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Exponential convergence of p-method $u'' + u = 2 \sin(x)$

If your problem is to know $n$ such that $$||u - p_{n-1}||_{L^{\infty}} \leq \frac{1}{2^{n-1}n!} \left( \frac{\pi}{2}\right)^n =10^{-k}$$ i means that you want to solve for $n$ the equation $$n!=4^{- ...
Claude Leibovici's user avatar
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Eigenvalue problems $y''(x) + λy(x) = 0, 1 < x < 2, y(1) = y'(2) = 0$

You are looking for trigonometric functions as solutions of $$-y'' = \lambda^2 y,$$ with a zero at $x=1$ and a zero of the derivative ar $x=2.$ The lowest eigenfunction is the sine of a quarter period ...
Roland F's user avatar
  • 2,276
0 votes

Eigenvalue problems $y''(x) + λy(x) = 0, 1 < x < 2, y(1) = y'(2) = 0$

You probably learned how to solve this in differential equations class, Calculus II or whatever you called it. But, it seems unfamiliar in the setting of linear algebra. So, how would you solve this ...
user317176's user avatar
  • 11.3k
3 votes
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The Variational form of a biharmonic PDE

Assume that $u$ and $v$ are smooth and let's do some formal computations first (we'll worry about the correct function spaces once we have an idea of what the weak formulation looks like). Observe ...
Giovanni's user avatar
  • 6,386
1 vote

The Variational form of a biharmonic PDE

Assume wlog that $u_D = 0$. Then let the space that $u$ lives in be $H^2_0(\Omega)$. For $u,v \in H^2_0(\Omega)$, define $$a(u,v) := \int_{\Omega} \Delta u \Delta v - \int_{\partial \Omega} \Delta u \...
Laithy's user avatar
  • 463
1 vote

Maximum principle of strong solution of linear parabolic equation in $\mathbb{R}\times [0,T]$

I think the integration by part of the second order derivative should be $$\int^R_{-R}\partial_x^2 (u-u^*)(u-u^*)^+dx= [\partial_x^2 (u-u^*)(u-u^*)^+]^R_{-R} -\int^R_{-R}|\partial_x(u-u^*)^+|^2dx.$$ I ...
mnmn1993's user avatar
  • 437
0 votes

Uniqueness and continuous dependence on the data of Heat equation.

To address the lack of connectedness when applying the strong maximum principle: Since $U\subset \mathbb R^m$ is open and $\mathbb R^m$ is connected, it follows that $U$ is locally connected from ...
K.defaoite's user avatar
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2 votes
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Is there another way of deriving the Fourier transform of 1?

Yes, we can derive the inverse Fourier Transform of $1$ using the approach you outlined. First, we let $F(k)=1$. Then, we have for any $\phi\in \mathbb{S}$, $$\begin{align} \langle \mathscr{F^{-1}}\{...
Mark Viola's user avatar
  • 180k
1 vote

Maximum principle of strong solution of linear parabolic equation in $\mathbb{R}\times [0,T]$

Let me make an attempt and please correct me if I am wrong. I try to argue like in Chapter 2 "Godlewski, Raviart - Hyperbolic systems of conservation laws", which is a beautiful but rare ...
Hyperbolic PDE friend's user avatar
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Example of an integrable function $f$ such that $|\int(1+x^2)f(x)\,dx|<|\int f(x)\,dx|$

For $h_1(x)=1_{[0,1]}(x)$ and $h_2(x)={1\over 2}1_{(1,3]}(x)$ let $$f(x)=(1+x^2)^{-1}[h_1(x)-h_2(x)]$$ Then $$\int (1+x^2)f(x)\,dx = \int h_1(x)\,dx- \int h_2(x)\,dx =0$$ $$\int f(x)\,dx =\int h_1(...
Ryszard Szwarc's user avatar
2 votes

Is $ W^{k,p}(\mathbb{R}^n) $ equal to $ W^{k,p}_{loc}(\mathbb{R}^n) $?

These spaces $W^{k,p}_{loc}(\Omega)$ and $W^{k,p}(\Omega)$ are never equal for open sets $\Omega$. The problem is that for these loc-spaces the behavior close to the boundary (and at infinity) cannot ...
daw's user avatar
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1 vote

Example of an integrable function $f$ such that $|\int(1+x^2)f(x)\,dx|<|\int f(x)\,dx|$

Let $f(x) = 1_{[0, 1)}(x) - 1_{[1, \frac{3}{2}]}(x)$. It is clearly in $L^1$. The LHS equals $\frac{5}{12}$ while the RHS equals $\frac{1}{2}$.
David Gao's user avatar
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0 votes

d'Alembert's solution to the wave equation via Fourier Transforms

\begin{align} \frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{\hat g(k)}{k}\sin (kt) e^{ikx}dk &=\frac{1}{4\pi i}\int_{-\infty}^{\infty}\frac{\hat g(k)}{k}\left[e^{ik(x+t)}-e^{ik(x-t)}\right]dk \\ &...
Gonçalo's user avatar
  • 9,447
0 votes

