A different approach for the fractional chemical model

2021 ◽  
Vol 68 (1 Jan-Feb) ◽  
Author(s):  
Khaled Saad
Keyword(s):  
Numerical Solutions ◽  
Polynomial Interpolation ◽  
Absolute Error ◽  
Mass Action ◽  
Algebraic Equations ◽  
Mass Action Kinetics ◽  
Chemical Equations ◽  
Newton Polynomial ◽  
The Absolute ◽  
Raphson Method

This article analyzes and compares the two algorithms for the numerical solutions of the fractional isothermal chemical equations (FICEs) based on mass action kinetics for autocatalytic feedback, involving the conversion of a reactant in the Liouville-Caputo sense. The first method is based upon the spectral collocation method (SCM), where the properties of Legendre polynomials are utilized to reduce the FICEs to a set of algebraic equations. We then use the well-known method like Newton-Raphson method (NRM) to solve the set of algebraic equations. The second method is based upon the properties of Newton polynomial interpolation (NPI) and the fundamental theorem of fractional calculus. We utilize these methods to construct the numerical solutions of the FICEs. The accuracy and effectiveness of these methods is satisfied graphically by combining the numerical results and plotting the absolute error. Also, the absolute errors are tabulated, and a good agreementfound in all cases.

scholarly journals A Pseudospectral Algorithm for Solving Multipantograph Delay Systems on a Semi-Infinite Interval Using Legendre Rational Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
E. H. Doha ◽  
D. Baleanu ◽  
A. H. Bhrawy ◽  
R. M. Hafez
Keyword(s):  
Numerical Solutions ◽  
Rational Functions ◽  
Absolute Error ◽  
Delay Systems ◽  
Infinite Interval ◽  
Algebraic Equations ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
Linear And Nonlinear ◽  
Pseudospectral Scheme

A new Legendre rational pseudospectral scheme is proposed and developed for solving numerically systems of linear and nonlinear multipantograph equations on a semi-infinite interval. A Legendre rational collocation method based on Legendre rational-Gauss quadrature points is utilized to reduce the solution of such systems to systems of linear and nonlinear algebraic equations. In addition, accurate approximations are achieved by selecting few Legendre rational-Gauss collocation points. The numerical results obtained by this method have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively limited nodes used, the absolute error in our numerical solutions is sufficiently small.

Numerical Simulation of Fractional Delay Differential Equations Using the Operational Matrix of Fractional Integration for Fractional-Order Taylor Basis

2021 ◽  
Vol 6 (1) ◽  
pp. 10
Author(s):  
İbrahim Avcı 
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Absolute Error ◽  
Algebraic Equations ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

In this paper, we consider numerical solutions for a general form of fractional delay differential equations (FDDEs) with fractional derivatives defined in the Caputo sense. A fractional integration operational matrix, created using a fractional Taylor basis, is applied to solve these FDDEs. The main characteristic of this approach is, by utilizing the operational matrix of fractional integration, to reduce the given differential equation to a set of algebraic equations with unknown coefficients. This equation system can be solved efficiently using a computer algorithm. A bound on the error for the best approximation and fractional integration are also given. Several examples are given to illustrate the validity and applicability of the technique. The efficiency of the presented method is revealed by comparing results with some existing solutions, the findings of some other approaches from the literature and by plotting absolute error figures.

scholarly journals Some Dynamical Models Involving Fractional-Order Derivatives with the Mittag-Leffler Type Kernels and Their Applications Based upon the Legendre Spectral Collocation Method

2021 ◽  
Vol 5 (3) ◽  
pp. 131
Author(s):  
Hari M. Srivastava ◽  
Abedel-Karrem N. Alomari ◽  
Khaled M. Saad ◽  
Waleed M. Hamanah
Keyword(s):  
Collocation Method ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Spectral Collocation Method ◽  
Hybrid Technique ◽  
Algebraic Equations ◽  
Spectral Collocation ◽  
Finite Differences Method ◽  
Nonlinear Algebraic Equations ◽  
Raphson Method

Fractional derivative models involving generalized Mittag-Leffler kernels and opposing models are investigated. We first replace the classical derivative with the GMLK in order to obtain the new fractional-order models (GMLK) with the three parameters that are investigated. We utilize a spectral collocation method based on Legendre’s polynomials for evaluating the numerical solutions of the pr. We then construct a scheme for the fractional-order models by using the spectral method involving the Legendre polynomials. In the first model, we directly obtain a set of nonlinear algebraic equations, which can be approximated by the Newton-Raphson method. For the second model, we also need to use the finite differences method to obtain the set of nonlinear algebraic equations, which are also approximated as in the first model. The accuracy of the results is verified in the first model by comparing it with our analytical solution. In the second and third models, the residual error functions are calculated. In all cases, the results are found to be in agreement. The method is a powerful hybrid technique of numerical and analytical approach that is applicable for partial differential equations with multi-order of fractional derivatives involving GMLK with three parameters.

