Constructive polynomial approximation with \textit{a priori} error bounds for nonlinear initial value differential problems
The authors obtain an approximate solution to the initial value problem \[ y'=f(x,y), \quad y(0)=y_0, \] where \(f\) is continuous and is Lipschitzian with respect to each argument when the other is fixed. First, \(f(x,y)\) is approximated by a sequence \(\{B_n(f; x,y)\}\) of Bernstein polynomials in two variables of an appropriate degree according to a prescribed accuracy. Then an exact series solution to \(y'=B_n (f;x,y)\), \(y(0)=y_0\), is constructed by the Frobenius method. Finally, the infinite series solution is truncated in order to obtain an explicit polynomial whose error with respect to the exact solution of the original problem is less than the prescribed accuracy.
scientific article; zbMATH DE number 1429527
scientific article