Collocation methods based on quintic splines are fonnulated and analyzed for the secondorder two-point boundary value problems with mixed boundary conditions. These methods determine quintic spline approximation to the solution of the boundary value problem. by forcing lhe approximating solution to satisfy the given operator equation, or a perturbed one at the nodes, the boundary conditions and auxiliary end conditions. The methods that are based on the initial operator equation produce not optimal approximation. as compared to lhe corresponding interpolation procedures. This paper derives appropriate perturbations of lhe initial differential equation, such. that the application of the collocation procedure leads 10 optimal approximating schemes. The theoretical behavior of the method has been verified numerically on a variety of benchmark problems found in !.he literature. L INTRODUCTION In !.his paper we consider a collocation of the solul..ion u of the second order two-point boWldary problem Lu == 02U(S) + pes) Du(s) + q(s) u(s) '" [(s) , aSs S b • subject to boWldary conditions '. . Bu == L (Oij DJu(a) + bij OJu(b» = g;, i == O. I i"' (Ua)
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