Why Use Orthogonality Properties in Special PDE Problems?

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SUMMARY

The discussion centers on the necessity of using orthogonality properties in solving specific partial differential equation (PDE) problems, particularly highlighted in exercise 13-3 from a provided PDF. The participant, Jonathan, shares insights on the limitations of using Fourier Series for derivatives, concluding that the derivative of a function does not equate to the derivative of its Fourier Series. This realization underscores the importance of orthogonality in obtaining correct solutions for PDEs.

PREREQUISITES
  • Understanding of partial differential equations (PDEs)
  • Familiarity with Fourier Series and their properties
  • Knowledge of orthogonality in function spaces
  • Basic calculus, particularly differentiation techniques
NEXT STEPS
  • Study the application of orthogonality properties in solving PDEs
  • Explore the implications of Fourier Series on function derivatives
  • Review advanced topics in functional analysis related to orthogonal functions
  • Examine case studies involving PDEs and orthogonality in engineering contexts
USEFUL FOR

Mathematicians, physicists, and engineers dealing with partial differential equations, as well as students seeking to deepen their understanding of orthogonality and Fourier analysis in applied mathematics.

jgthb
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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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