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

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0
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2answers
339 views

Solving a PDE: basic first-order hyperbolic equation $u_t = -u_x$

So I have to solve the first-order hyperbolic equation $u_t = -u_x$. It is a PDE, since there is the time and spatial variable, but I'm overwhelmed by the maths given in books of how this works. Could ...
2
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1answer
108 views

An equality that holds with $v_t \in L^2(0,T;L^2(\Omega))$ but its proof requires $v_t \in L^2(0,T;H^1(\Omega))$

Let $Q=(0,T)\times \Omega$. For all $\varphi \in C_c^\infty(Q)$ such that $0 \leq \varphi \leq 1$, the following holds $$\int_Q \varphi^2 (\Delta v)v_t = \int_Q |\nabla v|^2 \varphi \varphi_t - ...
2
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0answers
226 views

Laplace Transform of the Wave Equation

I am given a damped wave equation $u_{tt}(t,x)+2u_t(t,x)=u_{xx}(t,x); \forall t>0$ Now I know the laplace transform of this given the initial conditions, $u(0,x)=\sin x, u_t(0,x)=0;$ is ...
1
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1answer
48 views

Is $\lVert u \rVert_{L^2(M)} + \lVert \Delta u \rVert_{L^2(M)}$ equivalent to $\lVert u \rVert_{H^2(M)}$?

On a bounded Riemannian manifold without boundary, is it true that the norms $$\lVert u \rVert_{L^2(M)} + \lVert \Delta u \rVert_{L^2(M)}$$ is equivalent to the full $H^2$ norm $\lVert u ...
3
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2answers
75 views

Elliptic Operators and Continuity

I am reading a book on Hodge theory (Ref: http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/hodge-smf.pdf) or for english ...
0
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1answer
69 views

How to show $\int_Q \varphi^2 (\Delta v)v_t = \int_Q |\nabla v|^2 \varphi \varphi_t - 2\int_Q (\nabla v \cdot \nabla \varphi) (\varphi v_t)$

Let $Q=\bigcup_{t \in (0,T)}\Omega \times \{t\}$. I have seen this identity for all $\varphi \in C_c^\infty(Q)$ such that $0 \leq \varphi \leq 1$: $$\int_Q \varphi^2 (\Delta v)v_t = \int_Q |\nabla ...
0
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2answers
38 views

If $f(x,y,z,a,b)=0$, then how come $f_x+\frac{\partial z}{\partial x} f_z=0$?

My partial differential equations textbook by Tyn Myint-Yu says the following: We consider a system of geometrical surfaces described by the equation $f(x,y,z,a,b)=0$, where $a$ and $b$ are ...
0
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1answer
32 views

Commutator of two PDO.

Let $\partial_0=\partial_t$ and $g^{00}=1$. Consider the following quasilinear PDO: \begin{align} Lu=\sum_{j,k=0}^{n}g^{jk}(t,x)\partial_j\partial_ku+\sum_{j=0}^{n}b^j(t,x)\partial_ju+au \end{align} ...
0
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1answer
22 views

Does $u(x,t)=0$ on $(x,t)\in(-\delta,\delta)\times{\Bbb R}$ imply $u\equiv 0$ for wave equations?

Consider the wave equation $$ u_{tt}=u_{xx}. $$ Suppose $u(x,t)=0$ on $(x,t)\in(-\delta,\delta)\times{\Bbb R}$ for some $\delta>0$. Do I have $u\equiv 0$? I thought that this can be done ...
0
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1answer
44 views

Evaluating $x\frac{\partial}{\partial y}\left(y\frac{\partial u}{\partial y}\right)$

What is $$x\frac{\partial}{\partial y}\left(y\frac{\partial u}{\partial y}\right)$$ I'm getting $\displaystyle{x\frac{\partial u}{\partial y}+xy\frac{\partial^2 u}{\partial y^2}}$. But the book says ...
0
votes
1answer
64 views

Partial differential equation .

I'm new to the forum and I hate to be the guy who asks for people to solve my homework but I've been trying to solve the follow exercise and I've been stuck for hours :(. Partial equation is probably ...
-1
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2answers
71 views

MATLAB AND PDE (Programmation)

I am not specialized in applied mathematics. therefore i found it very difficult to program the heat equation in matlab. you can find the equation in image form. can someone help me. cordially
0
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1answer
44 views

For a PDE $u' + Au = f$, if $f$ and $u'$ are smooth does it mean $Au$ is also smooth?

Suppose I have a solution $u \in L^2(0,T;H^1(\Omega))$ with $u' \in L^2(0,T;H^{-1}(\Omega))$ of the PDE $$u' + Au = f$$ where $A:L^2(0,T;H^1(\Omega)) \to L^2(0,T;H^{-1}(\Omega))$ is an elliptic ...
1
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1answer
70 views

what is the solution of this question on partial differential equation

what is the solution of this $$\frac{1}{D^2_x-D^2_y}\sin(x-y)$$ please solve, should I move using $\exp(x-y)$ or any other method is there to solve this for particular integral
2
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1answer
73 views

Validating a PDE problem solution

I have the following problem, which I have tried to solve myself and I would like someone to verify that my answer is valid. The problem is the following: By separation of variables, derive the ...
1
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1answer
92 views

Solutions of a Poisson equation on an ellipsoid, with Neumann boundary condition.

