Questions tagged [linear-pde]
This tag is for questions relating to linear partial differential equations, in which the dependent variable (and its derivatives) appear in terms with degree at most one
540
questions
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1
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45
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Sequences of solutions of Poisson equation [closed]
this semester I'm taking my first course in Partial Differential Equations and I'm having a problem with the following exercise. It's from a guide of exercises that our teacher's assistant gave us.
...
-1
votes
0
answers
32
views
Try solve $(-2 x+y-u) u_x-2(x-u) u_y=-5 u, with \quad u(0,-x)=-2$
I tried solve this PDE mediant the characteristic method:
$(-2 x+y-u) u_x-2(x-u) u_y=-5 u,$ with $\quad u(0,-x)=-2$
Then I haved this sistem:
$$\frac{dx}{-2x+y-u} = \frac{dy}{-2x-2u} = \frac{du}{-5u} ...
- 33
1
vote
0
answers
58
views
Uniqueness for the mild solution of a heat equation on the torus
Consider the following linear parabolic equation in one spatial dimension for $u=u(x,t)$ on the one-dimensional flat torus $\mathbb{T}^1,$ meaning $x \in \mathbb{T}^1$ and $t \in (0, T]:$
$$ \...
- 291
1
vote
0
answers
11
views
Unitary Transformation for a Self-Adjoint Elliptic Operator in one dimension
Let $a\colon \mathbb{R}\to \mathbb{R}_+$ be bounded, bounded from below by a strictly positive constant and Lipschitz-continuous. Consider the self-adjoint linear operator $L:=-\partial_x a(x)\...
- 903
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votes
1
answer
50
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Solution for linear first order PDE using method of characteristics
For my thesis I'm trying to find a general solution for this PDE:
$$R\left(1-\frac{3}{2}\tau\right)^{-1/3}\frac{\partial t}{\partial \tau}-\left(1-\frac{3}{2}\tau\right)^{-2/3}\frac{\partial t}{\...
0
votes
0
answers
25
views
Reduction of periodic boundary conditions to Dirichlet boundary conditions
Let us consider the following heat equation with periodic boundary conditions, which is nothing but the problem on the flat torus:
$$ \partial_t u(x,t) - \partial_x^2 u(x,t) = f(x,t),(x,t) \in [0,1] \...
- 291
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votes
0
answers
22
views
Divergence Constraints in Parabolic PDEs
Consider the following system of heat equations in 2D with free divergence conditions:
\begin{equation}
\frac{\partial b^1}{\partial t} = \frac{\partial^2 b^1}{\partial x^2} + \frac{\partial^2 b^1}{\...
- 1,543
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votes
0
answers
14
views
Positiveness of a solution of a system of elliptical equations
Given $u \in H^1(\mathbb{R}^N)$, suppose there exists a unique $B(u) \in H^1(\mathbb{R}^N)$ weak solution of
$$
-\Delta v + V(x) v = u, \mathbb{R}^N,
$$
where $V$ is a suitable weight. Let $N \geq 3$ ...
- 1,273
0
votes
0
answers
56
views
Wave Equation with Non Homogenous Boundary Conditions and Force
With a wave equation like this
$$\frac{\partial^2 y}{\partial t^2}=c^2 \frac{\partial y^2}{\partial x^2}+\beta x$$
Where
\begin{gathered}
y(0, t)=A \\
y(L, t)=B \\
y(x, 0)=f(x) \neq 0 \\
\frac{\...
- 1
3
votes
0
answers
93
views
Well-posedness result for a linear parabolic equation on torus
Consider the following linear parabolic equation in one spatial dimension for $u=u(x,t)$ on the one-dimensional torus $\mathbb{T}^1,$ meaning $x \in \mathbb{T}^1$ and $t \in (0, T]:$
$$ \partial_t u- ...
- 291
0
votes
1
answer
63
views
Well-posedness of $\partial_t^3 u = \Delta u$
I am interested in the well-posedness of the following PDE:
$$\partial_t^3 u = \Delta u.$$
It resembles both the heat equation and the wave equation, but with a third-order time derivative.
Although ...
- 785
-3
votes
1
answer
36
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Failure of Schauder estimates in $L^{\infty}$ [closed]
How should I go proving that (say) there are no estimates on the hessian of $u$ of the form $\|\nabla^2u\|_{\infty} \leq C \|f\|_{\infty}$? (Here one can consider $f$ continuous on the closure of the ...
- 1
2
votes
1
answer
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Evans - existence of parabolic PDE, why does $B(u_m,v)\to B(u,v)$?
