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Zbl 1042.65053
Vigo-Aguiar, JesÃºs; Ramos, Higinio
Dissipative Chebyshev exponential-fitted methods for numerical solution of second-order differential equations.
(English)
[J] J. Comput. Appl. Math. 158, No. 1, 187-211 (2003). ISSN 0377-0427

Chebyshev interpolation is employed to produce an algorithm for $n$th-order approximate soluton of the ordinary oscillatory differential equation $$y''- 2gy'+ (g^2+ w^2)y= f(x,y),\quad y= y(x),\quad x_0\le x\le\infty.\tag1$$ The mapping $s= x+{1\over 2} h(\alpha+ 1)$ takes $-1\le \alpha\le 2\xi-1$ to $x\le s\le x+\xi h$, $\xi\in [0,1]$. Expanding in Chebyshev polynomials in $\alpha$ the solution $y$ of (1) satisfies $$y(x+\xi h)= 2\exp(g\xi h)y(x)\cos(w\xi h)- \exp(2g\,\xi h) y(x-\xi h)+ \sum^\infty_{k=0} (a^+_k R^+_k+ a_k' R^-_k),\tag2$$ $$R^{\pm}_k= (h/2w) \int^{2\xi-1}_{-1} \exp(gh(\xi\mp \textstyle{{1\over 2}}(\alpha+ 1))\,T_k(\alpha)\sin (h(\xi- \textstyle{{1\over 2}} (\alpha+1))\,d\alpha.$$ 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.
[J. B. Butler jun. (Portland)]
MSC 2000:
*65L05 Initial value problems for ODE (numerical methods)
65L60 Finite numerical methods for ODE
34A34 Nonlinear ODE and systems, general

Keywords: second-order ordinary differential equations; oscillatory problems; exponentially fitted methods; numerical examples; comparison of methods; Chebyshev interpolation; algorithm

Cited in: Zbl 1117.65106

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