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
425
questions
0
votes
1
answer
66
views
+50
General solution of PDEs using Green's function
I have a question regarding the form of the general solution to a PDE in terms of its Green's function. For example, consider the heat equation:
\begin{equation}
\frac{\partial u}{\partial t}-\Delta u=...
1
vote
1
answer
43
views
Exercise on transport equation
I have to solve the following exercise:
\begin{equation}
\begin{cases}
\partial_t u(t,x)-7\partial_x u(t,x)=0\\
u(t=0,x)=e^{-x^6}
\end{cases}
\end{equation}
And I found the classical solution $u(t,x)=...
- 447
0
votes
0
answers
43
views
Uniform norm of eigenfunctions of laplacian
Let $\{\phi_i\}$ the sequence of the eigenfunctions of laplacian operator on a domain $\Omega$, that is, considering $\{\lambda_i\}$ the respective eigenvalues, we have
$$
\int \nabla \phi_{i} \cdot \...
- 301
0
votes
0
answers
17
views
How many positive harmonic functions are there on a half-space for a vanishing boundary condition
Given a space $\mathbb{R}^n$, can I say all functions
$$u: \{x_1\geq 0\}\to \mathbb{R}$$
s.t. u is harmonic on $\{x_1\geq 0\}$, u is non-negative and u vanishes on the boundary $\{x_1=0\}$ is exactly $...
- 63
0
votes
2
answers
48
views
How to prove that two PDE's are related?
Say that I have PDE a)
$U_x+U_y=\alpha U$
then I have PDE b)
$U_{xx}+U_{yy}=\beta U$
It is obvious that the first and the second are related by that they are composed of two operators which differ by ...
- 963
1
vote
0
answers
30
views
A characterization for the kernel of an elliptic operator
Consider $\Omega \subset \mathbb{R}^{N}$ a smooth domain and Let $\lambda$ an eigenvalue of $-\Delta$. Define the operator
$$
Lu = -\Delta u - \lambda u.
$$
Now I will use the Theorem 3, page 319 from ...
- 301
0
votes
0
answers
24
views
Partial Integration in solving a PDE [duplicate]
Background:
I've been self-studying a book on Partial Differential Equations by Walter Strauss and ran across a particularly challenging problem in a section on First-Order Linear Equations.
The ...
- 1
1
vote
1
answer
57
views
Lax-Wendroff scheme stability analysis for a linear system of conservation laws
I hope you can agree that the Lax-Wendroff scheme for the $3$-dimensional wave equation (or consider any system of $4$ linear conservation laws ) can be written as:
\begin{align*}
Q_{i,j,k}^{n+1}&...
- 650
0
votes
0
answers
91
views
Well-posedness of vector-valued linear transport equation on time-varying domains
I have a vector-valued transport equation on a smoothly time-varying domain $\Omega = \Omega (t) \subset \mathbb{R}^2$ for the variable $u(x,t)$ assuming values in $\mathbb{R}^2$:
$\dfrac{\partial u}{\...
- 11
1
vote
1
answer
30
views
Regularity theorem for bounded coefficients
Consider the fallowing result:
Assume $a^{ij} \in C^{1}(U)$, $f \in L^{2}$ and let $u \in H^{1}(U)$ be a weak solution of
$$
-\sum_{i,j=1}^{n}(a^{ij}u_{x_i})x_j = f, \text{ in }U
$$
Then $u \in H^{2}_{...
- 301
0
votes
1
answer
27
views
Necessity of boundeness of domain in Strong Maximum Principle
The Strong Maximum Principle for $c \geq 0$ is as fallowing:
Assume $u \in C^{2}(U) \cap C(\overline{U})$ and
$$
c \geq 0 \text{ in } U.
$$
Supose also $U$ is connected.
i) If
$$
Lu \leq 0 \text{ in } ...
- 301
0
votes
0
answers
36
views
Every $H^1_0$ function is bounded?
Let $U$ be a bounded open set with smooth boundary. We know that a function $u$ in $C^{\infty}_0(U)$ is bounded, because it has compact support and is continous. But is it true that a function $u \in \...
- 301
1
vote
0
answers
31
views
Eigenvalues decreasing to zero
Let $(H, (\cdot,\cdot))$ be infinit dimensional separable Hilbert space. Also considerer $T : H \rightarrow H$ a non-null compact, self-adjoint operator such that
$$
(T(v),v) \geq 0, \forall v \in H.\...
