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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 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]

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 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]