Estimates of solutions of two-point problems for systems of first-order ordinary differential equations and the method of lines

Abstract

In the paper estimates are established on the solution of systems of first-order differential equations

subject to two-point conditions of the form

which enable us, in particular, to obtain an estimate of the order of uniform convergence of the method of lines for solving periodic boundary-value problems for second-order nonlinear parabolic partial differential equations of form

For problems with boundary conditions containing the derivatives

where the functions

satisfy, in a small neighborhood of the solutionu(t,x) being examined of Eq. (3), the inequalities

, for which the approximation is only of the first order relative to the net steph, the uniform convergence of the approximate solutions to the exact one is established to the second order relative toh.