# Heat equation PDE for spherical case

• FiberOptix
In summary, The person is looking for help with solving the heat equation PDE with specific initial conditions involving a spherical mass of water and finding resources for solutions. They mention trying separation of variables and finding some resources through googling, but ultimately need more help and guidance.
FiberOptix
Hello, I believe this is my first post. I would like to solve the heat equation PDE with some special (but not complicated) initial conditions, my scenario is as follows:

A perfectly spherical mass of water, where the outer surface is at some particular temperature at t=0 (but not held at that temperature).

It seems simple enough but for some reason I have not been able to find all that much satisfying those conditions. Any help would be sincerely appreciated, even if it's only to point out a resource.

Thank you,

FiberOptix

Come now, it's almost been a week! I'm interested in 1D and 2D solutions as well if that makes it easier. I hope to have something soon, and if this really is a big problem for most people I'll write it up in latex and post a link.

Hi,
have you tried separation of variables? the solution should look similar to the l=0 free wave solution of spherically symmetric Schroedinger equation with t replaced by -it.
This help?

mheslep said:

Actually if you look at my initial post you may find that there are a few details that make the generalization from these resources a little difficult, hence my plea for help on this forum. I even found a few better resources than those by googling but not until I got Carslaw & Jeager's book from the library here was I really on the right track.

FiberOptix said:
Actually if you look at my initial post you may find that there are a few details that make the generalization from these resources a little difficult, hence my plea for help on this forum. I even found a few better resources than those by googling but not until I got Carslaw & Jeager's book from the library here was I really on the right track.
Yes, sorry, so you did. IIRC the often missed trick for full sphere problems is forcing the boundary condition u(t,theta=0) == u(t,theta=2pi). Same with the first derivative BC's.

## 1. What is the heat equation PDE for the spherical case?

The heat equation PDE for the spherical case is a partial differential equation that describes the distribution of heat within a spherical object over time. It is given by the equation: ∂u/∂t = α(1/r^2)∂/∂r(r^2∂u/∂r), where u represents the temperature, t represents time, and r represents the distance from the center of the sphere.

## 2. What are the boundary conditions for the heat equation PDE in the spherical case?

The boundary conditions for the heat equation PDE in the spherical case depend on the specific problem being solved. However, some common boundary conditions include prescribed temperature at the surface of the sphere, insulated boundary (no heat transfer), and convective boundary (heat transfer due to fluid flow).

## 3. How is the heat equation PDE solved for the spherical case?

The heat equation PDE for the spherical case can be solved using various numerical methods, such as finite difference, finite element, or spectral methods. These methods involve discretizing the spherical domain into smaller elements and solving the resulting system of equations using numerical algorithms.

## 4. What are some real-life applications of the heat equation PDE for the spherical case?

The heat equation PDE for the spherical case has a wide range of applications in various fields, including physics, engineering, and geophysics. Some examples include modeling the temperature distribution within planets, heat transfer in nuclear reactors, and thermal analysis of spherical objects in industrial processes.

## 5. What are the limitations of the heat equation PDE for the spherical case?

The heat equation PDE for the spherical case assumes certain simplifications, such as a homogeneous and isotropic medium and steady-state conditions. Additionally, it does not take into account external factors such as sources of heat and changes in material properties due to temperature. These limitations may affect the accuracy of the model in certain scenarios.

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