Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Extended two-step P-stable methods for periodic initial-value problems
Neural, Parallel & Scientific Computations
Quintic C2-spline integration initial value equation problems
Journal of Computational and Applied Mathematics - Proceedings of the 8th international congress on computational and applied mathematics
On some 4-point spline collocation methods for solving second-order initial value problems
Applied Numerical Mathematics
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We propose global collocation methods for second-order initial-value problems y''=f(x, y) and y''=f(x, y, y'). The present methods are based on quintic C2-splines S(x) with three collocation points [image omitted] , j=1,..., 3 in each subinterval [xi-1, xi], i=1,..., N. It is shown that the method (c1=(5-√5)/10, c2=(5+√ 5)/10) has a convergence of order six, while in the remaining cases (c1, c2∈(0, 1), with c1≠c2) the order is five. The absolute stability properties appear that for all c1, c2∈[0.8028, 1) with c1≠c2, the methods are A-stable independent of the particular choice of the collocation points, while the sixth-order method has a large region of absolute stability. Moreover, the sixth-order method has a phase-lag of order six with actual phase-lag (3/25(8!))v6, and it possesses (0, 37.5)∪(60, 122.178) as the interval of periodicity and absolute stability. The superiority of the obtained methods is demonstrated by considering periodic stiff problems of practical interest.