I want to find the number of solutions of the following problem: Fix $n \in \mathbb{N}$, with $n \geq 2$. Define the domain $\Omega$ as the ellipsoid $$\Omega = \left\{x = (x_1,...,x_n) \in ...
1
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1answer
48 views

Clarifying the PDE notation C^1([0,T], X).

In studying nonlinear hyperbolic PDE, I've come across the following spaces: $C([0,T],H^s(\mathbb{R}^n))$. $C^1([0,T], H^s(\mathbb{R}^n))$. $L^p([0,T],H^s(\mathbb{R}^n))$. I presume that $(1)$ ...
0
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1answer
50 views

is this intengral bounded?

Im just stuck beacause I dont know if this integral in bounded, I was trying to make a change of variable but I cant get to anything: (edited what need is that f is bounded for a fixed x) ...
5
votes
2answers
528 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 ...
1
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0answers
66 views

About the comparison principle for harmonic function in the weak sense

I believe that the following result is true: Let $u,v \in H^{1}(\Omega)$, where $\Omega$ is open bounded set of $R^{n} $. Supoose that $u,v$ is harmonic in $\Omega$ in the weak sense, that is $$ ...
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0answers
69 views

Non linear PDEs and Minimal surfaces

Let $M$ be a compact riemannian manifold with boundary. If we consider harmonic functions in this manifold it may be imposible to build a sequence of harmonic functions which will converge (in some ...
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1answer
91 views

Sign of Laplacian at critical points of $\mathbb R^n$

Suppose we are in $\mathbb R^n$. What can we say about the sign of $\Delta u(\vec x)$ if u($\vec x$) has a local max/min at $\vec x$? I've tried looking at the reverse of the second partial derivative ...
2
votes
1answer
93 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 ...
0
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1answer
37 views

How to separate $x^2X''v+xX'v+Xv_{yy}+x^2Xv_{zz} = 0$

I have the following partial differential equation: $$x^2X''v+xX'v+Xv_{yy}+x^2Xv_{zz} = 0,$$ where $X = X(x)$ and $v = v(y,z)$. How to separate it? Thank you for any help! From the book: ...
2
votes
1answer
450 views

Use the method of characteristics to solve nonlinear first order pde.

I find this problem challenging: Use the method of characteristics to solve $u_t+u_x^2=t$ with $u(x,0)=0$. I know I'm supposed to let $p=u_x$ and $q=u_t$. Then I get $F(x,t,u,p,q)=p^2+q-t=0$. But ...
2
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0answers
128 views

aubin-lion lemma

Let $\Omega \subset \mathbb{R}$ and we have $$u_n \rightarrow u \mbox{ in } L^{\infty}(0,T;H^2(\Omega)) \mbox{ weak star }$$ and $$\frac{\partial u_n}{\partial t} \rightarrow \frac{\partial ...
1
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2answers
46 views

PDE course question

What courses do I need for a course in partial differential equations? My university has a prereq of Multivariate Calculus and Ordinary Differential Equations. However, I opened up a book on pdes in ...
0
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1answer
51 views

Separation of variables with three independent variables

I have the following differential equations problem: Derive sets of three ordinary differential equations from the following partial differential equation by separation of variables: ...
2
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1answer
93 views

Why should the diffusivity matrix (of elliptic operator) map tangent space to itself?

I have seen that an elliptic operator $A$ on a hypersurface $\Gamma$, written as $$Au=-\nabla_\Gamma \cdot (M(x)\nabla_\Gamma u)$$ (where $\nabla_\Gamma$ is the tangential or surface gradient) is ...
1
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1answer
55 views

Understanding partial differential equation requirement

I'm reading about separation of variables in my Fourier series book and there is one requirement in a problem I don't understand. Here it is: I don't understand, why can't the $\sqrt{-A}$ take a ...
1
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1answer
39 views

Solve the initial value problem $u_x^2u_t-1=0$, $u(x,0)=x$.

Solve the initial value problem $u_x^2u_t-1=0$, $u(x,0)=x$. This becomes $u_x^2u_t=1$, $u(x,0)=x$. I was thinking that this was a nonlinear wave equation at first, but the $x$ component is ...
1
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2answers
68 views

Partial Differential Equation help with delta function boundary conditions

I need help with a differential equation, the trouble is I don't think it's separable and I have tried and failed to apply the method of characteristics to figure it out. z is also bound between zero ...
2
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1answer
137 views

Use Fourier's method of separation of variables to solve the boundary value problem

Use Fourier's method of separation of variables to solve the boundary value problem comprising the following PDE and BC: PDE: $x \sin(y) u_x + \cos(y) u_y = -2 \sin(y) u $, $u = u(x,y)$ Boundary ...
8
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2answers
315 views

Compute $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$.

