A new family of 𝒜− acceptable nonlinear methods with fixed and variable stepsize approach

Abstract

Solving stiff, singular, and singularly perturbed initial value problems (IVPs) has always been challenging for researchers working in different fields of science and engineering. In this research work, an attempt is made to devise a family of nonlinear methods among which second‐ to fourth‐order methods are not only 𝒜− stable but 𝒜− acceptable as well under order stars' conditions. These features make them more suitable for solving stiff and singular systems in ordinary differential equations. Methods with remaining orders are either zero‐ or conditionally stable. The theoretical analysis contains local truncation error, consistency, and order of accuracy of the proposed nonlinear methods. Furthermore, both fixed and variable stepsize approaches are introduced wherein the latter improves the performance of the devised methods. The applicability of the methods for solving the system of IVPs is also described. When used to solve problems from physical and real‐life applications, including nonlinear logistic growth and stiff model for flame propagation, the proposed methods are found to have good results.

A Novel Optimized Exponentially Fitted Algorithm with Time-Efficiency

“…It may be noted that the two-dimensional system in (27) is attempted to be solved with the variable stepsize approach by the proposed exponentially fitted optimal method and by a couple of other competitive methods available in literature while taking different values of the tolerance (ε). For details of the variable stepsize approach, one can see the recently published research in [22,23,28] and the references cited therein. While simulating the system (27) with the methods under consideration, the lower order method is chosen to be the implicit version of the Trapezoidal second-order method.…”