scholarly journals A Legendre Wavelet Spectral Collocation Method for Solving Oscillatory Initial Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
A. Karimi Dizicheh ◽  
F. Ismail ◽  
M. Tavassoli Kajani ◽  
Mohammad Maleki
Keyword(s):  
Differential Equation ◽  
Spectral Method ◽  
Initial Value Problems ◽  
Algebraic Equations ◽  
Initial Value ◽  
Numerical Examples ◽  
Runge Kutta Method ◽  
Collocation Points ◽  
Oscillatory Initial Value Problems ◽  
Exponentially Fitted

In this paper, we propose an iterative spectral method for solving differential equations with initial values on large intervals. In the proposed method, we first extend the Legendre wavelet suitable for large intervals, and then the Legendre-Guass collocation points of the Legendre wavelet are derived. Using this strategy, the iterative spectral method converts the differential equation to a set of algebraic equations. Solving these algebraic equations yields an approximate solution for the differential equation. The proposed method is illustrated by some numerical examples, and the result is compared with the exponentially fitted Runge-Kutta method. Our proposed method is simple and highly accurate.

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 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.

EXPONENTIALLY AND TRIGONOMETRICALLY FITTED EXPLICIT ADVANCED STEP-POINT (EAS) METHODS FOR INITIAL VALUE PROBLEMS WITH OSCILLATING SOLUTIONS

2003 ◽  
Vol 14 (02) ◽  
pp. 175-184 ◽  
Author(s):  
G. PSIHOYIOS ◽  
T. E. SIMOS
Keyword(s):  
Numerical Solution ◽  
Initial Value Problems ◽  
Initial Value ◽  
Numerical Examples ◽  
Oscillating Solutions ◽  
Exponentially Fitted ◽  
Predictor Corrector ◽  
Area Of Application

In this paper, an exponentially fitted and trigonometrically fitted predictor–corrector class of methods is developed. These methods represent a totally new area of application for the explicit advanced step-point or EAS methods developed by Psihoyios and Cash. Numerical examples show that the newly developed procedure is much more efficient than well-known methods for the numerical solution of initial value problems with oscillating solutions.

Ultraspherical Differentiation Method for Solving System of Initial Value Differential Algebraic Equations

2009 ◽  
Vol 9 (3) ◽  
pp. 226-237 ◽  
Author(s):  
M. El-kady ◽  
M.A. Ibrahim
Keyword(s):  
Nonlinear Programming ◽  
Differential Equations ◽  
Spectral Method ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Initial Value ◽  
Numerical Examples ◽  
Ultraspherical Polynomials ◽  
Differentiation Method ◽  
Wide Range

AbstractIn this paper, we introduce a new spectral method based on ultraspherical polynomials for solving systems of initial value differential algebraic equations. Moreover, the suggested method is applicable for a wide range of differential equations. The method is based on a new investigation of the ultraspherical spectral differentiation matrix to approximate the differential expressions in equations. The produced equations lead to algebraic systems and are converted to nonlinear programming. Numerical examples illustrate the robustness, accuracy, and efficiency of the proposed method.

Quasi Trefftz Spectral Method for Stokes Problem

1997 ◽  
Vol 07 (08) ◽  
pp. 1187-1212 ◽  
Author(s):  
S. A. Lifits ◽  
S. Yu. Reutskiy ◽  
G. Pontrelli ◽  
B. Tirozzi
Keyword(s):  
Free Boundary ◽  
Spectral Method ◽  
Initial Value Problems ◽  
Moving Boundary ◽  
Stokes Problem ◽  
Boundary Value ◽  
Elliptic Operators ◽  
Initial Value ◽  
Primitive Variables

A new numerical Quasi Trefftz Spectral Method (QTSM) which was earlier suggested for solving boundary value and initial value problems with elliptic operators is applied to linear stationary hydrodynamic problems. The primitive variables [Formula: see text] are used. The method has been found to work well for different problems, including free boundary ones. The problem of the Stefan type in the domain with moving boundary is also considered. The possibilities of further developments of QTSM are discussed.

scholarly journals FORMULATION OF BLOCK SCHEMES WITH LINEAR MULTISTEP METHOD FOR THE APPROXIMATION OF FIRST-ORDER IVPS

2020 ◽  
Vol 4 (3) ◽  
pp. 313-322
Author(s):  
Sunday Obomeviekome Imoni ◽  
D. I. Lanlege ◽  
E. M. Atteh ◽  
J. O. Ogbondeminu
Keyword(s):  
Basis Function ◽  
Initial Value Problems ◽  
Hermite Polynomials ◽  
Multistep Method ◽  
Linear Multistep Method ◽  
Initial Value ◽  
Numerical Examples ◽  
First Order ◽  
Block Scheme ◽  
Multi Step Methods

ABSTRACT In this paper, formulation of an efficient numerical schemes for the approximation first-order initial value problems (IVPs) of ordinary differential equations (ODE) is presented. The method is a block scheme for some k-step linear multi-step methods (and) using the Hermite Polynomials a basis function. The continuous and discrete linear multi-step methods (LMM) are formulated through the technique of collocation and interpolation. Numerical examples of ODE have been examined and results obtained show that the proposed scheme can be efficient in solving initial value problems of first order ODE.

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