Compute the Fourier series for $x^3$ and use it to compute the value of $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$. I determined the coefficients of the Fourier series, which are $$a_0 = ...
2
votes
1answer
126 views

Burgers equation with initial data $u(x,0) = x^2$

I have the PDE $$u_t + u u_x = 0, t>0$$ $$u(x,0) = f_0(x) = x^2$$ Reading this answer we arrive at the solution $$u(x, t) = f_0(x-ut) = (x-ut)^2$$ $$u = x^2 - 2xut + u^2 t^2 =0$$ $$u^2 t^2 ...
2
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0answers
53 views

Prove hyperbolic operator L is simply the wave operator in a coordinate system moving with velocity $-B/2A$

This exercise comes from A First Course in Partial Differential Equations - H.F. Weinberger 8.2) Show that if the operator $L[u] = A(\frac{d^2u}{dt^2})+B(\frac{d^2u}{dxdt})+C(\frac{d^2u}{dx^2})$ is ...
0
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1answer
36 views

Average temperature of an object with a non-uniform temperature distribution [closed]

We place some solid three-dimensional object on a planar surface that heats the object until some equilibrium situation is reached (the object is also cooling and radiating heat in the air) where any ...
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1answer
189 views

Biharmonic Equation on a square (fourier series solution needed)

$\nabla^{4}f=1$ for $f$ defined in a square from $x\in[-1,1]$ and $y\in[-1,1]$ The boundary conditions are: $f=0,f_{xx}=0$ on $x=\pm 1$ $f=0,f_{yy}=\mp 1$ on $y=\pm 1$ I intend to solve this ...
19
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1answer
647 views

Was Euler right?

We have a differential equation $$ y + y' = f(x) $$ and assume $f$ is infinitely differentiable. And we want to find particular solution. Then,I set $$ y_p = f(x)-f'(x)+f''(x)...., $$ i.e., ...
1
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0answers
192 views

Prove that $\vec x = \vec0$ is an asymptotically stable fixed point for this linear system.

Consider the linear vector field $\vec x' = A\vec x$, $\vec x \in R^2$ , where A is an $2$ x $2$ constant matrix. Suppose all eigenvalues of A have negative real parts. Prove that $\overrightarrow x = ...
5
votes
3answers
309 views

Laplace's equation on a square domain with a central point reservoire

Could someone please tell me the solution to this problem. I have $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$$ on the square domain $-L<x<L, -L<y<L$ with ...
1
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1answer
52 views

Behaviour of nonlinear heat equation

I am working with the following nonlinear heat equation $$ \begin{cases} u_t - u_{xx} + \mu u = u f(u_x), & t \in \mathbb R, x \in (0,1) \\ u (x,0) = u_0 (x) \\ ...
0
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1answer
54 views

A Specific Example about Parabolic PDE

I am solving a PDE numerical problem. And I have already had a algorithm. However, it seems to be hard to find a specific example to test my solution. Could you please give me one? The equation ...
1
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1answer
46 views

Positive solutions to a PDE equation

In this question I am concerned with nonlinear positive harmonic solutions to the following problem $$Δu(x,y)=0, (x,y)∈(a,b)×ℝ$$ $$u(x,y)=0, (x,y)∈{a₀}×ℝ$$ where $a₀$ is a real constant in the ...
2
votes
1answer
35 views

The localization of smooth boundary

Let $\Omega$ be an open set in $\mathbb R^n$ with smooth boundary and $p \in \partial\Omega$ a fixed point. For any $0<r<R$, can we find an open set $\Omega_1$ with smooth boundary such that ...
1
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0answers
31 views

Reducing the space dimension of a 3D pde using integration?

I have had this idea regarding reducing a 3D PDE to a 2D or even 1D PDE using integration; I am sure there is a more official and thorough method, so please let me know. Suppose you have the simple ...
0
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1answer
39 views

Inequalities in weak maximum uniformly elliptic operator

As is well known that if $L=\sum_{i,j}a_{ij}(x)\partial_{ij}+\sum_ib_i(x)+c(x)$ such that $a_{ij}$'s satisfy uniformly elliptic conditions and $c(x)\geq0$ on $U$, and if $u\in C^2(U)\cap C(\bar{U})$ ...
1
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0answers
603 views

Proof of strong maximum principle for harmonic functions

Let $u\in\mathscr C^2(U)\cap\mathscr C(\bar U)$ be harmonic in the non-empty open and connected set $U\subset\mathbb R^n$. If there exists a Point $x_0\in U$, so that $u$ has a local Maximum at $x_0$, ...
3
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0answers
170 views

Simultaneous Orthogonal basis for $L^2(\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$

Given a smooth bounded set $U\subset \mathbb{R}^n$, there is a simultaneous orthogonal basis for $L^2(U)$ and $H^1_0(U)$ by the existence of eigenvectors to the Laplacian in a bounded domain, which ...
1
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1answer
103 views

What does $\nabla ^\perp$ mean?

As in the title, I am reading a paper and they use the notation $\nabla ^\perp$ without explaining what it means. Is this standard notation for something? I have not seen it before! Thanks in ...