PDE with initial conditions

Using the Laplace transform $$ s^2\hat u(x,s) - \hat u_{xx}(x,s) = su(x,0) + u_t(x,0) = s $$ Now solving this ode with boundary conditions $\hat u(x,s) = \hat u_x(1,s) = 0$ we have $$ \hat u(x,s) = \...
Cesareo's user avatar
  • 33.4k
1 vote

Best way to decouple a system of second-order ODE

Adding both equations as real and imaginary part gives $$ z''=x''+iy''=\Omega(y'-ix')+\omega_0^2(x+iy) \\=-i\Omega z'+\omega_0^2z $$ The term coupling the real and imaginary parts is the first-order ...
Lutz Lehmann's user avatar
1 vote
Accepted

Help in solving PDE with charasteristic equation problem

Let $\xi=-e^x$ and $\eta=-\frac{1}{2}y^2$; then $x=\ln(-\xi)$ and $y=\pm\sqrt{-2\eta}$, so $$ f(-e^x,-\frac{1}{2}y^2)=y\ln x \implies f(\xi,\eta)=\pm\sqrt{-2\eta}\ln\ln(-\xi). \tag{1} $$ Therefore, $$ ...
Gonçalo's user avatar
  • 9,447
1 vote

Help in solving PDE with charasteristic equation problem

You can approach this problem by looking for a new system of coordinates, $X(t), Y(t)$ that are time-dependent, and a function $U(t,X,Y)$ such that $$U(t,X(t),Y(t))=u(t,x,y)$$ Note that, $$\frac{d U}{...
Delansi's user avatar
  • 61
0 votes

Understanding the Extension Problem and the Positivity of an Operator

To prove $T>0$ you can apply integration by parts that for every nonzero $f\in C_c^2(\mathbb R^n)$, $$(-\Delta f,f)=-\sum_{j=1}^n\int\partial_{jj}f\cdot\bar f=\sum_{j=1}^n\int\partial_jf\cdot\...
Liding Yao's user avatar
  • 1,815
0 votes
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Piecewise Continuity enough to ensure finite Fourier coefficients?

The function $s(x)=\dfrac{1}{x^2}$ is not piecewise continuous on $[-\pi,\pi]$, because it is not even defined at $x=0$. Even if one defined $s(0)=c$ for some $c\in\mathbb{R}$, this function is still ...
Julio Puerta's user avatar
  • 4,931
2 votes

Fundamental solution of heat equation in a ball with homogeneous Dirichlet boundary condition

For the standard heat equation ($\partial_t u-\Delta u=0$), there is an explicit solution formulated in Gegenbauer polynomial when dimension is not less than 3. See Theorem 2 in Hsu, P. (1986). ...
Boey C's user avatar
  • 46
0 votes

On exact differential equation

You are looking for the Poincare lemma and you also need the assumption that you are trying to solve that exact DE on a star-shaped (convex or simply connected will also do the trick). Let us assume ...
F. Conrad's user avatar
  • 4,360
0 votes

Why isn't the integral of the total differential of a function equal to the function?

Expanding on the comment of user317176 When you partially differentiated the original equation with respect to x and y you found two results: $$ \frac{\partial w}{\partial x}\hspace{1.5mm} and\hspace{...
Rich's user avatar
  • 226
0 votes

Finding mean first passage time for reflecting Brownian motion

Define $\tau_x=\inf \{t\ge 0: X_t\in \partial B_{\epsilon}(0)|X_0=x\}$ and $G(x,t)=P(\tau_x>t)$. Using the backward equation involving the transition kernel for a time-homogenous process, i.e., $p(...
Fellow InstituteOfMathophile's user avatar
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Does the inequality $\iint_{\mathbb R^2} \frac{|f(x)|^2}{|x-y|^a} \ dy\,dx\le k ||f||_{L^2(\mathbb R)}^2$ hold?

The claim as stated is not true. Suppose that $f$ can be reduced to a single variable function, i.e $\exists \phi:\mathbb R_{\geq 0}\to\mathbb R$ such that $f(x)=\phi(|x|).$ Now see that $$\Vert f\...
K.defaoite's user avatar
  • 12.5k
1 vote
Accepted

How to prove an adapted Feynman Kac Formula for $v_t + \frac{1}2 \sigma ^2 (t,y) v_{yy} + b(t,y) v_y - \delta(t,y) v + h(y) = 0$ using SDE techniques?

