# Boundary conditions for the Heat Equation

• A
• Leonardo Machado
In summary, the conversation discusses studying the heat equation in polar coordinates via separation of variables. This leads to three ODEs, and the boundary conditions depend on the physical situation being described. The speaker has solved the problem by using an inhomogeneous boundary condition and separating the solution into a steady state and a solution of the heat equation.
Leonardo Machado
Hello guys.

I am studying the heat equation in polar coordinates

$$u_t=k(u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta})$$

via separation of variables.
$$u(r,\theta,t)=T(t)R(r)\Theta(\theta)$$

which gives the ODEs

$$T''+k \lambda^2 T=0$$
$$r^2R''+rR+(\lambda^2 r^2-\mu^2)R=0$$
$$\Theta''+\mu^2\Theta=0$$

but i can't properly think about the boundary conditions to this problem. I see every where people resolving it with

$$|u(0,\theta,t)|<\inf \mapsto |R(0)| < \inf$$

and

$$u(r*,\theta,t)=0 \mapsto R(r*)=0$$

being r* the border of the disc.

But i understand the radial condition as a termal bath at zero temperature and i really want to change it for a finite value but i don't know how to procede without the zeros...

Any suggestions?

The boundary conditions depend on what physical situation you wish to describe. Can you be more specific about this?

The boundary condition at r = 0 should be zero radial temperature gradient.

Time equation should be
$$T' + k\lambda^2T = 0$$

Chestermiller said:
The boundary condition at r = 0 should be zero radial temperature gradient.
This is correct only in the case of rotational symmetry of the problem. The general solution will contain contributions from ##J_1##, which has non-zero derivative at ##r = 0##.

You should have the inner boundary condition:
$$\frac{\partial u}{\partial r}\Bigg|_{r=0}=0$$
This is the proper symmetry condition. The outer boundary condition is physics dependent however and can be absolutely anything.

I have solved it guys!

To operate with inhomogeneous bondary conditions I've used

$$u(r,\theta,t)=v(r,\theta,t)+u_E(r,\theta)$$

being u_E the steady state and "v" the solution of the heat equation.

## What is the Heat Equation?

The Heat Equation is a partial differential equation that describes how heat is distributed and transferred in a given system over time. It is commonly used in physics and engineering to model heat flow in various materials and systems.

## What are boundary conditions?

Boundary conditions are the set of constraints or requirements that must be satisfied at the edges or boundaries of a system. In the context of the Heat Equation, they specify the temperature or heat flux at the boundaries of the system.

## Why are boundary conditions important in the Heat Equation?

Boundary conditions are crucial in solving the Heat Equation because they provide the necessary information to determine a unique solution. Without them, the equation would have an infinite number of solutions, making it impossible to accurately model heat flow in a system.

## What are the types of boundary conditions for the Heat Equation?

There are three main types of boundary conditions for the Heat Equation: Dirichlet, Neumann, and Robin. Dirichlet boundary conditions specify the temperature at the boundary, Neumann boundary conditions specify the heat flux at the boundary, and Robin boundary conditions combine both temperature and heat flux specifications.

## How do boundary conditions affect the solution of the Heat Equation?

The specific boundary conditions chosen for a given system can greatly affect the behavior and outcome of the Heat Equation. Different boundary conditions can result in different temperature distributions and heat transfer rates, making it important to carefully consider and accurately model them in order to obtain meaningful results.

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