Numerical Solutions of the Nonlinear Fractional-Order Brusselator System by Bernstein Polynomials

In this paper we propose the Bernstein polynomials to achieve the numerical solutions of nonlinear fractional-order chaotic system known by fractional-order Brusselator system. We use operational matrices of fractional integration and multiplication of Bernstein polynomials, which turns the nonlinear fractional-order Brusselator system to a system of algebraic equations. Two illustrative examples are given in order to demonstrate the accuracy and simplicity of the proposed techniques.

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

Fractal and Fractional ◽

10.3390/fractalfract5010008 ◽

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.

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The Bernstein Operational Matrices for Solving the Fractional Quadratic Riccati Differential Equations with the Riemann-Liouville Derivative

Abstract and Applied Analysis ◽

10.1155/2013/461970 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7 ◽

Cited By ~ 8

Author(s):

Dumitru Baleanu ◽

Mohsen Alipour ◽

Hossein Jafari

Keyword(s):

Differential Equation ◽

Approximate Analytical Solution ◽

Fractional Integration ◽

Bernstein Polynomials ◽

Operational Matrix ◽

Operational Matrices ◽

Algebraic Equations ◽

Riccati Differential Equation ◽

Riccati Differential Equations ◽

Liouville Derivative

We obtain the approximate analytical solution for the fractional quadratic Riccati differential equation with the Riemann-Liouville derivative by using the Bernstein polynomials (BPs) operational matrices. In this method, we use the operational matrix for fractional integration in the Riemann-Liouville sense. Then by using this matrix and operational matrix of product, we reduce the problem to a system of algebraic equations that can be solved easily. The efficiency and accuracy of the proposed method are illustrated by several examples.

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Numerical Solution of Time-Fractional Order Telegraph Equation by Bernstein Polynomials Operational Matrices

Mathematical Problems in Engineering ◽

10.1155/2016/1683849 ◽

2016 ◽

Vol 2016 ◽

pp. 1-6 ◽

Cited By ~ 5

Author(s):

M. Asgari ◽

R. Ezzati ◽

T. Allahviranloo

Keyword(s):

Linear System ◽

Numerical Solution ◽

Fractional Order ◽

Original Problem ◽

Bernstein Polynomials ◽

Telegraph Equation ◽

Operational Matrices ◽

Algebraic Equations ◽

Numerical Examples ◽

Fractional Differential

We present a new method to solve time-fractional order telegraph equation (TFOTE) by using Bernstein polynomials. By implementation of Bernstein polynomials operational matrices of fractional differential on TFOTE, we reduce the original problem to a linear system of algebraic equations. Also, we prove the convergence analysis. In order to show the efficiency of the proposed method, we present two numerical examples.

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Approximate Solutions of Differential Equations by Using the Bernstein Polynomials

ISRN Applied Mathematics ◽

10.5402/2011/787694 ◽

2011 ◽

Vol 2011 ◽

pp. 1-15 ◽

Cited By ~ 5

Author(s):

Y. Ordokhani ◽

S. Davaei far

Keyword(s):

Differential Equations ◽

Bernstein Polynomial ◽

Numerical Solutions ◽

Bernstein Polynomials ◽

Approximate Solutions ◽

High Accuracy ◽

Operational Matrices ◽

Algebraic Equations ◽

Linear Algebraic Equations ◽

Bernstein Polynomial Basis

A numerical method for solving differential equations by approximating the solution in the Bernstein polynomial basis is proposed. At first, we demonstrate the relation between the Bernstein and Legendre polynomials. By using this relation, we derive the operational matrices of integration and product of the Bernstein polynomials. Then, we employ them for solving differential equations. The method converts the differential equation to a system of linear algebraic equations. Finally some examples and their numerical solutions are given; comparing the results with the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method.

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CONTINUOUS-TIME FRACTIONAL ORDER LINEAR SYSTEMS IDENTIFICATION USING CHEBYSHEV WAVELET

EurasianUnionScientists ◽

10.31618/esu.2413-9335.2020.6.77.1002 ◽

2020 ◽

Vol 6 (8(77)) ◽

pp. 23-28

Author(s):

Shuen Wang ◽

Ying Wang ◽

Yinggan Tang

Keyword(s):

Linear Systems ◽

Fractional Order ◽

Continuous Time ◽

Fractional Integration ◽

Operational Matrices ◽

Computation Complexity ◽

Order Linear ◽

Fractional Integration Operator ◽

Systems Identification ◽

Chebyshev Wavelet

In this paper, the identification of continuous-time fractional order linear systems (FOLS) is investigated. In order to identify the differentiation or- ders as well as parameters and reduce the computation complexity, a novel identification method based on Chebyshev wavelet is proposed. Firstly, the Chebyshev wavelet operational matrices for fractional integration operator is derived. Then, the FOLS is converted to an algebraic equation by using the the Chebyshev wavelet operational matrices. Finally, the parameters and differentiation orders are estimated by minimizing the error between the output of real system and that of identified systems. Experimental results show the effectiveness of the proposed method.