Error Estimation For The Methods Of Correlated colour Temperature Calculation

2021 ◽  
pp. 70-77
Author(s):  
Sergey V. Prytkov ◽  
Maxim V. Kolyadin
Keyword(s):  
Numerical Solutions ◽  
Search Algorithm ◽  
Absolute Error ◽  
National Standards ◽  
Software Developers ◽  
Colour Temperature ◽  
Universal Approach ◽  
Binary Search Algorithm ◽  
The Absolute

To date, a lot of methods have been developed for calculating correlated colour temperature (CCT). There are both numerical solutions (Robertson’s method, Yoshi Ohno method, binary search algorithm) and analytical (Javier Hernandez-Andres’s method, McCamy’s method). At the same time, the information about their accuracy is of a segmental fragmentary nature, therefore, it is very difficult to develop recommendations for the application of methods for certain radiators. In this connection, it seems extremely interesting to compare the error of the most well-known CCT calculating methods, using a single universal approach. The paper proposes an algorithm for researching the error of the methods for calculating correlated colour temperature, based on the method for plotting lines of constant CCT of a given length. Temperatures corresponding to these lines are taken as true, and the chromaticity lying on them are used as input data for the researched method. The paper proposes an approach when first the distribution of the error in the entire range of determination of CCT is determined, followed by bilinear interpolation for the required chromaticity. Using this approach, the absolute errors of the following methods for calculating CCT: McCamy, Javier Hernandez, Robertson, and Yoshi Ohno were estimated. The error was estimated in the range occupied by quadrangles of possible values from ANSI C78.377 chromaticity standard, developed by American National Standards Institute for LED lamps for indoor lightning. The tabular and graphical distribution of the absolute error for each investigated method was presented in the range of (2000–7000) K. In addition, to clarify the applicability of the methods for calculating CCT of the sky, the calculation of the distribution of the relative error up to 100000 K was performed. The results of the study can be useful for developers of standards and measurement procedures and for software developers of measuring equipment.

scholarly journals A Comparative Study of the Fractional-Order Clock Chemical Model

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1436 ◽  
Author(s):  
Hari Mohan Srivastava ◽  
Khaled M. Saad
Keyword(s):  
Comparative Study ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Absolute Error ◽  
Chemical Model ◽  
Kutta Method ◽  
Integer Order ◽  
Runge Kutta Method ◽  
The Absolute ◽  
Better Than

In this paper, a comparative study has been made between different algorithms to find the numerical solutions of the fractional-order clock chemical model (FOCCM). The spectral collocation method (SCM) with the shifted Legendre polynomials, the two-stage fractional Runge–Kutta method (TSFRK) and the four-stage fractional Runge–Kutta method (FSFRK) are used to approximate the numerical solutions of FOCCM. Our results are compared with the results obtained for the numerical solutions that are based upon the fundamental theorem of fractional calculus as well as the Lagrange polynomial interpolation (LPI). Firstly, the accuracy of the results is checked by computing the absolute error between the numerical solutions by using SCM, TSFRK, FSFRK, and LPI and the exact solution in the case of the fractional-order logistic equation (FOLE). The numerical results demonstrate the accuracy of the proposed method. It is observed that the FSFRK is better than those by SCM, TSFRK and LPI in the case of an integer order. However, the non-integer orders in the cases of the SCM and LPI are better than those obtained by using the TSFRK and FSFRK. Secondly, the absolute error between the numerical solutions of FOCCM based upon SCM, TSFFRK, FSFRK, and LPI for integer order and non-integer order has been computed. The absolute error in the case of the integer order by using the three methods of the third order is considered. For the non-integer order, the order of the absolute error in the case of SCM is found to be the best. Finally, these results are graphically illustrated by means of different figures.

scholarly journals Solutions of First-Order Volterra Type Linear Integrodifferential Equations by Collocation Method

