“Heat Conduction Equation”的概念、定义、习题解题方法及翻译-数学辞典

单词

Heat Conduction Equation

释义

Heat Conduction Equation

A diffusion equation of the form

(1)

Physically, the equation commonly arises in situations where

is the thermal diffusivity

and

the temperature.

The 1-D heat conduction equation is

(2)

This can be solved by Separation of Variables using

(3)

Then

(4)

Dividing both sides by

gives

(5)

where each side must be equal to a constant. Anticipating the exponential solution in

, we have picked a negativeseparation constant so that the solution remains finite at all times and

has units of length. The

solutionis

(6)

and the

solution is

(7)

The general solution is then



			(8)

If we are given the boundary conditions

(9)

and

(10)

then applying (9) to (8) gives

(11)

and applying (10) to (8) gives

(12)

so (8) becomes

(13)

Since the general solution can have any

(14)

Now, if we are given an initial condition

, we have

(15)

Multiplying both sides by

and integrating from 0 to

gives

(16)

Using the Orthogonality of

and


	(17)

(18)

If the boundary conditions are replaced by the requirement that the derivative of the temperature be zero at the edges,then (9) and (10) are replaced by

(19)

(20)

Following the same procedure as before, a similar answer is found, but with sine replaced by cosine:

(21)

where

(22)

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