In Evans book, chapter 7.1, he establishes existence of weak solutions of
$$\partial_t u + Lu = f$$
where $L$ is a uniformly elliptic differential operator.
He first shows that for any $m$, the ...
- 120
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votes
1
answer
42
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Nonhomogeneus PDE function requires expansion in sines?
I'm studying about the solution to the PDE:
$$
\Delta u(x,y)=-f(x,y)\\ u(0,y)=u(a,y)=0 \\ u(x,0)=g(x) \ \ , \ \ u(x,b)=h(x)
$$
And the first step is to start solving it like a homogeneus equation with ...
- 401
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votes
0
answers
28
views
High order finite difference schemes for boundary value problems on a finite interval
I have some questions. I'm going to assume everything is in 1d with a Laplacian operator. If I discretize the Laplacian operator using $p = 2a+1$ grid points with periodic boundary conditions, I ...
- 543
2
votes
0
answers
66
views
Longtime behaviour of the heat kernel on the real line for bounded initial conditions
Let $u(t,x)$ be the fundamental solution to the heat equation $u_t = \frac{1}{2}u_{xx}$ with initial condition $u(0,\cdot)$. That is, $u(t,x) = \int_{\mathbb{R}}p_{t}(x-y)u(0,y)\mathrm{d}y$ where $p_t(...
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vote
0
answers
61
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Quantum Ergodic Theorem: why $\sqrt{-\Delta}$ is used instead of $-\Delta$?
I'm studying the proof of Quantum Ergodic Theorems in the book Partial Differential Equations II: Qualitative Studies of Linear Equations (3rd edition) by Michael E. Taylor. The book includes the ...
- 323
1
vote
1
answer
73
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How to use fourier transform in PDE
I’m trying to understand the Fourier transform in the context of PDE. I can follow most of the calculation and proofs, but I have come up with some “weird” cases and I’d like some clarification.
I ...
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41
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How to classify and solve this PDE?
Consider the following equation
$$
\partial_t f(t, x) = C_1 \partial_x f(t, x) - x g(t) f(t, x) + 2 g(t) \int_{x}^{\infty} f(t, y) dy
$$
for $t>0$, $x\in\mathbb{R}$, some function $g(t)>0$ and ...
- 3,465
0
votes
1
answer
36
views
How do I solve this system of PDE'S?
I am having trouble solving this system of PDE's that I got from a big problem
$$\partial_\phi B(\phi) + \sin^2(\theta) \partial_\theta C(\theta, \phi) = 0$$
$$\partial_\phi C(\theta, \phi) + B(\phi) ...
- 3,184
0
votes
0
answers
33
views
Understanding a Passage about Linear Partial Differential Equations with Non-constant Coefficients
A passage (translated by me) from Tello del Castillo's Ecuaciones en Derivadas Parciales reads
We now study equations of the form
$$\frac{\partial u}{\partial t} + \sum_{i=1}^Na_i(t,x)\frac{\partial ...
- 5,302
6
votes
1
answer
145
views
warping functions obey diff eq. does that imply $g_t$ obeys same diff eq?
Consider $(M,g_{t})$ equipped with a $1$-parameter family of warped metrics for real parameter $t>0$
$$g_{t} = \frac{1}{\phi_t(u)^{2}}\ du^{2} + \phi_t(u)\ dv^{2}$$
and suppose that the warping ...
- 104
0
votes
0
answers
52
views
Explicit solution of a ODE in $\mathbb{R}$
Let $g : (0, +\infty) \to (0, +\infty)$ continuous. Given $h \in L^2( \mathbb{R})$, I know there exists a unique $f \in H^1(\mathbb{R})$ weak solution of
$$
(1) \hspace{5mm} -\left( f''(s) + \frac{1}{...
- 1,273
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votes
0
answers
32
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Solving First Order Linear PDE's Using Seperation of Variables
been trying to figure out whether or not we are able to solve linear, semi-linear and quasi-linear PDE's using Separation of Variables instead of Characteristics. take for instance the PDE $u_x+xu_y=y$...
1
vote
0
answers
24
views
Positive part of the solution - relation between $B(u)$ and $B(u^+)$
Given $u \in L^2(\mathbb{R}^N)$, by Riesz Representation Theorem there exists a unique $B(u) \in H^1(\mathbb{R}^N)$ weak solution of
$$
-\Delta v + v = u, \quad \mathbb{R}^N.
$$
Given $u \in L^2(\...
