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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Completes of an orthonormal set in a Hilbert space

Let numbers $\{\rho_n\}_{n\ge 0}, \quad \rho_n \neq \rho_k,\quad (n\neq k)\quad$ of the form $\quad \rho_n = n + \frac{a}{n}+\frac{\kappa_n}{n}, \quad \{\kappa_n\} \in \ell^2 \quad $ be given. Then $$ ...
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Existence of solution based on convergence of finite-difference scheme

Consider the following boundary value problem $$ \begin{cases} y''(x)=p(x)y'(x)+q(x)y(x)+r(x)\\ y(0)=\alpha,\quad y(1)=\beta. \end{cases} $$ for continuous $p(x)$, $q(x)$, $r(x)$ and positive $q(x)>...
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what more can I say using Sturm's comparison theorem?

Can I use Sturm's comparison theorem to say something about the average? Let $f(t)$ be a continuous function such that $\lim_{t \to \infty } f(t) = \infty$. Let us consider the following function $$ \...
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How to derive a key equation in Sturm-Liouville Theory

Recently bombed a quiz on Sturm-Liouville theory and orthogonal polynomials in my math methods for physics class, and I'm trying to go through the chapter on the theory and plug up the holes in the ...
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Brezis' exercise 8.31.2: if $\int_I f=0$ then $\|u\|_{L^2(I)} \leq \frac{1}{(1+\pi^2)} \|f\|_{L^2(I)}$

Let $I$ be the open interval $(0, 1)$. I am trying to solve a problem in Brezis' Functional Analysis Exercise 8.31 Consider the Sturm-Liouville operator $A u=-u^{\prime \prime}+u$ on $I$ with Neumann ...
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Third-order PDE and Fourier series

I am wondering how to find a Fourier series solution of the following PDE: $$u_{t} + i u_{xxx} = 0$$ $$u(0, x) = u_{0}(x)$$ where $t$ is in $(0, \infty)$ and $x$ is in the torus $\mathbb{T} = \mathbb{...
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Sturm-Liouville problem with singular weight

I have the following eigenvalues problem: $ xy'' + \lambda y = 0 $, $0 \leq x \leq 1$, $y(0)=y(1)=0$ that if we rewrite becomes $y''+ \lambda \frac{1}{x} y = 0$, a Sturm-Liouville problem with weight $...
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Completeness of system of eigenfunctions of Hermitian and Sturm-Liouville operators

My professor in the course "Math for Physicists" mentioned that the eigenfunction of a Hermitian operator or a Sturm-Liouville operator is an orthonormal basis for the function space on ...
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Identifying eigenvalue and eigenfunctions of Sturm Liouville Problem

Here is the problem: $-y''=\lambda y$ with boundary conditions of $y(0)=0$ and $y(\pi)=-y'(\pi)$ My attempt is below: General solution is $$y(x) = a \sin(\sqrt\lambda x)+b\cos(\sqrt\lambda x).$$ ...
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Solve and integral equation with symmetric kernel [closed]

I have the following integral equation with symmetric kernel $$ x(t)=\sin(\pi t)+\pi \cos (\pi t) +\lambda \int_{0}^{1} k(t,s)x(s)\,ds $$ where $k(x,t)$ is a symmetric kernel given by $$k(t,s)= \...
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Find the Green’s function corresponding to the Sturm-Liouville problem

My Sturm-Liouville problem was $y^{\prime\prime}+\lambda y=0$, $y(0)=0$, $y(1)=0$. I found the eigenvalues to be $(n\pi)^2$ and the associated eigenfunction to be $\sin(n\pi x)$. I now have to find ...
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Brezis' exercise 8.26.6: if $k_1 \ge k_2 \ge 0$ then $S_{k_2} - S_{k_1} \ge 0$

Let $I$ be the open interval $(0, 1)$. Assume that $p \in C^1(\bar I)$ and $q \in$ $C(\bar I)$ such that $p(x) \geq \alpha>0$ and $q(x) \geq \alpha>0$ for all $x \in \bar I$. Here $\alpha$ is a ...
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The well-posedness of $-(p u')' + q u=f$ on $(0, 1)$ with boundary conditions $u'(0)= u'(1)=0$

