Finding values for a harmonic function

In summary, the conversation discusses a function u that is harmonic in a domain containing a closed disc defined as D(x,y),r) where r=x^2+y^2 ≤ 1. The values of u on the boundary of the disc are given in terms of the polar angle # by sin# + cos#. The task is to find the value of u at the center of the disc and its maximum and minimum values on the closed disc. It is suggested to use the mean value property for harmonic functions to solve for a, and the maximum and minimum principle for b. It is also noted that u must be continuous for the maximum and minimum values to be found on the boundary.
  • #1
Stephen88
61
0
A function u is harmonic in a domain containing the closed disc x^2+y^2 ≤ 1. Its values on the boundary
are given in terms of the polar angle # by sin# + cos#. Without finding u find
a. its value at the centre of the disc
b. its maximum and minimum values on the closed disc.
The disc is probably defined as D(x,y),r) where r=x^2+y^2 ≤ 1 and I think that the max and minimum can be find on the boundary of the set.But I think that u must also be continuous for that to happen.
Can someone give me a detailed explanation or an example on how to solve this problem?
 
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  • #2
What is the best way to solve a and b?
 
  • #3
For a use the mean value property for harmonic functions. For b use the maximum and minimum principle.
 

FAQ: Finding values for a harmonic function

What is a harmonic function?

A harmonic function is a type of mathematical function that satisfies the Laplace's equation. This equation states that the sum of the second-order partial derivatives of a function is equal to zero. In simpler terms, a harmonic function is a function whose value at a point is equal to the average of the values of its neighboring points.

How do you find values for a harmonic function?

There are several methods for finding values for a harmonic function. One way is to use the Laplacian operator, which is a differential operator that allows us to find the Laplace's equation for a given function. Another method is to use boundary conditions, where we specify the values of the function at certain points in order to solve for the values at other points.

What is the importance of harmonic functions?

Harmonic functions have many applications in various fields such as physics, engineering, and mathematics. They are used to model physical phenomena such as electric and magnetic fields, fluid flow, and heat distribution. They also play a crucial role in solving boundary value problems and understanding the behavior of complex systems.

Can a harmonic function have multiple solutions?

Yes, a harmonic function can have multiple solutions. This is because the Laplace's equation is a linear equation, which means that any linear combination of solutions is also a solution. Additionally, for some boundary conditions, there may be an infinite number of solutions for a given harmonic function.

Are there any real-world examples of harmonic functions?

Yes, harmonic functions can be found in many real-world situations. For example, a guitar string can vibrate in a way that is described by a harmonic function. The distribution of temperature in a room can also be modeled using a harmonic function. Additionally, the flow of water in a river can be approximated by a harmonic function.

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