Ridwanulahi Iyanda Abdulganiy

There are numerous models in the form of ordinary differential equations that require numerical treatment, for they have no closed-form solutions due to some reasons, including non-linearity, stiffness, singularity, and stability. Having in mind the increasing demand for efficient and cost-effective algorithms yet straightforward to code; an attempt is made in this research paper to introduce a new time-efficient optimal exponentially fitted numerical algorithm whose qualitative analysis unfolds its local truncation error, zero-stability, A-stability, consistency, convergence, and accuracy (at least third-order). The proposed algorithm is also implemented in an adaptive step-size mode, whereas both constant and adaptive approaches outperform several existing algorithms when tested upon differential models taken from applied sciences. The efficiency curves further confirm the time-effectiveness of the proposed exponentially fitted one-step optimal method.

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 family of Simpson Block Method (SBM) is proposed for the numerical integration of Delay Differential Equations (DDEs). The methods are developed through multistep collocation technique using constant step width. The convergence and accuracy of the methods are established through some standard DDEs in the reviewed literature. Keywords— Block Method, Collocation Technique, Delay Term, Delay Differential Equation, Self Starting.

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.