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Two matrix methods for solution of nonlinear and linear Lane-Emden type equations with mixed condition by operational matrix

International Journal of Applied Mathematical Research ◽

10.14419/ijamr.v4i3.4052 ◽

2015 ◽

Vol 4 (3) ◽

pp. 420 ◽

Cited By ~ 1

Author(s):

Behrooz Basirat ◽

Mohammad Amin Shahdadi

Keyword(s):

Bernstein Polynomials ◽

Operational Matrix ◽

Numerical Procedure ◽

Operational Matrices ◽

Algebraic Equations ◽

Mixed Condition ◽

Matrix Methods ◽

Numerical Examples ◽

Linear Algebraic Equations ◽

The Given

<p>The aim of this article is to present an efficient numerical procedure for solving Lane-Emden type equations. We present two practical matrix method for solving Lane-Emden type equations with mixed conditions by Bernstein polynomials operational matrices (BPOMs) on interval [<em>a; b</em>]. This methods transforms Lane-Emden type equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equations. We also give some numerical examples to demonstrate the efficiency and validity of the operational matrices for solving Lane-Emden type equations (LEEs).</p>

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

Fractal and Fractional ◽

10.3390/fractalfract6010010 ◽

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.

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Using Fractional Bernoulli Wavelets for Solving Fractional Diffusion Wave Equations with Initial and Boundary Conditions

Fractal and Fractional ◽

10.3390/fractalfract5040212 ◽

2021 ◽

Vol 5 (4) ◽

pp. 212

Author(s):

Monireh Nosrati Sahlan ◽

Hojjat Afshari ◽

Jehad Alzabut ◽

Ghada Alobaidi

Keyword(s):

Fractional Order ◽

Fractional Derivatives ◽

Wave Equations ◽

Bernoulli Polynomials ◽

Fractional Diffusion ◽

Computational Time ◽

Operational Matrices ◽

Algebraic Equations ◽

Diffusion Wave ◽

Klein Gordon

In this paper, fractional-order Bernoulli wavelets based on the Bernoulli polynomials are constructed and applied to evaluate the numerical solution of the general form of Caputo fractional order diffusion wave equations. The operational matrices of ordinary and fractional derivatives for Bernoulli wavelets are set via fractional Riemann–Liouville integral operator. Then, these wavelets and their operational matrices are utilized to reduce the nonlinear fractional problem to a set of algebraic equations. For solving the obtained system of equations, Galerkin and collocation spectral methods are employed. To demonstrate the validity and applicability of the presented method, we offer five significant examples, including generalized Cattaneo diffusion wave and Klein–Gordon equations. The implementation of algorithms exposes high accuracy of the presented numerical method. The advantage of having compact support and orthogonality of these family of wavelets trigger having sparse operational matrices, which reduces the computational time and CPU requirements.

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An Integral Operational Matrix of Fractional-Order Chelyshkov Functions and Its Applications

Symmetry ◽

10.3390/sym12111755 ◽

2020 ◽

Vol 12 (11) ◽

pp. 1755

Author(s):

M. S. Al-Sharif ◽

A. I. Ahmed ◽

M. S. Salim

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Fractional Differential Equations ◽

Fractional Integral ◽

Operational Matrix ◽

Operational Matrices ◽

Algebraic Equations ◽

Science And Engineering ◽

Numerical Examples ◽

Fractional Differential

Fractional differential equations have been applied to model physical and engineering processes in many fields of science and engineering. This paper adopts the fractional-order Chelyshkov functions (FCHFs) for solving the fractional differential equations. The operational matrices of fractional integral and product for FCHFs are derived. These matrices, together with the spectral collocation method, are used to reduce the fractional differential equation into a system of algebraic equations. The error estimation of the presented method is also studied. Furthermore, numerical examples and comparison with existing results are given to demonstrate the accuracy and applicability of the presented method.

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Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration

Mathematics ◽

10.3390/math8010096 ◽

2020 ◽

Vol 8 (1) ◽

pp. 96 ◽

Cited By ~ 1

Author(s):

İbrahim Avcı ◽

Nazim I. Mahmudov

Keyword(s):

Differential Equations ◽

Numerical Solutions ◽

Fractional Integration ◽

Operational Matrix ◽

Main Idea ◽

System Of Equations ◽

Algebraic Equations ◽

Fractional Order Differential Equations ◽

Fractional Differential ◽

The Given

In this article, we propose a numerical method based on the fractional Taylor vector for solving multi-term fractional differential equations. The main idea of this method is to reduce the given problems to a set of algebraic equations by utilizing the fractional Taylor operational matrix of fractional integration. This system of equations can be solved efficiently. Some numerical examples are given to demonstrate the accuracy and applicability. The results show that the presented method is efficient and applicable.

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