Conventionally, the method of solving fourth-order initial value problems of an ordinary differential is to first reduce it to a system of first-order differential equations. This approach affects the effectiveness and convergence of the numerical method as a result of the transformation. This paper comprises of the derivation, analysis, and implementation of a new hybrid block method which is derived by collocation and interpolation of an assumed basis function. The basic properties of the block method including zero stability, error constants, consistency, order, and convergence were analyzed. From the analysis, the block method derived was found to be zero-stable, consistent, and convergent. Also, the block method was tested on some numerical examples and the result computed shows that the derived schemes are more accurate than existing methods in the literature.

T α ω n ∗ is the star - like transformation semigroup of the finite set X n . Basic extraction of elements were done using | X n + 1 − λ X n | ≤ | X n − λ X n + 1 | from Full Transformation Semigroup provided n≥1. The five equivalence Green’s relation of star - like semigroup were solved using inline with the general method and definitions were given. The elements of order preserving star-like semigroup in this study were listed from full transformation semigroup, some tables were formed and sequences were created using these tables alongside L ,H, R , J and D equivalence relations. The patterns of the arrangement and tables were examined and also properly generalized .