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.


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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