Dissipative Chebyshev exponential-fitted methods for numerical solution of second-order differential equations.
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.
second-order ordinary differential equations
scientific article; zbMATH DE number 1981531
scientific article