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

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Partial DE Question

I have a quick question. Question 1). Hence deduce that $$ u(x,t)=F(x+at)+G(x-at) $$ satisfies the original PDE for any twice-differentiable functions $F$ and $G$. c) Hence find the solution of ...
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50 views

Laplacian $\Delta u$ in spherical coordinates

The Laplacian $\Delta u$ in spherical coordinates is $$\Delta u=\frac{\partial^2u}{\partial\rho^2}+\frac{2}{\rho}\frac{\partial ...
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77 views

Characteristcs and blow up time of a PDE

I am trying to picture the characteristic projections of $$\dfrac{\partial u}{\partial t}+u^2\dfrac{\partial u}{\partial x}=0 \text{ with }x\in\mathbb{R} \text{ and } t>0$$ with the initial data ...
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70 views

An inequality from PDEs: $|v|^\alpha v -|w|^\alpha w \leq C(|v|^\alpha + |w|^\alpha)|v-w|$

In using Stritcharz estimates to prove well-posedness of the nonlinear Schrödinger equation $$i\partial_tu = \Delta u - \lambda|u|^\alpha u$$ where $\alpha>0$, one requires the following inequality ...
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47 views

The existence of a subsequence of harmonic functions that converges pointwise

Let $u_{n}$ be a family of harmonic functions on $\mathbb{R}^n$, and there exists a point $x_{0}$ such that $\{u_{n}(x_{0})\}$ is bounded. Then does it exist a subsequence of $u_{n}$ that converges ...
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73 views

Proof of Lemma 4.2 in [G-T] pg 55

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n (n\geq 3)$ and let $\Omega_0$ be any domain containing $\Omega$ for which the divergence theorem is true. Let $f$ be bounded and locally Holder ...
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684 views

Double Fourier Series coefficient derivation

We are looking at the rectangular membrane problem. We're assuming that $f(x,y)$ is a smooth function defined on the rectangle $\{(x,y) : 0 \leq x \leq a, 0 \leq y \leq b \}$. We want to expand $f$ ...
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689 views

Solutions to Laplace's Equation in the plane in polar coordinates

I'm trying to solve Laplace's equation in the disc $r\leq a$ using separation of variables (so relatively simple stuff, compared to what's often on here), and I've proceeded like so thus far: Let ...
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198 views

Eigenfunction expansion of homogeneous pde

Use eigenfunction expansion to solve $$u_{t}=u_{xx}$$ $$u_{x}(0,t)=u_{x}(1,t)=0$$ $$u(x,0)=\pi$$ for $$0<x<\frac{1}{2}$$ and $$u(x,0)=0$$ for $$\frac{1}{2}<x<1$$ I'm not well versed in ...
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36 views

Max of laplacian in polar coordinates

$$ \Delta u=0 $$ on $$ x^2+y^2<4 $$ $$ u=3\cos(2 \theta)+1 $$ on $$ x^2+y^2=4 $$ don't find the specific solution. what is $u$ at the origin? what is the max of u and its location on the disk ...
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140 views

Cauchy Data query.

Let $$\ y \dfrac {\partial u}{\partial x} - \ x\dfrac {\partial u}{\partial y}=0.$$ The characteristic curve obtains the function $y^2+x^2=\eta^2 $, i.e., a circle of radius $\eta$ and centre $(0,0)$. ...
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225 views

Fourier transform of $g(x)=x\frac{\partial f}{\partial x}$?

I have a problem with the Fourier transform of the function $g(x)=x\frac{\partial f}{\partial x}$. I need the transform to be itself a function of the Fourier transform of $f(x)$ and I don't know how ...
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44 views

conditions for uniqueness of a quasi linear pde?

