Rapid convergence of approximate solutions for first order nonlinear boundary value problems - Alberto Cabada; Juan J. Nieto; Seppo Heikkilä (International Journal of Mathematics and Mathematical Sciences)
In this paper we study the convergence of the approximate solutions for the following first order problem u′(t)=f(t,u(t));t∈[0,T],au(0)−bu(t0)=c,a,b≥0,t0∈(0,T]. Here f:I×ℝ→ℝ is such that ∂kf∂uk exists and is a continuous function for some k≥1. Under some additional conditions on ∂f∂u, we prove that it is possible to construct two sequences of approximate solutions converging to a solution with rate of convergence of order k.