High accuracy quintic spline solution of fourth order two-point boundary value problems

Abstract

It has been known for a long time that quintic spline collocation for fourth order two-point boundary value problems: y ⁽⁴⁾+p(x)y=q(x), a<x<b, y(a)=α ₀,y(b)=α ₁, y′(a)=β ₀, y′(b)=β ₁, provides only 0(h ²) approximations; an order h ² method using quintic splines was described by Usmani [8]. But quintic spline interpolation is an 0(h ⁶) process and therefore it is natural to look for an alternative method using quintic splines which would provide sixth order approximations. In the present paper we present a method using quintic splines which provides 0(h ⁶) uniformly convergent approximations for the solution of fourth order two-point boundary value problems.

Keywords:

• Two-point boundary value problems
• fourth order differential equations
• quintic spline
• collocation