Numerical solution for a class of fractional optimal control problems using the fractional-order Bernoulli functions

The main purpose of this paper is to provide an efficient method for solving some types of fractional optimal control problems governed by integro-differential and differential equations, and because finding the analytical solutions to these problems is usually difficult, a numerical method is proposed. In this study, the fractional-order Bernoulli functions (F-BFs) are applied as basis functions and a new operational matrix of fractional integration is constructed for these functions. In the first step, the problem is transformed into an equivalent variational problem. Then the F-BFs, the constructed operational matrix, the Gauss quadrature formula, and necessary conditions for optimization are used to convert the problem into a system of algebraic equations. Finally, with the aid of Newton’s iterative method, the system of algebraic equations is solved and the approximate solution of the problem is obtained. Several numerical examples have been analysed for illustrating the efficiency and accuracy of the proposed method, and the results have been compared with the exact solutions and the results of other methods. The results show that the method provides accurate solutions.

Dual System Least-Squares Finite Element Method for a Hyperbolic Problem

This study investigates the dual system least-squares finite element method, namely the LL∗ method, for a hyperbolic problem. It mainly considers nonlinear hyperbolic conservation laws and proposes a combination of the LL∗ method and Newton’s iterative method. In addition, the inclusion of a stabilizing term in the discrete LL∗ minimization problem is proposed, which has not been investigated previously. The proposed approach is validated using the one-dimensional Burgers equation, and the numerical results show that this approach is effective in capturing shocks and provides approximations with reduced oscillations in the presence of shocks.

Design and optimization of a small horizontal axis wind turbine using BEM theory and tip loss corrections

In this paper, an optimization approach of a small horizontal axis wind turbine based on BEM theory including De Vries and Shen et al. tip loss corrections is proposed. The optimal blade geometry was obtained by maximizing the power coefficient along the blade using the optimal angle of attack and the optimal tip speed ratio. The Newton’s iterative method applied to axial induction factor was used to solve the problem. This study was conducted for a NACA4418 small wind turbine, at low wind velocity. Among the two used tip loss corrections, the De Vries correction was found to be the most suitable for this blade optimization method. The optimal design was obtained for a tip speed ratio of 5 and has recorded a power coefficient equal to 0.463.

New wavelet method for solving boundary value problems arising from an adiabatic tubular chemical reactor theory

This paper displays an efficient numerical technique of realizing mathematical models for an adiabatic tubular chemical reactor which forms an irreversible exothermic chemical reaction. At a steady-state solution for an adiabatic rounded reactor, the model can be diminished to a conventional nonlinear differential equation which converts into a system of the nonlinear equation that can proceed numerically utilizing Newton’s iterative method. An operational matrix of coordination is derived and is utilized to decrease the model for an adiabatic tubular chemical reactor to an arrangement of algebraic equations. Simple execution, basic activities, and precise arrangements are the fundamental highlights of the proposed wavelet technique. The numerical solutions attained by the present technique have been contrasted and compared with other techniques.

Optimization of technological parameters for cyclic steam stimulation of oil reservoirs

Depletion of oil reserves leads to need to develop unconventional and hard-to-recover reserves, including high-viscosity oil fields. An effective way to do this is to use thermal enhanced oil recovery methods. Existing models do not consider the actual displacement of the heating front with convective flows. Therefore, the actual tasks are to model the physical processes occurring in the reservoir and to optimize the technological parameters of the development during cyclic steam stimulation. This article is a continuation of earlier research and offers to consider a different version of movement of boundary of heating front. Clarification of the development of thermal field in reservoir is associated with setting the shape of boundary considering gravitational forces, in contrast to the previously proposed model, where the assumption of the frontal propagation of the thermal front is accepted. The aims of the article are to determine the production rate for cyclic steam stimulation with described geometry; calculation steam injection time using real data, optimization of production. The research methodology is based on the use of a system of conservation laws. The main equations are solved analytically, and the flow rate is calculated using Newton’s iterative method. Thus, this article offers the first integrated physical-mathematical model of cyclic steam stimulation, considering the presence of convective and gravitational forces in the formation of heated zone profile. Problem of production optimization is solved using real data. The characteristic times are consistent with the real data. These calculations help to choose the most rational development strategy.

An algorithm for constructing a direct and inverse operator of a real process

Consider the task of building a mathematical model of the real process, which translates the data at the entrance to a certain result at the output. Considered the case when severaldata is submitted to the entrance, and the output result is only one. The direct operator of the real process makes it possible to determine (provide) the result at the exit based on the known data at the entrance. The reverse operator on a known result on the way out of the real process allows you to find the necessary input. Operators of the real process are modeled with algebraic polynom to some extent. The degree of algebraic polynomic and its coefficients depend on a specific real process. Since input and output are known with some error in real-world processes, we take into account input and output errors when building operators. The task of building such operators is incorrect on Adamar, so we use the method of regularization of Tikhonov. This method allows you to build sustainable approach (taking into account the error of the input and output data) the right operators. The article examines in detail the algorithm for building a reverse operator. The direct operator algorithm is reviewed in the authors' previous article (link [2] in this article). Building a reverse operator comes down to solving a non-linear equation in an incorrect setting. The non-linear equation is solved by Newton's iterative method. The software implementation of the algorithm has been carried out. Three test examples are considered, which confirm the correctness of the algorithm and program. The algorithm can be summarized in case there are several data (at least two) at both the entrance and exit.

Numerical Solution of Fractional Sine-Gordon Equation Using Spectral Method and Homogenization

In this paper, a new method for the numerical solution of fractional sine-Gordon (SG) equation is presented. Our method consists of two steps, in first step: the main equation is converted to a homogeneous one using interpolation. In second step: two-dimensional approximation of functions by shifted Jacobi polynomials is used to reduce the problem into a system of nonlinear algebraic equations. The archived system is solved by Newton’s iterative method. Our method is stated in general case on rectangular [a,b] × [0,T] which is based upon Jacobi polynomial by parameters (α,β). Several test problems are employed and results of numerical experiments are presented and also compared with analytical solutions. Also, we verify the numerical stability of the method, by applying a disturbance in the problem. The obtained results confirm the acceptable accuracy and stability of the presented method.

Stability Analysis of Jacobian-Free Newton’s Iterative Method

It is well known that scalar iterative methods with derivatives are highly more stable than their derivative-free partners, understanding the term stability as a measure of the wideness of the set of converging initial estimations. In multivariate case, multidimensional dynamical analysis allows us to afford this task and it is made on different Jacobian-free variants of Newton’s method, whose estimations of the Jacobian matrix have increasing order. The respective basins of attraction and the number of fixed and critical points give us valuable information in this sense.

Efficient numerical treatments for a fractional optimal control nonlinear Tuberculosis model

In this paper, the general nonlinear multi-strain Tuberculosis model is controlled using the merits of Jacobi spectral collocation method. In order to have a variety of accurate results to simulate the reality, a fractional order model of multi-strain Tuberculosis with its control is introduced, where the derivatives are adopted from Caputo’s definition. The shifted Jacobi polynomials are used to approximate the optimality system. Subsequently, Newton’s iterative method will be used to solve the resultant nonlinear algebraic equations. A comparative study of the values of the objective functional, between both the generalized Euler method and the proposed technique is presented. We can claim that the proposed technique reveals better results when compared to the generalized Euler method.