# Heat equation with non homogeneous BCs

• jackkk_gatz
In summary, the conversation discusses the use of a change of variable, theta, and the resulting equations after the change. The individual proposing the solution has already tried interpolation and is now attempting to use the method of separation of variables to obtain the general solution. The problem lies in finding the correct form of w, as the boundary conditions are non-homogeneous. Wolfram suggests the use of Bessel functions.
jackkk_gatz
Homework Statement
$$\frac{1}{r}\frac{\partial }{\partial r}(r\frac{\partial T}{\partial r})+\frac{\partial^2T}{\partial z^2}=0$$

$$\left.-k\frac{\partial T}{\partial r}\right\rvert_{r=R}=h[T(R,z)-T_{\infty}]$$

$$\left.k\frac{\partial T}{\partial z}\right\rvert_{z=H}+h[T(r,H)-T_{\infty}]=q_s$$

$$\left.-k\frac{\partial T}{\partial z}\right\rvert_{z=0}=0$$

$$\left.-k\frac{\partial T}{\partial r}\right\rvert_{r=0}=0$$

$where \ T_0 \ ,T_{\infty} \ and \ q_s \ are \ constants$
Relevant Equations
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I did a change of variable $$\theta(r,z) = T(r,z)-T_{\infty}$$ which resulted in

$$\frac{1}{r}\frac{\partial }{\partial r}(r\frac{\partial \theta}{\partial r})+\frac{\partial^2 \theta}{\partial z^2}=0$$

$$\left.-k\frac{\partial \theta}{\partial r}\right\rvert_{r=R}=h\theta$$

$$\left.k\frac{\partial \theta}{\partial z}\right\rvert_{z=H}+h\theta=q_s$$

$$\left.-k\frac{\partial \theta}{\partial z}\right\rvert_{z=0}=0$$

$$\left.-k\frac{\partial \theta}{\partial r}\right\rvert_{r=0}=0$$

After that I proposed $$\theta(r,z)=v(r,z)+w(r,z)$$ where w(r,z) should must satisfy

$$\left.-k\frac{\partial w}{\partial r}\right\rvert_{r=R}=hw(R,z)$$

$$\left.k\frac{\partial w}{\partial z}\right\rvert_{z=H}+hw(r,H)=q_s$$

$$\left.-k\frac{\partial w}{\partial z}\right\rvert_{z=0}=0$$

$$\left.-k\frac{\partial w}{\partial r}\right\rvert_{r=0}=0$$

I already tried interpolation, doesn't work. I don't know how w(r,z) should look like in order to satisfy the above equations. Is there an easier method?
In short, everything I have been trying has failed and I don't know what to do anymore, I have looked for books on PDEs, all the ones I have found deal with very simple cases, which are of no use to me. I have almost no experience solving this kind of equations to know what to do or to guess how w might look like

Last edited:
I just noticed I asked my question in the wrong section

Have you tried obtaining the general solution by the method of separation of variables? It looks like it will work in this case but I have not solved the problem.

What is ##x=0## doing in the boundary condition ##~\left.-k\dfrac{\partial T}{\partial r}\right\rvert_{x=0}=0~##. Is it a typo?

I reported this thread and it should be moved to the Advanced Homework forum by a mentor at some point in time.

kuruman said:
Have you tried obtaining the general solution by the method of separation of variables? It looks like it will work in this case but I have not solved the problem.

What is ##x=0## doing in the boundary condition ##~\left.-k\dfrac{\partial T}{\partial r}\right\rvert_{x=0}=0~##. Is it a typo?

I reported this thread and it should be moved to the Advanced Homework forum by a mentor at some point in time.
Yes it was a typo, fixed it already. And yes I have tried to get the the general solution by the method of separation of variables, the thing is I know how to apply it but with homogeneous BC where I do some things with Sturm-Liouville. The thing is Sturm-Liouville only works with homogeneous BCs, I know a method to transform the non homogeneous BCs to homogeneous, which is the one I already wrote where I have to guess the form of w(r,z)

And thanks for helping to move my question to the right section

jackkk_gatz said:
I have tried to get the the general solution by the method of separation of variables,
Wolfram gives me Bessel functions.

## What is the heat equation with non-homogeneous boundary conditions?

The heat equation with non-homogeneous boundary conditions refers to a partial differential equation (PDE) that models the distribution of heat (or temperature) in a given region over time, where the boundary conditions specify temperatures or heat fluxes that are not constant (i.e., not zero) along the boundaries of the region.

## How do you solve the heat equation with non-homogeneous boundary conditions?

To solve the heat equation with non-homogeneous boundary conditions, one typically uses methods such as separation of variables, Fourier series, or integral transforms. The solution often involves breaking the problem into simpler parts, solving for each part, and then combining the solutions. Superposition and transformation techniques are also commonly employed to handle the non-homogeneous terms.

## What are some common techniques to handle non-homogeneous boundary conditions?

Common techniques to handle non-homogeneous boundary conditions include the method of eigenfunction expansions, the use of Green's functions, and the method of images. Another approach is to transform the non-homogeneous problem into a homogeneous one by introducing an auxiliary function that satisfies the non-homogeneous boundary conditions.

## Can the heat equation with non-homogeneous boundary conditions be solved numerically?

Yes, the heat equation with non-homogeneous boundary conditions can be solved numerically using methods such as finite difference methods, finite element methods, and finite volume methods. These numerical techniques discretize the PDE and the boundary conditions, allowing for approximate solutions to be computed on a grid or mesh.

## What are some real-world applications of the heat equation with non-homogeneous boundary conditions?

Real-world applications of the heat equation with non-homogeneous boundary conditions include thermal engineering, climate modeling, material science, and biological systems. For example, it can be used to model the cooling of electronic components, the temperature distribution in a building, the heat treatment of metals, and the spread of heat in biological tissues.

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