Questions tagged [sturm-liouville]

The Sturm–Liouville equation is a particular second-order linear differential equation with boundary conditions that often occurs in the study of linear, separable partial differential equations.

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What are all the generalizations needed to pass from finite dimensional linear algebra with matrices to fourier series and pdes?

I've studied Linear Algebra on finite dimensions and now I'm studying fourier series, sturm-liouville problems, pdes etc. However none of our lecturers made any connection between linear algebra an this. I think this is a very big mistake because I…
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Can Sturm-Liouville theory actually solve ODEs?

My teacher talked about Sturm-Liouville theory, and we learned that any second order differential equation can be put into the self-adjoint form. What is this? Well, the book says is something in the…
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Is the Schwarzian Derivative a connection?

The Schwarzian derivative is the quadratic differential $$ S(f) = \Bigg( \frac{f'''}{f'} - \frac{3}{2} \Big( \frac{f''}{f'} \Big)^2 \Bigg) (dz)^2 .$$ Bill Thurston, in his Math overflow answer here and his paper "Zippers and Univalent Functions"…
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Solving a second-order linear ODE: $\frac{d^2 y}{dx^2}+(x+1)\cdot \frac{dy}{dx}+5x^2\cdot y=0$

Recently, a friend challenged me to find the general solution of the following differential equation: $$\frac{d^2 y}{dx^2}+(x+1)\cdot \frac{dy}{dx}+5x^2\cdot y=0 \tag{1}$$ This is a second-order linear ordinary differential equation. I have tried…
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Condition for Self-Adjoint Sturm-Liouville Operator

Consider the Sturm-Liouville operator $$L(u) = -(pu')' + qu$$ where, $p \in C^1[a,b]$ and $q \in C[a,b]$ with $p(t) \neq 0$ for $t \in [a,b]$ are complex valued functions, with boundary conditions: \begin{align*}\alpha u(a) + \beta u'(a) = 0 \\…
r9m
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What is the motivation for the equation of the Sturm-Liouville problem?

In a course on PDE's, one would typically come across some treatment of Sturm-Liouville problems, which are roughly in the form $$\frac{d}{dx} \left [p(x) \frac{dy}{dx} \right ]+y \left (q(x)+\lambda w(x)\right )=0$$ I'll omit the details for each…
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How to see that the eigenfunctions form a basis for the function space?

We have a Sturm-Liouville operator $$ L=\frac{1}{w(x)}\left[\frac{d}{dx}\left(p(x)\frac{d}{dx}\right)+q(x)\right] $$ and consider $$ \frac{\partial c}{\partial t}=Lc, $$ with homogeneous boundary conditions. If we are now searching for solutions,…
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Separation of variables of 2nd order PDE yields 1st order ODEs

Consider the telegraph or Klein-Gordon equation on a rectangle*, $$ \begin{align} \left(\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}\right)\psi(x,y)=\gamma^2\psi(x,y), \end{align} $$ with $\gamma$ some arbitrary (maybe complex)…
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Inverse spectral problem: How to recover the function $ q(x) $?

The forward problem is a second order Sturm-Liouville operator $$ - \frac{d^{2}}{dx^{2}}y(x)+q(x)y(x)=zy(x) $$ with the boundary conditions $ y(0)=0=y(\infty) $. If I know the spectral measure function $ \sigma (x) =\sum_{\lambda_{n} \le x}1 $, then…
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Proof for Sturm Liouville eigenfunction expantion pointwise convergence theorem

In "Elementary Partial Differential Equation" by Berg and McGregor, the following theorem is given without proof: Let $f(x)$ be piecewise smooth on the interval $[a,b]$ and let $\{\varphi_n(x)\}$ be the eigenfunctions of a self-adjoint regular…
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Linearization which is a Sturm-Liouville problem: Stability questions

Consider the scalar phase equation $$ \theta_t=\theta_{xx}+f(\theta),\qquad f(\theta+2\pi).\qquad (1) $$ Traveling waves profiles $\theta(x-ct)$ can be found using phase-plane analysis for $$ \theta_{\xi\xi}=-c\theta_{\xi}-f(\theta).\qquad…
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The Node Theorem - an argument from physics

In the mathematics literature, the theorem falls under the Oscillation Theory of the Sturm-Liouville equation. It doesn't seem to have a special name in the mathematics literature, but it is well-known as the Node Theorem in the physics literature,…
Benjamin T
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Sturm-Liouville differential equation eigenvalue problem

If we have a Sturm-Liouville differential equation of the form $$ \frac{d}{dx}[p(x)\frac{dy}{dx}]+q(x)y=-\lambda w(x)y $$ and define the linear operator $L$ as $$L(u) = \frac{d}{dx}[p(x)\frac{du}{dx}]+q(x)u $$ then we get the equation…
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Precise conditions on Sturm-Liouville Theorems

In Sturm-Liouville (SL) theory (https://en.wikipedia.org/wiki/Sturm-Liouville_theory), there are three fundamental theorems concerning the solutions of the SL differential equation, $…
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Question about the solution to the heat equation in spherical coordinates

I was solving the heat equation in spherical coordinates with standard boundary conditions: temperature held at 0 at the boundary $r=\alpha$. I was able to find all eigenvalues and eigenfunctions. I'm not going to show all my work because it's…
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