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

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0
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1answer
98 views

How to solve: $xu_{x}+u_{y}=1,u(x,0)=2\ln(x)$, $x>1.$

I have been trying to solve the following problem: Let $u(x,y)$ be the solution to the Cauchy Problem $$xu_{x}+u_{y}=1,\;\;u(x,0)=2\ln(x),\quad x>1.$$ Then $u(e,1)=?$ I was trying to solve it ...
0
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1answer
57 views

meaning of “doubly inward-pointing”

I am currently trying to understand the $b$-calculus developed by R. Melrose. an important part of the theory is the stretched product of a manifold $X$ with boundary $\partial X$. looking at the ...
2
votes
2answers
346 views

Elliptic equation and barrier estimate.

I have trouble solving the following Evans' PDE problem. I would appreciate it if someone could help me solving it. Thank you very much in advance. Let $U\subset \mathbb{R}^n$ be an bounded domain ...
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2answers
640 views

Question from Evans' PDE book

How do you do the second part of question 8, chapter 5, of Evans' PDE book (first edition)? I have proven the inequality for smooth, compactly supported functions using integration by parts, and I ...
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0answers
42 views

Show inequality for modified bilinear form

Let $\Omega_h$ denote to the domain that is bounded by a polygon, and $V_h$ to the space of all $c\in C^0(\Omega)$ such that $v_{|T}$ is linear on any (curved) triangle T and $v=0$ in the vertices of ...
1
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2answers
132 views

Partial Differential Equation

Please help me solving the following PDE: $\partial_t f=\sin (t)\,\partial_x f+\lambda \,\partial_{xx} f$ with initial condition $f(x,0)=1$ for all $x$
1
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1answer
72 views

Simple Harmonic estimate

I know this is simple, and I see all the pieces of the puzzle are there, but I can't seem to get it. Let $u$ be a solution of $$\Delta u = f \;\;\; x \in B_4 $$ Then if we can bound $$\int_{B_4} ...
0
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1answer
95 views

Why this inequality yields at most exponential growth?

Let $\Omega=\mathbf{R}^{n-j}\times\omega$, where $\omega\subset\mathbf{R}^j$ is a smooth bounded domain. Consider a function $u:\overline\Omega\rightarrow\mathbf{R}$ that satisfies $$u(x,y)+k\leq ...
0
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1answer
106 views

Laplace transform in solving 2d wave equation

I have the following wave equation $\dfrac{\partial^2 u}{\partial x^2}+\dfrac{\partial^2 u}{\partial y^2}=\dfrac{1}{c^2}\dfrac{\partial^2 u}{\partial t^2}$ with boundary conditions at $x=0,\ ...
0
votes
3answers
111 views

The initial value problem $u_{x}+u_{y}=1,u(s,s)=1,0\leq s\leq1,$

I was thinking about the problem that says: The initial value problem $u_{x}+u_{y}=1,u(s,s)=1,0\leq s\leq1,$ has (a)two solutions, (a)a unique solution, (a)no solution, (a)infinitely many ...
0
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1answer
94 views

Consider the wave equation

Consider the wave equation $\frac{d^2}{dt^2} u(x,t)= \frac{d^2}{dx^2} u(x,t)$ -$a\leq x \leq a$ ,$t \geq 0$ subject to the initial conditions $u_t (x,0)= 0$, $u(x,0)=x$. Find a solution using the ...
2
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2answers
71 views

Solving Elliptical PDE

I have been trying to solve the following PDE: $Au_{xx} + Bu_{xy} + Cu_{yy} = 0$ In my case, I know that this is an elliptical PDE, that is $B^2 - 4AC < 0$. Does there exists an analytic ...
0
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1answer
46 views

How to argue that a positive solution of a elliptic problem is bounded? (Particular case)

Let $\Omega\in\mathbf{R}^N$ an unbounded domain and $u\in C^2(\Omega)\cap C(\overline\Omega)$, $u>0$ such that $$\Delta u + f(u)=0, \ \ \ \mbox{em} \ \ \Omega,$$ where $f$ is a bounded lipschitz ...
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0answers
302 views

Relationship between integral equations and partial differential equations

In my functional analysis class, we did a lot of problems involving integral equations such as proving existence & uniqueness using spectral theory and the Banach fixed point theorem. I've never ...
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0answers
78 views

Laplace equation

I've been asked to compute the fundamental solution (in the distributional case) of the Laplacian. I have reach $u(r)=c\cdot r+c'$ for $r>0$ but i don't know how to prove that $\Delta ...
3
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1answer
197 views

Question on proof in Evans PDE

This is on page 542 of Evans PDE book. The last inequality states that $$\int_{U}{C(|Du|+1)|u|dx} \leq \frac{1}{2}\int_{U}|Du|^2dx + C\int_{U}{|u|^2+1 \ dx}$$ Where is this coming from? I think ...
3
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1answer
254 views

Uniqueness existence for a PDE

I'm trying to solve a set of exercises in order to prepare myself for a test. This question verses about energy method on partial differential equations and I would like to ask for help on that, and, ...
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0answers
496 views

