parameters. Also, closed form analytical expressions are derived for the steady-state current response and the current ratio at potential steps for various values of diffusion coefficients, rate constant, and rotation rate. The derived approximate analytical results are shown to be in satisfactory agreement with numerical simulation results. [ABSTRACT FROM AUTHOR]

A mathematical model of the magnetohydrodynamic free convective flow of a viscous incompressible uid, which is based on a system of coupled steady-state nonlinear deferential equations, is discussed. A new approach of the homotopy perturbation method is employed to derive analytical expressions of the uid velocity, uid temperature, and species concentration. The efficiency and accuracy of the derived results are tested against highly accurate and widely used numerical methods. The obtained analytical expressions are employed to study the effects of the magnetic field, chemical reaction, and other relevant flow parameters on uid velocity, uid temperature, and species concentration. Sensitivity analysis of these parameters is also presented. [ABSTRACT FROM AUTHOR]

• Analytical expressions for the electric potential for planar, spherical and cylindrical geometries are derived used a simple series method. • The proposed mathematical approach is simple and reliable and can be easily implemented by researchers from various science disciplines. • A study of the effect of various parameters such as grain size, grain shape and doping level is conducted. • Analytical formula for the density of oxygen ion is derived. • The maximum surface potential is determined. In this article, the mathematical model presented by Barami and Ghafarinia (Sensors and Actuators B: Chemical 293 (2019) 31–40) for metal oxide grains is discussed. The nonlinear governing model, which is a Poisson-Boltzmann-type equation, is solved by a simple and efficient method. Analytical expressions for the electrical potential and oxygen ion density within the planar, cylindrical and spherical metal oxide grains are expressed in terms of the absolute value of the electric potential and grain thickness. In addition, an investigation of the effects of parameters such as grain diameter, grain thickness, and uniform distribution of single donors on the performance. The validity of the derived analytical expression is established by direct comparison with computer-generated numerical simulations in addition to previously available limiting cases' results (low and high oxygen adsorption on the grain). [ABSTRACT FROM AUTHOR]