Rapid convergence of approximate solutions for first order nonlinear boundary value problems
Alberto Cabada,1Juan J. Nieto,1and Seppo Heikkilä2
Received20 Aug 1996
Abstract
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.