Let $I$ be the open interval $(0, 1)$. Assume that $p \in C^1(\bar I)$ and $q \in$ $C(\bar I)$ such that $p(x) \geq \alpha>0$ and $q(x) \geq \alpha>0$ for all $x \in \bar I$. Here $\alpha$ is a ...
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How to estimate eigenvalues of Sturm-Liouville ODE whose solutions are not known

Suppose $p, q$ are functions that satisfy the properties for $$\frac{d}{dx} \Big( p(x) \frac{d\phi}{dx}\Big) + q(x)\phi + \lambda w(x) \phi = 0 \tag{1}$$ to be a Sturm-Liouville problem on some ...
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Substitution on a Temperature Problem

Given real parameters $A,B,C$ consider the temperature problem with non-homogeneous boundary conditions: $$u_t=ku_{xx}, \;\;\; u=u(x,t), \;\;\;0\leq x\leq \pi,\;\;\; t,k> 0$$ $$u_x(0,t)=u(0,t)+...
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Help Solving and Understanding a Temperature Problem

Consider the following temperature problem: $$u_t(x,t)=ku_{xx}(x,t), \;0\leq x \leq \pi,\;\; t,k >0$$ with boundary conditions: $$u_x(0,t)=u(0,t)$$ $$u_x(\pi,t)=u(\pi,t)$$ $$u(x,0)=f(x)$$ I know ...
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Brezis' exercise 8.21.3: if $k \neq k_0$ and $f \in L^2(I)$ then problem $(2)$ admits a unique solution $u \in H^2(I)$

Let $I$ be the open interval $(0, 1)$. I'm trying to solve a problem in Brezis' Functional Analysis, i.e., Exercise 8.21 Assume that $p \in C^1(\bar I)$ and $q \in$ $C(\bar I)$ such that $p(x) \geq \...
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The existence of a solution to $-(pU')' + qU = f$ on $(0, 1)$ with boundary condition $U(1) = 0$

Let $I$ be the open interval $(0, 1)$. Assume that $p \in C^1(\bar I)$ and $q \in$ $C(\bar I)$ such that $p(x) \geq \alpha>0$ and $q(x) \geq 0$ for all $x \in \bar I$. Let $f \in L^2 (I)$. In the ...
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Brezis' hint for exercise 8.21.1: how to obtain $1 \leq \|v_0' \|_1$?

I'm trying to solve a problem in Brezis' Functional Analysis, i.e., Exercise 8.21 Assume that $p \in C^1([0,1])$ with $p(x) \geq \alpha>0 \quad \forall x \in[0,1]$ and $q \in$ $C([0,1])$ with $q(x)...
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Second Order Differential Equation Sturm Liouville Problem, eigenvalue properties

I have the spectral problem $X'' + (\alpha(x) - \lambda)X = 0$, with boundary conditions $X(0) = 0 = X(1)$. This is a Sturm Liouville problem, and the book I am reading says that clearly $$ ||\alpha(x)...
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Boundary Value Problem has no solutions

Given the folllowing ODE, find its eigenvalues and eigenfunctions: $$ x^2y''-xy'+\lambda y=0,y(1)=y(L)=0,L >1 $$ Doing the analysis for each condition of $\lambda, i.e.,\lambda=0, \lambda>1, \...
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Eigenfunction Uniqueness Theorem

Consider the following Sturm-Liouville boundary value problem: $\text{Given parameters }c > 0 \text{ and } \beta > 0, \text{ let } y=y(x) \text{ for } 0 \leq x \leq c. \text{ We have }$ $$y''+ \...
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Legendre's Polynomial and spherical harmonics

The differential equation that is satisfied by the Legendre's polynomials is: $$(1-x^2)y'' - 2xy' + \lambda y = 0 (*)$$ I have also been told that the Legendre's polynomial with the parameter $x = \...
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Help Understanding Sturm-Liouville Boundary Value Problem

Consider the following Sturm-Liouville boundary value problem: $\text{Given parameters }c > 0 \text{ and } \beta > 0, \text{ let } y=y(x) \text{ for } 0 \leq x \leq c. \text{ We have }$ $$y''+ \...
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Vibration of second order linear ODE

