Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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51 views

On spectrum of periodic boundary value problem

Consider the following boundary value problem on the infinite strip $(-\infty,\infty)\times[0,1]$ w/periodic conductivity $\gamma(x,y)=\gamma(x+2\pi,y)>0$: $$\begin{cases} ...
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26 views

Question about the proof of $W^{1,p}_0(\Omega) \Rightarrow u=0 $ on $\partial \Omega$

I am reading the proof of Theorem 9.17 in the book Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. The theorem says: Suppose that $\Omega$ is of class ...
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1answer
25 views

Domain of dependence and energy in the wave equation

Where does the second summand come from in the derivative expression? I am fairly certain it has to do with the fact that the boundary is dependent on $t$, but can someone provide either intuitive ...
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0answers
26 views

Existence and uniqueness of solutions for a system of first order PDEs

Which results can be applied and which conditions are needed, to ensure the existence and uniqueness of the solutions of the first order of PDEs: A$\dfrac{\partial}{\partial ...
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10 views

A version of Hörmander multiplier theorem

Let $m>n/2$ be an integer. Let $h\in H^m_{loc}(\mathbb{R}^n)$ satisfy that $\displaystyle \exists M>0,\forall R>0,\sum_{|\alpha|\le m}\int_{\frac R2\le|w|\le2R}R^{2|\alpha|}|\partial^\alpha ...
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1answer
31 views

Intuition behind boundary conditions for PDE

Suppose that I am trying to model the spread of heat through a $1$ dimensional rod of length $L$ in meters with $$\frac{\partial u}{\partial t} = D\frac{\partial^2 u}{\partial x^2}$$ where $u(x,t)$ ...
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30 views

Find the general solution of this pde?

I have a final in my PDE Class and needs some help with a review problem. Find the general solution of the equation for $v=v(x,y)$: $$x\frac{\partial v}{\partial x}+y\frac{\partial v}{\partial ...
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1answer
34 views

Meaning of generators in Lie Algebra of PDE

Consider some PDE involving a scalar function $u(x,t)$ with two independent variables $x$ and $t$. Assume that this PDE has a Lie Algebra spanned by the following generators, $X_1=\partial_x,\quad ...
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1answer
22 views

Fourier decomposition of solutions of the wave equation with respect to the spatial variable

Say I have a wave equation of the form $$\nabla^{2}f(t,\mathbf{x})=\frac{1}{v^{2}}\frac{\partial^{2}f(t,\mathbf{x})}{\partial t^{2}}$$ which is clearly a partial differential equation (PDE) in ...
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1answer
25 views

Equivalent ways to study perturbed abstract Cauchy problem

I am considering the following abstract Cauchy problem on Banach space $X$: \begin{cases} u'(t)=Au(t)+\big(f(t)+Bu(t)\big),&t\in[0,T],\\ u(0)=x_0, \end{cases} Suppose $A$ generates a $C_0$ ...
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14 views

heat equation-calculate the temperature of an bar

let an bar of lenght 50 cm, and temperature on t=0 is 100 degree. The question is calculate the degree on the middle of the bar. So i try to write the heat equation: $\dfrac{\partial u}{\partial t}= ...
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1answer
31 views

Solving PDE $v_t(t,x) + \frac{1}{2} v_{xx} (t,x) = 0$

We are given a PDE with $$v_t(t,x) + \frac{1}{2} v_{xx} (t,x) = 0$$ $$ v(T,x) = x^2 $$ for $0 < t \leq T$ and $x \in \mathbb{R}$ So far I have found that using the Feynmann-Kac equation, we get ...
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0answers
32 views

Computing a Green's function - where did I go wrong?

This is from a homework problem that was recently returned to me in a numerical analysis course. The grader even noted that he didn't know where I went wrong but the solution was marked as incorrect. ...
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0answers
27 views

Importance of Sobolev Spaces

Why Sobolev spaces are so important in study of partial differential equations? What could have light up the mind of researchers to use these spaces to analyze PDEs?
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20 views

How to prove the sufficient condition for a PDE has positive eigenvalues

Strauss' textbook says it is $\left[f(x)f'(x)\vphantom{\dfrac11}\right]_{x=a} ^{x=b}<0$, or $=0$. the assumption is $f''(x)=-\lambda f(x)$ for real $\lambda$. they told me to use Green's first ...
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1answer
39 views

Singular Perturbation Theory problem for slightly unstable pde

I'm learning about singular perturbation theory for solving an approximate solution to a pde and I'm a little confused on how to apply it. The problem I'm trying to work on is $\frac{\partial ...
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0answers
49 views
1
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1answer
50 views

Solve: $xu_x + yu_y+ \frac{1}{2}(u^2_x+u^2_y) =u, u(x, 0) =\frac{1}{2}(1−x^2)$

In the plane find two solutions of the initial-value problem $xu_x + yu_y+ \frac{1}{2}(u^2_x+u^2_y) =u, u(x, 0) =\frac{1}{2}(1−x^2)$. I think we get to use the method of characteristics But I am not ...
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1answer
23 views

Integrals with functions as bounds

How to calculate integral such as $$\int_{g(χ)}^{φ(χ)} f(s) \, ds$$ where $F'(s)=f(s)$ Integrals like this appear often in PDE's .I'd like to know the whole theory i mean if there is a formula how ...
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0answers
6 views

When is a symmetric hyperbolic system a second order scalar equation?

