One parameter quadratic C~1-spline collocation method for solving first order ordinary initial value problems

摘要

The convergence and stability analysis of a "variable" quadratic C~1-spline collocation method for solving the initial value problem y'(x)=f(x,y), y(0)=yo,x∈[o,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 β1:2 and it turns out that the method is a continuous extension of the well-known mid-point and trapezoidal methods, if?=1:2 and β=1, respectively. Moreover, a wide 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∈~3[o,b], together with illustrative examples will also be presented.