Diffusion Phenomena Numerical Analysis

**References**

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**Chapter 7**

**Abstract**

**1. Introduction**

**115**

Equations

*Gerald Tendayi Marewo*

A Modified Spectral Relaxation

Method for Some Emden-Fowler

In this chapter, we present a modified version of the spectral relaxation method for solving singular initial value problems for some Emden-Fowler equations. This study was motivated by the several applications that these equations have in Science. The first step of the method of solution makes use of linearisation to solve the model problem on a small subinterval of the problem domain. This subinterval contains a singularity at the initial instant. The first step is combined with using the spectral relaxation method to recursively solve the model problem on the rest of the problem domain. We make use of examples to demonstrate that the method is reliable, accurate and computationally efficient. The numerical solutions that are obtained in this chapter are in good agreement with other solutions in the literature.

**Keywords:** Emden-Fowler equations, Lane-Emden equations, singular initial value

*dx* <sup>þ</sup> *r x*ð Þ¼ , *<sup>y</sup> s x*ð Þ, 0 <sup>&</sup>lt;*x*, *<sup>γ</sup>* <sup>&</sup>gt;<sup>0</sup>

(1)

*dx* ð Þ¼ <sup>0</sup> 0, *<sup>m</sup>* <sup>∈</sup> (2)

*dx* ð Þ¼ <sup>0</sup> 0, *<sup>α</sup>* <sup>∈</sup>

for the Lane-Emden Eq. (1) models several phenomena such as the thermal behaviour of a spherical cloud of gas acting under the mutual attraction of its molecules [1], the temperature variation of a self gravitating star, the kinetics of combustion [2], thermal explosion in a rectangular slab [3] and the density distribution in isothermal gas spheres [4]. Moreover, Eq. (1) has been used many a time

A particular case of Eq. (1) is the Emden-Fowler equation of the first kind:

As mentioned in [5], Eq. (2) represents the dimensionless form of the governing equation for the gravitational potential of a Newtonian self-gravitating, spherically

*dx* <sup>þ</sup> *<sup>y</sup><sup>m</sup>* <sup>¼</sup> 0, *<sup>y</sup>*ð Þ¼ <sup>0</sup> 1, *dy*

problem, spectral relaxation method, numerical method

*y*ð Þ¼ 0 *α*,

*dy*

The singular initial value problem

as a benchmark for new methods.

*d*2 *y dx*<sup>2</sup> <sup>þ</sup>

2 *x dy*

*d*2 *y dx*<sup>2</sup> <sup>þ</sup> *<sup>γ</sup> x dy* **Chapter 7**
