Rapid convergence of approximate solutions for first order nonlinear boundary value problems (Q1392699)

Summary: The authors study the convergence of approximate solutions to the first-order problem \[ u'(t)= f(t,u(t));\;t\in[0,T],\;au(0)- bu(t_0)= c, a, b\geq 0,\;a+b> 0,\;t_0\in (0,T]. \] Here, \(f: I\times \mathbb{R}\to\mathbb{R}\) is such that \({\partial^kf\over \partial u^k}\) exists and is a continuous function for some \(k\geq 1\). Under some additional conditions on \({\partial f\over\partial u}\), the authors prove that it is possible to construct two sequences of approximate solutions converging to a solution with rate of convergence of order \(k\).