Why Use Orthogonality Properties in Special PDE Problems?

In summary, the conversation discusses the use of orthogonality properties in a specific type of PDE problem. The person has encountered this problem multiple times and has provided examples and solutions in various files. They have also realized that their previous method may not work due to the difference between the derivative of a function and the derivative of its Fourier Series. They seek further insight on this issue.
  • #1
jgthb
8
0
Hello

I have been wondering for some time about, why I have to use orthogonality properties in a special kind of PDE problem I have encountered a few times now.

As an example see exercise 13-3 in this file:
http://www.student.dtu.dk/~s072258/01246-2009-week13.pdf"

I have described my thoughts on this in this file:
http://www.student.dtu.dk/~s072258/ortho_comments.pdf"

And I have solved the problem in this file (the right way by use of orthogonality properties):
http://www.student.dtu.dk/~s072258/orthogonality_solution.pdf"

The last file is in danish, but part b) (which I comment on in the second file) almost only consist of calculations, so it shouldn't be a problem.


Some insight would be much appreciated :)
Jonathan
 
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FAQ: Why Use Orthogonality Properties in Special PDE Problems?

What is the concept of orthogonality in science?

The concept of orthogonality in science refers to the property of two entities being perpendicular or at right angles to each other. In mathematics and physics, it is often used to describe the relationship between vectors, where two vectors are considered orthogonal if their dot product is zero.

How is orthogonality used in data analysis?

In data analysis, orthogonality is utilized to simplify complex data and make it easier to interpret. It is used to identify and eliminate redundant or irrelevant variables in a dataset, reducing the dimensionality of the data and improving the accuracy of statistical models.

What is the significance of orthogonality in signal processing?

In signal processing, orthogonality is crucial for separating and isolating different components of a signal. By using orthogonal transformations, such as Fourier Transform or Wavelet Transform, signals can be broken down into individual components and analyzed separately, allowing for better understanding and manipulation of the signal.

How does orthogonality play a role in quantum mechanics?

In quantum mechanics, orthogonality is a fundamental concept used to describe the behavior of subatomic particles. The wave functions of particles are considered orthogonal to each other, meaning they do not interfere with each other. This allows for the concept of superposition, where particles can exist in multiple states simultaneously.

What are some practical applications of orthogonality in engineering?

Orthogonality has numerous practical applications in engineering, such as in antenna design, control systems, and image processing. In antenna design, orthogonal antennas are used to minimize interference and improve signal reception. In control systems, orthogonal signals are used for accurate measurement and control of various parameters. In image processing, orthogonal transformations are used to compress and store images without losing important information.

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