The method of successive interpolations solving initial value problems for second order functional differential equations (Q2871100)

Consider the initial value problem: NEWLINE\[NEWLINE \begin{aligned} &x''(t) = f(t, x(t), x(\varphi(t))), \quad t\in [0, a ],\\& x(0)= x_0, \quad x'(0)= \nu_0, \end{aligned}NEWLINE\]NEWLINE where \( a>0, x_0, \nu_0 \in \mathbb{R} \) and \( \varphi: [0,a]\rightarrow \mathbb{R} \) such that \( 0\leq \varphi(t)\leq a\) for all \(t \in [0,a] \). The authors present a new numerical method to solve the initial value problem. The introduced method bases on Picard's sequence of successive approximations and improvements of the first author's previous work.