Solving the 3-d heat equation

Other suggestions for solving this problem could be to use a partial Fourier transform or to simplify the problem by considering specific boundary conditions. In summary, there is no analytic way to solve this 3-d problem, but there are various numerical methods and techniques that can be used to solve it. It is recommended to use a computer program or software package for efficient and accurate solutions.
  • #1
nickthequick
53
0
Hi,

I've been trying to solve

[itex]\frac{\partial \vec{u}(x,y,z,t)}{\partial t} = \nu \nabla^2 \vec{u}(x,y,z,t)[/itex]

Where the Laplacian is 3-d. My initial conditions are

[itex]\vec{u}(x,y,z,0)=\vec{u_o}(x,y,z)[/itex].

And my BC is that

[itex](u_z,v_z,w)= 0 [/itex] at z=0. where [itex]\vec{u} = (u,v,w)[/itex]

I want to solve for [itex]\vec{u}(x,y,0,t)[/itex] i.e. at z=0.

Here, [itex] (x,y) \in \mathbb{R}, z \in (-\infty,0), t\in \mathbb{R}^+ [/itex].

Is there an analytic way to do this? So far I have only come across numerical schemes or examples for lower dimensions.

Any suggestions would be appreciated,

Nick

Nick
 
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  • #2
,Unfortunately, there is no analytic way to solve this 3-d problem. The numerical schemes you are referring to are the most commonly used methods used to solve such problems. Some of the popular numerical schemes are finite difference methods, finite element methods, and boundary element methods. However, these methods require a lot of numerical calculations, so it would be best to use a computer program or software package to solve the problem.
 

1. How is the 3-D heat equation solved?

The 3-D heat equation is solved using numerical methods, such as finite difference or finite element methods, to discretize the equation and solve for the temperature at discrete points in the 3-D space. These methods use iterative algorithms to approximate the solution to the equation.

2. What is the significance of solving the 3-D heat equation?

The 3-D heat equation is an important tool in understanding heat transfer in three-dimensional systems, such as in engineering and physics. It allows scientists to predict and analyze the temperature distribution and heat flow in complex systems, which is crucial for many practical applications.

3. What are the key assumptions made in solving the 3-D heat equation?

The 3-D heat equation assumes that the temperature in the system is continuous and smooth, and that there are no external heat sources. It also assumes that the system is in steady-state, meaning that the temperature is not changing with time. Additionally, it assumes that the thermal properties of the materials in the system are constant.

4. How accurate are the solutions obtained from solving the 3-D heat equation?

The accuracy of the solutions obtained from solving the 3-D heat equation depends on the numerical methods used and the level of discretization of the system. Generally, the solutions become more accurate as the number of discrete points increases, but it also increases the computational cost.

5. Can the 3-D heat equation be solved analytically?

In some simple cases, the 3-D heat equation can be solved analytically using mathematical techniques. However, for most real-world problems, the equation is too complex to be solved analytically, and numerical methods are required.

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