Given a linear symmetric operator of the form $$A^\mu \partial_{\mu}v+Bv=F$$ where $v\in C^{\infty}(\mathbb{R}^m ,\mathbb{R}^n)$ is a solution of the system and $A^\mu, B$ are $n \times n$ matrices ...
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0answers
12 views

Solution to Laplaces

I am trying to show that the following is a solution to Laplaces equation. $v(x) = c_2*Ln(|x|), x \in R^2$ where $c_2$ is an arbitrary constant. And $x \neq 0$ . Im just stuck on how to get started ...
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16 views

Question about asympotic behavior of $\frac{1}{s}\int_0^s u(x,t) dt$ .

I am just reading a paper, in the final theorem, the author wants to prove that $u(x,t)$ converges to some $v(x)$ in the $L^2$ norm as $t$ $\to$ $\infty$. But in the proof, he defines a ...
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37 views

Seeking help with finding the general solution of this differential equation

I am trying to find the general solution of the following equation. $$\int_0^\infty \frac{\partial f(x, t)} {\partial t} \sin(x \xi) \, dx = \xi \int_0^\infty f(x, t) \cos(x \xi) \, dx -\alpha \xi ^2 ...
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13 views

Good numerical method for finding the eigenvalues and eigenfunctions of the Dirichlet-Laplacian?

Let us confine ourselves to 2D. What is the best numerical method for solving the eigenvalues and eigenfunctions of the Dirichlet-Laplacian operator? Possibly, it depends on the shape of the domain? ...
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1answer
17 views

Solve transport equations by using Laplace transform

I'm trying to solve rather formally one-dimensional transport equation: $$ u_{t}+cu_{x}=0\quad\text{in $(0,\infty)\times(-\infty,\infty)$} $$ with an initial data $u_{0}$, which is bounded and ...
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1answer
19 views

Estimation of gradients in Poisson's equation

I am trying to show the following result. Let $D\subset\Bbb R^3$ be a bounded open set with smooth boundary. For any $f\in H^{-1}(D)$, let $\phi$ be the unique weak solution to the following ...
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15 views

Diffusion/Heat equation, weak maximum principle

Consider the problem: $$u_t -div(A(x) \nabla u) +a(x) u = f $$ $$u(\sigma,t)=0 $$ $$u(x,0)=g(x) $$ and its variational formulation $$ \lt \dot{u(t)},v \gt_* + B(u(t),v;t) =(f,v)_{L^2} \quad \forall ...
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1answer
17 views

Does uniformly boundness in $W^{1,1}$ implies strong convergence in $L^{1}$?

Suppose $f_i$ is uniformly bounded in $W^{1,1}$. My question is, can we conclude that there exists a sub-sequence of $f_i$ convergent strongly to some $f$ in $L^{1}$? I am just reading a paper, this ...
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1answer
46 views

Kato's inequality

Let u be a smooth function defined in a Riemannian manifold $(M,g)$. The well known Kato's inequality states $$|∇|∇u||^2≤|∇^2u|^2$$ where $∇^2$ represents the Hessian operator of $M$. I would like ask ...
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27 views

Solving $4u_{tt}-3u_{xt}-u_{xx}=0$

Solving $\begin{cases} 4u_{tt}-3u_{xt}-u_{xx}=0\tag1\\u(x,0)=x^2\quad\text{and}\quad u_t(x,0)=e^x\end{cases}$ in $\mathbf R\times\mathbf R_{>0}$ First I factorized and get for the first line; ...
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0answers
9 views

unicity solution of partial differential equation

let the following problem: $$ \begin{cases} \dfrac{\partial^2 u}{\partial t^2}= a^2 \dfrac{\partial^2 u}{\partial x^2}\\ u(0,t)=u(l,t)=0\\ u(x,0)=f(x)\\ \dfrac{\partial u}{\partial x}(x,0)=g(x) ...
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0answers
16 views

Poincare type inequality on unit square: can you improve on my constants?

