Heat Equation with Periodic Boundary Conditions

In summary, the periodic function that satisfies the condition that ##f(\theta)=f(\theta+2\pi)## is the exponential function.
  • #1
StretchySurface
6
0
Homework Statement
Find General Solution of Heat Equation on a Ring.
Relevant Equations
See below
I'm solving the heat equation on a ring of radius ##R##. The ring is parameterised by ##s##, the arc-length from the 3 o'clock position. Using separation of variables I have found the general solution to be:

$$U(s,t) = S(s)T(t) = (A\cos(\lambda s)+B\sin(\lambda s))*\exp(-\lambda^2 kt)$$.

Here is my confusion:
Periodicity demands that ##U(s',t) = U(s' + 2\pi R, t).## This means that
$$\cos(\lambda s') = cos(\lambda s' + \lambda*2\pi R)$.$

This finally gives condition that ##\lambda = k/R%## where ##k## is an integer. This is the correct answer as told by my professor/textbook.

However, periodicity can also demands ##U(s',t) = U(s' + 2\pi nR, t)##, i.e revolving around the ring ##n## complete times. But this leads to the condition ##2\pi \lambda nR = 2 \pi m## where ##m## and ##n## are arbitrary integers. This gives us ##\lambda = m/(nR)##. This doesn't seem to be the correct answer but I don't understand why and rational number divided by ##R## is not an eigenvalue.
 
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  • #2
Something with a period of ##4\pi## does not necessarily satisfy the condition that ##f(\theta)=f(\theta+2\pi)## !
 
  • #3
Does that mean the following is untrue: We require ##U(s′,t)=U(s′+4πR,t)##?
 
  • #4
It is a true statement, but:
It's a necessary but insufficient condition.
 
  • #5
Okay, so I don't really see the point you're trying to convey and how it relates to my question. Below, I'll write out a set steps in reasoning to see if you can pinpoint the error I'm making.

Let's only consider the case ##n = 2## i.e traveling around the ring twice.
Periodicity demands that ##U(s',t)=U(s'+4πR,t)##.
This means that ##2\lambda R = k##, for any integer ##k##.
Thus ##\lambda_{k} = k/(2R)## which includes some values not mentioned in the notes I'm working from.
 
  • #6
StretchySurface said:
some values not mentioned
Yes. And those do NOT satisfy the condition ##U(s′,t)=U(s′+2\pi R,t)##. Do you propose to drop that condition ?
 
  • #7
Put slightly differently, 2pi periodic functions are automatically 4pi periodic but 4pi periodic functions are not necessarily 2pi periodic.
 
  • #8
Ah okay, now I understand! Thank you very much!
 
  • #9
Btw if ##\lambda## is the inverse of a length, and ##k## is an integer (I suppose dimensionless), that exponential doesn't look good to me.

Edit: ok the ##k## in the exponential is the diffusion coefficient (I suppose)... my bad
 
  • #10
dRic2 said:
Btw if ##\lambda## is the inverse of a length, and ##k## is an integer (I suppose dimensionless), that exponential doesn't look good to me.

Edit: ok the ##k## in the exponential is the diffusion coefficient (I suppose)... my bad
I think you can be excused as the OP used ##k## both for the diffusion coefficient as well as an integer ... :wink:
 

Related to Heat Equation with Periodic Boundary Conditions

1. What is the Heat Equation with Periodic Boundary Conditions?

The Heat Equation with Periodic Boundary Conditions is a mathematical model used to describe the flow of heat through a given material or system. It takes into account factors such as temperature, time, and the physical properties of the material to determine how heat is transferred and distributed within the system.

2. What are Periodic Boundary Conditions?

Periodic Boundary Conditions are a set of conditions applied to a system that assumes the system is repeated infinitely in all directions. This means that any heat flow or temperature changes at the boundary of the system are mirrored on the opposite side, creating a periodic pattern within the system.

3. Why are Periodic Boundary Conditions important in the Heat Equation?

Periodic Boundary Conditions are important in the Heat Equation because they allow for the accurate modeling of systems that have periodic properties, such as materials with repeating structures or systems with recurring patterns. They also simplify the mathematical calculations involved in solving the Heat Equation.

4. How are Periodic Boundary Conditions applied in the Heat Equation?

Periodic Boundary Conditions are typically applied by setting the temperature or heat flux at one boundary of the system to be equal to the temperature or heat flux at the opposite boundary. This creates a continuous and periodic flow of heat within the system, allowing for the accurate modeling of the heat transfer process.

5. What are some real-world applications of the Heat Equation with Periodic Boundary Conditions?

The Heat Equation with Periodic Boundary Conditions has many practical applications in various fields, including engineering, physics, and materials science. It is commonly used to study heat transfer in materials with repeating structures, such as crystals or polymers, and to model heat flow in systems with periodic properties, such as electronic circuits or heat exchangers.

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