Efficient Jacobi-Gauss collocation method for solving initial value problems of Bratu type

2013 ◽  
Vol 53 (9) ◽  
pp. 1292-1302 ◽  
Author(s):  
E. H. Doha ◽  
A. H. Bhrawy ◽  
D. Baleanu ◽  
R. M. Hafez
Keyword(s):  
Collocation Method ◽  
Initial Value Problems ◽  
Initial Value

A Multiple Interval Chebyshev-Gauss-Lobatto Collocation Method for Ordinary Differential Equations

2016 ◽  
Vol 9 (4) ◽  
pp. 619-639 ◽  
Author(s):  
Zhong-Qing Wang ◽  
Jun Mu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
High Order Accuracy ◽  
Nonlinear Ordinary Differential Equations ◽  
Initial Value ◽  
Order Accuracy ◽  
Long Time

AbstractWe introduce a multiple interval Chebyshev-Gauss-Lobatto spectral collocation method for the initial value problems of the nonlinear ordinary differential equations (ODES). This method is easy to implement and possesses the high order accuracy. In addition, it is very stable and suitable for long time calculations. We also obtain thehp-version bound on the numerical error of the multiple interval collocation method underH1-norm. Numerical experiments confirm the theoretical expectations.

scholarly journals A Multiple-Step Legendre-Gauss Collocation Method for Solving Volterra’s Population Growth Model

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Majid Tavassoli Kajani ◽  
Mohammad Maleki ◽  
Adem Kılıçman
Keyword(s):  
Population Growth ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Initial Conditions ◽  
Algebraic Equations ◽  
Infinite Domain ◽  
Initial Value ◽  
Step Size ◽  
Integral Term ◽  
Multiple Step

A new shifted Legendre-Gauss collocation method is proposed for the solution of Volterra’s model for population growth of a species in a closed system. Volterra’s model is a nonlinear integrodifferential equation on a semi-infinite domain, where the integral term represents the effects of toxin. In this method, by choosing a step size, the original problem is replaced with a sequence of initial value problems in subintervals. The obtained initial value problems are then step by step reduced to systems of algebraic equations using collocation. The initial conditions for each step are obtained from the approximated solution at its previous step. It is shown that the accuracy can be improved by either increasing the collocation points or decreasing the step size. The method seems easy to implement and computationally attractive. Numerical findings demonstrate the applicability and high accuracy of the proposed method.

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.

A Collocation Method for Direct Numerical Integration of Initial Value Problems in Higher Order Ordinary Differential Equations

2011 ◽  
Vol 57 (2) ◽  
pp. 311-321
Author(s):  
R. Adeniyi ◽  
M. Alabi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Higher Order ◽  
Initial Value ◽  
Integration Schemes ◽  
Collocation Technique ◽  
Direct Numerical Integration

A Collocation Method for Direct Numerical Integration of Initial Value Problems in Higher Order Ordinary Differential Equations This paper concerns the solution of initial value problems (IVPs) in ordinary differential equations (ODEs) of orders higher than unity. The Chebyshev polynomials is hereby adopted as basis function in a multi-step collocation technique for the derivation of continuous integration schemes for direct solution of these ODEs without recourse to the conventional approach of first reducing such to their equivalent first order differential systems.

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.

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