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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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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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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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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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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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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