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

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2
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1answer
124 views

Finding a value a for topologically conjugacy between two flows

Let A be a hyperbolic matrix such that all solutions of $\overrightarrow x' = A \overrightarrow x $ tend to the origin at t goes to infinity, and suppose B = $\begin{bmatrix}a-3 & 5 \\ -2 & ...
2
votes
2answers
113 views

Differential operators confussion

I want to solve this PDE: $$u_t-6uu_x+u_{xxx} = 0\,(1)$$ with the Inverse Scattering Method. This method is based on showing that the above equation can be expressed as $$L_t=LB-BL,\,(2)$$ where $L$ ...
2
votes
2answers
132 views

Solve the pde $u_t(x,t)=u_{xx}(x,t)-bu(x,t)+q(t)$ for $u(x,t)$

I have the example pde $u_t(x,t)=u_{xx}(x,t)-b(t)u(x,t)+q_0$, where $b(t)$ is a function of only $t$ and $q_0$ is a constant, $0<x<\pi$, $t>0$. The subscripts denote derivatives. I also have ...
2
votes
1answer
214 views

A problem about convergence…

I am studying an article of Berestychi-Caffarelli-Niremberg - Monotonicity for elliptic equations in unbounded Lipschitz domains, and I don't understand a convergence in the demonstration of the lemma ...
1
vote
1answer
86 views

a vector calculus problem when coping with problem 9 chapter 5 evans

Hi I was trying to understand the solution given by Ray Yang in the post question 9 - chap 5 evans PDE It gets to the sort of things I am quite bad at... When I get to the point ...
1
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1answer
30 views

Laplace Operator in $3D$

I am looking to find the radial part of Laplace's operator in three dimensions. I looked up Laplace's operator in spherical coordinates and from there I guess the radial part is: ...
1
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2answers
98 views

about weak convergence in $L^2(0,T;H)$.

Exercise Suppose $H$ is Hilbert space and $u_k$ converges weakly to $u$ in $L^2(0,T;H)$. Suppose further we have the uniform bounds $\mathrm{esssup}_{0≤t≤T} ||u_k(t)||≤C$. Then ...
1
vote
1answer
265 views

Express partial derivatives of second order (and the Laplacian) in polar coordinates

$z=f(x,y)$ where $x=rcosθ$ and $y=rsinθ$ Find $ \frac{\partial z}{\partial x}$ and $ \frac{\partial^2 z}{\partial x^2}$ I'm having big troubles with using chain rule, in particularly the second ...
1
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0answers
42 views

Inequality $\Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le C\|f(x)\|_{L^\infty(\mathbb R)}^2 \|f(x)\|_{H^1(\mathbb R)}$

For complex-valued functions $f_1, f_2, f_3:\mathbb R\to\mathbb C$, I want to know that the following inequality holds: $$ \Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le ...
1
vote
1answer
62 views

Products of distributions ill-defined

This question concerns distributions, as often encountered in PDE theory, which are defined as continuous linear functionals on the space $C_0^{\infty}(\Omega)$ of test functions. The product of two ...
1
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2answers
59 views

Interior $H^2$ regularity - inequalities over regions $U$ and $W \subset U$

This question is a direct continuation of my previous question. However, this one is requesting only for a relatively simple explanation. Note: $W \subset U \subset \mathbb{R}^n$. On page 330 of PDE ...
1
vote
0answers
76 views

Explicit solution of the nonlinear Schrödinger equation

Consider the linear Schrödinger equation, $$ (LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases} $$ $x\in \mathbb R^{n}.$ Taking the Fourier transform with ...
1
vote
2answers
163 views

How can I solve these pde's?

Three different problem I got: 1.. $xu_x+2x^2u_y-u=x^2e^x$ and $u(x,x^2+x)=xe^x+x^2$ 2.. $yu_{xx}+(x+y)u_{xy}+xu_{yy}=0, \quad x\neq y$ 3.. $(y+xu)u_x+(x+yu)u_y=u^2-1$ Couldnt even start. Could ...
1
vote
1answer
101 views

$C_c^{\infty}$ is dense in $W^{k,p}(\mathbb R^n)$

As the title say, I want to prove that $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$ i.e. $\displaystyle{ W^{k,p} (\mathbb R^n) = W_0^{k,p}(\mathbb R^n) \quad (\star)}$. In a book ...
1
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0answers
194 views

Separation of variables - Facing difficulty in application of Orthogonality Condition

I am trying to model transient cooling of two concentric cylinders sharing an interface along the length. Heat will flow in radial direction only. Radius of inner cylinder (or inner radius of outer ...
1
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1answer
1k views

Solving Laplace's equation using finite differences

I having coding in MatLab to approximate solutions to Laplace's equation in 2D using finite differences. I was able to do it without much problem. I learnt about how to implement this using this: ...
1
vote
1answer
141 views

The differential equation $xu_{x}+yu_{y}=2u$ satisfying the initial condition $y=xg(x),u=f(x)$

I was thinking about the following problem : Find out which of the following option(s) is/are correct? The differential equation $xu_{x}+yu_{y}=2u$ satisfying the initial condition $y=xg(x),u=f(x)$, ...
1
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0answers
308 views

Integrating pde backwards in time

I need to solve a partial differential equation backwards in time. In other words, I have the discrete partial differential equation $$ \begin{align} p_\text{new} & = p_j + ...
0
votes
2answers
83 views

Does the following have a solution for f(x,y)?

