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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 nth-order approximate soluton of the ordinary oscillatory differential equation

y '' -2gy ' +(g 2 +w 2 )y=f(x,y),y=y(x),x 0 x·(1)

The mapping s=x+1 2h(α+1) takes -1α2ξ-1 to xsx+ξh, ξ[0,1]. Expanding in Chebyshev polynomials in α the solution y of (1) satisfies

y(x+ξh)=2exp(gξh)y(x)cos(wξh)-exp(2gξh)y(x-ξh)+ k=0 (a k + R k + +a k ' R k - ),(2)
R k ± =(h/2w) -1 2ξ-1 exp(gh(ξ1 2(α+1))T k (α)sin(h(ξ-1 2(α+1))dα·

Truncating the series (2) after n terms and choosing ξ=ξ j =1 2(α j +1) leads to an implicit algorithm relating the values y(x±ξ j h) where α j are the extremal nodes of T n (α), j=1,,n. Numerical results are presented for four specific linear examples. These compare well with results obtained by other methods.

65L05Initial value problems for ODE (numerical methods)
65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
34A34Nonlinear ODE and systems, general