- 301
0
votes
0
answers
24
views
Constant solution for an elliptic equation
Let $U$ be a open and connected subset of $\mathbb{R}^n$ with regular boundary. Consider the fallowing elliptic problem
$$
\Delta u + c(x) u = u^3, U\\
\hspace{2cm}u = 0, \partial U,
$$
where $c$ may ...
- 301
1
vote
1
answer
64
views
Reference for $L^p$ estimates
My PDE professor showed the following result:
Let $f \in L^{p}(\Omega)$, for $1 < p < \infty$. Also consider $u \in L^{1}_{loc}(\Omega)$ a solution of
\begin{align}
-\Delta u + a(x)u&= f, \...
- 301
1
vote
1
answer
58
views
How do I continue? Which method should I relearn? $0.5u_{yy}+xu_y+x^2u=x^4$
this question requires me to solve the PDE using methods from ODEs, unfortunately - I’ve managed to forgot the methods required to solve this.
what I did so far:
$$0.5u_{yy}+xu_y+x^2u=x^4$$
$u=e^{ry}$...
- 19
1
vote
0
answers
25
views
Fokker Planck Equation with strongly convex potential function
Let $\rho_t$ be the (weak) solution of Fokker-Planck equation
$$
\partial_t\rho_t = \nabla\cdot(\rho_t\nabla(\log\rho_t + V))
$$
with initial condition $\rho_0$ and no-flux boundary condition $\rho_t(\...
- 11
0
votes
1
answer
64
views
How do I approach $ a\frac{\partial I}{\partial a} + b\frac{\partial I}{\partial b} + c\frac{\partial I}{\partial c} = \frac{\pi}{2}?$
How do I solve this partial differential equation:
$$ a\frac{\partial I}{\partial a} + b\frac{\partial I}{\partial b} + c\frac{\partial I}{\partial c} = \frac{\pi}{2} ?$$
Is there a way to approach ...
- 47
0
votes
0
answers
29
views
$W^{2,2} \subset W^{1,p}$ for same $p > 2$?
Consider the following problem:
Let $U \subset \mathbb{R}^{n}$ be an bounded open set. Find conditions on $p$ for which $W^{2,2}(U) \subset W^{1,p}(U)$.
What I have done: I have proved that $W^{2,2}(U)...
- 301
0
votes
2
answers
65
views
Explicit solution for the linear ODE system $p'_n = p_{n+1} + p_{n-1} - 2\,p_n$?
For the one-dimensional heat equation $\partial_t \rho = \partial_{xx} \rho$ with initial condition $\rho(x,0) := \rho_0(x) \geq 0$ satisfying $\int_0^\infty \rho_0(x)\,\mathrm{d}x = 1$ and some ...
- 2,346
1
vote
0
answers
14
views
Boundary Caccioppoli's inequality for parabolic equation.
Let $ \Omega $ be a $ C^1 $ bounded domain in $ \mathbb{R}^d $. $ A(x,t):\mathbb{R}^d\to\mathbb{R}^{d\times d} $ is a matrix valued function such that
\begin{eqnarray}
\mu|\xi|^2\leq A(x,t)\xi_i\xi_j\...
2
votes
0
answers
151
views
Heat equation with Dirichlet and Neumann boundary conditions
The Problem
Let $\Omega\subseteq\mathbb{R}^d$ be open and bounded with $C^2$-boundary. Let $T>0$ and $u\in C^{1,2}\left((0,T)\times\Omega\right)$ be a solution of the heat equation
$$\partial_t u-\...
- 648
0
votes
1
answer
31
views
On the existence of global classical non-zero solutions of a linear elliptic equation
Does the equation
$$-\Delta u +u=0$$ have any non-zero classical, i.e., $C^2$, solutions on $\mathbb{R}^d$? How about if $\mathbb{R}^d$ is replaced with half-space? How about solutions of polynomial ...
- 245
0
votes
0
answers
28
views
Solution of Poisson's equation with a linear term
Consider the partial differential equation
$$ \Delta u = au - b,~~~~~ x \in \Omega \subset \mathbb{R}^{n},~a,b>0$$
$$ \frac{\partial u}{\partial n} = 0~~ \text{on}~ \partial \Omega.$$
Using some ...
- 1,065
0
votes
0
answers
36
views
Condition for existence of solutions of a system of first order PDE
Consider (locally) the following system of linear first order PDEs:
$$
\forall\,1\leq j<k\leq n,\quad \frac{\partial F_k}{\partial x_j}-\frac{\partial F_j}{\partial x_k} = \nu_{jk}(x),
$$
for the ...
