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

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2
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1answer
590 views

Numerical techniques for solving systems of first-order semi-linear hyperbolic PDEs in two variables

I need to solve such systems of PDEs numerically and I'm wondering what the "standard" method (if there is one) is. The systems I'm interested in have the form: $\frac{\partial F_i}{\partial t} + ...
1
vote
2answers
28 views

Solving Simple Partial Differential Equation

I don't remember how i can solve this simple partial differential equation. Can someone help me? $$x\frac{\partial \phi}{\partial x}+y\frac{\partial \phi}{\partial y}+ (\alpha+1-x)\phi =0$$ Please, ...
0
votes
1answer
15 views

Verification of some conditions and facts of the Laplacian on a Riemannian manifold

I came across the following introduction to a paper I was reading: "Let $L$ be a second-order, elliptic, differential operator, with smooth coefficients, on a Riemannian manifold $M$. Assume $-L$ is ...
2
votes
1answer
62 views

What does the term “regularity” mean?

When I was an undergraduate, I took a course on regularity theory for nonlinear elliptic systems. This included topics such as the direct method of calculus of variations, mollifiers, integration over ...
2
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1answer
75 views

uses of Riemannian geometry for questions not related to Riemannian geometry

Poincaré conjecture (whose formulation does not make use of notions from Riemannian geometry: "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere") was eventually proved using ...
7
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1answer
291 views

References on the Nash-Moser implicit function theorem

To learn, the Nash-Moser implicit function theorem, I tried the document Hamilton (1982) The Inverse Function Theorem of Nash and Moser, but the article is very encyclopedic. I have a ...
0
votes
0answers
6 views

Boundary conditions for a radiative heat transfer problem

Consider the heat equation $$ \frac{\partial T}{\partial t} - a\Delta T + \mathbf v \cdot \nabla T = S $$ where $S$ is a source term dependent of the radiation intensity $I$ and the temperature $T$. ...
0
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0answers
7 views

Choosing between SOV/Green's functions/Laplace transform for solving PDE - Guideline for choosing the most appropriate method?

Forgive me if this questions seems silly, but I have a question which is keeping me busy. I'm not really looking for a mathematical proof (but it is welcome), however I'm more looking for guided ...
0
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0answers
15 views

How to scale a problem involving heat and the flow of a viscous fluid?

I recently got set this problem and I was wondering if anyone would be able to give me some help on the later parts. An incompressible thermal conducting fluid is contained between two infinite ...
0
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0answers
19 views

Regularity of elliptic PDE in terms of weighted Sobolev space

There seems to be a whole industry of weighted ver. of the Poincaré's inequality (Weighted Poincare Inequality) I wonder if there are results like, weighted $L^2$ equivalent of (interior/boundary) ...
0
votes
1answer
23 views

Reflection from the upper half ball to the whole ball is harmonic

I have a question about problem 9(b) in Chapter 2 of Evans' PDE book. It says if we have $u$ is harmonic in the open upper half ball $U^+$ and $u\in C^2(U^+)\cap C(\bar{U^+})$, $u=0$ for $x\in ...
2
votes
1answer
41 views

Solving Bessel's ODE problem with Green's Function

If we have an inhomogeneous boundary value problem $x^2 y'' + xy' + (x^2 -1)y = x,$ $y(0) = y(b) = 0,$ where $b>0$ How to use Green's Funtion to Solve this problem. I am facing issues with ...
0
votes
1answer
18 views

Solution of $xu_x + yu_y = 0$

I have the first oder PDE $$xu_x + yu_y = 0 \; \text{on} \; \mathbb{R}^2$$ and I found the solution of that PDE is $$u(x,y) = f\left(\frac{y}{x}\right) = e^C = K$$ which is a constant solution. So, ...
1
vote
1answer
21 views

