I solved the following Sturm-Liouville Problem: $\begin{matrix} w^{\prime \prime}(x) = \mu w(x), \\ w^{\prime}(0) = > w(1) = 0 \end{matrix}$ and found that the eigenvalues were $\displaystyle \mu_{n}= \left( \frac{\pi(2n-1)}{2} \right)^{2}$, $n=0,1,\cdots$ and that the corresponding eigenfunctions were $\displaystyle w_{n}=\cos \left(\frac{\pi(2n-1)}{2} \right)x$, $n = 0, 1, \cdots$ Now, in addition, I am being asked to […]

Use the appropriate engenfunction expansion to represent the best solution. $$u”=f(x), u'(0)=\alpha, u'(1)=\beta$$ I use the function $$\phi”+\lambda\phi=0$$ to get the eigenfunction is $$\phi=A\cos x\sqrt{\lambda}+B\cos x\sqrt{\lambda}$$ but how should I decide A and B? Is it by system$$\phi'(0)=\alpha, \phi'(1)=\beta$$ or it should be $$\phi'(0)=0, \phi'(1)=0$$ and why? After getting eigenvalue and eigenfunctions, what should I […]

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 Sturm-Liouville problem. Then for each value of $x$ in $[a,b]$, the Fourier series of $f(x)$ relative to $\{\varphi_n(x)\}$ converges, and […]

Find the eigenvalues of $$y” + \lambda y = 0, \; y'(0) = 0, y(1) = 0$$ For $\lambda >0$, $$y(x) = c_1 \cos(\sqrt{\lambda} x) + c_2 \sin(\sqrt{\lambda}x)$$ We get that $y'(0) = 0 \implies c_2 = 0$, but when I try to solve for $\lambda$ when doing $y(1) = 0$, I run into trouble. […]

There are two matrices $A$ and $B$ which can not be diagonalized simultaneously. Is it possible to block diagonalize them? What if the matrices have an special pattern? Physics of the problem is that, I have $2$ layers of atoms where $A$ is connectivity within the layer $1$ itself and $B$ is connectivity between layer […]

Consider the Laplace operator $$\Delta: H^{k+2}(\mathbb{T}^2) \to H^k(\mathbb{T}^2)$$ where $\mathbb{T^2}$ is the two-dimensional torus (which is a compact manifold without boundary), so that $$ H^k(\mathbb{T}^2) \simeq H^k_{per}([-\pi,\pi)).$$ Here, $H^k$ denotes the Sobolev space $W^{k,2}$ equipped with the norm of your choice. I have two questions: 1) Is there an orthonormal basis $\lbrace v_n \rbrace_{n \in […]

Given a sequence of functions orthogonal over some interval, which satisfy Dirichlet boundary conditions at that Interval, can we construct a Sturm Liouville problem that gives these as its eigenfunctions? For example, if we take the sequence of scaled Bessel’s functions $J_n (\zeta_i x)$ for all positive integral values of $i$ where $\zeta_i$s are roots […]

Let $V$ be the linear space of all real polynomial $p(x)$ of degree $\leq n$. If $p \in V$, define $q=T(p)$ to mean that $q(t)=p(t+1)$ for all real $t$. Prove that $T$ has only the eigenvalue $1$. What are the eigenfunctions belonging to this eigenvalue? It is obvious that constant polynomials are eigenfunctions with eigenvalues […]

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