A new three-point sixth-order THAGE iteration method for mildly nonlinear two-point boundary value problems with enginee
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ORIGINAL ARTICLE
A new three‑point sixth‑order THAGE iteration method for mildly nonlinear two‑point boundary value problems with engineering applications Pinaki Ranjan Mohanty1 Received: 31 July 2020 / Accepted: 25 August 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract We study a new sixth-order compact discretization using uniform three-grid point for the mildly nonlinear differential equation 𝜙�� = g(t, 𝜙) , subject to the values of ϕ given at two end points of the regular solution interval. We also discuss three-step AGE (THAGE) iteration method as an application to the resulting difference equation as a powerful numerical computation device. In this algorithm, the common term is evaluated first to save the CPU time in comparison with the corresponding two-step algorithm. In addition, the error analysis is studied. Numerical performance is compared with the exact solution, and with the two-step AGE and SOR iteration methods. Keywords THAGE algorithm · Sixth-order compact uniform discretization · Forced oscillation equation · Halm equation · Emden–fowler equation · The root mean square error Mathematics Subject Classification 65L10 · 65L12 · 65L20
1 Introduction First, we consider the two-point nonlinear boundary value problem (BVP) of the form
𝜙�� = g(t, 𝜙), t0 < t < tJ+1
(1)
The two-point boundary conditions are given by ( ) ) ( 𝜙 t0 = 𝛾0 , 𝜙 tJ+1 = 𝛾1 ,
(2)
where 𝛾0 and 𝛾1 are two boundary values. For t0 ≤ t ≤ tJ+1 , −∞ < 𝜙 < ∞, we assume that (i) 𝜙(t) and g(t, 𝜙) are continuous, 𝜕g (ii) 𝜕𝜙 exist and continuous,
(iii)
𝜕g > 0. 𝜕𝜙
* Pinaki Ranjan Mohanty [email protected]; [email protected] 1
Department of Computer Science, College of Science, Purdue University, West Lafayette, IN 47907‑2107, USA
Above existence and uniqueness conditions for the BVP (1)-(2) have been studied by Keller [1]. Many physical problems in physics and engineering science are related with nonlinear two-point BVPs (1)-(2), the solution of which requires huge computational work and sometime leads distraction in depths understanding of rigorous numerical algorithms. There are some numerical methods available in literature to solve such BVPs, such as the collocation method, the shooting method, the finite difference method, etc. Depending upon the various nonlinearities involved and the nature of the problem, it is not always possible to apply directly these methods. In present studies, an efficient iterative algorithm for the numerical solution of BVPs (1)-(2) by the help of embedded function technique is discussed. Originally, the group explicit algorithm for the large linear system of equations was studied by Evans [2]. Evans [3] also studied alternating group explicit (AGE) iteration algorithm, which is suitable for use on parallel computers. Further, Evans and Ahmad [4] obtained the solution of differential equation and compared the performance of AGE method with the corresponding successive over relaxation (SOR) iteration method. Sukon and Evans [5, 6] have introduced two-
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