Domain monotonicity of eigenvalues

Let $\Omega_{1}$, $\Omega_{2}$ be subsets of $\mathbb{R}^{2}$ with smooth boundary and $\Omega_{1} \subsetneq \Omega_{2}$. Let $-\lambda_{1}$ and $-\lambda_{2}$ be the smallest (in magnitude) ...
3
votes
1answer
20 views

Duals of Sobolev Spaces vanishing on parts of the boundary

I am revising for a Finite Elements course and have the following question about the definition of $H^{-1}$. Let $D\subseteq \mathbb{R}^2$ be bounded Lipschitz domain and let $\Gamma_0 \subseteq ...
1
vote
1answer
19 views

Poisson functional on bounded domain

I was wondering if it is actually clear that on bounded domains the Poisson integral is bounded from below: $$I[u]=\int_{\Omega} \left( \frac{1}{2}\lvert \nabla u \rvert^2 - u\rho \right)\, dx,$$ I ...
3
votes
2answers
221 views

Can I follow a graduate course in PDE without having studied ODE

Hi I am considering taking the first course on Partial differential equations at my university next semester. I have already taken a first course on functional analysis . I haven't taken a proof based ...
6
votes
3answers
309 views

Weakly differentiable but classically nowhere differentiable

Is there any example of a function which is weakly differentiable but none of its versions are classically nowhere differentiable (or differentiable only on a set of measure 0) ? Thanks
1
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0answers
13 views

PDE Singularity and time of shock

Given the following PDE: $$∂p/∂t+p*∂p/∂x=0$$ $$p=p(x,t)$$ $$p(x,0)=e^{-x^2}$$ It is requested to find its solution, show that it is not well behaved by verifying what happens to $∂p/∂t$ and $∂p/∂x$ ...
1
vote
1answer
40 views

Use the Laplace Transform to solve the following PDE.

I need to use the Laplace Transform to solve the following PDE, but I don't think I'm doing it correctly. $u_{t}(y,t)=\nu\nabla^2 u(y,t)$ with $u(0,t)=u_{0}$ and $u(y,0)=0$. What I have so far: ...
0
votes
1answer
20 views

Damped wave equation on $\mathbb{R}^{2}/2\pi\mathbb{Z}^{2}$

Let $a \in (0, 1)$ and let $u$ satisfy \begin{align*} u_{tt} - \Delta_{x}u + au_{t} &= 0\\ u(x,0) &= 0\\ u_{t}(x, 0) &= f(x) \end{align*} with $t \geq 0$, $x \in ...
3
votes
0answers
24 views

Coupled partial differential equation, with boundaries specification

Please, help me to find a books or samples to learn how to solve such coupled equations $$\begin{eqnarray} \frac{\partial T_1(x,t)}{\partial t}&=& \alpha_1 \frac{\partial^2 T_1(x,t)}{ ...
4
votes
2answers
60 views

How to simplify $ \int_{\Bbb{R}^2}\Delta\varphi(x)\log|x|^2\ dx $ using Green's indentity?

Let $\varphi\in C_c^\infty(\Bbb{R^2})$ (infinitely differentiable functions with compact support) and consider $$ I=\int_{\Bbb{R}^2}\Delta\varphi(x)\log|x|^2\ dx, $$ the existence of which is ...
0
votes
0answers
14 views

PDE Variable transform (integral) Chain Rule

Can somebody help me with the variable transform where the new variable is integral variable? This is a moving boundary problem where the radius of particle, $R(t)$ changes with time. The equations ...
2
votes
1answer
44 views

solution of 1st order PDE

Find the solution of PDE, $$u_xu_y = u$$ with the initial condition $u(x,0) = 0$ in the domain $x \geq 0$ and $y \geq 0$. I have try the method of characteristic, but it seems like not working for ...
0
votes
0answers
15 views

Existence and uniqueness of Dirichlet problem

Let $U= \{x \in \Bbb R^n: |x|>1 \}$. Suppose $u \in C^2 (U) \cap C(\bar U)$ is a bounded solution of the following Dirichlet problem: $\Delta u=0 \in U$ and $u=\phi$ on $\Gamma=\{x \in \Bbb R^n: ...
-1
votes
1answer
26 views