I'm studying the problem $u_x+u_t=x(1-u^2), u=g\in \mathcal{C}^1(\Omega)$ at $\partial\Omega$ , $g$ bounded, where $\Omega$ is $\{(x,t)\in\mathbb{R}^2|t>0\}$. I found the solution, but I don't know ...
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40 views

if $U(x,y)$ dont satisfy in laplacian equation then how to prove $\nexists Q$: $\nabla Q=(-U_{y},U_{x}) $?

assume $U:\mathbb R^2\to\mathbb R$ such that $U\in\mathbb C^2$ and $U_{xx}+U_{yy}\neq0$ how prove there is no function like $Q:\mathbb R^2\to\mathbb R$ such that $$\nabla Q=(-U_{y},U_{x}) $$ $\nabla ...
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157 views

Energy method for the following for the following PDE

I wish to prove that the following PDE has at most one solution: $ \Delta (u) - u^3 = f $ in $\Omega $ $ u = \phi $ on $\partial \Omega $ f is continuous in $ \Omega $ Well, I tried to apply energy ...
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514 views

Prove solutions to the reduced Helmholtz equation are unique

As the title says, the question is to prove the solutions to the Neumann problem of the reduced Helmholtz equation $$\Delta u - ku = 0 ,k>0$$ in a bounded domain $D$ , are unique. I was able ...
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90 views

Alternative complete bases for Fourier Series.

Knowing that $$\left\{ \sin\left(kx\right)\right\} _{k\in\mathbb{N}}$$ and $$\left\{ \cos\left(kx\right)\right\} _{k\in\mathbb{N\cup}\left\{ 0\right\} }$$ are complete systems in $L^2(0,\pi)$. How ...
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416 views

Maximum Principle for Poisson Equation

For a smooth $u(x)$, $x \in \mathbb{R}^n$, satisfying: $\Delta u = -f$ for $||x||<1$ , $u=g$ on $||x||=1$ I want to show that there exists a constant $C$ such that: $$\max\{|u|:||x||\leq 1\} ...
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141 views

Solving PDE using Hopf-Lax formula

How can I solve this pde by using the Hopf-Lax formula? $$\frac{\partial u}{\partial t}\cdot \frac{\partial u}{\partial x}=1,\; u(x,0)=x$$ Thanks lot!
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247 views

When is this solution unique? - Method of Characteristics [duplicate]

I have a problem where I am supposed to solve a PDE and then tell where in the plane that it has a unique solution, I found an almost identical question but didn't understand the answer that was ...
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121 views

Show that the given elliptic equation has a radially symmetric solution.

I have the following problem (Gilbarg, Trudinger: "Elliptic PDEs of second order", problem 3.8): Consider the equation \begin{equation} L_n u \equiv a^{ij}D_{ij} u = 0,\quad a^{ij} = \delta^{ij} + ...
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198 views

Weak solution and Rankine-Hugoniot condition

I know from PDE lectures that in case of the following Cauchy problem: $$ \left\{ \begin{array}{l} u_{t}+uu_{x}=0\\ u(x,0)=g(x)\\ \end{array} \right. $$ if a piecewise $C^{1}$ function ...
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71 views

Show $f(x^2 + y^2 , \ln(\frac{x}{y}))$ is a solution to a partial differential equation?

Given function is $z = f(x^2 + y^2, \ln(x/y))$ Let $z = f(u,v)$, let $u = x^2 + y^2, v = \ln(x/y)$, show that $z$ satisfies the equation $$x \frac{\partial z}{\partial x} + y\frac{\partial ...
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52 views

Generalization of zero-diagonal square matrices to linear operators

Which linear operators in Banach or Hilbert spaces (e.g., partial differential operators or some other operators in functional spaces) are generalizations of square matrices $A=(a_{ij})$ such that ...
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104 views

How can I modify the Eikonal equation to have smooth iso-contours?

I am solving the Eikonal Equation in 2D: $ | \nabla T(x,y)|=1/V(x,y) $ for the traveltime, T(x,y), from a starting point: $ T(x_0,y_0) = 0$. The curves $ T(x,y)=C$ forms closed contours around the ...
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126 views

How to derive this existence result from Rabinowitz's book?

