One parameter quadratic C¹ -spline collocation method for solving first order ordinary initial value problems

The convergence and stability analysis of a "variable" quadratic C¹-spline collocation method for solving the initial value problem y^′(x)=f(x,y),y(0)=y₀,x∈[0,b] will be considered. Letting the interior (non-nodal) collocation point x_k+β=x_k+βh be dependent on some parameter β∈(0,1], it will be shown that the proposed method is strongly unstable if β<12 and it turns out that the method is a continuous extension of the well-known mid-point and trapezoidal methods, if β=12 and β=1, respectively. Moreover, a wider region of absolute stability is achieved if β→1^-. Error bounds in the uniform norm for s⁽ⁱ⁾-y⁽ⁱ⁾,i=0,1 if y∈C³[0,b], together with illustrative examples will also be presented.