Help in understanding the notation

I am reading the paper in this link https://dl.dropboxusercontent.com/u/20327748/99-16.ps.pdf Please help me in the notation used in page 5, $(M \vee \phi_n)\wedge M$ it is in line 2 of page 5. ...
0
votes
0answers
15 views

How do I find out the Partial Differential Equation for the given expression? [on hold]

$$z=x^2f(y)+y^2g(x)$$ Well, I found out that $$ z_x=2xf(y)+y^2g'(x)\\ \ z_y=x^2f'(y)+2yg(x).$$ How do I relate them to get the partial differential equation?
2
votes
1answer
14 views

Let $u(x)$ be harmonic for $x \in \Bbb R^3$. Suppose that $\nabla u \in L^2 (\Bbb R^3)$. Prove that $u$ is a constant.

Let $u(x)$ be harmonic for $x \in \Bbb R^3$. Suppose that $\nabla u \in L^2 (\Bbb R^3)$. Prove that $u$ is a constant. I tried with Green's equation but it didn't work. Could anyone tell me which ...
5
votes
1answer
76 views

Integration by parts for general measure?

Let $\mu$ be a general measure, suppose $f,g$ has compact support on $\mathbb{R}$, when does the integration by parts formula hold $$\int f'g d\mu = - \int g'fd\mu?$$ I know in general this is false, ...
1
vote
1answer
20 views

Some clarification on nonlinear PDEs

In Strauss' book $\textit{Partial Differential Equations: An Introduction}$, one of the important PDEs listed on p. $2$ is the shock wave equation given by $$u_x + uu_y= 0 \text{.}$$ It is nonlinear, ...
1
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0answers
34 views

Solving Heat Equation with Laplace Transform

I am trying an alternative method to separation of variables to the following equation $$ \begin{cases} u_{xx} =4u_t , 0 < x < 2, t>0\\ u(0,t)=0, u(2,t)=0, t>0\\ u(x,0)=2\sin(\pi x), 0 ...
3
votes
2answers
109 views

Going from a weak formulation to a pointwise a.e statement; don't understand text (PDEs, sobolev spaces)

I just read this: For $u \in H^1(Q)$ where $Q=\cup_{t \in (0,T)}\Omega \times \{t\}$, we have that $$\int_{\Omega}u_tv +\nabla u \cdot \nabla v = \int_{\Omega}fv\quad\text{for all $v \in ...
1
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0answers
22 views

Show that boundary layers diffuse out from the plate with speed $\sqrt{\frac{\nu}{t}}$

I was wondering if somebody would be able to help me with this problem. I know how to solve it using dimension arguments but I'm unsure what is meant by 'transform techniques'. Any help would be ...
1
vote
1answer
450 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 ...
1
vote
2answers
30 views

Difference between two solution of inhomogeneous linear equation

Show that the difference between two solutions of an inhomogeneous linear equation $Lu =g$ with the same $g$ is the solution of the homogenous equation $Lu=0$ I know the definition of linearity, but ...
4
votes
1answer
42 views

1D Wave PDE with “strange” Boundary Conditions

I've just arrived home from an exam and cannot come to terms with the fact I couldn't solve the following question: Find a solution to $$ \left\{ \begin{array}{ll} u_{tt} - u_{xx}, & ...
2
votes
1answer
37 views

Why does the limit $ \lim_{\varepsilon\to 0+}\int_{\varepsilon}^M \frac{\varphi(x)-\varphi(0)}{x}\ dx$ exist for smooth $\varphi$?

