Purpose

The numerical solution of third-order boundary value problems (BVPs) has a great importance because of their applications in fluid dynamics, aerodynamics, astrophysics, nuclear reactions, rocket science etc. The purpose of this paper is to develop two computational methods based on Hermite wavelet and Bernoulli wavelet for the solution of third-order initial/BVPs.

Design/methodology/approach

Because of the presence of singularity and the strong nonlinear nature, most of third-order BVPs do not occupy exact solution. Therefore, numerical techniques play an important role for the solution of such type of third-order BVPs. The proposed methods convert third-order BVPs into a system of algebraic equations, and on solving them, approximate solution is obtained. Finally, the numerical simulation has been done to validate the reliability and accuracy of developed methods.

Findings

This paper discussed the solution of linear, nonlinear, nonlinear singular (Emden–Fowler type) and self-adjoint singularly perturbed singular (generalized Emden–Fowler type) third-order BVPs using wavelets. A comparison of the results of proposed methods with the results of existing methods has been given. The proposed methods give the accuracy up to 19 decimal places as the resolution level is increased.

Originality/value

This paper is one of the first in the literature that investigates the solution of third-order Emden–Fowler-type equations using Bernoulli and Hermite wavelets. This paper also discusses the error bounds of the proposed methods for the stability of approximate solutions.

中文翻译：

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使用小波方法求解三阶 Emden-Fowler 型方程

目的

三阶边值问题 (BVPs) 的数值求解因其在流体动力学、空气动力学、天体物理学、核反应、火箭科学等领域的应用而具有重要意义。 本文的目的是开发两种基于 Hermite 的计算方法小波和伯努利小波用于求解三阶初始/BVPs。

设计/方法/方法

由于奇点的存在和强非线性特性，大多数三阶 BVP 不占据精确解。因此，数值技术在求解此类三阶 BVP 时发挥着重要作用。所提出的方法将三阶 BVP 转换为代数方程组，并在求解它们时获得近似解。最后,通过数值模拟验证了所开发方法的可靠性和准确性。

发现

本文讨论了使用小波求解线性、非线性、非线性奇异（Emden-Fowler 型）和自伴随奇异摄动奇异（广义 Emden-Fowler 型）三阶 BVP。已经给出了所提出方法的结果与现有方法的结果的比较。随着分辨率级别的提高，所提出的方法可将精度提高到小数点后 19 位。

原创性/价值

这篇论文是文献中最早使用 Bernoulli 和 Hermite 小波研究三阶 Emden-Fowler 型方程解的文献之一。本文还讨论了所提出的近似解稳定性方法的误差界限。