I have the following equations: \begin{equation} {1\over f(x,y)} {\partial f(x,y) \over \partial x} \alpha(x,y) + {1\over f(x,y)} {\partial f(x,y) \over \partial y} \beta(x,y) = \gamma(x,y) ...
0
votes
1answer
790 views

Solve the partial differential equation $u_t + uu_x=0$ [duplicate]

Solve the following partial differential equation $u_t + uu_x=0$ with $u=u(x,t)$ and $u(x,0)=x$. I am having trouble in applying the SIDE CONDITION. The Characteristics are $dx/dt$=$u$, here u is ...
0
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1answer
230 views

Show $au_x+bu_y=f(x,y)$ gives $u(x,y)=(a^2+b^2)^{-\frac{1}{2}}\int_{L}fds +g(bx-ay)$ if $a\neq 0$.

For my homework I am asked to do the following: Solve $au_x+bu_y=f(x,y)$, where $f(x,y)$ is a given function. If $a\neq 0$ write the solution in the form $$u(x,y)=(a^2+b^2)^{-\frac{1}{2}}\int_{L}fds ...
0
votes
1answer
80 views

I need to find a specific maximum principle

I need a maximum principle that says: If $L$ is an elliptic operator and $u$ is a positive function ($u\in C^2(\Omega)$, with $\Omega\subset\mathbb{R}^n$ an unbounded domain) such that $Lu\geq0$ ...
-1
votes
1answer
96 views

Diffusion equation problem

Heat conduction is described by the diffusion equation $$u_t = k u_{xx},$$ where $u(x,t)$ is the temperature at $x$ at time $t$ and $k$ is some constant. A homogeneous thin pipe occupying the region ...
110
votes
5answers
8k views

What do modern-day analysts actually do?

In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about ...
34
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2answers
3k views

Why are mathematician so interested to find theory for solving partial differential equations but not for integral equation?

Why are mathematician so interested to find theory for solving partial differential equations (for example Navier-Stokes equation) but not for integral equations?
12
votes
1answer
485 views

Do eigenfunctions of elliptic operator form basis of $H^k(M)$?

We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$. If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ form a basis for ...
14
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5answers
3k views

Blow up of a solution

What exactly does blow up mean, when people say, for example, that a solution (to a pde (say)) blows up. Thanks.
12
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2answers
2k views

A Problem in Evans' PDE

Problem 7 in §6.6 states as follows: Let $u\in H^1(\mathbb{R}^n)$ have compact support and be a weak solution of the semilinear PDE $$-\Delta u+c(u)=f\,\,\text{ in } \mathbb{R}^n,$$ where ...
11
votes
2answers
727 views

Relation of Brownian Motion to Helmholtz Equation

one can obtain solutions to the Laplace equation $$\Delta\psi(x) = 0$$ or even for the Poisson equation $\Delta\psi(x)=\varphi(x)$ in a Dirichlet boundary value problem using a random-walk approach, ...
9
votes
1answer
476 views

$C^\infty_{0}(\bar\Omega)$ dense in $W^{k,p}(\Omega)$: the closure is necessary?

It is a fundamental result of Sobolve space that Let $\Omega = \mathbb{R}^d$ or $\mathbb{R}^d_+$, then $C^\infty_{0}(\bar\Omega)$ is dense in $W^{k,p}(\Omega)$. However, in some literatures, ...
8
votes
2answers
504 views

what is separation of variables

I was pushing my way through a physics book when the author separated the variables of the Schrödinger equation and I lost the plot: $$\Psi (x, t) = \psi (x) T(t)$$ can someone please explain how ...
6
votes
1answer
912 views

A PDE exercise from Gilbarg Trudinger. problem 2.2

Prove that if $\Delta u=0$ in $\Omega\subset \mathbb{R}^n$ and $u=\partial u/\partial\nu=0$ on an open smooth portion of $\partial \Omega$, then $u$ is identically zero. I found a proof here, but I'm ...
4
votes
1answer
2k views

What kinds of PDE can't be solved by separation of variables?

