Block hybrid linear multistep method was proposed to overcome the Dahl Quist order barrier for linear multistep methods. This research aims to answer questions relating to the convergence, accuracy, and effectiveness of the block hybrid method when utilized to obtain the solution of Initial Value Problems (IVPs). In this research, an order (k+3) block hybrid method applicable to obtain the direct solution of IVP’s of ordinary differential equations (ODEs) is presented. Collocation and interpolation of power series at finely selected grid points were used to improve the method’s consistency, convergence, accuracy and zero stability. Linear problems were solved to show the accuracy and efficiency of the proposed method, and the error obtained from the comparison of exact and approximate results shows that the proposed method is effective in solving the class of problem.
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 .
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.
Conventionally, the method of solving fourth order initial value problems of ordinary differential is to first reduce it to a system of first order differential equation. 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 literature.
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