A One-Point Third-Derivative Hybrid Multistep Technique for Solving Second-Order Oscillatory and Periodic Problems

Abstract

This paper describes a third-derivative hybrid multistep technique (TDHMT) for solving second-order initial-value problems (IVPs) with oscillatory and periodic problems in ordinary differential equations (ODEs), the coefficients of which are independent of the frequency omega and step size h . This research is significant because it has numerous applications to real-life phenomena such as chaotic dynamical systems, almost periodic problems, and duffing equations. The current method is derived from the collocation of a derivative function at the equidistant grid and off-grid points. The TDHMT obtained is a continuous scheme for obtaining simultaneous approximations to the solution and its derivative at each point in the x 0 , x N interval integration. The presence of high derivatives increases the order of the method, which increases the accuracy method’s order and the stability property, as discussed in detail. Finally, the proposed method is compared to existing methods in the literature on some oscillatory and periodic test problems to demonstrate the technique’s effectiveness and productivity.

Unveiling the Power of Implicit Six-Point Block Scheme: Advancing numerical approximation of two-dimensional PDEs in physical systems

“…The function (17) has coefficients that are the first derivative of ( 10) and (11), which are as follows,…”

“…Partial Differential Equations (PDEs) are a useful tool for the mathematical expression of many natural phenomena and are useful in the solution of physical and other issues requiring functions of several variables. The transmission of heat/sound, fluid movement, turbulent flow, heat transfer analysis, elasticity, electrostatics, and electrodynamics are a few examples of these issues; see Ahsan et al [ 1 ], Wang and Guo [ 2 ], Arif et al [ 3 , 4 ], Adoghe et al [ 5 ], Nawaz et al [ 6 ], Animasaun et al [ 7 ], Devnath et al [ 8 ], Ahsan et al [ 9 ], Wang et al [ 10 ], Rufai et al [ 11 ], Nawaz and Arif [ 12 ], Ramakrishna et al [ 13 ], El Misilmani et al [ 14 ]). According to Quarteroni and Valli [ 15 ], numerical approximation techniques for partial differential equations (PDEs) constitute a cornerstone in diverse scientific and engineering disciplines.…”

Comparison of Approximate Analytical and Numerical Solutions of the Allen Cahn Equation

“…In last few years, approximate analytical techniques attract attention of many researchers to fnd the approximate solutions of nonlinear PDE's due to their simplicity and easy approach. Series of methods have been established by mathematicians and researchers such as the homotopy perturbation method (HPM) [18,19], homotopy analysis method (HAM) [20][21][22], variational iteration method (VIM) [23][24][25], Adomian decomposition method (ADM) [26][27][28], diferential transform method (DTM) [29][30][31], tanh method (THM) and extended tanh method [32][33][34], use of trigamma function [35], Catalan-type numbers [36], and polynomials [37][38][39]. Traveling wave solutions are also prominent techniques in fnding the closed form of solutions.…”