Dissipative Chebyshev exponential-fitted methods for numerical solution of second-order differential equations.
type: Item
instance of: scholarly article
title: Dissipative Chebyshev exponential-fitted methods for numerical solution of second-order differential equations.
zbMATH Open document ID: 1042.65053
DOI: 10.1016/S0377-0427(03)00473-4
author: Higinio Ramos
author: Jesús Vigo-Aguiar
published in: Journal of Computational and Applied Mathematics
publication date: 2003-09-15T00:00:00Z
review text: Chebyshev interpolation is employed to produce an algorithm for \(n\)th-order approximate soluton of the ordinary oscillatory differential equation \[ y''- 2gy'+ (g^2+ w^2)y= f(x,y),\quad y= y(x),\quad x_0\leq x\leq\infty.\tag{1} \] The mapping \(s= x+{1\over 2} h(\alpha+ 1)\) takes \(-1\leq \alpha\leq 2\xi-1\) to \(x\leq s\leq x+\xi h\), \(\xi\in [0,1]\). Expanding in Chebyshev polynomials in \(\alpha\) the solution \(y\) of (1) satisfies \[ y(x+\xi h)= 2\exp(g\xi h)y(x)\cos(w\xi h)- \exp(2g\,\xi h) y(x-\xi h)+ \sum^\infty_{k=0} (a^+_k R^+_k+ a_k' R^-_k),\tag{2} \] \[ R^{\pm}_k= (h/2w) \int^{2\xi-1}_{-1} \exp(gh(\xi\mp \textstyle{{1\over 2}}(\alpha+ 1))\,T_k(\alpha)\sin (h(\xi- \textstyle{{1\over 2}} (\alpha+1))\,d\alpha. \] Truncating the series (2) after \(n\) terms and choosing \(\xi= \xi_j= {1\over 2}(\alpha_j+ 1)\) leads to an implicit algorithm relating the values \(y(x\pm \xi_j h)\) where \(\alpha_j\) are the extremal nodes of \(T_n(\alpha)\), \(j= 1,\dots, n\). Numerical results are presented for four specific linear examples. These compare well with results obtained by other methods.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L60
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 1981531
zbMATH Keywords: second-order ordinary differential equations
zbMATH Keywords: oscillatory problems
zbMATH Keywords: exponentially fitted methods
zbMATH Keywords: numerical examples
zbMATH Keywords: comparison of methods
zbMATH Keywords: Chebyshev interpolation
zbMATH Keywords: algorithm
reviewed by: John B. Butler jun.
MaRDI profile type: Publication
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full work available at URL: s0377-0427(03)00473-4
OpenAlex ID: W2007665571
description: scientific article; zbMATH DE number 1981531