I am now considering the behavior of a class of second order ODE as $x$ is large enough. $$y''+\alpha(x)y'+y=0 \qquad (1)$$ Clearly, when $\alpha$ is a small constant, the equation changes into the ...
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Verification: Eigenfunction expansion using products of bessel and spherical harmonics

Preface This is a can you please check my work post. Maybe these are frowned upon. However, I have been working on this for a couple of days now. I can use some help. Context I am solving a physics ...
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Systematic way to determine the ansatz and sign in eigenvalue problem resulting form separation of variables fo Laplace's equation

To solve $$ \nabla^2 u = 0$$ in a rectangle $(0 \leq x \leq L; 0 \leq y \leq H)$ using separation of variables, we assume the solution is of the form $$u(x,y) = h(x)\phi(y)$$ or $$u(x,y) = h(y)\phi(x)$...
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Uniqueness of Sturm-Liouville like problem

the following is an exercise taken from the written exam of the functional analysis course that I am following Let $f \, : \, [0,1] \times \mathbb{R} \to \mathbb{R}$ be a $C^1$ function that ...
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Asymptotic Expansion of eigenvalues

I would like to answer the following question Show that the large eigenvalues of the system $$ \frac{d^2 w}{d x^2}=\left(x^4+x^2-\lambda^2\right) w, \quad w(-\infty)=w(\infty)=0, $$ are given by $$ \...
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The zero of a nontrivial solution of Sturm Liouville problem is isolated and vanishes

I read Coddington, theory of ordinary differential equations. And there is a statement that "A zero of a nontrivial solution of $$ (p(t)x(t)')'+g(t)x(t)=0,\quad p(t)>0,\quad p, p', g\in \...
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Sturm-Liouville problem: where did we use $p' \in C(\bar{I})$?

I'm reading section 8.4 Some Examples of Boundary Value Problems in Brezis' Functional Analysis. Example 2 (Sturm-Liouville problem). Consider the problem $$ (18) \quad \left\{\begin{array}{l} -\left(...
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Converting to Sturm-Liouville using integration factor [closed]

I know this type of question has been asked before, but I cannot find information on this specific form I am supposed to be using. We are being asked to convert a second-order DE \begin{equation*}\...
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Bounds on Sturm-Liouville eigenfunctions and its derivatives

Suppose we have the self-adjoint Sturm-Liouville problem $$ (p(x)y')'+q(x)y=\lambda y \\ Ay(a)+By'(a)=0 \\ Ay(b)+By'(b)=0 $$ whose eigenvalues are $0<|\lambda_1|<|\lambda_2|<\cdots$ and $\{\...
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Showing that the solution of the ODE has at least k zeros

Consider the differential equation: $$x''(t) + q(t) x(t) = 0 \qquad , a\leq t\leq b$$ , where $q$ is a continuous function. If $\min_{t\in [a,b]} q(t) > k^2 \pi^2 (b-a)^{-2}$ for some positive ...
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Conjugate symmetry in inner products and the intuition behind projections

Before I begin, I understand why inner products need to maintain conjugate symmetry as seen in this post. This is not what my question is directly about. My motivation is to derive complex Fourier ...
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Derivation of Complex Fourier Series coefficients through inner products (and swapping arugments)

I am trying to derive the complex Fourier series coefficients given by:$$f(x)=\sum_{n=-\infty}^{\infty}{c_n}e^{inx}$$ with coefficients:$$c_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}{f(x)}e^{-inx}dx$$ I am ...
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Inhomogeneous Sturm-Liouville problems

For the following 3 similar Sturm-Liouville problems, I'm struggling to understand how to get to the solutions. I know the solutions already but wolfram alpha gives others, maybe more general ones. ...
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PDE - separation of variables in a rectangle

I'm trying to solve a PDE problem, and got this S-L problem $ \nabla^2 u_2(x,y)=0,\,\, x \in (-1, 1),\,\, y\in (0, \pi) \\ u_2(x,0) = \cos(0.5\pi x),\,\, x\in [-1,1] \\ u_2(x,\pi) = 2\sin(\pi x),\,\, ...
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Re-writing an ODE by Multiplying with a Function

