A hybrid second derivative three-step method of order 7 is proposed for solving first order stiff differential equations. The complementary and main methods are generated from a single continuous scheme through interpolation and collocation procedures. The continuous scheme makes it easy to interpolate at off-grid and grid points. The consistency, stability, and convergence properties of the block formula are presented. The hybrid second derivative block backward differentiation formula is concurrently applied to the first order stiff systems to generate the numerical solution that do not coincide in time over a given interval. The numerical results show that the new method compares favorably with some known methods in the literature.

One of the well-known schemes for the direct numerical integration of second-order initial-value problems is due to Falkner. This paper focuses on the construction of a family of adapted block Falkner methods which are frequency dependent for the direct numerical solution of second-order initial value problems with oscillatory solutions. The techniques of collocation and interpolation are adopted here to derive the new methods. The study of the properties of the proposed adapted block Falkner methods reveals that they are consistent and zero-stable, and thus, convergent. Furthermore, the stability analysis and the algebraic order conditions of the proposed methods are established. As may be seen from the numerical results, the resulting family is efficient and competitive compared to some recent methods in the literature.

A second derivative trigonometrically fitted block backward differentiation formula (SDTFBBDF) based on the collocation technique is proposed in this paper. The method is specifically considered for the numerical solution of oscillatory problems. The SDTFBBDF which has one main method and one complementary method as by-products of the continuous second derivative backward differentiation formula (CSDBDF) whose coefficients depend on the frequency and step size. The stability and the convergence of SDTFBBDF are established and the performance of the proposed method is verified on some numerical examples to show its accuracy and efficiency.