Let $D(\Bbb{R}):=C_c^\infty(\Bbb{R})$ and $$ p.v.(1/x)(\varphi):=\lim_{\varepsilon\to 0}\int_{|x|>\varepsilon}\frac{\varphi(x)}{x}\ dx. $$ for $\varphi\in D(\Bbb{R})$. I'm trying to understand ...
2
votes
1answer
39 views

Fundamental solution Laplace-Poisson equation

Let $\Phi:\mathbb{R}^n\setminus\{0\}\rightarrow\mathbb{R}$ be the fundamental solution of the Laplace equation (see e.g. in the book of Evans). For a function $f\in\mathcal{C}_c^2(\mathbb{R}^n)$ we ...
1
vote
2answers
427 views

Find the general solution of the PDE

Find the general solution of the PDE $ xu_x-xyu_y-y=0 $ for all $u(x,y)$ and find the parametric form of the solution of the PDE which follows the side condition $ **u(s^2,s)=s^3** $ I got part ...
5
votes
0answers
160 views

Doubts relating to Spaces of type $\mathcal{S}$

I have doubts in the following two questions : 1) What is the Laplace transform of $[x^k\varphi(x)]^{(q)}$, where $\varphi\in \mathcal{S}_\alpha^\beta$ and $-\infty<x<\infty$ , ...
8
votes
1answer
312 views

Heat Kernel Property

Let $\phi$ be the Heat Kernel in $\mathbb{R}^n$, i. e. $$\phi (x,t)={(4\pi t)}^{-n/2}\exp\left( - \frac{\mid x \mid ^2}{4t}\right)$$ and let $u$ satisfy Heat equation. Show that: ...
0
votes
1answer
29 views

Transport equations with constant coefficients

Let $X$ be the vector field given by $X = b \cdot \nabla_x + \partial_t$ where $b \in \mathbb{R}^n$ is fixed. Let $f \in C^1(\mathbb{R}^{n+1})$. Assume that $u \in C^1(\mathbb{R}^n \times ...
2
votes
1answer
29 views

System with arbitrary function of an unknown

How can I solve the following system $$ (u_x)^2 - (u_t)^2 = 1 \\ u_{xx} - u_{tt} = f(u) $$ where $f$ is an arbitrary function of $u$, $u$ and $f$ to be determined. I don't know any approach, ...
1
vote
1answer
18 views

Bounded linear functionals in solving PDE

Many theorems in functional analysis, Rietz, Hahn-Banach, for example, are used to find linear functionals in certain spaces. But why are bounded linear functionals useful in solving PDE? This ...
1
vote
1answer
21 views

Partial Differential Equation Formation (Arbitrary Functions) [closed]

Form the partial differential equation by eliminating the arbitrary functions from: $$z=f(x^2+y^2)+x+y$$
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0answers
12 views

Steklov eigenvalue on unit ball [closed]

Show that the eigenvalues of the Dirichlet-to-Neumann map of the unit ball $B^n$ of the n- dimensional euclidean space $R^n$ are 0, 1, 2, ... . Furthermore, the eigenspace of k is given by space of ...
0
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0answers
20 views

Existence of solution to a linear second order PDE

I want to show the existence and the uniqueness of a solution to the following partial differential equation: \begin{equation} ...
0
votes
0answers
27 views

Banach contraction mapping to find unique solution of $y(x) = \int_0^1 G(x,s) f(s,y(s)) ds$

Problem statement: Use the Banach contraction mapping theorem to show that $$y(x) = \int_0^1 G(x,s)\ f(s,y(s))\ ds$$ has a unique, continuous solution $y = y(x)$ if (1) $G(x,s) \ge 0$ for $0 \le x, ...
2
votes
1answer
30 views

Evans pde book: details on an bound for a Sobolev norm in the proof of the Meyers-Serrin theorem

Let $U$ be an open subset of $\mathbb{R}^n$ and $f\in W^{m,p}(U)$. Suppose that $$\|f\|_{W^{m,p}(V)}\leq\delta\tag{1}$$ for all $V\subset\subset U$ (that is, all $V$ such that $V\subset\overline{V} ...