The initial temperature (at t = 0) of a thin homogeneous plate

in the xy-plane is given by the function f = f(x, y). The faces of the plate are supposed completely isolating. After the moment t = 0 the boundaries of A are held in the temperature 0. Determine the temperature function

on A (where t is the time).

Since it’s a question of a two-dimensional heat flow, the heat equation gets the form

∇2u ≡ u′′ xx + u′′yy = u′t.

(1)

One have to find for (1) a solution function u which satisfies the initial condition

u(x, y, 0) = f(x, y) in A

(2)

and the boundary condition

u(x, y, t) = 0 on boundary of A for t > 0.

(3)

For finding a simple solution of the differential equation (1) we try the form

u(x, y, t) := X(x)Y (y)T(t),

(4)

whence the boundary condition reads

X(0) = Y (0) = X(a) = Y (b) = 0.

(5)

Substituting (4) in (1) and dividing this equation by XY T give the form

+ = ⋅.

(6)

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:

= −k12, = −k22, ⋅ = −k2,

(7)

where k² = k ₁² + k ₂². We soon explain why these constants are negative. Because the equations (7) may be written

the general solutions of these ordinary differential equations are

(8)

Now we remark that if the right side of the third equation (7) were +k², then we had T = Ce^k2c2t which is impossible, since such a T and along with this also the temperature u = XY T would ascend infinitely when t →∞. 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 C₁ = C₂ = 0, and then the two last conditions (5) require that

If we had D₁ = 0 or D₂ = 0, then X or Y would vanish identically, which cannot occur. Thus we have

whence only the eigenvalues

are possible for the obtained X and Y . Considering the equation k² = k ₁² + k ₂² we may denote

qmn := k2c2 = c2

(9)

for all m, n ∈ ℤ₊.

Altogether we have infinitely many solutions

of the equation (1), where the coefficients c_mn are, for the present, arbitrary constants. These solutions fulfil the boundary condition (3). The sum of the solutions, i.e. the double series

u(x, y, t) := ∑ m=1∞∑ n=1∞c mne−qmnt sin sin ,

(10)

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 expansion of f(x, y) in the rectangle A, and therefore we have the expression

cmn := ∫ 0a∫ 0bf(x, y) sin sin dxdy

(11)

for the coefficients.

The result of calculating the solution of our problem is the temperature function (10) with the formulae (9) and (11).

References