Sturm–Liouville theory | Wikipedia audio article
Channel: wikipedia tts
Date: 27/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory


00:03:51 1 Sturm–Liouville form
00:04:24 1.1 Examples
00:04:32 1.1.1 The Bessel equation
00:05:38 1.1.2 The Legendre equation
00:06:55 1.1.3 An example using an integrating factor
00:10:26 1.1.4 The integrating factor for a general second-order differential equation
00:13:12 2 Sturm–Liouville equations as self-adjoint differential operators
00:18:20 3 Second order differential equation with boundary conditions
00:21:34 4 Example
00:22:09 5 Application to normal modes
00:22:38 6 Representation of solutions and numerical calculation
00:24:07 6.1 Construction of a nonvanishing solution
00:24:10 7 Application to partial differential equations
00:28:33 8 See also
00:29:53 9 References
00:30:24 10 Further reading
00:30:33 0, L1 or y
00:33:04 Representation of solutions and numerical calculation
00:34:58 0; it does not need to be an eigenvalue) and any solution y0 of (1) with λ
00:37:36 c0u0 + c1u1 satisfies the first boundary condition (2). This is simple to do since X(n)(a)
00:37:49 0, for n  0.  The values of X(n)(b) and X̃(n)(b) provide the values of u0(b) and u1(b) and the derivatives u′0(b) and u′0(b), so the second boundary condition (3) becomes an equation in a power series in λ.  For numerical work one may truncate this series to a finite number of terms, producing a calculable polynomial in λ whose roots are approximations of the sought-after eigenvalues.When λ
00:38:45 Construction of a nonvanishing solution
00:39:03 μqy; i.e., q, w, and λ are replaced in (1) by 0, −q, and μ respectively. Then the constant function 1 is a nonvanishing solution corresponding to the eigenvalue μ0
00:39:35 u0 + iu1 will never vanish because two linearly-independent solutions of a regular S–L equation cannot vanish simultaneously as a consequence of the Sturm separation theorem. This trick gives a solution y0 of (1) for the value λ0
00:40:12 Application to partial differential equations
00:48:30 See also



Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.

Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain

Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.

Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.9936996022690276
Voice name: en-US-Wavenet-A


"I cannot teach anybody anything, I can only make them think."
- Socrates


SUMMARY
=======
In mathematics and its applications, a classical Sturm–Liouville theory, named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882), is the theory of a real second-order linear differential equation of the form

where y is a function of the free variable x. Here the functions p(x), q(x), and w(x)  0 are specified at the outset. In the simplest of cases all coefficients are continuous on the finite closed interval [a,b], and p has continuous derivative. In this simplest of all cases, this function y is called a solution if it is continuously differentiable on (a,b) and satisfies the equation (1) at every point in (a,b). In addition, the unknown function y is typically required to satisfy some boundary conditions at a and b.  The function w(x), which is sometimes also called r(x), is called the "weight" or "density" function.
The value of λ is not specified in the equation; finding the values of λ for which there exists a non-trivial solution of (1) satisfying the boundary conditions is part of the Sturm–Liouville (S–L) problem.
Such values of λ, when they exist, are called the eigenvalues of the boundary value problem defined by (1) and the prescribed set of boundary conditions. The corresponding solutions (for each such λ) are the eigenfunctions of this problem. Under normal assumptions on the coefficient functions p(x), q(x), and w(x) above, they induce a Hermitian differential operator in some function space defined by boundary conditions. The resulting theory of the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and their completeness in a suitable function space became known as Sturm–Liouville theory. This theory is important in applied m ...