We see $$\begin{aligned}dv(t,Y_t)-\delta(t,Y_t)v(t,Y_t)dt&=-h(Y_t)dt+v_y(t,Y_t)\sigma(t,Y_t)dW_t\\ \implies d(v(t,Y_t)e^{-\int_0^t\delta(s,Y_s)ds})&=-h(Y_t)e^{-\int_0^t\delta(s,Y_s)ds}dt+v_y(t,...
Snoop's user avatar
  • 15.3k
2 votes
Accepted

Meaning of "distributional version" of a system

At the start of Theorem 9 of the paper, there is the condition $$ \lim_{\epsilon\rightarrow 0}f_i=\rho_i^0\chi_i. $$ Now, consider both the equations in the same limit $$ \lim_{\epsilon\rightarrow 0}(...
Jon's user avatar
  • 5,492
2 votes

Does the inequality $\iint_{\mathbb R^2} \frac{|f(x)|^2}{|x-y|^a} \ dy\,dx\le k ||f||_{L^2(\mathbb R)}^2$ hold?

Your estimate is a special case of the weak Young-inequality (used in Potential theory) which follows from the Hardy-Littlewood-Sobolev inequality, see the first answer of https://mathoverflow.net/...
Jacob Körner's user avatar
6 votes

Does the inequality $\iint_{\mathbb R^2} \frac{|f(x)|^2}{|x-y|^a} \ dy\,dx\le k ||f||_{L^2(\mathbb R)}^2$ hold?

For $f\neq 0$ the left hand side is equal $\infty,$ as $$\int\limits_{-\infty}^\infty|f(x)|^2\left [\int\limits_{-\infty}^\infty{dy\over |x-y|^\alpha}\right ]\,dx\\ =\int\limits_{-\infty}^\infty|f(x)|^...
Ryszard Szwarc's user avatar
1 vote
Accepted

If $f\in W^{1,p}(\Omega)\cap C(\overline{\Omega})$ then can we say that $\mu(\operatorname{supp}(f)-\{f\ne 0\})=0$?

No. Take $F=$ a fat Cantor set in $\Omega=[0,1]$. Since $F$ is closed one can find a smooth compactly supported function $f$ such that $f=0$ on $F$. Now assuming by support you mean the closure of the ...
kieransquared's user avatar
2 votes

Question about the proof of Theorem 5 in the subsection 5.6.2 of Evans' PDE book (page 283 in the 2nd version)

(Since we want to prove a.e. equality in $U$ and $\bar u = u$ in $U$, I will just replace $\bar u$ with $u$ in the following) Here is a way to see it : first, note that since $U$ is a bounded set, we ...
Stratos supports the strike's user avatar
8 votes
Accepted

How does one convert the mean curvature equation into a homogeneous linear elliptic P.D.E?

Yes, it follow from the fundamental theorem of calculus and it is 'well-known' amongst members of the PDE community. Let $\Omega \subset \mathbb R^n$ be open, and $a^{ij},b \in C^1( \Omega \times \...
JackT's user avatar
  • 7,034
1 vote

Proving $\int\limits_{Q} u^2 dx ≤ C(n) \int\limits_{Q} (u^2_{x1} + · · · + u^2_{xn}) dx$ over a cube $Q$

Let $V(x)=x=(x_1, \dots, x_n)$, then we get by the divergence theorem and the fact that $u$ vanishes on $\partial Q$ $$\int_Q u(x)^2dx = n^{-1} \int_Q u(x)^2 \text{div}(V)(x)dx =- \int_Q \nabla(u(x)^2)...
Severin Schraven's user avatar
0 votes

Finding "Solution" of the partial diffrential equation $z=px+qy+f(p,q)$

You are right about the complete integral and the singular solution. To obtain the general solution, however, you must eliminate $a$ from the system $$ \begin{cases} g(x,y,z,a,\phi(a))=0, \\ \...
Gonçalo's user avatar
  • 9,447
1 vote
Accepted

Principal eigenfunction of $\Delta$ on rectangular patch of $S^d$

As a not so trivial example take hyperspherical coordinates in $\mathbb R^4$. I use Mathematica for short $$\text{SeparatedLaplacian}(1)=\text{Expand}\left[ \frac{1}{R(r)\ \Theta (\theta ) R(r) \Psi (\...
Roland F's user avatar
  • 2,276
0 votes

Proving $\int\limits_{Q} u^2 dx ≤ C(n) \int\limits_{Q} (u^2_{x1} + · · · + u^2_{xn}) dx$ over a cube $Q$

One possible proof would be via developing $u$ into its Fourier series and then using simple formula for second norm (so called Parselval's theorem). In case of $n=1$, say $u = \sum a_ke^{2{\pi}kx}$, ...
Salcio's user avatar
  • 117
5 votes

What can semigroup theory do better in the study of PDEs compared to alternative methods?

