Fourier Series: One Dimensional Equation of Heat Conduction - Zero boundary values (Temperature or temperature gradients)

ONE DIMENSIONAL EQUATION OF HEAT CONDUCTION: Zero boundary values (Temperature or temperature gradients)

Problems with zero boundary values (Temperature or temperature gradients)

Example 1: A rod of length l with insulated side is initially at a uniform temperature f(x). Its ends are suddenly cooled to 0° C and are kept at the temperature. Find the temperature function u (x, t).

 (OR)

Solve the equation  subject to the conditions u (0, t)=0, u (1, t) = 0 and u(x, 0) = f(x).

Solution: The temperature function u (x, t) satisfies the one dimensional heat equation is

From the given problem, we get the following boundary and initial conditions.

(i) u (0, t) = 0 for all t≥0

(ii) u (l, t) = 0 for all t≥0

(iii) u (x, 0) = f(x)

Now, the suitable solution which satisfies our boundary conditions is given by

u (x, t) = (A cos px + B sin px)  ……… (1)

Applying condition (i) in equation (1), we get

To find B_n expand f (x) in a half‒range Fourier sine series in the interval (0, l)

Substitute, B_n value in equation (4), we get the general solution.

Example 2: Solve  subject to the conditions

 (i) u (0, t) = 0 for all t ≥ 0

 (ii) u (l, t) = 0 for all t≥ 0

 (iii) u (x, 0) = 

Solution:

The temperature function u (x, t) satisfies the one dimensional heat equation is

From the given problem, the boundary and initial conditions are

 (i) u(0, t) = 0 for all t≥0

 (ii) u (l, t) = 0 for all t≥0

 (iii) u (x, 0) = 

Now, the suitable solution which satisfies our boundary conditions is given by

u (x, t) = (A cos px + B sin px)       …….. (1)

Equations from (1) to (4) is same as Example No. 1

Apply condition (iii) in equation (4), we get

To find В_n expand the given f(x) in a half range Fourier sine series in the interval [0, 1]

Substitute, B_n value in equation (4), we get

Example 3: A homogeneous rod of conducting material of length l has its ends kept at zero temperature. The temperature at the centre is T and falls uniformly to zero at the two ends. Find u (x, t).

Solution: The temperature function u (x, t) satisfies the one dimensional heat equation is

From the given problem we get the following boundary and initial conditions

(i) u (0, t) = 0 for all t≥0

(ii) u (l, t) = 0 for all t≥0

Since the temperature at the centre is T and falls uniformly to zero at the two ends, its distribution at t=0 is as given in the figure.

The equation of OB is

Now, the suitable solution which satisfies our boundary conditions is given by

To find B_n expand f (x) in a half range Fourier sine series in the interval [0, 1]

EXERCISE

1. A rod l cm long with insulated lateral surface is initially at temperature u₀, at an inner point distance x cm from one end. If both ends are kept at zero temperature. Find the temperature function at any point of the rod at any time t.

2. Solve the boundary value problem :

 a² = u_xx = u_t with conditions u(0, t) = 0, u (l, t) = 0

and u(x, 0) = lx ‒ x², 0 ≤ x ≤l