Rabinowitz puts forth the following example from his book "Minimax Methods in Critical Point Theory with Applications to Differential Equations" p.25 I will copy below the statement exactly as ...
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96 views

The characterization of Sobolev space

If $\Omega$ is a bounded open set in $\mathbb R^n$ and $u$ is a distribution with supp$u \subset \subset \Omega$. For any $s \in \mathbb R$, if ${(I - \Delta )^{\frac{s}{2}}}u \in L_{loc}^2(\Omega ...
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89 views

Behavior of the pointwise norm of the gradient w.r.t. to boundary conditions in elliptic PDEs

Let $B\subset \mathbb{R}^2$ be some open ball in the interior of a (nice) domain $\Omega$ and $y_i\in H_0^1(\Omega)\cap H^2(\Omega)$ for $i=1,2$. If I know that \begin{align} &\bullet\quad ...
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40 views

Necessary Condition for $C^{2}$ Regularity of this Function

If I define $$u(x,t):=\frac{1}{4\pi}\int_{B(x,t)}\frac{f(y,t-|x-y|)}{|x-y|}\;dy$$ for $(x,t)\in\mathbb{R}^{3}\times(0,\infty)$, what regularity of $f$ is required so that $u$ is at least $C^{2}$ ...
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226 views

Advection Equation with $f(x)\cdot u(x,t)$ source term

I'm working through solving a form of the advection equation for a course I'm taking. Without going into the specifics of my particular problem, the PDE has the general form shown below. I've ...
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66 views

Solutions of $u_{xx}+u_{yy}=u^3$

Does anyone know a function $u(x,y)$ other than $(x^2+y^2)^{-1/2}$ and 0 that satisfies the equation $$u_{xx}+u_{yy}=u^3.$$
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488 views

Solving a PDE wave equation with initial conditions and boundaries (from Strauss's PDE, exercise 3.2.6)

The problem is: Solve $u_{tt}=c^2u_{xx}$ in $0<x<\infty, 0\leq t<\infty,u(x,0)=0,u_t(x,0)=V$, $u_t(0,t)+au_x(0,t)=0$. So I get how $u(x,t)=tV$ for $0<ct<x$. However, I can't seem to ...
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264 views

Heat equation separation of variables with different boundary conditions

I need to solve the following heat equation using separation of variables $ \left\{\begin{matrix} u_{t}=c^{2}u_{xx} \; \; ,0<x<L, t>0 \\ u(0,t)= 0 \\ u_{x}(L,t)=0 \\ u(x,0)=f(x) ...
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623 views

Solving a semilinear partial differential equation

My trouble is in finding the solution $u = u(x,y)$ of the semilinear PDE $$x^2u_x +xyu_y = u^2$$ passing through the curve $u(y^2,y) = 1.$ So I started by using the method of characteristics to ...
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64 views

Show that each of the following equations has a solution of the form $u(x,y) = f(ax+by) $ for a proper choice of constant $a,b$.

Find the constant for each example. (a) $u_x + 3u_y =0$ (b) $3u_x - 7u_y = 0$ (c) $2u_x + \pi u_y =0$
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68 views

Solve the following boundary value problems

$$ \frac{\partial^{2} u}{\partial x \partial y} (x,y) =3x^{2} , u(x,0) = x^n (n > 0) , u(0,y) = 0 $$ This is what I've done so far... $$ \frac{\partial u}{\partial y} (x,y) = x^3 + f(y) $$ ...
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93 views

PDE method of characteristics $u(t)+u^2u(x)=0,\quad u(x,0)=x$

I'm confused on how to include the $u^2$ expression in the solution process $$u(t)+u^2u(x)=0,\quad u(x,0)=x$$ where $u(t)$ and $u(x)$ denote the partial of u with respect to those variables I'm ...
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102 views