I have not yet understood what does the theory offer for the study of PDEs, that other methods do not provide? A very important aspect is study of the asymptotic behavior by estimating the resolvent ...
Pedro's user avatar
  • 18.9k
1 vote

Implementing Boundary Conditions at Infinity in Numerical Method

The way to approach this numerically is to implement a perfectly matched layer boundary condition. When you are doing a graphical analysis of your numerical solution, you can only do it in a finite ...
FriendlyNeighborhoodEngineer's user avatar
1 vote

weakest condition on $\Omega$ such that $W^{1,p}_0(\Omega)= \lbrace u: \tilde{u} \in W^{1,p}(\mathbb{R}^N)\rbrace$

This is too long for a comment. Initially, I thought that this holds for all open (and bounded) sets. However, the following is a counterexample. Let $\Omega = (-1,1)^2 \setminus \{0\}\times (0,1)$. ...
gerw's user avatar
  • 31.4k
1 vote
Accepted

Solving the Poisson equation $-\Delta u=f$ on a domain $G$

Yes, you are completely right. However, I think there is an issue that $f \in C_c^1(\mathbb{R}^n)$ and not $f \in C_c^2(\mathbb{R}^n)$. If the latter was the case, you could look up all the details in ...
F. Conrad's user avatar
  • 4,360
0 votes

semi group and perturbation trick

I guess you will need some regularly for $A$ (for $\mathrm{div}$?) to ensure the well-posedness. The classical strategy for the proofs is as follows: Existence: you prove that the governing operator ...
S. Maths's user avatar
  • 2,248
2 votes
Accepted

Existence of an extension operator $E: W_0^{s,p}(\Omega) \rightarrow W^{s,p}(\mathbb{R}^d)$

This is a very nice question. It appears to me that the answer is in general, no. Moreover, it appears to be an open problem determining geometric conditions on $\Omega$ for which such an extension ...
JackT's user avatar
  • 7,034
0 votes

Non-linear first order PDE

Here is an alternative solution, using the method of characteristics. Let's rewrite the PDE as $$ F:=xp^2+q-y=0 \qquad(p=u_x,\, q=u_y). \tag{1} $$ Solving the Lagrange-Charpit equations, we obtain \...
Gonçalo's user avatar
  • 9,447
0 votes

How to find the characteristic form of coupled PDEs?

Just follow example 1, p. 117. Let $a^{2} = A$. Then the linear combination \begin{align} l_{1} (k_{t} + \omega_{0}' k_{x} + \omega_{2} A_{x}) + l_{2} (A_{t} + \omega_{0}' A_{x} + \omega_{0}'' A k_{x})...
Matthew Cassell's user avatar
0 votes

$[a,b]$ has smooth boundary???

Yes, it is smooth. To see this, you can use one of the classical characterizations of smooth sub-manifolds of $\mathbb R^n$. Then, the set $\{a,b\}$ is a smooth ($C^\infty$) manifold of dimension $0$.
S. Maths's user avatar
  • 2,248
4 votes
Accepted

How can I find out if this 1st order quasi-linear pde has *no solution*

The answer from Toby Saunders-A'Court begins with an assumption. The assumption is found valid at the end. So his answer is correct and this is by far the simplest way to solve the problem. More ...
JJacquelin's user avatar
  • 66.3k
1 vote
Accepted

Calculating the Divergence of a Function on Unit Ball

The divergence, denoted here as $\operatorname{div}$, is the divergence on $\mathbb S^{n-1}$. Let $M^{n-1}$ be an oriented hypersurface in $\mathbb R^n$ with unit normal $\nu$ and $F:M \to \mathbb R^n$...
JackT's user avatar
  • 7,034
0 votes

A question about reflection method of the wave equation

Yes, by the usual restrictions on start and boundary conditions at $$(x=0,x=\infty, x=0 \ \forall \ t>0 )$$ the solution is odd in in $x$ in order to fix the zero at $x=0$, if extended to $x\in \...
Roland F's user avatar
  • 2,276
0 votes

Finding solutions $u_n$ for the boundary value problem $u_{rr}+\frac{u_r}{r}+\frac{u_{\theta \theta}}{r^2}=0$

Given the Laplace equation in polar coordinates $$\Delta u(r,\theta )=\frac{u^{(0,2)}(r,\theta )}{r^2}+\frac{u^{(1,0)}(r,\theta )}{r}+u^{(2,0)}(r,\theta )=0$$ Going to separate the variables with the ...
gpmath's user avatar
  • 966
1 vote
Accepted

Clarification of estimates of $H^1$ norm and $H^{-1}$ norm.

More precise assumptions on $A$ and $c$ are needed, for $(2)$ and $(3)$ do not hold for general complex-valued coefficients $A$ and $c$. My answer below relies on the following (customary) assumptions:...
user1302959's user avatar

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