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

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
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.

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

Symmetry ◽  
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.

scholarly journals On Bernstein Polynomials Method to the System of Abel Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
A. Jafarian ◽  
S. Measoomy Nia ◽  
Alireza K. Golmankhaneh ◽  
D. Baleanu
Keyword(s):  
Linear System ◽  
Numerical Solution ◽  
Integral Equations ◽  
Bernstein Polynomials ◽  
Singular System ◽  
Special Kind ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Abel Integral Equations ◽  
The Given

This paper deals with a new implementation of the Bernstein polynomials method to the numerical solution of a special kind of singular system. For this aim, first the truncated Bernstein series polynomials of the solution functions are substituted in the given problem. Using some properties of these polynomials, the solution of the problem is reduced to solve a linear system of algebraic equations. In order to confirm the reliability and accuracy of the proposed method, some weakly Abel integral equations systems with comparisons are solved in detail as numerical examples.

Bernstein polynomials for solving fractional heat- and wave-like equations

2012 ◽  
Vol 15 (4) ◽  
Author(s):  
Davood Rostamy ◽  
Kobra Karimi
Keyword(s):  
Numerical Analysis ◽  
Numerical Solution ◽  
Fractional Derivatives ◽  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Fractional Differential

AbstractIn this paper, a novel numerical analysis is introduced and performed to obtain the numerical solution of the fractional heat- and wave-like equations. A general formulation for the Bernstein fractional derivatives operational matrix is given. In this approach, a truncated Bernstein series together with the Bernstein operational matrix of fractional derivatives are used to reduce the solution of fractional differential problems to the solution of a system of algebraic equations. Numerical examples are considered aiming to demonstrate the validity and applicability of the proposed techniques and to compare with the existing results.

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

2015 ◽  
Vol 4 (3) ◽  
pp. 420 ◽  
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>

scholarly journals Numerical solution of linear stochastic Volterra integral equations via new basis functions

Filomat ◽  
2019 ◽  
Vol 33 (18) ◽  
pp. 5959-5966
Author(s):  
Tofigh Cheraghi ◽  
Morteza Khodabin ◽  
Reza Ezzati
Keyword(s):  
Linear System ◽  
Numerical Solution ◽  
Integral Equations ◽  
Orthogonal Basis ◽  
Volterra Integral Equations ◽  
Basis Functions ◽  
New Method ◽  
Algebraic Equations ◽  
Numerical Examples

In this article, we use a new method based on orthogonal basis functions for the numerical solution of stochastic Volterra integral equations of the second kind (SVIE). By using this method, a SVIE can be reduced to a linear system of algebraic equations. Finally, to show the efficiency of the proposed method, we give two numerical examples.

scholarly journals Numerical Solution of Nonlinear Fredholm Integrodifferential Equations by Hybrid of Block-Pulse Functions and Normalized Bernstein Polynomials

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
S. H. Behiry
Keyword(s):  
Numerical Solution ◽  
Numerical Method ◽  
Present Method ◽  
Bernstein Polynomials ◽  
Integrodifferential Equations ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Hybrid Functions ◽  
Block Pulse Functions ◽  
Nonlinear Algebraic Equations

A numerical method for solving nonlinear Fredholm integrodifferential equations is proposed. The method is based on hybrid functions approximate. The properties of hybrid of block pulse functions and orthonormal Bernstein polynomials are presented and utilized to reduce the problem to the solution of nonlinear algebraic equations. Numerical examples are introduced to illustrate the effectiveness and simplicity of the present method.

Numerical solution for the fractional-order one-dimensional telegraph equation via wavelet technique

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kumbinarasaiah Srinivasa ◽  
Hadi Rezazadeh
Keyword(s):  
Analytical Solution ◽  
Numerical Solution ◽  
Convergence Analysis ◽  
Collocation Method ◽  
Fractional Order ◽  
Numerical Technique ◽  
Telegraph Equation ◽  
Algebraic Equations ◽  
One Dimensional ◽  
Wavelet Collocation Method

AbstractIn this article, we proposed an efficient numerical technique for the solution of fractional-order (1 + 1) dimensional telegraph equation using the Laguerre wavelet collocation method. Some examples are illustrated to inspect the efficiency of the proposed technique and convergence analysis is discussed in terms of a theorem. Here, the fractional-order telegraph equation is converted into a system of algebraic equations using the properties of the Laguerre wavelet, and solutions obtained by the proposed scheme are more accurate and they are compared with the analytical solution and other method existed in the literature.

scholarly journals Optimal Solution of the Fractional Order Breast Cancer Competition Model

2021 ◽  
Author(s):  
H. Hassani ◽  
J. A. Tenreiro Machado ◽  
Zakieh Avazzadeh ◽  
Elaheh Safari ◽  
S. Mehrabi
Keyword(s):  
Breast Cancer ◽  
Fractional Order ◽  
Legendre Polynomials ◽  
General Procedure ◽  
Optimal Solution ◽  
Competition Model ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Multipliers Method

Abstract This paper discusses the fractional order breast cancer competition model (F-BCCM), which considers population dynamics among cancer stem, tumor and healthy cells, as well as the effects of excess estrogen and the body’s natural immune response on the cell populations. Generalized shifted Legendre polynomials and their operational matrices are presented in the scope of a general procedure for the solution of the F-BCCM. The application of the Lagrange multipliers method transforms the F-BCCM into a system of algebraic equations. Additionally, the convergence analysis of the method and two illustrative numerical examples complement the study.Mathematics Subject Classification: 97M60; 41A58; 92C42.

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

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Hasib Khan ◽  
Hossein Jafari ◽  
Rahmat Ali Khan ◽  
Haleh Tajadodi ◽  
Sarah Jane Johnston
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Bernstein Polynomials ◽  
Operational Matrices ◽  
Algebraic Equations

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.

scholarly journals NUMERICAL TREATMENT OF THE SPACE–TIME FRACTAL–FRACTIONAL MODEL OF NONLINEAR ADVECTION–DIFFUSION–REACTION EQUATION THROUGH THE BERNSTEIN POLYNOMIALS

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040001 ◽  
Author(s):  
M. H. HEYDARI ◽  
Z. AVAZZADEH ◽  
Y. YANG
Keyword(s):  
Numerical Solution ◽  
Fractional Derivatives ◽  
Bernstein Polynomials ◽  
Space Time ◽  
Diffusion Reaction ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Reaction Equation ◽  
System A ◽  
Advection Diffusion

In this paper, the nonlinear space–time fractal–fractional advection–diffusion–reaction equation is introduced and a highly accurate methodology is presented for its numerical solution. In the time direction, the fractal–fractional derivative in the Atangana–Riemann–Liouville concept is utilized whereas the fractional derivatives in the Caputo and Atangana–Baleanu–Caputo senses are mutually used in the space variable to define this new class of problems. The presented method utilizes the Bernstein polynomials (BPs) and their operational matrices of fractional and fractal–fractional derivatives (which are generated in this study). To this end, the unknown solution is expanded by the BP and is replaced in the equation. Then, the generated operational matrices and the collocation method are employed to generate a system of algebraic equations. Eventually, by solving this system a numerical solution is obtained for the problem. The validity of the designed method is investigated through three numerical examples.

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