time-dependent example of heat equation
The initial temperature (at ) of a plate
in the -plane is given by the function . The faces of the plate are supposed completely isolating. After the the boundaries of are held in the temperature . Determine the temperature function
on (where is the time).
Since it’s a question of a two-dimensional heat , the heat equation gets the form
One have to find for (1) a solution function which satisfies the initial condition
and the boundary condition
For finding a simple solution of the differential equation (1) we try the form
whence the boundary condition reads
Substituting (4) in (1) and dividing this equation by give the form
It’s easily understood that such a condition requires that the both addends of the left side and the right side ought to be constants:
where . We soon explain why these constants are negative. Because the equations (7) may be written
Now we remark that if the right side of the third equation (7) were , then we had which is impossible, since such a and along with this also the temperature would ascend infinitely when . And since, by symmetry, the right sides the two first equations (7) must have the same sign, also they must by (6) be negative.
The two first boundary conditions (5) imply by (8) that , and then the two last conditions (5) require that
If we had or , then or would vanish identically, which cannot occur. Thus we have
whence only the eigenvalues
are possible for the obtained and . Considering the equation we may denote
Altogether we have infinitely many solutions
provided it converges, is also a solution of the linear differential equation (1) and fulfils the boundary condition. In order to fulfil also the initial condition (2), one must have
But this equation presents the Fourier double sine series of in the rectangle , and therefore we have the expression
for the coefficients.
The result of calculating the solution of our problem is the temperature function (10) with the formulae (9) and (11).
- 1 K. Väisälä: Matematiikka IV. Hand-out Nr. 141. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).
|Title||time-dependent example of heat equation|
|Date of creation||2013-03-22 16:32:47|
|Last modified on||2013-03-22 16:32:47|
|Last modified by||pahio (2872)|
|Synonym||solving a time-dependent heat equation|