- 1,273
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votes
0
answers
37
views
Restriction of solution operator of a elliptical equation in $\mathbb{R}^N$
For a given $u \in L^2(\mathbb{R}^N)$, we know by Riez Representation Theorem that there exists a unique $B(u) \in H^1(\mathbb{R}^N)$ such that
$$
\int_{\mathbb{R}^N} \nabla B(u) \nabla \varphi + B(u) ...
- 1,273
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votes
0
answers
31
views
Positiveness of the solution of a system of elliptical equations
Given $u \in L^2(\mathbb{R}^N)$, by Riesz Representation Theorem we know there exists a unique $v=B(u)\in H^1(\mathbb{R}^N)$ weak solution of
$$
(1) \,\,-\Delta v + v = u, \mathbb{R}^N.
$$
Suppose ...
- 1,273
0
votes
2
answers
31
views
PDE with initial conditions
I need to solve the initial-boundary value problem:
$$ u_{tt} = u_{xx}, 0<x<1, t>0 $$
With the conditions
$ u(0, t) = 0, t > 0 $
$u_x(1, t) = 0, t > 0$
$u(x, 0) = 1, 0 < x < 1$...
- 446
0
votes
0
answers
9
views
A inequality involving an exponential non linearity of a PDE
Let $f : \mathbb{R} \to \mathbb{R}$ a continuous function and suppose there exists $\alpha_0 > 0$ such that
$$
\lim_{|s| \to +\infty} \frac{|f(s)|}{e^{\alpha |s|^{N/(N-1)}}} = 0, \quad \forall \...
- 1,273
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votes
0
answers
31
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First Order linear PDE with complex variable coefficients.
Consider the following first order linear equations:
$$\partial_t u(t,x) = a(t,x)\nabla u(t,x)+b(t,x)u(t,x),u(0,x)=u_0(x).$$
If $a,b$ are real functions, this can be solved by characteristic method. ...
- 1
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votes
0
answers
53
views
A simple question about $b(x)\cdot \nabla u(x)=0$
I want to find solutions $u$ to the equation
$$b(x,y,z)\cdot\nabla u(x,y,z)=0\quad (x,y,z)\in D$$
$$u(1,0,z)=1,\, u(0,y,0)=0$$
where $D$ is the domain (a simplex) defined by
$$D=\{(x,y,z)\in[0,1]^3: 0\...
- 53
2
votes
1
answer
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views
Is it possible to solve a system of first-order linear PDE's through a matricial approach?
Consider the following ODE:
$$P_n\frac{d f_n(x)}{d x} = \sum_{n' = 1}^NQ_{n,n'}f_{n'}(x),$$
for $n$ and $n' = 1, 2, ..., N$, and in which $P_n$ and $Q_{n,n'}$ are real constant parameters. If we ...
- 31
2
votes
0
answers
68
views
Help understanding how to apply IMEX methods to one-dimensional PDEs
I need to compute a solution of the following PDE:
$$\frac{\partial u}{\partial t} + v \frac{\partial u}{\partial x} = 0$$
For didactic purposes, I need to use an IMEX method. The point is no one ever ...
- 21
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votes
0
answers
36
views
Higher regularity for linear parabolic equation with time depndent coefficient
I am looking for a higher regularity result for the solution of the problem
$$\partial_t u+div(-A(t,x)\nabla u)=f$$
in a bounded smooth domain $\Omega$ with ...
0
votes
0
answers
76
views
Existence and unique solution to a linear PDE
I'm doing an exercise with no solution, the question says $u_x+xu_y=0$ for $x,y\in \mathbb{R}$, with initial value $u(x,0)=f(x)$, where $f$ is a real function.
Now the question ask me to impose some ...
- 1
2
votes
1
answer
288
views
Prove that a Lagrangian that Induces a Linear Elliptic Euler-Lagrange PDE has Unique Form
I am asking if existence, taken as an assumption for a solution $L$ to the linear operator equation $$\mathcal{E}L = F$$ with conditions on $F$, implies further conditions on $F$ and uniqueness of a ...
- 601
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votes
0
answers
26
views
Wave equation on the half-line: is there always a $C^2$ solution?
This question has been asked again here but no answer was given.
The problem is the following:
\begin{align*}
&u_{tt}=u_{xx}, \ \ \ \ \ \ \ \ \ \ \ \ \text{in}\ \mathbb{R}_+ \times (0,\infty) \\...
- 449
1
vote
0
answers
35
views
A function with weak derivatives in two dimension also has weak derivative in one dimension?
