An implicit semi-analytical technique: Development, analysis and applications

Solving differential equations over extended temporal domains is pivotal for many scientific and engineering endeavors. The progression of numerous phenomena in the natural world occurs over prolonged durations, necessitating a comprehensive understanding of their temporal dynamics to facilitate precise prognostications and judicious decision-making. Existing methodologies such as homotopy perturbation, Adomian decomposition, and homotopy analysis are noted for their precision within limited temporal scopes. An optimized decomposition method proposed by Odibat extends this accuracy over a longer term, yet its solutions become untenable over substantial timeframes. This paper proposes a novel semi-analytical method to yield solutions over greater temporal extents. Moreover, this work integrates a convergence control parameter to bolster the proposed method’s accuracy and operational efficiency. A rigorous theoretical convergence analysis is delineated, substantiating the method’s validity. Three illustrative examples are numerically examined and discussed to validate the proposed approaches empirically.