Solving the vector Laplace equation in cylindrical coordinates

So I have a problem which (in one case) leads me to the following vector Laplace equation: $\nabla^2 \mathbf{A} = 0$ with $\mathbf{A}$ the magnetic vector potential, whereon I have imposed the ...
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2answers
746 views

Fundamental solution of Heat equation

For every bounded function $u_0\in C(\mathbb R)$ the function $$u(t,x)=\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty}e^{-\frac{-(x-y)^2}{4t}}u_0(y)dy$$ is a solution of $\dfrac{\partial v}{\partial ...
2
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0answers
68 views

Convergence of a series in the $C^\infty$ topology

I am struggeling to understand the following argument given by F.Treves in the book "Introduction to Pseudodifferential Operators". The motivating problem for this is to find an approximate kernel ...
2
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2answers
187 views

fourier series and heat equation

Let $v$ a solution of he heat equation, given by $\frac{\partial v}{\partial t}(t,x)=\frac{\partial^2v}{\partial x^2}(t,x)$ for $t>0,x\in\mathbb R$ with the following properties $v(0,x)=u_0(0) ...
1
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1answer
108 views

Find all functions of class $C^2$, $f:\mathbb R^2\to\mathbb R$ such that $\frac{\partial^2f}{\partial x \partial y} = 0$

Please can you help me to find all functions of class $C^2$, $f:\mathbb R^2\to\mathbb R$ such that $\frac{\partial^2f}{\partial x\partial y} = 0$. Thank you so much!
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0answers
82 views

Separation of variables linear PDE

I am stuck with the separation of variables for the following PDE: $$ -Ay^{2}\partial _{y}^{2}f(x,y)-y^{2}\partial _{x}^{2}f(x,y)+iBy \partial _{x} f(x,y)+C= \lambda _{n}f(x,y)$$ Here, $A, B, C$ are ...
6
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1answer
240 views

A Priori Estimates for p-Laplace Equation

Suppose $\Omega\subset\mathbb{R}^n$ is a bounded domain and $f\in L^q$ with $q\in (1,\infty)$. If $u\in H_0^1(\Omega)$ satisfies $$\int_\Omega \nabla u\nabla v=\int_\Omega fv,\ \forall\ v\in ...
5
votes
3answers
1k views

Laplace's equation via Fourier transformation

I have a Laplace equation with some data along the $y$-axis: $$ \begin{cases} u_{xx} + u_{yy} &= 0 \\ u(0,y) &= f(y) \\ u_x(0,y) &= g(y). \end{cases} $$ There is no information of any ...
18
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2answers
728 views

Applications of Pseudodifferential Operators

I am very interested in just about anything that has to do with PDE's, and inevitably pseudodifferential operators comes up. Its obvious that such a novel way of looking at PDE's would be important, ...
5
votes
2answers
508 views

How do I solve a PDE with a Dirac Delta function?

I have a PDE in the form of $$ \frac{\partial u}{\partial t} + \frac{\partial u}{\partial x} + u = \delta(x-1), $$ with initial condition $u(x,0)=100$. I'm trying to solve it numerically, but I have ...
7
votes
1answer
181 views

Is $C_c^{\infty}$ dense in $X_0^{\alpha}$?

While reading papers on fractional Laplacian, I always meet space $X_0^{\alpha}(\mathcal{C}_{\Omega})$ which is defined as following: $$X_0^{\alpha}(\mathcal{C}_{\Omega})=\{z\in ...
4
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1answer
181 views

differential operator on manifold

I am currently trying to understand the local expression of a (pseudo)differential operator $$ \int_{R^n} e^{(x - y)\cdot \xi} \sigma(x,\xi) \, d \xi $$ on a manifold $M$ (compact and boundaryless, ...
2
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0answers
87 views

A simple looking nonlinear PDE

A nonlinear PDE that has been bugging me for months, hoping someone has an idea, it looks so simple! Consider the following, for $u(x,t)$ $D \displaystyle\frac{\partial u}{\partial t} = u^2 ...
0
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1answer
47 views

Problem related to the type of pde

I was trying to solve the following problem: The partial differential equation $y^{3}u_{xx}-(x^{2}-1)u_{yy}=0$ is (a) parabolic in $\{(x,y):x<0\}$, (b) hyperbolic in ...
1
vote
1answer
114 views

Harmonic function product, Knowing that one is Harmonic implies something about the other?

Hello I have been fighting with this problems for a couples of days and cant get anything better, I will thanks a lot for your help, for little hint, some clue or idea : Let $A$ be an Harmonic ...
0
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3answers
244 views

How to prove this partial derivative?

Consider $u:\mathbf{R}\times\omega\rightarrow\mathbf{R}$, where $\omega\subset\mathbf{R}^{n-1}$ is a bounded domain. For each $y\in\omega$ and each $\lambda>0$, consider ...
7
votes
2answers
542 views

Why is Korn's inequality not trivial?

Let $\Omega \subset \subset \mathbb{R}^N$ have smooth boundary, $N \geqslant 2$ and $$ \mathcal{E} ( v) := \int_{\Omega} \sum_{i, j} \varepsilon_{i j} ( v) \varepsilon_{i j} ( v) \mathrm{d} x := ...
1
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1answer
142 views

How do harmonic function approach boundaries?