- 312
2
votes
1
answer
34
views
Is there a name for the PDE resembles a "reversed" heat/diffusion equation $\frac{\partial^{2} u}{\partial t^{2}} = \frac{\partial u}{\partial x}$?
Is there a name for the second-order linear Partial Differential Equation of the form $\frac{\partial^{2} u}{\partial t^{2}} = \frac{\partial u}{\partial x}$ which resembles the heat/diffusion ...
- 21
0
votes
1
answer
28
views
Solving Poisson's equation subject to a non-linear constraint
Problem
Let the function $f \in (L^4 \cap H^1)(\Omega)$ satisfy the following pair of equations:
$$-\Delta f = 2 \nabla \theta \cdot (J[\theta] \nabla \theta)\quad \text{ on }\Omega,$$
$$\nabla f \...
- 21
0
votes
0
answers
24
views
Condition for no solution and infinite no of solutions to a first order PDE
Let $P,Q$ and $R$ are continuous functions of $x,y,u$ and consider the first order PDE:
$$
Pu_x+Qu_y=R,
$$
such that $u=g$ on a curve $\Gamma$ where $u=u(x,y)$ solves the above PDE.
Is there any ...
- 409
1
vote
0
answers
66
views
Why is Neumann boundary condition in linear elasticity prescribed as traction?
Normally, when you look at what a Neumann BC is, you get that it's a prescribed derivative of the solution in the direction of the outward surface normal at the boundary. However, in linear elasticity ...
- 33
1
vote
0
answers
45
views
Decay of solution to transport diffusion equation
Let $\Omega\subset\mathbb{R}^3$ be a bounded smooth domain and consider the following initial boundary value problem
\begin{align}
\partial_tu-\Delta u+b\cdot\nabla u&=0\\
u_{|\partial\Omega}&=...
- 954
2
votes
0
answers
43
views
Long time behavior of solution to heat equation
Say $\Omega\subset\mathbb{R}^3$ is a bounded, connected domain with smooth boundary. Consider the initial-boundary value problem
\begin{align}
\partial_t u&=\Delta u \text{ for } x\in\Omega, t>...
- 954
0
votes
1
answer
19
views
Can we find a uniform bound of the solution of a series of linear partial differential equations related to a parameter.
Let $\sigma \in[0,1]$,we consider following series of linear partial differential equations related to the parameter $\sigma$,for example
$$
\left\{\begin{aligned}
\Delta \Phi &=\sigma f(x, y) \...
- 183
1
vote
0
answers
4
views
Is it possible to approximate some PDE semigroups by explicit methods? [duplicate]
I'm concerned with numerical methods for the approximation of semigroup associated to following Cauchy problems (which typically involves unbounded operators):
$$
\begin{cases}
\dfrac{du}{dt} + Au = ...
1
vote
0
answers
24
views
Solve a coupled PDE in a rectangle
We consider a coupled PDE in a rectangle $\Omega=(-1,1)\times(-1,1)$. For the simplicity, we assume the functions are periodic in $x_{1}$ direction.
\begin{equation*}
\left\lbrace
\begin{split}
\nabla\...
- 1,264
2
votes
0
answers
26
views
How does one see currents as solutions to continuity equations?
I often hear people say that the solution to the continuity equation is a current or can be viewed as a current but I never understood how exactly one can rewrite a continuity equation to make use of ...
- 1,494
1
vote
1
answer
74
views
Is there a name for this differential equation?
I am studying the temperature $T$ of a wire that generates heat due to current flow. The electric resistivity is temperature dependent, so I end up with the equation:
$$\Delta T + \lambda T = f $$
If $...
- 137
0
votes
1
answer
76
views
Solution of Poisson equation for $f(x,y)=1$
Consider Poisson equation
$$
\Delta u=f
$$
To be more specific,
$$
\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=f(x,y)
$$
What would be the analytical(Exact) solution of the above ...
- 2,171
0
votes
1
answer
84
views
Cosh and Sinh in the Wave equation.
So im trying to solve an equation
$$
u_{xx}+4u_{tt}=0, \ \ \ \ \ 0<x < \pi, \ \ \ \ 0 < t < 2
$$
$$
u(0,y)=u(\pi,t)=0, \ \ \ \ \ 0 \le t \le 2
$$
After seperation of variables i get that:
$...
