“Lane-Emden Differential Equation”的概念、定义、习题解题方法及翻译-数学辞典

单词

Lane-Emden Differential Equation

释义

Lane-Emden Differential Equation

A second-order Ordinary Differential Equation arising in the study of stellar interiors. It is given by

(1)

(2)

It has the Boundary Conditions

			(3)
			(4)

Solutions

for

, 1, 2, 3, and 4 are shown above. The cases

, 1, and 5 can be solved analytically(Chandrasekhar 1967, p. 91); the others must be obtained numerically.

For (), the Lane-Emden Differential Equation is

(5)

(Chandrasekhar 1967, pp. 91-92). Directly solving gives

(6)

(7)

(8)

(9)

(10)

(11)

The Boundary Condition

then gives

and

, so

(12)

and

is Parabolic.

For (), the differential equation becomes

(13)

(14)

which is the Spherical Bessel Differential Equation

(15)

with

and

, so the solution is

(16)

Applying the Boundary Condition

gives

(17)

where

is a Spherical Bessel Function of the First Kind (Chandrasekhar 1967, pp. 92).

For , make Emden's transformation

			(18)
			(19)

which reduces the Lane-Emden equation to

(20)

(Chandrasekhar 1967, p. 90). After further manipulation (not reproduced here), the equation becomes

(21)

and then, finally,

(22)

References

Chandrasekhar, S. An Introduction to the Study of Stellar Structure. New York: Dover, pp. 84-182, 1967.

随便看

数学辞典收录了8975条数学词条，基本涵盖了常用数学知识及数学英语单词词组的翻译及用法，是数学学习的有利工具。