Suppose that $D$ is a domain in $\mathbb{R}^n$ (that is, an open, bounded and connected subset), and that $u$ is an harmonic function on $D$. Let $x_0$ be a point at the boundary of $D$. Question ...
1
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1answer
273 views

Solving $u_t=4u_{xx}$

How do we solve the folllowing diffusion problem? $u_t=4u_{xx}$ $u(0,t)=0$ $u(3,t) = 0$ $u(x,0)=\sin(2\pi x/3)-2\sin(\pi x)+7\sin(5\pi x/3)$
2
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3answers
654 views

Help with Evans PDE problem

I have trouble proving the following problem (Evans PDE textbook 5.10. #15). Could anyone kindly help me solving the problem? I know that I should somehow use Poincare inequality but I still cannot ...
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0answers
70 views

Harmonic Function in $\Omega$ that is continuous in $\overline{\Omega}$ except at a point on the boundary

My problem is the following. Let $x_{0}\in\partial\Omega$ and $\Omega\subseteq\mathbb{R}^{2}$ open and connected domain. Suppose there exists $R\in\mathbb{R}$ such that $\Omega\subseteq B_{x_{0},R}$. ...
5
votes
1answer
107 views

Specific solution to the $1D$ wave equation

So my current solution to the $1D$ wave equation is (with my given boundary and initial conditions): $$y(x,t) = \sum_{n=1}^\infty C_n\cdot \sin\frac{n \pi x}{2 l}\cdot\cos\frac{n \pi c t}{2 l}$$ ...
1
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0answers
127 views

Should I get the absolute value of the result of the inverse discrete fourier transform?

The result of equation 36 can be positive and negative.And if I don't get the absolute value of it,the ocean surface tend to be very regular.But according to the paper,the author never get the ...
4
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1answer
256 views

A fundamental solution for the Laplacian from a fundamental solution for the heat equation

Here is a heuristic reasoning. Suppose that the function $u(x, t)$ solves $$\partial_t u = \Delta u.$$ Integrating in $t$ we can define a new function $v$: $$v(x)=\int_0^\infty u(x, t)\, dt.$$ ...
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1answer
420 views

Fourier Transform of a Homogeneous Heat Equation with a Source

Consider the heat equation $$\color{blue}{\begin{align} u_t&=ku_{xx}-bt^2u,\quad-\infty<x<\infty,\quad t>0,\\ u(x,0)&=\exp\left[-x^2\right]. \end{align}}$$ I am asked to solve it ...
0
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1answer
149 views

How to prove that this is an Harmonic funtion?

Let $u$ be an Harmonic function in $B(0,a)$ in $R^3$ we define $I(x)=x\dfrac{a^2}{|x|^2} $ Let $w(x) = u(I(x))$. Is there a way to prove that $w$ is harmonic without making too much computation? ...
0
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1answer
290 views

damped wave equation

For $t>0$, $x$ in a compact Riemannian manifold $(M,g)$, and $a\in C^\infty(M)$, $a\geq0$, $(\partial_t^2+a\partial_t-\Delta_g)u=0$ is called the damped wave equation. My question is...why is the ...
1
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0answers
88 views

the maximum principle in some weird function.

I'm taking a course of particial diferencial equation, I think it was a mistake xD, but now here I'm and I have some troubles with the following problems: Let $u(x_0,y_0)$ be a point of the boundary ...
2
votes
3answers
2k views

Wave Equation with One Non-Homogeneous Boundary Condition

Consider the following wave equation: $$\begin{align} u_{tt}&=u_{xx},\quad x\in(0,\pi),\quad t>0,\\ u(x,0)&=0,\quad u_t(x,0)=0,\\ \color{red}{u(0,t)}&\color{red}{=\phi(t)},\quad ...
0
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1answer
84 views

Solve the following problem

Solve $$\frac{\partial u}{\partial t} = 2\frac{\partial^2 u}{\partial x^2}$$ with $0 < x < 3, t > 0$, given that $u(0,t) = u(3,t) = 0$, and $$u(x,0) = 5\sin 4\pi x - 3\sin 8\pi x + 2\sin 10 ...
0
votes
2answers
121 views

solve the boundary value problem

Solve the boundary value problem $$\begin{cases} \displaystyle \frac{\partial u}{\partial t} = 2 \frac{\partial^2 u}{\partial x^2} \\ \ \\u(0,t) = 10 \\ u(3,t) = 40 \\ u(x, 0) = 25 \end{cases}$$
2
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1answer
108 views

Initial Value Method of Characteristics Question

Help me to solve the following Partial differential equation: $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=-ku, \;\;u(x,0)=2, \;\; k>0 \;\text{is a constant}\;\; \text{and} \;\; ...
1
vote
1answer
348 views

Complex results in inverse Fourier transform for simulating ocean water

I don't understand the equation37 in simulate ocean water by Jerry Tessendorf.The result is all complex number, how to be the slope.Even if I compute the magnitude of it,the result is just positive ...