Obtain the temperature distribution u( x,t ) in a literally insulated rod of length L if both ends on the rod are kept at 0°c and the initial temperature distribution in the bar is :
u(x,0) = 6sin[πx/L]
To obtain the temperature distribution u(x,t) in the insulated rod, we can use the heat equation. The heat equation describes how the temperature changes over time in a given medium.
The heat equation is given by:
∂u/∂t = α ∂^2u/∂x^2
where u(x,t) is the temperature distribution, t is the time, x is the position along the rod, and α is the thermal diffusivity of the material.
To solve the heat equation, we need to apply the given boundary conditions and the initial condition. In this case, the boundary conditions are that both ends of the rod are kept at 0°C, which means u(0,t) = 0 and u(L,t) = 0. The initial condition is given by u(x,0) = 6sin[πx/L].
To solve the heat equation, we can separate variables by assuming a solution of the form u(x,t) = X(x)T(t). Substituting this into the heat equation, we get:
X(x)T'(t) = αX''(x)T(t)
Dividing both sides by X(x)T(t) gives:
T'(t)/T(t) = αX''(x)/X(x)
Since the left side of the equation only depends on t and the right side only depends on x, both sides must be equal to a constant. Let's call this constant -λ^2.
T'(t)/T(t) = -λ^2
X''(x)/X(x) = -λ^2
The above two differential equations can be solved separately. Solving the first equation gives:
T(t) = e^(-λ^2t)
Solving the second equation gives:
X(x) = A sin(λx) + B cos(λx)
where A and B are constants to be determined.
Applying the boundary conditions u(0,t) = 0 and u(L,t) = 0:
u(0,t) = X(0)T(t) = (A sin(0) + B cos(0))e^(-λ^2t) = 0
This implies B = 0.
u(L,t) = X(L)T(t) = (A sin(λL))e^(-λ^2t) = 0
This implies sin(λL) = 0, which gives λL = π, 2π, 3π, ...
So, we have λ = π/L, 2π/L, 3π/L, ...
Now, using the initial condition u(x,0) = 6sin[πx/L], we can substitute x = 0 and t = 0 into the separated solution:
u(x,0) = X(x)T(0) = (A sin(λx) + B cos(λx))e^(-λ^2 * 0) = A sin(λx)
Comparing this with the given initial condition, we see that A = 6. Therefore, the separated solution is:
u(x,t) = 6 sin(πx/L) e^(-π^2 t/L^2)
This is the temperature distribution in the insulated rod at any position x and time t.