2017 ◽  
Vol 2017 ◽  
pp. 1-5 ◽  
Author(s):  
Olumuyiwa A. Agbolade ◽  
Timothy A. Anake
Keyword(s):  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Absolute Error ◽  
Integrodifferential Equations ◽  
Analysis Method ◽  
Volterra Type ◽  
First Order ◽  
Collocation Points ◽  
Homotopy Analysis ◽  
The Absolute

The numerical solutions of linear integrodifferential equations of Volterra type have been considered. Power series is used as the basis polynomial to approximate the solution of the problem. Furthermore, standard and Chebyshev-Gauss-Lobatto collocation points were, respectively, chosen to collocate the approximate solution. Numerical experiments are performed on some sample problems already solved by homotopy analysis method and finite difference methods. Comparison of the absolute error is obtained from the present method and those from aforementioned methods. It is also observed that the absolute errors obtained are very low establishing convergence and computational efficiency.

scholarly journals Numerical solution of the Bagley–Torvik equation using shifted Chebyshev operational matrix

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Tianfu Ji ◽  
Jianhua Hou ◽  
Changqing Yang
Keyword(s):  
Numerical Scheme ◽  
Numerical Solutions ◽  
Order Differential Equation ◽  
Operational Matrix ◽  
Absolute Error ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Maximum Absolute Error ◽  
Fractional Order Differential Equation ◽  
Numerical Examples

AbstractIn this study, an efficient numerical scheme based on shifted Chebyshev polynomials is established to obtain numerical solutions of the Bagley–Torvik equation. We first derive the shifted Chebyshev operational matrix of fractional derivative. Then, by the use of these operational matrices, we reduce the corresponding fractional order differential equation to a system of algebraic equations, which can be solved numerically by Newton’s method. Furthermore, the maximum absolute error is obtained through error analysis. Finally, numerical examples are presented to validate our theoretical analysis.

scholarly journals Numerical Analysis of Viscoelastic Rotating Beam with Variable Fractional Order Model Using Shifted Bernstein–Legendre Polynomial Collocation Algorithm

2021 ◽  
Vol 5 (1) ◽  
pp. 8
Author(s):  
Cundi Han ◽  
Yiming Chen ◽  
Da-Yan Liu ◽  
Driss Boutat
Keyword(s):  
Numerical Analysis ◽  
Fractional Order ◽  
Governing Equation ◽  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Matrix Product ◽  
Algebraic Equations ◽  
Rotating Beam ◽  
The Matrix ◽  
The Time Domain

This paper applies a numerical method of polynomial function approximation to the numerical analysis of variable fractional order viscoelastic rotating beam. First, the governing equation of the viscoelastic rotating beam is established based on the variable fractional model of the viscoelastic material. Second, shifted Bernstein polynomials and Legendre polynomials are used as basis functions to approximate the governing equation and the original equation is converted to matrix product form. Based on the configuration method, the matrix equation is further transformed into algebraic equations and numerical solutions of the governing equation are obtained directly in the time domain. Finally, the efficiency of the proposed algorithm is proved by analyzing the numerical solutions of the displacement of rotating beam under different loads.

scholarly journals Training an asymmetric signal perceptron through reinforcement in an artificial chemistry

2014 ◽  
Vol 11 (93) ◽  
pp. 20131100 ◽  
Author(s):  
Peter Banda ◽  
Christof Teuscher ◽  
Darko Stefanovic
Keyword(s):  
State Of The Art ◽  
Chemical Species ◽  
Mass Action ◽  
Mass Action Kinetics ◽  
Dna Strand Displacement ◽  
Autonomous Adaptation ◽  
Biochemical Systems ◽  
Dna Strand ◽  
Logic Functions ◽  
Application Specific

State-of-the-art biochemical systems for medical applications and chemical computing are application-specific and cannot be reprogrammed or trained once fabricated. The implementation of adaptive biochemical systems that would offer flexibility through programmability and autonomous adaptation faces major challenges because of the large number of required chemical species as well as the timing-sensitive feedback loops required for learning. In this paper, we begin addressing these challenges with a novel chemical perceptron that can solve all 14 linearly separable logic functions. The system performs asymmetric chemical arithmetic, learns through reinforcement and supports both Michaelis–Menten as well as mass-action kinetics. To enable cascading of the chemical perceptrons, we introduce thresholds that amplify the outputs. The simplicity of our model makes an actual wet implementation, in particular by DNA-strand displacement, possible.

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