Purpose

The purpose of this paper is to propose an efficient computational technique, which uses Haar wavelets collocation approach coupled with the Newton-Raphson method and solves the following class of system of Lane–Emden equations: −(tk1y′(t))′=t−ω1f1(t,y(t),z(t)), −(tk2z′(t))′=t−ω2f2(t,y(t),z(t)),where t > 0, subject to the following initial values, boundary values and four-point boundary values: y(0)=γ1, y′(0)=0, z(0)=γ2, z′(0)=0, y′(0)=0, y(1)=δ1, z′(0)=0, z(1)=δ2, y(0)=0, y(1)=n1z(v1), z(0)=0, z(1)=n2y(v2),where n1,n2,v1,v2∈(0,1) and k1≥0, k2≥0, ω1<1, ω2<1, γ₁, γ₂, δ₁, δ₂ are real constants.

Design/methodology/approach

To deal with singularity, Haar wavelets are used, and to deal with the nonlinear system of equations that arise during computation, the Newton-Raphson method is used. The convergence of these methods is also established and the results are compared with existing techniques.

Findings

The authors propose three methods based on uniform Haar wavelets approximation coupled with the Newton-Raphson method. The authors obtain quadratic convergence for the Haar wavelets collocation method. Test problems are solved to validate various computational aspects of the Haar wavelets approach. The authors observe that with only a few spatial divisions the authors can obtain highly accurate solutions for both initial value problems and boundary value problems.

Originality/value

The results presented in this paper do not exist in the literature. The system of nonlinear singular differential equations is not easy to handle as they are singular, as well as nonlinear. To the best of the knowledge, these are the first results for a system of nonlinear singular differential equations, by using the Haar wavelets collocation approach coupled with the Newton-Raphson method. The results developed in this paper can be used to solve problems arising in different branches of science and engineering.

中文翻译：

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非线性奇异微分方程组的Haar小波配置方法

目的

本文的目的是提出一种有效的计算技术，该方法将Haar小波配置方法与Newton-Raphson方法结合使用，并解决了以下Lane-Emden方程组系统： -（Ťķ1个ÿ′（Ť））′=Ť-ω1个F1个（Ť，ÿ（Ť），ž（Ť））， -（Ťķ2ž′（Ť））′=Ť-ω2F2（Ť，ÿ（Ť），ž（Ť）），其中t > 0，但要遵循以下初始值，边界值和四点边界值： ÿ（0）=γ1个， ÿ′（0）=0， ž（0）=γ2， ž′（0）=0， ÿ′（0）=0， ÿ（1个）=δ1个， ž′（0）=0， ž（1个）=δ2， ÿ（0）=0， ÿ（1个）=ñ1个ž（v1个）， ž（0）=0， ž（1个）=ñ2ÿ（v2），哪里 ñ1个，ñ2，v1个，v2∈（0，1个） 和 ķ1个≥0， ķ2≥0， ω1个<1个， ω2<1个，γ ₁，γ ₂，δ ₁，δ ₂是实常数。

设计/方法/方法

为了处理奇异性，使用了Haar小波，对于处理在计算过程中出现的非线性方程组，使用了Newton-Raphson方法。还建立了这些方法的收敛性，并将结果与​​现有技术进行了比较。

发现

作者提出了三种基于统一Haar小波逼近和Newton-Raphson方法的方法。作者获得了Haar小波配置方法的二次收敛性。解决了测试问题，以验证Haar小波方法的各种计算方面。作者观察到，仅需很少的空间划分，作者就可以针对初始值问题和边值问题获得高精度的解决方案。

创意/价值

本文提供的结果在文献中不存在。非线性奇异微分方程组不但因为奇异而且不易处理。据了解，这是通过使用Haar小波配置方法和Newton-Raphson方法结合的非线性奇异微分方程组的第一个结果。本文得出的结果可用于解决科学和工程学不同学科中出现的问题。