How Can a 3D PDE Be Simplified to a 1D Equation in Theta?

In summary, the conversation discusses a PDE involving cylindrical coordinates, with constraints on z and r. The question is raised on how to simplify the 3D PDE into a 1D PDE in teta without using separation of variables. It is suggested to use the Laplasian in cylindrical coordinates and discard the d/dr and d/dz terms. The speaker hopes for assistance in finding a solution.
  • #1
Ido
2
0
I got the following PDE:
Laplasian[F]+a*d(F)/d(teta)=E*F

I worked with cylindrical coordinates (r,teta,z)
(teta is the angle between the x-axis and the r vector (in xy plane))
a,E are constants

I got the constrains: z=0 r=a , so the whole problem is on a simple ring

How can I make the 3D PDE just 1D in teta?
Can I use the Laplasian in cylindrical coordinates, and throw away the d/dr and d/dz?

I hope someone can help me.
 
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  • #2
Without separation of variables (which I know you don't want to do), I don't see a way.
 
  • #3


The given PDE can be simplified to a 1D PDE in the teta direction by making use of the constraints given. Since the whole problem is on a simple ring, we can assume that the function F is only dependent on the teta coordinate. Therefore, the Laplacian operator in cylindrical coordinates can be simplified to just the second derivative with respect to teta, i.e. Laplacian[F] = (1/r)*d/dr(r*dF/dteta).

Furthermore, since the constraints state that z=0 and r=a, any terms containing the z or r coordinates can be eliminated. This leaves us with the simplified 1D PDE:

(1/a)*d/dteta(a*dF/dteta) + a*E*F = 0

This can be solved using standard methods for solving 1D PDEs. It is important to note that the solution obtained will only be valid for the specific constraints given, i.e. for the problem on a simple ring. If the constraints were to change, the PDE would need to be re-evaluated accordingly.

In summary, to make the 3D PDE just 1D in teta, we can use the Laplacian in cylindrical coordinates and eliminate any terms containing the z or r coordinates based on the given constraints. This will result in a simplified 1D PDE that can be solved using appropriate methods.
 

FAQ: How Can a 3D PDE Be Simplified to a 1D Equation in Theta?

1. What is a Constrained PDE?

A Constrained PDE (Partial Differential Equation) is a mathematical equation that describes the relationship between multiple variables in a system, subject to certain constraints. These constraints can be physical, such as boundary conditions, or mathematical, such as initial conditions. Constrained PDEs arise in many fields of science and engineering, and can be used to model a wide range of physical phenomena.

2. What does 3D -> 1D mean in the context of Constrained PDEs?

The notation 3D -> 1D refers to the dimensionality of the problem being solved. In the case of Constrained PDEs, it means that the problem involves three spatial dimensions (x, y, z) and one temporal dimension (t). This means that the system being modeled is changing over time in three-dimensional space.

3. How are Constrained PDEs solved?

Constrained PDEs are typically solved using numerical methods, such as Finite Difference or Finite Element methods. These methods involve discretizing the problem domain into a grid of points, and then using iterative techniques to approximate the solution at each point. The accuracy of the solution depends on the size of the grid and the specific numerical method used.

4. What are some real-world applications of Constrained PDEs?

Constrained PDEs are used in a variety of fields, including physics, engineering, and biology. They can be used to model fluid dynamics, heat transfer, electromagnetics, and many other physical phenomena. For example, Constrained PDEs are used to simulate weather patterns, design airplanes, and study the behavior of biological cells.

5. What are the challenges of solving Constrained PDEs?

Solving Constrained PDEs can be computationally intensive, especially for problems with complex geometries or high-dimensional systems. The accuracy of the solution also depends on the numerical method used and the size of the grid, which can be challenging to optimize. Additionally, interpreting and visualizing the results of a Constrained PDE can be difficult due to the high-dimensional nature of the problem.

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