## Dissipative Chebyshev exponential-fitted methods for numerical solution of second-order differential equations.(English)Zbl 1042.65053

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.

### MSC:

 65L05 Numerical methods for initial value problems involving ordinary differential equations 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems
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