The Cauchy problem for singularly perturbed weakly nonlinear second-order differential equations: an iterative method (Q1706036)

This paper proposes an algorithm for constructing an asymptotic sequence (approximate solution) for the Cauchy problem for a second-order singularly perturbed weakly nonlinear differential equations. The constructed sequence converges to the classical solution of the Cauchy problem, when the diffusion parameter \(\varepsilon \to 0\). To prove the convergence, the Banach fixed-point theorem for contraction maps of complete metric spaces is used. This is an asymptotical sequence because the deviation (in the sense of the norm of the space of continuous functions) of its \(n\)th element from the solution to the problem is proportional to the \((n + 1)\)th power of the perturbation parameter.