Approximation of the solution of the Cauchy problem for nonlinear ODE systems by means of Fourier series in functions orthogonal in the sense of Sobolev

I. I. Sharapudinov

Daghestan Electronic Mathematical Reports, Tome 7 (2017), pp. 66-76.

Résumé

Consider the systems of functions ${\varphi}_{r,n}(x)$ $(r=1,2,\ldots, n=0,1,\ldots)$ orthonormal with respect to a Sobolev-type inner product of the form $\langle f,g\rangle= \sum_{\nu=0}^{r-1}f^{(\nu)}(a)g^{(\nu)}(a)+\int_{a}^{b}f^{(r)}(x)g^{(r)}\rho(x)(x)dx$ generated by a given orthonormal system of functions ${\varphi}_{n}(x)$ $( n=0,1,\ldots)$. It is shown that the Fourier series in the system ${\varphi}_{r,n}(x)$ $(r=1,2,\ldots, n=0,1,\ldots)$ and their partial sums are a convenient and very effective tool for the approximate solution of the Cauchy problem for ordinary differential equations (ODEs).

Détail

Keywords: the Cauchy problem, Fourier series, Sobolev orthogonal functions.

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