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An electrochemical photobioreactor with a packed bed containing transparent gel granules and immobilized photosynthetic bacterial cells is shown with a one-dimensional two-phase flow and transport model. We consider the biological/chemical events in the electrochemical photobioreactor, the intrinsically connected two-phase flow and mass transport, and other factors. This model is based on a system of nonlinear equations. This paper applies Akbari-Ganji's and Taylor series methods to find analytical solutions to nonlinear differential equations that arise in an immobilized-cell electrochemical photobioreactor. Approximate analytical expressions of the concentration of glucose and hydrogen are obtained in liquid and gas phases for different parameter values. Numerical simulations are presented to validate the theoretical investigations. [ABSTRACT FROM AUTHOR]

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]

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]

The main purpose of this study was to produce abundant new types of soliton solutions for the Radhakrishnan-Kundu-Lakshmanan equation that represents unstable optical solitons that emerge from optical propagations through the use of birefringent fibers. These new types of soliton solutions have behaviors that are bright, dark, W-shaped, M-shaped, periodic trigonometric, and hyperbolic and were not realized before by any other method. These new forms have been detected by using four different techniques, which are, the extended simple equation method, the Paul-Painlevé approach method, the Ricatti-Bernoulli-sub ODE, and the solitary wave ansatz method. These new solitons will be arranged to create a soliton catalog with new impressive behaviors and they will contribute to future studies not only for this model but also for the optical propagations through birefringent fiber. [ABSTRACT FROM AUTHOR]

This work aims to provide the numerical performances of the computer epidemic virus model with the time delay effects using the stochastic Levenberg-Marquardt backpropagation neural networks (LMBP-NNs). The computer epidemic virus model with the time delay effects is categorized into four dynamics, the uninfected S(x) computers, the latently infected L(x) computers, the breakingout B(x) computers, and the antivirus PC's aptitude R(x). The LMBP-NNs approach has been used to numerically simulate three cases of the computer virus epidemic system with delay effects. The stochastic framework for the computer epidemic virus system with the time delay effects is provided using the selection of data with 11%, 13%, and 76% for testing, training, and verification together with 15 neurons. The proposed and data-based Adam technique is overlapped to execute the LMBP-NNs method's exactness. The constancy, authentication, precision, and capability of the LMBP-NNs scheme are perceived with the analysis of the state transition measures, regression actions, correlation performances, error histograms, and mean square error measures. [ABSTRACT FROM AUTHOR]

The theoretical model for a packed porous catalytic particle of the slab, cylindrical, and spherical geometries shape in fixed-bed electrochemical reactors is discussed. These particles have internal mass concentration and temperature gradients in endothermic or exothermic reactions. The model is based on a nonlinear reaction–diffusion equation containing a nonlinear term with an exponential relationship between intrinsic reaction rate and temperature. The porous catalyst particle's concentration is obtained by solving the nonlinear equation using Akbari-Ganji's method. A simple and closed-form analytical expression of the effectiveness factor for slab, cylindrical, and spherical geometries was also reported for all values of Thiele modulus, activation energy, and heat reaction. The accordance with results of a reliable numerical method shows the good accuracy that their approximate solution yields. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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.

A mathematical model of reaction-diffusion problem with Michaelis-Menten kinetics in catalyst particles of arbitrary shape is investigated. Analytical expressions of the concentration of substrates are derived as functions of the Thiele modulus, the modified Sherwood number, and the Michaelis constant. A Taylor series approach and the Akbari-Ganji's method are utilized to determine the substrate concentration and the effectiveness factor. The effects of the shape factor on the concentration profiles and the effectiveness factor are discussed. In addition to their simple implementations, the proposed analytical approaches are reliable and highly accurate, as it will be shown when compared with numerical simulations. [ABSTRACT FROM AUTHOR]

A thorough study of the effects of ambient temperature and surface emissivity on the heat transfer of a conductive-radiative fin is presented in this article. A semi-analytical method based on Green's function coupled with fixed point iterative schemes will be employed to solve a nonlinear energy equation that governs a straight rectangular fin. The method will be shown to converge for both Picard's and Mann's fixed point iteration schemes. The proposed method's efficiency and accuracy will be tested against some of the widely used numerical approaches found in the literature. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]

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]

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]

The mathematical model of Oldham J. Phys. Chem. B 2000 , 104 , 4703 for rotating disc electrode in an unsupported system is discussed. This article presents a new analytical method for the calculation of concentration at a rotating disc electrode controlled by diffusion, convection, and migration. This model contains a steady-state nonlinear differential equation in a three-ion system under the assumption that all the ions have the same diffusivity. The homotopy perturbation method is employed to solve the nonlinear governing equation, where the ionic concentration and current are obtained analytically, in terms of charge numbers. The comparison between the analytical results of this study and previous studies confirms that the result of the proposed method is in stronger agreement with numerical simulations than other analytical methods.