scholarly journals Solution for the System of Lane–Emden Type Equations Using Chebyshev Polynomials

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 181
Author(s):  
Yalçın ÖZTÜRK
Keyword(s):  
Approximate Solution ◽  
Nonlinear System ◽  
Numerical Method ◽  
Collocation Method ◽  
Chebyshev Polynomials ◽  
Matrix Equation ◽  
Algebraic Equations ◽  
Chebyshev Series ◽  
Solution Function ◽  
The Given

In this paper, we use the collocation method together with Chebyshev polynomials to solve system of Lane–Emden type (SLE) equations. We first transform the given SLE equation to a matrix equation by means of a truncated Chebyshev series with unknown coefficients. Then, the numerical method reduces each SLE equation to a nonlinear system of algebraic equations. The solution of this matrix equation yields the unknown coefficients of the solution function. Hence, an approximate solution is obtained by means of a truncated Chebyshev series. Also, to show the applicability, usefulness, and accuracy of the method, some examples are solved numerically and the errors of the solutions are compared with existing solutions.

scholarly journals A Numerical Method for Lane-Emden Equations Using Hybrid Functions and the Collocation Method

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Changqing Yang ◽  
Jianhua Hou
Keyword(s):  
Differential Equation ◽  
Numerical Method ◽  
Collocation Method ◽  
Chebyshev Polynomials ◽  
Initial Value Problems ◽  
Algebraic Equations ◽  
Initial Value ◽  
Numerical Examples ◽  
Hybrid Functions ◽  
Block Pulse Functions

A numerical method to solve Lane-Emden equations as singular initial value problems is presented in this work. This method is based on the replacement of unknown functions through a truncated series of hybrid of block-pulse functions and Chebyshev polynomials. The collocation method transforms the differential equation into a system of algebraic equations. It also has application in a wide area of differential equations. Corresponding numerical examples are presented to demonstrate the accuracy of the proposed method.

scholarly journals Solution of the Falkner–Skan Equation Using the Chebyshev Series in Matrix Form

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Abdelrady Okasha Elnady ◽  
M. Fayek Abd Rabbo ◽  
Hani M. Negm
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Numerical Method ◽  
Matrix Representation ◽  
Point Of View ◽  
Finite Domain ◽  
Algebraic Equations ◽  
Chebyshev Series ◽  
Infinite Domain ◽  
Computational Point

A numerical method for the solution of the Falkner–Skan equation, which is a nonlinear differential equation, is presented. The method has been derived by truncating the semi-infinite domain of the problem to a finite domain and then expanding the required approximate solution as the elements of the Chebyshev series. Using matrix representation of a function and their derivatives, the problem is reduced to a system of algebraic equations in a simple way. From the computational point of view, the results are in excellent agreement with those presented in published works.

Numerical solution of Riccati equation using operational matrix method with Chebyshev polynomials

2015 ◽  
Vol 08 (02) ◽  
pp. 1550020
Author(s):  
Yalçın Öztürk ◽  
Mustafa Gülsu
Keyword(s):  
Approximate Solution ◽  
Riccati Equation ◽  
Chebyshev Polynomials ◽  
Matrix Method ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Chebyshev Series ◽  
Numerical Examples ◽  
Operational Matrix Method

In this paper, we present numerical technique for solving the Riccati equation by using operational matrix method with Chebyshev polynomials. The method consists of expanding the required approximate solution as truncated Chebyshev series. Using operational matrix method, we reduce the problem to a set of algebraic equations. Some numerical examples are given to demonstrate the validity and applicability of the method. The method is easy to implement and produces very accurate results.

scholarly journals A Novel and Efficient Numerical Algorithm for Solving 2D Fredholm Integral Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
H. Bin Jebreen
Keyword(s):  
Approximate Solution ◽  
Numerical Method ◽  
Convergence Analysis ◽  
Integral Equations ◽  
Numerical Algorithm ◽  
Numerical Experiments ◽  
Operational Matrix ◽  
Fredholm Integral Equations ◽  
Scaling Functions ◽  
Algebraic Equations

A novel and efficient numerical method is developed based on interpolating scaling functions to solve 2D Fredholm integral equations (FIE). Using the operational matrix of integral for interpolating scaling functions, FIE reduces to a set of algebraic equations that one can obtain an approximate solution by solving this system. The convergence analysis is investigated, and some numerical experiments confirm the accuracy and validity of the method. To show the ability of the proposed method, we compare it with others.

scholarly journals Numerical solutions of multi-order fractional differential equations by Boubaker polynomials

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 226-230 ◽  
Author(s):  
A. Bolandtalat ◽  
E. Babolian ◽  
H. Jafari
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Boubaker Polynomials ◽  
Fractional Differential ◽  
The Given

AbstractIn this paper, we have applied a numerical method based on Boubaker polynomials to obtain approximate numerical solutions of multi-order fractional differential equations. We obtain an operational matrix of fractional integration based on Boubaker polynomials. Using this operational matrix, the given problem is converted into a set of algebraic equations. Illustrative examples are are given to demonstrate the efficiency and simplicity of this technique.

