Interpreting a PDE for Heat/Diffusion with Vanishing Neumann Boundary Conditions

In summary, the given equation \triangle u = f(x,y,z) on a body with vanishing neumann boundary conditions can be interpreted in terms of heat and diffusion. The lack of a time term in the equation may be confusing, but it suggests that u represents the concentration of heat or substance within the body, and the equation states that no heat or substance will escape the body. This process is time invariant and the laplacian of u is defined to be equal to the function f. This reasoning can provide physical intuition for the distribution of heat or substance within the body.
  • #1
PhDorBust
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Given [tex] \triangle u = f(x,y,z)[/tex] on a given body with vanishing neumann boundary conditions. I'm asked to interpret it in terms of heat and diffusion.

Since heat/diffusion take the form [tex] u_t = k \triangle u[/tex], I am a little confused as I there is no time term here. I think the answer is that u denotes the concentration of heat/substance and the PDE is saying that no heat/substance will escape the body. The process is time invariant, so the PDE is just defining the distribution of heat/substance inside the body to follow this strange rule that its laplacian is f?

Is this reasoning correct? Can I assign some physical intuition to the distribution of heat/substance inside the body following the rule that its laplacian be f?
 
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  • #2
bump.. is something unclear?
 

1. What is a Partial Differential Equation (PDE)?

A Partial Differential Equation (PDE) is a mathematical equation that involves multiple variables and their partial derivatives. It is used to describe the relationship between these variables and their rates of change or how they vary with respect to each other. PDEs are commonly used in physics, engineering, and other scientific fields to model and analyze complex systems.

2. What is the difference between a PDE and an ordinary differential equation (ODE)?

The main difference between a PDE and an ODE is that the former involves multiple independent variables, while the latter only involves one. In other words, PDEs describe systems that are dependent on more than one physical dimension, while ODEs describe systems that are only dependent on a single physical dimension.

3. What is the role of boundary conditions in solving a PDE?

Boundary conditions are additional information or constraints that are used in conjunction with a PDE to obtain a unique solution. These conditions specify the behavior of the solution at the boundaries of the domain in which the PDE is being solved. Without boundary conditions, a PDE may have an infinite number of solutions.

4. What are some numerical methods used to solve PDEs?

There are various numerical methods that can be used to approximate solutions to PDEs. Some commonly used methods include finite difference, finite element, and finite volume methods. These methods discretize the PDE into a system of algebraic equations, which can then be solved using computational techniques.

5. How are PDEs used in real-world applications?

PDEs have numerous applications in various fields of science and engineering. They are used to model and analyze a wide range of physical phenomena, such as heat transfer, fluid dynamics, electromagnetism, and quantum mechanics. PDEs are also used in computer graphics and image processing, as well as in financial mathematics for option pricing.

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