This is the boundary condition for laterally insulated rod having ends maintained at certain temperatures.
Governing equation for the following condition
∂T(x,t)/∂t=k∂2T(x,t)/∂x2 ------------- (ii)
For separation of variable T(x,t)=f(x).g(t) ---------- (iii)
Substituting equation (iii) in equation (ii);
we get g′(t)/g(t)=kf′′(x)/f(x)=l ----- (iv)
g′(t)/g(t)=l
d(g(t))/g(t)=ldt
Integrating both sides, we get
ln(g(t)/c2)=lt
g(t)/c2=exp(lt)
g(t)=c2exp(lt) ----(v)
kf′′(x)/f(x)=l
∂2f(x)/∂x2=(l/k)f(x)
∂2f(x)/∂x2=−m2f(x)
l/k=−m2 -----(vi)
Solution to this differential equation is
f(x)=c1sin(mx)+c2cos(mx)
T(x,t)=(c2exp(lt))(c1sin(mx)+c3cos(mx))
Using equation (i)
T(0,t)=(c2exp(lt))(c1sin(0)+c3cos(0))=0
T(0,t)=c3c2exp(lt)=0
c3=0
Using equation (i)
T(2,t)=(c2exp(lt))(c1sin(m.2))=0
T(2,t)=c1c2exp(lt)sin(m.2)=0
We obtain 2.m=nπ
m=nπ/2
Using equation (vi)
l/k=−m2=−n2π2/4
l=−n2π2k/4
So the equation becomes
T(x,t)=∑n=1∞(c2exp(−n2π2kt/4))(c1sin(nπx/2))
T(x,t)=∑n=1∞(cn(sin(nπx/2)exp(−n2π2kt/4))
cn=c1.c2
We have to find cn now
T(x,0)=∑n=1∞(cn(sin(nπx/2)exp(−n2π2k.0/4))
∑n=1∞(cn(sin(nπx/2)=100(2x−x2)
It is a sine Fourier series.we know;
cn=2×∫02(100(2x−x2)(sin(nπx/2))dx
solving the integration using by parts and then putting the limits we get,
cn=−3200/(n3π3)((−1)n−1)
Answer:
T(x,t)=∑n=1∞(cn(sin(nπx/2)exp(−n2π2kt/4)),
where cn=−3200/(n3π3)((−1)n−1)
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