Let $\Omega = (-1,1) \times (-1,1) \subset \mathbb{R}^{2}$. In $\mathbb{R}^{2}$ we use the coordinates $(r,s)$. For a given $u \in L^1_{loc}(\Omega)$, suppose it has all weak derivatives of first ...
- 1,273
0
votes
0
answers
58
views
Verifying Solution of Laplace Equation
I am attempting to find a unique solution of the following Dirichlet Problem:
$$\begin{cases} u_{rr} + \frac{1}{r} u_r + \frac{1}{r^2} u_{\theta \theta}=0, \quad r \in [0, 1) \\
u(1, \theta) = 4 - 9 \...
- 319
0
votes
0
answers
45
views
Finite Difference method, ADI Scheme of Douglas and Rachford
I am trying to implement the ADI scheme of Douglas and Rachford.
For $p(X,Z,t)$, there is:
$$
\begin{gathered}
A=p^{n-1}+\Delta t_n\left[F_0\left(p^{n-1}, t_{n-1}\right)+F_1\left(p^{n-1}, t_{n-1}\...
2
votes
0
answers
53
views
Solving linear second order hyperbolic PDE $\nabla \cdot \left( M \nabla u\right)=0$
Let $\Omega \subset \mathbb{R}^{2}$ be a bounded and connected domain with a smooth boundary $\Gamma$. Furthermore, let $M$ be a matrix-valued function, where the entries $m_{ij}$ are smooth and ...
- 139
2
votes
1
answer
109
views
Reference to a Theorem (or book) about parabolic PDEs
I am studying free-boundary problems, and I see that many authors face the following standard problem from the theory of parabolic PDEs:
Given a domain $[x_1, x_2] \times [t_0, t_1)$, find a function $...
- 198
3
votes
0
answers
150
views
Uniqueness of the solution to systems of first-order linear PDEs
Context:
Let $\Omega \subset \mathbb{R}^p$ be an domain.
For functions $A_{jk}^i : \Omega \to \mathbb{R}$ and $B_k^i : \Omega \to \mathbb{R}$ with some regularity, I am interested in the following ...
- 157
1
vote
0
answers
73
views
Solution to simple system of PDEs - Did I prove that only trivial solutions exist?
I am trying to solve a system of first-order linear PDEs. In the best case, I would want to solve it explicitly but proving that there exists a (unique) solution would also be helpful.
Let $\Omega \...
- 139
4
votes
0
answers
201
views
Solution to linear second order PDE
I am trying to prove the existence (and uniqueness) of a weak solution for a specific PDE. First, let me formulate the problem.
Let $\Omega \subset \mathbb{R}^{2}$ be a bounded and connected domain ...
- 139
0
votes
0
answers
71
views
Solutions with disjoint support to the continuity equation
Suppose $f,g. u \in C^{1}((0,T) \times \mathbb{R})$ with$$ \partial_{t}(f+g) + \partial_{x}((f+g)u) = 0, \quad \text{on } (0,T) \times \mathbb{R}$$ where for each $t \in (0,T)$ we have $\text{spt}(f(t,...
- 1,235
1
vote
0
answers
256
views
What hypothesis can I assume in order to obtain infinity solutions for this problem?
It is related to the classic theory of PDE.
Suppose that $U \subset \mathbb{R}^m$ is an open, connected and bounded set, $f : U \to \mathbb{R}$ and $g:\partial U\rightarrow \mathbb{R}$ and $c:\...
- 369
2
votes
1
answer
83
views
Solve $\nabla^2 u(x,y) = 1-y$ + B.C.
I am trying to solve:
$$\nabla^2 u(x,y) = \partial_{xx} u(x,y) + \partial_{yy} u(x,y) = 1-y$$
over the domain $[-1,1]\times [0,1]$, with the B.C.:
$$u(x,0) = 0 = u(\pm 1, y).$$
Is there any closed-...
- 3,524
7
votes
1
answer
135
views
Why can we pass limit under integral sign in proof of solving Poisson's equation? (Evans PDE)
On page 23 of Lawrence Evans' Partial Differential Equations text (2nd edition) he claims that
$$\frac{ f( x + he_i - y) - f( x-y)}{h} \to \frac{ \partial f}{ \partial x_i} ( x-y)$$
uniformly on $\...
- 312
1
vote
1
answer
133
views
Laplace equation in an annulus
Suppose we fix $0 < R_{0} < R$ and $f: \mathbb{R}^{3} \to \mathbb{R}$ is the solution of the Laplace equation $-\Delta f = 0$, in the annulus $|x| \in (R_{0},R]$. Suppose $f$ is radial. Is it ...
- 951