scholarly journals Rational Chebyshev functions with new collocation points in semi-infinite domains for solving higher-order linear ordinary differential equations

2015 ◽  
Vol 11 (7) ◽  
pp. 5403-5410 ◽  
Author(s):  
Mohamed Abdel -Latif Ramadan
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
Algebraic Equations ◽  
Infinite Domain ◽  
Order Linear ◽  
Linear Ordinary Differential Equations ◽  
Infinite Domains ◽  
Collocation Points ◽  
Rational Chebyshev

The purpose of this paper is to investigate the use of rational Chebyshev (RC) functions for solving higher-order linear ordinary differential equations with variable coefficients on a semi-infinite domain using new rational Chebyshev collocation points.  This method transforms the higher-order linear ordinary differential equations and the given conditions to matrix equations with unknown rational Chebyshev coefficients. These matrices together with the collocation method are utilized to reduce the solution of higher-order ordinary differential equations to the solution of a system of algebraic equations. The solution is obtained in terms of RC series. Numerical examples are given to demonstrate the validity and applicability of the method. The obtained numerical results are compared with others existing methods and the exact solution where it shown to be very attractive and maintains better accuracy.

scholarly journals A Comparison Study of Numerical Techniques for Solving Ordinary Differential Equations Defined on a Semi-Infinite Domain Using Rational Chebyshev Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Mohamed A. Ramadan ◽  
Taha Radwan ◽  
Mahmoud A. Nassar ◽  
Mohamed A. Abd El Salam
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Operational Matrices ◽  
Comparison Study ◽  
Infinite Domain ◽  
Linear Ordinary Differential Equations ◽  
Infinite Domains ◽  
Collocation Technique ◽  
Rational Chebyshev Functions ◽  
Rational Chebyshev

A rational Chebyshev (RC) spectral collocation technique is considered in this paper to solve high-order linear ordinary differential equations (ODEs) defined on a semi-infinite domain. Two definitions of the derivative of the RC functions are introduced as operational matrices. Also, a theoretical study carried on the RC functions shows that the RC approximation has an exponential convergence. Due to the two definitions, two schemes are presented for solving the proposed linear ODEs on the semi-infinite interval with the collocation approach. According to the convergence of the RC functions at the infinity, the proposed technique deals with the boundary value problem which is defined on semi-infinite domains easily. The main goal of this paper is to present a comparison study for differential equations defined on semi-infinite intervals using the proposed two schemes. To demonstrate the validity of the comparisons, three numerical examples are provided. The obtained numerical results are compared with the exact solutions of the proposed problems.

Solving systems of high-order linear ordinary differential equations with variable coefficients by means of exponential Chebyshev collocation method

2017 ◽  
Vol 8 (1-2) ◽  
pp. 40 ◽  
Author(s):  
Mohamed Ramadan ◽  
Kamal Raslan ◽  
Talaat El Danaf ◽  
Mohamed A. Abd Elsalam
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Collocation Method ◽  
Variable Coefficients ◽  
High Order ◽  
Matrix Equations ◽  
Algebraic Equations ◽  
Order Linear ◽  
Linear Ordinary Differential Equations ◽  
Linear Algebraic Equations

The purpose of this paper is to investigate the use of exponential Chebyshev (EC) collocation method for solving systems of high-order linear ordinary differential equations with variable coefficients with new scheme, using the EC collocation method in unbounded domains. The EC functions approach deals directly with infinite boundaries without singularities. The method transforms the system of differential equations and the given conditions to block matrix equations with unknown EC coefficients. By means of the obtained matrix equations, a new system of equations which corresponds to the system of linear algebraic equations is gained. Numerical examples are given to illustrative the validity and applicability of the method.

Remark on zeros of solutions of second-order linear ordinary differential equations

2016 ◽  
Vol 23 (4) ◽  
pp. 571-577
Author(s):  
Monika Dosoudilová ◽  
Alexander Lomtatidze
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Nontrivial Solution ◽  
Unique Solvability ◽  
Periodic Boundary ◽  
Periodic Boundary Value Problem ◽  
Order Linear ◽  
Linear Ordinary Differential Equations ◽  
Zeros Of Solutions

AbstractAn efficient condition is established ensuring that on any interval of length ω, any nontrivial solution of the equation ${u^{\prime\prime}=p(t)u}$ has at most one zero. Based on this result, the unique solvability of a periodic boundary value problem is studied.

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