Dzyadyk's technique for ordinary differential equations using Hermitian interpolating polynomials (Q2754815)

The author presents a modification of Dzyadyk's approximation-iteration method for solving the ordinary differential equation \(y'=f(x,y)\), where \(f\) is a sufficiently smooth function in the domain \(D=[x_0,x_0+h]\times[y_0-H,y_0+H]\) [\textit{V. K. Dzyadyk}, Approximation methods for solving differential and integral equations. Kiev: Naukova Dumka (1988; Zbl 0708.65067)]. This modification is based on the idea to replace the fundamental interpolating polynomials, used in Dzyadyk's technique, by the Hermitian ones. Namely, for the given nodes \(-1\leq\xi_0\leq\xi_1\leq\cdots \leq\xi_n\leq 1\) the values \(\{y_j\}_{j=0}^n\) of the approximate solution at points \(\{x_j\}_{j=0}^n\) are sought from the nonlinear systemNEWLINE\[NEWLINEy_j=y_0+\frac{h}{2}\sum_{i=0}^{n}a_{ij}f(x_i,y_i)+ \frac{h^2}{4}\sum_{i=0}^{n}b_{ij}[f_x'+ff_y'](x_i,y_i), \quad j=\overline{0,n},\tag{1} NEWLINE\]NEWLINE whereNEWLINE\[NEWLINEa_{ij}=\int_{-1}^{\xi_i}h_{2n+1,i}(\xi) d\xi,\quad b_{ij}=\int_{-1}^{\xi_i}\overline{h}_{2n+1,i}(\xi) d\xi,NEWLINE\]NEWLINE and \(h_{2n+1}(\xi), \overline{h}_{2n+1}(\xi)\) are the fundamental Hermitian polynomials. Explicit formula for the elements \(a_{ij}, b_{ij} \) are found in the cases where \(\xi_j=-\cos(j\pi /n)\) and \(\xi_j=-\cos((2j+1)\pi /(2n+2)\). NEWLINENEWLINENEWLINESystem (1) is solved by using simple successive iterations or Newton-Raphson iterations. An error estimation for this procedure is obtained. Examples are considered.