Implicit Three-Point Block Numerical Algorithm for Solving Third Order Initial Value Problem Directly with Applications

Reem Allogmany and Fudziah Ismail
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Reem Allogmany: Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah P.O. Box 344, Saudi Arabia
Fudziah Ismail: Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400 UPM, Malaysia

Mathematics, 2020, vol. 8, issue 10, 1-16

Abstract: Recently, direct methods that involve higher derivatives to numerically approximate higher order initial value problems (IVPs) have been explored, which aim to construct numerical methods with higher order and very high precision of the solutions. This article aims to construct a fourth and fifth derivative, three-point implicit block method to tackle general third-order ordinary differential equations directly. As a consequence of the increase in order acquired via the implicit block method of higher derivatives, a significant improvement in efficiency has been observed. The new method is derived in a block mode to simultaneously evaluate the approximations at three points. The derivation of the new method can be easily implemented. We established the proposed method’s characteristics, including order, zero-stability, and convergence. Numerical experiments are used to confirm the superiority of the method. Applications to problems in physics and engineering are given to assess the significance of the method.

Keywords: ODEs; third-order; Hermite interpolation; block; linear and nonlinear; IVPs (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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