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

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19 views

Find a function in terms of another

I need to express an in terms of f(x). I did it but I'm not sure if it is right. Consider \begin{cases} -u_{xx}+u=f(x), &0<x<\ell\\[0.5em] u_x(0)=0,\;u_x(l)=0& \end{cases} ...
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20 views

How to restore a differential equation knowing its solution?

How to restore a differential equation ( or system ) knowing its solution? Suppose I have a rational function $y=\frac{f(x)}{g(x)}$ and I want to know if there is an differential equation ...
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45 views

How to prove the inequality in Burgers equation and what is the relationship to energy dissipation

Consider Burgers equation: \begin{equation} u_t+uu_x-\epsilon u_{xx}=0 \quad , \quad u(x,0)=g(x) \quad , \quad u = 0 \mbox{ when } |x| \mbox{ large}, \end{equation} where ...
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31 views

Fourier series method

I have the following Boundary value problem $U_t-U_{xx}=0$ from zero to one; $t>0$ $ U(x,0)=x$ from zero to one $U(0,t)=U(1,t)=0$ I need to solve it using the Fourier series method, But I ...
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23 views

Orthogonal Level Sets and a generalization of harmonic functions

Forgive my ignorance of differential equations and analysis. I was playing around with orthogonal level curves of real valued functions in the plane, and realized this is one way a person could be ...
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25 views

Parameters in the Hamilton-Jacobi Equation

I'm reading through Gelfand and Fomin's 'Calculus of Variations', and they've just derived the Hamilton-Jacobi Equation: $$\frac{\partial S}{\partial x} + H \left(x, y_1, \ldots, y_n , \frac{\partial ...
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37 views

Strong solution of inviscid Burgers' equation with initial data $u(x,0)=x^2$

I'm studying for an exam and having trouble solving the following: Find the strong solution to the inviscid Burgers' equation $u_t+uu_x=0$ with initial data $u(x,0)=x^2$ Using the initial ...
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40 views

a priori estimates involving Sobolev norms

Let $\sigma,$ $f$ and $g$ be $C^{2}(\overline{\Omega})$ functions, with $0<\frac{1}{M} < \sigma < M.$ We have the Dirichlet problem: $\text{div}\sigma \nabla u=f, \hspace{3mm} \text{in} ...
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72 views

Poisson's eqution and radial symmetry

I don't understand why $\Phi$ does not depend on $\theta$ or $\phi$. On the boundary it makes sense because $\Phi$ on the boundary does not depend on any of the variables. I get that the sphere ...
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35 views

Using dummy variable to derive Nth partial Fourier sum

I am working on a problem$^{(1)}$ as follow: Using the standard formulas for the Fourier coefficients, show $$F_N(x) = \frac1{2\pi} \int_{-\pi}^{\pi} \left( 1+2 \sum_{n=1}^{N} ...
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3answers
30 views

Find the general solution of the equation $u_t(x,t)+t^{\frac{1}{3}}u_x(x,t)=u(x,t)$.

Find the general solution of the equation $u_t(x,t)+t^{\frac{1}{3}}u_x(x,t)=u(x,t)$. I'm having trouble starting this one. Any help would be greatly appreciated.
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26 views

Solve the damped quasilinear wave equation $u_t(x,t)+u(x,t)u_x(x,t)+u(x,t)=0$ with $u(x,0)=f(x)$.

Solve the damped quasilinear wave equation $u_t(x,t)+u(x,t)u_x(x,t)+u(x,t)=0$ with $u(x,0)=f(x)$. Determine if the solution breaks when $f$ satisfies the condition $f^\prime(x)>-1$ for all ...
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65 views

Weak formulation of a system of biharmonic pdes

Consider the system of pdes for functions $u,v$ $$\begin{cases} a_0\Delta ^2u+a_1u_{xx}v_{xx}+a_2u_{xy}v_{xy}+a_3u_{yy}v_{yy} = f\\ b_0\Delta ^2v+b_1u_{xx}v_{xx}+b_2u_{xy}v_{xy}+b_3u_{yy}v_{yy} = g ...
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2answers
72 views

Sobolev space $H_0^m(\Omega)$?

Let $\Omega \subset \mathbb{R}^n$ be open, then $H_0^m(\Omega) := \overline{C^{\infty}_c(\Omega)}$ with respect to the Sobolev norm. The problem is that I don't really see what kind of functions are ...
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24 views

Lower semicontinuity of ${\dot{H}}^1$ norm

I have a in $H^1(\mathbb{R^N})$ uniformly bounded sequence $u_n \in H^1$. I also know $u_n\to u$ in $L^p$ for every $2\leq p < 2^\ast$, where $\ast$ means the Sobolev exponent. Can I conclude that ...
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35 views

Example of maximal monotone operators in non-reflexive Banach spaces with applications in PDE

My question is about examples of maximal monotone operators that are defined in non-reflexive Banach spaces and have applications in PDEs, variational inequalities, etc (any application actually)? If ...
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1answer
27 views

