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2016 | OriginalPaper | Chapter

4. Sturm–Liouville Theory and Boundary Value Problems

Author : Jon H. Davis

Published in: Methods of Applied Mathematics with a Software Overview

Publisher: Springer International Publishing

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Abstract

Partial differential equation models with non-rectangular geometries, more independent variables, and variable material properties are introduced. Many examples of such problems have separated equations of Sturm-Liouville type, in turn leading to power series solutions. These topics are discussed with Bessel functions as a detailed example. More complicated problems involve inhomogeneous and forced system models. Many other examples have complicated domain shapes which make simple separation of variables methods inapplicable. For such cases, we introduce some numerical methods and emphasize the finite element approach.

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previous chapter Elementary Boundary Value Problems
next chapter Functions of a Complex Variable
Footnotes
1
This definition suffices to obtain the desired results and avoids the technicalities inherent in the rigorous definition of “adjoints” for mappings of the sort constructed above.
 
2
Here it is convenient to write the parameter as λ2 as used above, because it is possible for a finite number of the eigenvalues to be negative if there is no assumption on q.
 
3
Since the problem is singular, the Sturm–Liouville results of Section 4.3 do not apply directly, and completeness remains to be proven. Proofs of completeness for some singular problems appear in reference [9].
 
4
The HDF format developed by the National Center for Supercomputing Applications (NCSA) is supported for both import and export by Octave.
 
5
One might also evaluate the partial sums along the diagonal M = N and average only N rather than MN partial sums. The execution time is much lower, but the overshoot is evident in the plots.
 
6
E.g., the value at an infinite number of points, or an infinite number of Fourier coefficients.
 
7
If ∇2 u ≠ 0 in some region of \(\Sigma\), one can construct a smooth v vanishing outside of this region, and such that ∫ ∫ v∇2 u dA ≠ 0. Therefore ∇2 u must vanish in \(\Sigma\).
 
8
Look for IGES, CAD, FEM, and similar things on the world wide web.
 
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Metadata
Title
Sturm–Liouville Theory and Boundary Value Problems
Author
Jon H. Davis
Copyright Year
2016
Book
Methods of Applied Mathematics with a Software Overview

Print ISBN: 978-3-319-43369-1

Electronic ISBN: 978-3-319-43370-7

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-43370-7_4

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