Question

sample 1.A rod length l with insulated side initially at uniform temp.u0 it end are suddenly cooled to 0 and kept at temp.find temperature function u(x,t)

Answer 1

a sum of money is doubled itself 16 years find the rate of interest per annum

Answer 2

The temperature function u(x,t)=  $\frac{4u_{0} }{\pi }$ Σ $\frac{1}{n}$ Sin($\frac{n\pi x}{l}$) e $\frac{-c^{2}n^{2} \pi ^{2}t }{l^{2} }$.

Given :

A rod of length l with insulated sides is initially at a uniform temperature u₀. Its ends are suddenly cooled to 0⁰ C and kept at that temperature.

To Find :

The temperature function u(x,t).

Solution :

The differential equation u(x,t) satisfies the differential equation -

∂u/∂t = $c^{2}$ x  ∂u/∂$x^{2}$

The boundary conditions associated with the problem are -

u(0,t)=0, u(l,t)=0

The initial condition is u(x,0)= u₀

The solution is -

u(x,t)

= Σ uₙ(x,t)

= Σ aₙ Sin($\frac{n\pi x}{l}$) e^(-λ^2)t , λₙ= $\frac{n\pi c}{l}$

Since, u(x,0)= u₀, we have,

u₀= Σ aₙ Sin($\frac{n\pi x}{l}$)

aₙ= $\frac{2}{l}$ ∫ u₀ Sin($\frac{n\pi x}{l}$)dx=  { 0, when n is even and $\frac{4u_0}{n\pi }$, when n is odd }

Hence, the temperature function is,

u(x,t)= $\frac{4u_{0} }{\pi }$ Σ $\frac{1}{n}$ Sin($\frac{n\pi x}{l}$) e $\frac{-c^{2}n^{2} \pi ^{2}t }{l^{2} }$

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