# 3D Heat equation with elementary Dirichlet BC

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• Flo31
In summary, the conversation discusses a heat conduction problem with a semi-infinite domain, constant thermal diffusivity, and specific boundary conditions. The problem can be solved using Green's functions with Neumann boundary conditions, but the solution for Dirichlet conditions is not yet known. The use of door functions and Laplace transforms is considered, but the best approach is still uncertain.
Flo31
TL;DR Summary
3D Heat equation with dirichlet condition in semi infinite domain
Hi,

I am looking for the solution of the following heat conduction problem (see figure below):
• the geometry is the semi-infinite domain such that (x,y)∈R2 and z∈[0,∞[ ;
• the thermal diffusivity is constant;
• the domain is initially at a temperature of 0;
• At t>0, a small square of the surface ((x,y)∈R2) is instantaneously brought at a temperature of u0. The rest of the boundary is maintained at a temperature of 0.
The solution to such problem is fairly easy to get with Green's functions when Neumann boundary conditions are imposed(i.e. imposed heat flux on the surface z=0). I would be surprised that no analytical solution to the same problem with Dirichlet conditions does not exist, although I don't manage to find it. How such boundary can be treated?

I looked into the solutions given by A.V. Liukov Analytical Heat Diffusion Theory (1968), but nothing looks similar to this. The issue, here is the combination of having Dirichlet BC and that the value of temperature on z=0 depends on x and y. Using the properties of door function may help (similarly to the dirac distribution for describing a point source in Green's problem), but I am still not sure how to tackle the problem.

Thanks,
Florian

Why isn't taking Laplace transforms viable here?

## 1. What is the 3D heat equation with elementary Dirichlet BC?

The 3D heat equation with elementary Dirichlet BC is a mathematical model used to describe the flow of heat in a three-dimensional space, where the boundary conditions (BC) are specified as a constant value at the boundaries. It is commonly used in physics and engineering to study heat transfer in solid objects.

## 2. How is the 3D heat equation with elementary Dirichlet BC derived?

The 3D heat equation with elementary Dirichlet BC is derived from the fundamental laws of thermodynamics, specifically the Fourier's law of heat conduction and the conservation of energy. It takes into account the temperature gradient, thermal conductivity, and heat source/sink within the system.

## 3. What are the applications of the 3D heat equation with elementary Dirichlet BC?

The 3D heat equation with elementary Dirichlet BC has various applications, including but not limited to: predicting the temperature distribution in a solid object, analyzing heat transfer in buildings and structures, and designing efficient cooling systems for industrial processes.

## 4. What is the significance of the Dirichlet BC in the 3D heat equation?

The Dirichlet BC plays a crucial role in solving the 3D heat equation as it specifies the fixed temperature values at the boundaries of the system. This helps in determining the temperature distribution within the system and makes the problem well-posed, meaning that a unique solution can be obtained.

## 5. How is the 3D heat equation with elementary Dirichlet BC solved?

The 3D heat equation with elementary Dirichlet BC can be solved using various numerical methods, such as the finite difference method, finite element method, and spectral methods. These methods discretize the system into smaller elements and approximate the solution at each point using a set of equations derived from the 3D heat equation.

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