What kinds of PDE can't be solved by separation of variables? Except those without boundary and which are non-linear? Does it matters with the shape of solution domain? When should I use addition or ...
3
votes
2answers
564 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 ...
12
votes
1answer
590 views

The Helmholtz equation: How prove this $T\psi{(x)}\in\Omega$.

Let $\Omega\subset R^2$ be a simply connected bounded domain with infinitely differentiable boundary $\partial\Omega$and unit normal vector $v$ directed into the exterior of $\Omega$ ...
10
votes
2answers
259 views

Elliptic regularization of the heat equation

This is from PDE Evans, 2nd edition: Chapter 8, Exercise 3: The elliptic regularization of the heat equation is the PDE $$ u_t - \Delta u -\epsilon u_{tt}=0 \quad \text{in }U_T, \tag{$*$}$$ where ...
10
votes
1answer
601 views

Why no trace operator in $L^2$?

We have trace operator which allows us to define boundary values of an $H^1$ function. This is because of the fact that $C^\infty$ is dense in $H^1$ under the $H^1$ norm, I believe. I'm sure either ...
9
votes
1answer
188 views

Growth of Tychonov's Counterexample for Heat Equation Uniqueness

Define a function $\varphi$ on $\mathbb{R}_{+}$ by $$\varphi(t):=\begin{cases}e^{-1/t^{2}}, & {t>0}\\ 0, & {t\leq 0}\end{cases}\tag{1}$$ It is well-known that $\varphi$ is ...
9
votes
1answer
427 views

Origin of the name 'test functions'

This is a very simple question really: where did the name 'test functions', used nowadays when speaking of infinitely differentiable and compactly supported functions, come from? More to the point: is ...
7
votes
2answers
432 views

Definition of the principal symbol of a differential operator on a real vector bundle.

I'm trying to understand the construction of the dirac operator on a manifold, but actually I guess that doesn't really matter for the question at stake. I'm interested in understanding a definition ...
7
votes
2answers
522 views

Euler-Lagrange equations of the Lagrangian related to Maxwell's equations

Clarification on Lagrangian mechanics would be much appreciated: Suppose $$L(\phi,\,\,\phi_{,i},\,\,A_i, \dot A_i)=|\dot A+\nabla\phi|^2-|\nabla \times A|^2-c\phi+d\cdot A$$ Are the corresponding ...
6
votes
1answer
289 views

Why is it important to study the eigenvalues of the Laplacian?

Why is it important to study the eigenvalues of the Laplacian acting on regions in $\mathbb R^n$? What information does this give us? What problems does this information help us solve? In particular, ...
5
votes
2answers
584 views

Questions about weak derivatives

There are two definitions of generalized differentiation that seem relevant to the context of PDEs. (That is we generalize what objects can be differentiated but we stay in Euclidean space. There are ...
5
votes
1answer
1k views

Semi-infinite heat/diffusion equation with time-dependent B.C. at x=0

I have a problem in which I need to solve the diffusion equation: $$ \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} $$ on a semi-infinite domain from $x=0$ to $x=\infty$. The initial ...
4
votes
0answers
135 views

Heat equation, boundary gradient singularity

Consider the Cauchy-Dirichlet problem for the heat equation in a non-cylindrical region $\Omega \subset \mathbf{R}^+ \times \mathbf{R}$: $\Omega = \{ (t,x): \; 0 \leq t \leq 1, \; x \leq ...
4
votes
2answers
5k views

heat equation with Neumann B.C in matlab

$$ \frac{\partial u}{\partial t}=\alpha\frac{\partial^{2}u}{\partial x^{2}} \qquad u(x,0)=f(x)\qquad u_{x}(0,t)=0\qquad u_{x}(1,t)=2 $$ i'm trying to code the above heat equation with neumann b.c. ...
3
votes
1answer
89 views

Solving a first order nonlinear PDE

I have to solve this equation : $$ \frac{\partial u}{\partial x} \cdot \frac{\partial u}{\partial y} = xy $$ with initial condition $u(x,0) = x$. I know it is easy with separation of variables, but ...
3
votes
2answers
591 views

Prove an identity for the continuous integral solution of the conservation law

This is an exercise in Evans, Partial Differential Equations(1st edition), page 164, problem 13 Assume $F(0) = 0, u$ is a continuous integral solution of the conservation law: $$ \left\{ ...
2
votes
1answer
395 views

Elliptic operators on compact space are Fredholm

I have come across this fact in a reading of mine, but I cannot seem to prove it, and I cannot seem to find a proof of it. Mostly, I am confused why the range of an elliptic operator between ...
8
votes
1answer
216 views

question on translation of operator proof

Has anyone studied the book 'Nonlinear Partial differential equations with applications' by Tomas Roubicek? I am interested in discussing a point of interest in this book. Specifically, on page 52, ...