Dimensional analysis to retrieve constants in formula for PDE solution

Consider for example the PDE $$ \begin{cases} u_{tt}(x,t) - \Delta u(x,t) = 0 , & (x,t) \in \mathbb R^2,\\ u(x,0) = g(x),\\ u_{t}(x,t) = h(x), \end{cases} $$ which solution we know can be written ...
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2answers
22 views

Integral Representation of the solution to the advection-diffusion equation

So I am working on a problem in my PDE homework and I am stuck on one portion of the problem. I am given the following PDE IVP: $$u_t+au_x=u+\kappa u_{xx}, \ -\infty<x<\infty, \ t>0$$ ...
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25 views

PDE - bounded solution

Let $u\in\mathcal{C}^2(\Omega)\cap\mathcal{C}^0(\overline{\Omega})$ be a classical solution of $\Delta u=u^3-u$ on a bounded domain $\Omega\subset\mathbb{R}^2$. Assume that $u=0$ on $\partial\Omega$. ...
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50 views

Is the set of smooth functions with zero mean dense in the space of Sobolev functions with zero mean?

Given a Lipschitz domain $\Omega$ in $\mathbb{R}^n$, I know that the space of infinitely differentiable functions with compact support $C^\infty_0 (\Omega) $ is dense in the Sobolev space ...
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33 views

Conservative energy of the wave equation

If we take account of air resistance in the wave equation, we have an extra term proportional to speed, $ρu_{tt} − Tu_{xx} + ru_t = 0$ , Where $r > 0 $ Show that the energy of this system, $$ E = ...
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34 views

Question of $u\in L^p(U)$ does not have a trace on $\partial U$. [duplicate]

Let $U$ be bounded, with $C^1$ boundary. Show that a "typical" function $u\in L^p(U)$ does not have a trace on $\partial U$. More precisely, prove that does not exist a bounded linear operator ...
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57 views

Deduce the following estimate

Let $u$ be a smooth solution of the uniformly elliptic equation $Lu=-\sum_{i,j=1}^n a^{ij}(x)u_{x_i x_j}$ in $U$. Assume that the coefficients have bounded derivatives. Set $v:=|Du|^2+\lambda ...
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87 views

Using Markov Property in solving PDE/SDE

I am solving the PDE I used Feynman-Kac and eventually arrived at $F(t,x)$ $ = E[X_T^2|X_t = x]$ $ = E[(X_t \pm \sigma (W_T -W_t))^2|X_t = x]$ (iirc) So, I try to evaluate $E[(X_t \pm \sigma ...
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32 views

Solve the PDE by the method of characteristics.

I am trying to figure out where my solution went wrong. I am off by a factor of two. $$ u_x + u_y + u = e^{x+2y}$$ I first found that the characteristic curves are determined by $$\frac{dy}{dx} = 1 ...
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33 views

Uniqueness of solutions to the wave equation

we are given the problem $u_{tt}-c^2\Delta u=0$ with conditions $u(x,0)=u_0(x)$ and $u_t(x,0)=u_1(x)$ where $x\in\mathbb{R}^n$ and $u_0,u_1\in\mathcal{C}^1$ having compact supports. If a solution ...
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46 views

Stuck with this function

im trying to find the values of $\alpha \in \mathbb{R}$ for which the function $f: B_2(0,\frac{1}{2}) \rightarrow \mathbb{R} $, $ x \mapsto |\log(\|x\|_2)|^\alpha$ is in $L^2(B_2(0,\frac{1}{2}))$ and ...
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2answers
37 views

How should terms be scaled by finite dx and dt in numerical integration of 1D diffusion?

I am familiar with numerically integrating systems of ordinary-differential equations, but I feel that I am missing something important in terms of how numerically integrating ODEs differs from ...
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47 views

$\int_\Omega |\nabla u^+|^2 \, dx$ is not differentiable with respect to $u$ in $W_0^{1,2}(\Omega)$

Let $u \in W_0^{1,2}(\Omega)$, where $\Omega$ is some domain in $\mathbb{R}^N$, $N \geq 1$. Denote $u^+ := \max\{u, 0\}$. (It is know that $u^+$ also belongs to $W_0^{1,2}(\Omega)$ (see, e.g., ...
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31 views

Solution of a 2 dimensional laplace equation

I have the solution to $u_{xx}+u_{yy}=0$, $$u(x, y) = \sum^{\infty}_{n=1}\left(\left(\dfrac{2}{\pi}\int^{\pi}_{0}f(x)\sin{nx}\,{\rm ...
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32 views

System of four first-order partial differential equations

Question: Consider the wave equation: $$\frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 u}{\partial t^2}$$ for a smooth function $u(t,x)$ Letting $v=\frac{\partial u}{\partial x}$ and ...
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74 views

wave equation on a square domain

I'm stuck on the following problem. Let $u(x, y, t)$ denote a solution to the linear wave equation $k^2(u_{xx}+u_{yy}) = u_{tt}$ with $k = 2$ on a square domain with corners at (0, 0), (0, 1), ...
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36 views

A nonlinear Poisson equation problem. (Related to Laplace equation)

Let $\Omega\subset R^N$ be open bounded. Define \begin{cases} \Delta u = f(u) &x\in\Omega\\ u=1 & x\in\partial \Omega \end{cases} Q1: Suppose $f(u)=u^m$ where $m$ is odd. Prove that if there ...
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41 views

Fourier Integral for signum function.