PDE method of characteristics solving $e^{t^2}u_t+tu_x=0$ with $u(x,0)=x+2$

The equation is given as: $$e^{t^2}u_t+tu_x=0$$ with $u(x,0)=x+2$ I've got $x=\frac{1}{2}e^{t^2}+x_0$ but I'm not sure where to go from there
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154 views

Radially Symmetric Solutions of a Nonlocal Nonlinear Transport Equation

I'm reading a paper studying the following IVP of a density $\rho: \mathbb{R}^{n} \times \mathbb{R}^{\geq 0} \rightarrow \mathbb{R}$ $\rho_t + \nabla \cdot (\rho v) =0$ $\rho(\alpha,0) \equiv ...
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66 views

How to prove this uniformly bounded?

In the Theorem $3.9$ (page $41$) of the book: Gilbarg Trudinger, we have the estimate: $$\displaystyle\sup_\Omega\left[\operatorname{dist}(x,\partial\Omega)\cdot|\nabla u|\right]\leq ...
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82 views

pde, fourier series

I want to know of some sources where I can learn the methods to solve equations like this: $-d^2f/dx^2 = e^x$ on [0,2pi], $f \in V$ = {periodic, $C^{\infty} \cap L^2$}, $f(0) = 1$ The prof. talked ...
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180 views

PDE - (homogeneous) Heat equation - Solution?

today I have a question in PDE. It concerns the heat equation: Formulate the (homogeneous) heat equation for functions $f:(0,\infty)\times\mathbb{R^n} \longrightarrow\mathbb{C}$. Derive an equation ...
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90 views

Finding the general solution of the PDE $xu_x-xyu_y -u=0$ using a side condition

Find the general solution of the PDE: $$ xu_x-xyu_y -u=0 $$ I have found it to be: $u(x,y)=-xf(ye^x)$ This PDE has the property that $u(0,y)=0$. Therefore, $u(0,y)$ cannot be arbitrarily ...
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183 views

Asymptotic Expansion of a Multiscale Partial Differential Equation

I'm trying to understand how to solve $$-\nabla\cdot(K(\frac{x}{\epsilon})\nabla u(x,\frac{x}{\epsilon})=f \text{ in } \Omega$$ $$u(x,\frac{x}{\epsilon})=u_D \text{ in } \partial \Omega$$ where ...
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241 views

methods of characteristics for transport equation

I am solving by the method of characteristics the following equation: $$w_t+\frac{x}{T-t}w_x=0,\; t\in [0,T], x\in [0,1]$$ and some continuous initial data $w(x,0)=w_0(x)$ and I want to estimate the ...
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125 views

Laplace equation unit sphere

Laplec Equation is given by $\Delta u=0$ with $\Delta u=\sum_{i=1}^{n}u_{x_ix_i}$ Now I am confused with the following: What are solutions of Laplace Eq. for the inner of the unit sphere, which has ...
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315 views

Solve PDE using method of characteristics

The following problem I find challeging. Can you help me find a solution? The question is as follows: Determine the solution (in explicit form), using the method of characteristics, of the ...
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62 views

Linearization of Gross-Pitaevskii-Equation

Consider a PDE of the form $\partial_t \phi = A(\partial_\xi) \phi + c\partial_\xi \phi +N(\phi)$ where $N$ is some non-linearity defined via pointwise evaluation of $\phi$. If you want to ...
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155 views

Estimate a solution $f(x,t)$ to the heat equation with initial data $f(x,0) = \chi_{B(x_0,R)}$

For $x_0 \in \mathbb{R}^n$ and $R> 0$, set $B(x_0, R) = \{x \in \mathbb{R}^n \colon |x_0 - x| < R\}$. Let $\chi_{B(x_0, R)}$ denote the characteristic function for $B(x_0, R)$. The solution to ...
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45 views

Inequality from eigenvalue relations

This arises from the Saddle Point theorem as found in Rabinowitz "Minimax Methods in Critical Point Theory", but I'll just provide the relevant information to what I'm trying to understand. We ...