A simple NSC-RKHS algorithm for solving linear boundary value problems – Error bounds and implementations

Abstract

The simple algorithm is in essence a normal spline collocation method introduced by V.K. Gorbunov in 1989. The paper presents a reproducing kernel Hilbert space approach to the method, that gives rise to basic error inequalities. For a natural H2-class of mth order linear problems the approximation of any normal solution v is Cm-1-uniform, i.e. v(a)(t)-vn(a)(t)⩽caΔn2‖v‖Hm+2, for a<m, where Δn is diameter of the partition of the domain by nodes. Simplicity of the algorithm in Sobolev spaces is justified by explicit formulas for the kernel functions – discovered by Gorbunov and V.V. Petrishchev in 2003 and underestimated in the opinion of the author. As an application, numerical solutions in H5 and in H6 are compared with those computed by other authors. The simple method is then applied to a fifth-order nonlocal boundary value problem – in H6 and in H7.