scholarly journals A Nonclassical Radau Collocation Method for Nonlinear Initial-Value Problems with Applications to Lane-Emden Type Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Mohammad Maleki ◽  
M. Tavassoli Kajani ◽  
I. Hashim ◽  
A. Kilicman ◽  
K. A. M. Atan
Keyword(s):  
Orthogonal Polynomials ◽  
Numerical Method ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Algebraic Equations ◽  
Initial Value ◽  
Weighted Interpolation ◽  
Collocation Points ◽  
Present Solution ◽  
Nonlinear Algebraic Equations

We propose a numerical method for solving nonlinear initial-value problems of Lane-Emden type. The method is based upon nonclassical Gauss-Radau collocation points, and weighted interpolation. Nonclassical orthogonal polynomials, nonclassical Radau points and weighted interpolation are introduced on arbitrary intervals. Then they are utilized to reduce the computation of nonlinear initial-value problems to a system of nonlinear algebraic equations. We also present the comparison of this work with some well-known results and show that the present solution is very accurate.

CONVERGENCE ANALYSIS OF THE PSEUDOSPECTRAL METHOD FOR LINEAR DAEs OF INDEX-2

2013 ◽  
Vol 10 (04) ◽  
pp. 1350019 ◽  
Author(s):  
F. GHANBARI ◽  
F. GHOREISHI
Keyword(s):  
Approximate Solution ◽  
Chebyshev Polynomials ◽  
Numerical Experiments ◽  
Pseudospectral Method ◽  
Posteriori Error ◽  
Differential Algebraic Equations ◽  
A Posteriori ◽  
Algebraic Equations ◽  
A Posteriori Error ◽  
Index 2

This paper is concerned with the study of pseudospectral discretizations of the differential-algebraic equations (DAEs). Pseudospectral method based on Chebyshev polynomials are used to transcribe a given DAE into a system of algebraic equations. A posteriori error bound between the desired solution to the index 2, DAEs and the pseudospectral approximate solution of the problem is estimated in the weighted L2 norm. Some numerical experiments are considered to demonstrate the efficiency and the applicability of the method.

Approximate solution of Lane-Emden problem via modified Hermite operation matrix method

2021 ◽  
Vol 2 (2) ◽  
pp. 57-67
Author(s):  
Bushra Esaa Kashem ◽  
Suha SHIHAB
Keyword(s):  
Approximate Solution ◽  
Initial Value Problems ◽  
Operational Matrix ◽  
Mathematical Physics ◽  
Initial Value ◽  
Solution Function ◽  
Emden Equation ◽  
Physical Problems ◽  
Operational Matrix Of Integration ◽  
The Given

Lane-Emden equations are singular initial value problems and they are important in mathematical physics and astrophysics. The aim of this present paper is presenting a new numerical method for finding approximate solution to Lane-Emden type equations arising in astrophysics based on modified Hermite operational matrix of integration. The proposed technique is based on taking the truncated modified Hermite series of the solution function in the Lane-Emden equation and then transferred into a matrix equation together with the given conditions. The obtained result is system of linear algebraic equation using collection points. The suggested algorithm is applied on some relevant physical problems as Lane-Emden type equations.

A New Bernoulli Wavelet Operational Matrix of Derivative Method for the Solution of Nonlinear Singular Lane–Emden Type Equations Arising in Astrophysics

2016 ◽  
Vol 11 (5) ◽  
Author(s):  
S. Balaji
Keyword(s):  
Approximate Solution ◽  
Operational Matrix ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Operational Matrix Method ◽  
Wavelet Expansions ◽  
Collocation Points ◽  
Derivative Method ◽  
Bernoulli Wavelet ◽  
The Given

In this paper, a new method is presented for solving generalized nonlinear singular Lane–Emden type equations arising in the field of astrophysics, by introducing Bernoulli wavelet operational matrix of derivative (BWOMD). Bernoulli wavelet expansions together with this operational matrix method, by taking suitable collocation points, converts the given Lane–Emden type equations into a system of algebraic equations. Solution to the problem is identified by solving this system of equations. Further applicability and simplicity of the proposed method has been demonstrated by some examples and comparison with other recent methods. The obtained results guarantee that the proposed BWOMD method provides the good approximate solution to the generalized nonlinear singular Lane–Emden type equations.

A collocation method for numerical solutions of fractional-order logistic population model

2016 ◽  
Vol 09 (02) ◽  
pp. 1650031 ◽  
Author(s):  
Şuayip Yüzbaşı
Keyword(s):  
Approximate Solution ◽  
Fractional Derivative ◽  
Collocation Method ◽  
Fractional Order ◽  
Population Model ◽  
Numerical Solutions ◽  
Model Problem ◽  
Error Function ◽  
Approximate Solutions ◽  
Algebraic Equations

In this study, a collocation technique is presented for approximate solution of the fractional-order logistic population model. Actually, we develop the Bessel collocation method by using the fractional derivative in the Caputo sense to obtain the approximate solutions of this model problem. By means of the fractional derivative in the Caputo sense, the collocation points, the Bessel functions of the first kind, the method transforms the model problem into a system of nonlinear algebraic equations. Numerical applications are given to demonstrate efficiency and accuracy of the method. In applications, the reliability of the scheme is shown by the error function based on the accuracy of the approximate solution.

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