I have the following equation $$\frac{d}{dx} \bigg( \frac{1}{(1+ x^2)^2} \frac{dy}{dx} \bigg) + \lambda x^2 y =0 \ .$$ I'm trying to re-write it so that it looks more like a physically relevant ...
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Continuous dependence of eigenvalues for Sturm-Liouville problem

Let's say I have a certain Sturm-Liouville problem with Dirichlet initial conditions of the form $$ (p(x) y'(x))'+q_c(x)y(x)=\lambda w(x) y(x), \quad y(a)=y(b)=0 $$ where the function $q_c$ is a ...
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analytic solution for the convection-diffusion in the interval $[0,L]$

I want to solve the diffusion-convection equation for a system I need to simulate in my research. Below I describe how to solve the equation for reflecting boundary conditions on both sides and a ...
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Eigen values of BVP $(x^3y’)’+\lambda xy=0, y(1)=y(e)=0$ .

How to find eigen values of the Sturm Liouville problem $$(x^3y’)’+\lambda xy=0, y(1)=y(e)=0?$$ I only know that it’s Cauchy Euler equation as $$x^3y’’+3x^2y’+\lambda xy=0$$ which after dividing by $...
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Estimate for change in eigenvalues of truncated half line operator

Let $(t_n)_{n=1}^\infty\subset (0,\infty)$ be an infinite, discrete set of points. I use those to define a Sturm Liouville operator $H$ on the half line $[0,\infty)$. It acts on $H^2(0,\infty)$ as $$...
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Sturm-Liouville Problem $((x^2+1)y’)’+\frac{\lambda }{x^2+1}y=0, y(0)=y(1)=0. $

How to find eigenvalues and eigenfunctions of Sturm-Liouville Problem $$((x^2+1)y’)’+\frac{\lambda }{x^2+1}y=0, y(0)=y(1)=0 $$ In the question hint is given as $$\text{Let}~ x=\tan(t).$$ Now, as we ...
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Eigen Values of $(\frac{1}{3x^2+1}y’)’+\lambda (3x^2+1)y=0, y(0)=0, y(\pi)=0.$

How to find eigen values of Sturm Liouville Problem $$(\frac{1}{3x^2+1}y’)’+\lambda (3x^2+1)y=0, y(0)=0, y(\pi)=0?$$ I only know how to find eigen values of $y’’+\lambda y=0, y(0)=y(\pi)=0,$ because ...
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Do the Fourier sine and cosine series individually a complete basis by Sturm Liouville theorem?

I am confused by the Sturm-Liouville theorem implication on the Fourier sine and cosine series. Consider the simple ODE $$y''(x)=-k^2 y(x).$$ It is in Sturm-Liouville form. Let’s impose a boundary ...
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Non homogeneous Heat equation in polar coordinates with non homogeneous BC's

I'm trying to work around my way of this problem $$a\frac{1}{r}\frac{\partial }{\partial r}(r\frac{\partial u}{\partial r})+be^{-y(r-a)}=\frac{\partial u}{\partial t}$$ $$\left.\frac{\partial u}{\...
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Almost Sturm-Liouville operator on $L^2(\mathbb{R})$

I am used to deal with second order Sturm-Liouville operator on $L^2(\mathbb{R})$. What is useful for my problems is to identify the kernel, and count the number of zeroes of the eigenfunction. From ...
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Finding the spectrum of a Sturm-Liouville problem

I have the following Sturm-Liouville problem for $0 \le x\le \pi$ $$y'' + \lambda y=0, \qquad y(0) = 0, \qquad y(\pi)+y'(\pi) = 0 $$ How do I find the spectrum of the problem? And how will the ...
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Solving the 1D Heat Equation on [a,b] rather than [0,L]

Solve the 1D Heat Equation on $x \in [a,b]$ $$ \frac{\partial ^2 T}{\partial x^2} = \frac{1}{\alpha} \frac{\partial T}{\partial t}$$ $$ T(a,t) = T(b,t) = 0, T(x,0) = T_0(x) $$ Now, I know that ...
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