Heat Equation

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Shashank Dwivedi

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Determine the temperature distribution u(x,t) in a uniform rod of unit length whose one end is kept at 10 (degree celsius) and other end is insulated. Further it is given that:
u(x,0)=1-x 0<x<1.

I am trying to deduce the boundary conditions here. First boundary condition is u(0,t)=0 and second boundary condition seems to be partial derivative of u w.r.t x at (1,t) equal to zero. Am I going in the right direction or the second boundary condition is u(1,t)=0?
 
Shashank Dwivedi said:
Determine the temperature distribution u(x,t) in a uniform rod of unit length whose one end is kept at 10 (degree celsius) and other end is insulated. Further it is given that:
u(x,0)=1-x 0<x<1.

I am trying to deduce the boundary conditions here. First boundary condition is u(0,t)=0 and second boundary condition seems to be partial derivative of u w.r.t x at (1,t) equal to zero. Am I going in the right direction or the second boundary condition is u(1,t)=0?
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The insulated bdy. condition means heat-flow at that end zero?

What is the relationship between temperature (an/or the temperature distribution) and heat-flow?
 
I know the relation Q = mcT, where Q is the heat transfer, m is the mass, c is specific heat and T is change in temperature. Now as the end is insulated, the temperature change is zero and that makes the heat flow zero which is what I can deduce from this i.e partial derivative of u w.r.t to x at the other end should be zero as the heat flow as zero. Am I right in my explanation or am I missing something?
 
Shashank Dwivedi said:
I know the relation Q = mcT, where Q is the heat transfer, m is the mass, c is specific heat and T is change in temperature. Now as the end is insulated, the temperature change is zero and that makes the heat flow zero which is what I can deduce from this i.e partial derivative of u w.r.t to x at the other end should be zero as the heat flow as zero. Am I right in my explanation or am I missing something?
Click to expand...
What is the differential equation you are trying to solve?
 
Shashank Dwivedi said:
It is one dimensional heat equation.
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Fine, now please answer Prof Khan's question about which equation are you trying to solve. It is hard to help you solve a differential equation if we do not know which one you are trying to solve.
 
In this I have attached the differential equation along with my attempt. I was stuck in the beginning because of boundary conditions. I am unable to proceed so, please throw some light on how to proceed to reach to a solution of this heat equation.
 

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Shashank Dwivedi said:
Determine the temperature distribution u(x,t) in a uniform rod of unit length whose one end is kept at 10 (degree celsius) and other end is insulated. Further it is given that: u(x,0)=1-x 0<x<1.
First boundary condition is u(0,t)=10 and ..................................................................(edited - you have it correct in your hand written work )
second boundary condition \(\displaystyle [\frac{\partial u}{\partial x}]_{x=1, t=t} = 0\)............................. Correct
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\(\displaystyle \frac{\partial u}{\partial x}\) is proportional to the heat-flow normal to the cross-section (Fourier's Law)

[ref: https://www.google.com/search?q=Fourier's+law+of+heat+conduction&oq=Fourier's+law+of+heat+conduction&aqs=chrome..69i57.23116j0j1&sourceid=chrome&ie=UTF-8]

Instead 'u' to indicate temperature, it is preferable to write 'T' to indicate temperature - in Physics/Engineering/Chemistry. The symbol 'u' is generally reserved for "internal energy of a system". The given problem is "incomplete" - because it does not restrict itself to "conduction". We have to assume that only conductive heat-transfer is considered.
 
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