I am trying to bound $\int_{[0,1]^2} u^2$ in terms of its gradient and boundary integrals, $\int_{[0,1]^2} |\nabla u|^2$, $\int_{\partial[0,1]^2} u^2$, with the best possible constants. So far I ...
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0answers
14 views

Heat Equation stationary convergence

Consider the heat equation: $$u_t -\Delta u=0 \quad \text{in} \quad Q_T=\Omega \times(0,T) $$ $$u(\sigma,t)=0 $$ $$u(x,0)=g(x) \quad \text{in} \quad \Omega $$ a weak formulation is: find $u \in ...
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0answers
15 views

partial differentia equation and Laplace transformate

let the following differential equation: $$ \begin{cases} \dfrac{\partial^2 y}{\partial t^2}= a^2 \dfrac{\partial^2 y}{\partial x^2}, 0<x<l, t>0\\ y(0,t)=f(t)\\ y(x,0)=0\\ \dfrac{\partial ...
0
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1answer
33 views

Let $w = \log(u^{2} + v^{2})$ where $u=e^{(x^{2}+y)}$ and $v= e^{(x+y^{2})}$

Then $\frac{\partial w}{\partial x}$ for $(x=0,y=0)$ is ? I got answer as 0 since on partial differentiation I got, ...
3
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1answer
65 views

How can the Bessel function of the second kind be in the radial eigenfunction?

Let $0<a<b<\infty$. Consider the heat equations or wave equations on the annulus or the spherical layers $$\Omega:=\{x\in\mathbb{R}^d\mid a<\|x\|_2<b\},$$ ...
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1answer
46 views

Functional calculus: Does $A$ commute with $e^{iA^2}$?

Let $A$ be an unbounded self-adjoint operator. Is it then true that $A$ commutes with $e^{iA^2}$? This sounds natural to me but I have no clue whether this is true in general. The problem is that we ...
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0answers
22 views

How do we define $H^{-1}$? [duplicate]

In class we defined $H^{-n}$ on $\mathbb{R}^n$ via the Fourier transform of tempered distributions. But unfortuntely, on subset $\Omega \subset \mathbb{R}^n$ there are Schwartz functions. So let ...
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1answer
22 views

Strongly continuous group and generator commute, what about square roots?

Let $A$ be a positive self-adjoint operator, then $iA$ generates a unitary strongly continuous semigroup (Stone's theorem) $T$. Then from basic semigroup theory we know that $T$ and $A$ commute, but ...
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1answer
22 views

Stability (wrt parameters) of elliptic partial differential equation

consider the equation $$\mathcal Lu=f \quad \text{in } \Omega $$ With some appropriate boundary condition, $\Omega$ regoular as you like, $ \mathcal L$ to be defined by $$\mathcal ...
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1answer
20 views

where does the pdes mathematocal classification names come from?

PDEs are classified into hyperbolic, parabolic and elliptic. where do these names come from? Do they have anything to do their geometric shapes?
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1answer
9 views

How to find spacial periodicity

I am struggling with the following question. Consider $$U(x,t)=2\cos\left(kx-\sqrt{\frac{a}{b}}k^2+(\Delta k)^2)t\right)\cos\left((\Delta k)(x-2\sqrt{\frac{a}{b}}kt\right)$$ and suppose that $\Delta ...
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0answers
16 views

Bochner integrability and analytic semigroup

For a general strongly elliptic second order operator of the divergence form $$A=\partial_j\big(a^{ij}\partial_{i}\big)+b^i\partial_i,$$ with smooth enough coefficients on a smooth bounded domain ...
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15 views

linearized KDV equation

Consider the PDE $$u_t +4u_x +6u_{xxx} =0$$ for the function $u = u(x, t).$ (a) Determine for what wave speeds $c$ there exist bounded travelling wave solutions of $u$, and find the form of the ...
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17 views

Question about trace operator

From the general trace theorem we know for instance that if $f\in W^{1-\frac{1}{p},p}(\partial\Omega)$, then there exists a function $f\in W^{1,p}(\Omega)$ such that $f|_{\partial\Omega}=f$. But is it ...
3
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1answer
17 views

Poisson Equation: 'Dirichlet' type problem on all of $\mathbb{R}^N$

I've seen a great deal about solving the Dirichlet problem for Poisson's equation \begin{equation} \Delta u = f \quad \mbox{in} \quad \Omega \subset \mathbb{R}^N\\ u = 0 \quad \mbox{on} \quad ...
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25 views

PDEs on higher genus Riemann surfaces, e.g. Klein Curve

I'm trying to solve a PDE on compact Riemann surfaces of genus g > 1. Since these can be obtained as quotients of the upper half plane $\mathbb{H}_2$ by some Fuchsian group $\Gamma$, I suppose it's ...
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1answer
40 views

calculate solution of heat equation with method of separation of variables

let the following problem $$ \begin{cases} \dfrac{\partial u}{\partial t}= \dfrac{\partial^2 u}{\partial x^2}, 0<x<1, t>0\\ u(0,t)=0, u(1,t)+ \dfrac{\partial u}{\partial x}(1,t)=0\\ ...
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0answers
12 views

Solve Initial value poblem.

Utt=4uxx -infinite 0 U (x,0)=0 Ut (x,0)={1,-1 0, any where else , and sketch at t = 1 and t=4 That what I tried u= 1/4 integration g (z) dz and ...
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1answer
35 views

Numerical Method for KdV travelling waves

Can someone please direct me to the best NUMERICAL method or some references for solving $$-cu_x + uu_x + u_{xxx} = 0$$ with periodic boundary conditions. This governs travelling waves of the KdV ...