- 3
0
votes
0
answers
66
views
Intuition for Green's function for the Heat Equation
Here I am interested in the heat equation over the domain $\mathbb{R}_+\times\mathbb{R}^d$.
I read this question Green’s Function for the Heat Equation whereby for the heat equation
$$\partial_t u= \...
- 33
0
votes
0
answers
24
views
Inhomogeneous Laplace equation: the fourier series of the solution can't at the same time solve the PDE and satisfy the boundary condition
I have the following PDE
$$
V_1(y,z) = A + B \sum_{n, \text { odd }}^{\infty} \frac{1}{n^{3}}\left[1-\frac{\cosh \left(\frac{n \pi}{\alpha} y\right)}{\cosh \left(\frac{n \pi}{\alpha}\right)}\right] \...
- 7,131
1
vote
0
answers
29
views
References regularity of solutions to linear system of first order PDE
Consider a linear first order PDE:
$$\sum_i A_i(x)\frac \partial{\partial x_i}f(x) = B(x) f(x)$$
$f:\Omega\subset\mathbb{R}^n\to \mathbb{R}^m$ where $\Omega$ is bounded and $A_i,B :\Omega\to Hom(\...
0
votes
0
answers
37
views
Solving a linear non-separable first order PDE
I'm trying to solve a linear non-separable first order PDE of the form
$$ \nabla f(x,y,z) = g(x,y,z) \mathbf{v}, $$
where $g(x,y,z) = a_{11}x^2+a_{12}xy + \dots=\sum a_{ij}x_ix_j$ with $a_{ij}$ ...
- 101
1
vote
0
answers
54
views
1D wave equation - solving a specific initial/boundary value problem
I am trying to solve the following 1D wave equation problem with initial and boundary conditions.
$u_{tt} - c^2 u_{xx} = 0$
Initial and boundary conditions:
$u=f(x)\;, u_t=g(x) \;\;\; \text{for} \; t=...
0
votes
0
answers
23
views
Singular point of IVP
Given the IVP
$xu_x + tu_t = u+1, x \in \mathbb R, t\ge 0$
$u(x,t)= x^2, t^2 = x^2$
On solving I got the solution as:
$u= \frac{x^2 + t^2}{t} - 1$
How to illustrate/exhibit that the solution is ...
- 1,015
0
votes
0
answers
22
views
Solve transport equation when initial data is along $x = -t$.
Problem. Consider $c > 0$ and the PDE $u_t - cu_x =1$, $u(x,-x) = \phi(x)$. Solve, if possible, using the method of characteristics.
My Attempt. I write the system of ODEs so that $\frac{dt}{d\tau} ...
- 2,087
1
vote
1
answer
36
views
Are these two PDEs equivalent somehow?
I have the one dimensional wave equation for $f(x,y)$ $(1)$:
$$\frac{1}{c^2} \frac{\partial^2 f}{\partial t^2} = \frac{\partial^2 f}{\partial x^2}$$
and a system of PDE for $f(x,y)$ and $g(x,y)$ $(2)$:...
- 13
1
vote
0
answers
50
views
$l^{p}$ and $L^{p}$ applications in PDEs
I am starting to get familiar with PDEs theory.
As far as I know is the purpose is to apply functional analysis methods to study theses complicated problems. and this requires to define certain ...
- 183
1
vote
1
answer
97
views
Some questions about separation of variables and convergence in Laplace's equation
Consider the following problem for Laplace's equation on the square $\Omega = \{(x,y):0\leq x,y \leq 1\} \subseteq \mathbb{R}^2$:
$$P):\left\{\begin{aligned} &\Delta u = 0 \ \mathrm{in} \ \mathrm{\...
2
votes
3
answers
178
views
Linear first order PDE $u_x+u_t=u$ with the method of characteristics
I miss something so that I can understand how my teacher finds the solution. I will write the exercise as it is written first.
$$u_x+u_t=u, x\in \mathbb{R}, t>0$$ $$ u(x,0)=\cos x$$
Then I have:
$$\...
1
vote
1
answer
79
views
Schrödinger Operator (spectrum)
I am considering the stationary Schrödinger equation,
$$
\Psi_{xx}+(\lambda-u)\Psi=0
$$
with the Schrödinger Operator
$$
L=-\frac{\partial^2}{\partial x^2}+u.
$$
Why for $\lambda>0$ the spectrum is ...
- 215