Define the signum function, $\text{sgn}(x)$, by $$\text{sgn}(x)=\begin{cases} 1, & x>0\\0, & x=0\\-1, & x<0 \end{cases}$$ Establish the identity $$\dfrac{2}{\pi}\int_0^ \infty ...
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39 views

$L^2$ regularity of a convolution with Newtonian potential

I am reading Vorticity and incompressible flow (Bertozzi, Majda) and on page 71-72, we are concerned with recovering the velocity field of a flow from its vorticity. At some point we need to have the ...
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53 views

Solving a one dimensional wave equation

Consider the partial differential equation $$\frac{\partial^{2}u}{\partial t^{2}}=c^{2}\frac{\partial^{2}u}{\partial x^{2}}$$ For the region $0<x<\pi$ where $t>0$. With the boundary ...
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147 views

The rate of convergence for finite difference methods for Poisson's equation with piecewise constant data

I am solving the following PDE; $$ \nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \rho, $$ where $\rho(0.5,0.5) = 2$ (zero elsewhere), $0\leq x,y\leq1$ and the ...
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60 views

Equilibrium Temperature in insulated rod from BVP

this problem is giving me some trouble. It is a review problem given to us to study for our Final Exam this week. I would love some help in understanding how to solve it so that I can study it and ...
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30 views

Basic question about weak solution of p-Laplace equation

Let $p \geq 2$ and $B(x_0, d)$ an open ball in $R^n$ ($n \geq 2$). Let $u \in W^{1,p}(B(x_0,d))$ a weak solution of $\Delta_p u = f$ , $u = 0 , \ on \ \partial B(x_0,d)$, that is, $\int_{B(x_0,d)} ...
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33 views

Entropy Solution of $u_t+(u^2/2)_x=0$

Given the initial data $$ g(x)= \cases{ 1 & x< -1 \\ 0 & -1 < x< 0 \\ 2 & 0 < x< 1 \\ 0 & 1 < x \\ } $$ What is the entropy solution of $u_t+(u^2/2)_x=0$?
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34 views

Problem of Partial Differential Equations

For this question, I get stuck when I apply the second initial equation. My answer is $θ= Ae^-(kλ^2 t)\cos λx$, where $A$ is a constant. Would anyone mind telling me how to solve it?
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113 views

What concepts do I need to solve this nonlinear problem?

Problem Details: Let $L$ be a linear operator and $N(u)$ be a nonlinear function of $u$. Consider the IVP for the following nonlinear PDE: $$\partial_{t}u + L(u) = N(u)$$ defined on $(x,t) \in ...
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113 views

solving a PDE with with coordinates

First i need to find eigenfunctions in spherical coordinates with source term. I am having trouble with this . Can someone help guide me i also have 2 b.c.'s.
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55 views

Is this derivative somehow bounded?

I have a function $\phi: \mathbb{R} \rightarrow \mathbb{R}$ that is a test function and $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined by $f(x) = \phi(\frac{\|x\|}{n})$. Now if I take any ...
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105 views

Two exercises on Evans PDE book.

Those two problems bothers me for a while. I think I got most of it but I do want to have a nice and clean solution, so I post it here for discussion. All below I will use Einstein summation. The ...
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43 views

Heat equation fundamental solution

The following is from a book of PDEs and I have cannot seem to figure out a particular step in it with regard to the derivation of the fundamental solution of the heat equation. I have highlighted it ...
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29 views

Does this equation has a non-trivial solution

Suppose $u:[0,1]^2\to [0,1]$. The equation $$\int_{[0,1]}\frac{\partial u}{\partial u_1} (y,x)dx=\int_{[0,1]}\frac{\partial u}{\partial u_2} (x,y)dx, \phantom{00}\forall y\in[0,1]$$ has trivial ...
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58 views

Weak formulation of the Poisson equation with discontinuous source

We consider the equation $$ -\Delta u =f\chi_\Omega\quad \mbox{in }\mathbb{R}^3, $$ where $\Omega$ is a bounded and smooth domain, $\chi_\Omega$ is the charasteristic funtion of the set $\Omega$ and ...
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2answers
31 views

Which Sobolev-Space to use to formulate weak biharmonic equation, $H^2_0$ or $H_0^1\cap H^2$?

For the weak formulation of the biharmonic equation on a smooth domain $\Omega$ $$ \Delta^2u=0\;\text{in}\;\Omega\\ u=0, \nabla u\cdot \nu=0\; \text{on}\; \partial\Omega $$ why does one take ...
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63 views

Weak-* convergence in Sobolev spaces

Let's consider a sequence $\{f_n\}_n$ in the space $L^\infty(0,T;H^1(\mathbb{R}^n))$. What does it mean that $\{f_n\}_n$ converges weakly-* in $L^\infty(0,T;H^1(\mathbb{R}^n))$?