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

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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 ...
EleDan's user avatar
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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 ...
user1288096's user avatar
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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 ...
kkk's user avatar
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2 votes
1 answer
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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 ...
Barri's user avatar
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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) \\...
Plemath's user avatar
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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 ...
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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 \...
Jacob Ivanov's user avatar
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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}\...
THAT'S MY QUANT MY QUANTITATIV's user avatar
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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 ...
SebastianP's user avatar
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1 answer
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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 $...
george's user avatar
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3 votes
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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 ...
Paruru's user avatar
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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 \...
SebastianP's user avatar
4 votes
0 answers
175 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 ...
SebastianP's user avatar
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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,...
duelspace's user avatar
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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:\...
Silvinha's user avatar
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2 votes
1 answer
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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-...
anderstood's user avatar
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6 votes
1 answer
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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 $\...
kam's user avatar
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1 answer
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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 ...
InMathweTrust's user avatar
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General solution to linear PDE with mixed derivatives

Edit: I've reformulated the problem in a way that makes it easier to express the boundary and initial conditions. This involved expressing it in terms of a different function $g(x,t)$ (which was ...
Ciaran Harman's user avatar
1 vote
1 answer
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Linear kinetic PDE: Characteristics of the transport operator are given by the flow a Hamiltonian

I am trying to read and understand the article "Hypocoercivity for linear kinetic equations conserving mass." by Dolbeault, Mouhot, Schmeiser. doi: 10.1090/s0002-9947-2015-06012-7 (https://...
kumquat's user avatar
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2 votes
1 answer
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Quite general second order PDE

I know too little about solutions for PDEs in general, so I would be grateful if anyone has any idea if there is a solution. I am trying to find a function $h$ of $n$ coordinates defined on a cube of ...
Ygor Arthur's user avatar
-2 votes
1 answer
60 views

Solving an initial value problem using method of characteristics [closed]

I was given an initial value problem and I'm really struggling with how to solve it. My instructor hasn't taught any other method for solving yet apart from method of characteristics, so I guess one ...
user1170874's user avatar
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0 answers
109 views

Transforming a system of PDE's: linear and nonlinear

Consider the following system of partial differential equations: $$ \begin{cases} 2\sqrt{s}\dfrac{\partial}{\partial s} \sqrt{\mp \Omega_s(x)}=\sqrt{x}\sqrt{\pm\dfrac{\partial}{\partial x}\Omega_s(x)} ...
John Zimmerman's user avatar
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1 answer
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How to solve this first order PDE using method of characteristics, cannot find any resources anywhere online. [closed]

The PDE in question is $u_{x}u_{y}=1, u(x,0)=\sqrt{x}$. The process for solving a first order PDE using method of characteristics when the terms are summed is relatively well-established, but I cannot ...
deleted's user avatar
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1 answer
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How to solve this linear second order PDE in spherical coordinates

I have this simple looking linear PDE which I can't seem to solve:$$\psi_{rr}+\frac{\psi_{\theta \theta}}{r^2}-\cot \theta \frac{\psi_\theta}{r^2}=0$$ where $\psi (r,\theta)$ is the Stokes stream ...
Bob Stewart's user avatar
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Can you reverse time and space in differential equations?

Context: In my recent answer I proved that a certain map $T$ is linear, and smooth. In that answer I work with a Cauchy foliation $\Omega_s(x,y,z)$ of a Lorentzian manifold. In this question I will ...
John Zimmerman's user avatar
0 votes
1 answer
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Reference request: existence, uniqueness, and regularity of solutions to elliptic PDEs with periodic boundary conditions

I am interested in Theorems regarding existence, uniqueness, and regularity of solutions to linear 2nd order elliptic PDEs with periodic boundary conditions, e.g., $$\begin{cases} -u''+u=f & 0<...
Plutoro's user avatar
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2 votes
1 answer
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Is this a coincidece for harmonic equations.

Let $ \Omega $ be a smooth bounded domain in $ \mathbb{R}^d $. Consider the functional problem $$ \min_{v\in H_0^1(\Omega)}\int_{\Omega}|\nabla v|^2dx. $$ It is easy to calculate the E-L equation of ...
Luis Yanka Annalisc's user avatar
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0 answers
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Variational characterization of the first Steklov eigenvalue of an elliptic operator [duplicate]

Prove Let $I (u,v) = \langle u, \mathcal{L}v \rangle_{L^2 (\Sigma) } + \langle u , \mathcal{B} v \rangle_{L^2(\partial \Sigma)} $ then $\rho$ is frist eigenfunction of $\mathcal{I}$ if \begin{align} ...
Ricardo Melo's user avatar
4 votes
1 answer
118 views

PDE: Is the solution unique?

Consider the partial differential equation: $$ t\frac{\partial^2}{\partial t^2} \sum_{n=1}^k \Phi_n(x,t)=-x\frac{\partial}{\partial x}\sum_{n=1}^k \Phi_n(x,t)+\sum_{n=2}^{k-1}a(n)\Phi_n(x,t) $$ where $...
John Zimmerman's user avatar
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0 answers
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A reference for solution of non homogeneous heat equation on bounded domain

In PDE's book from Evans, is said that $$ u(x,t) = \int_0^t \int_{\mathbb{R}^N} \frac{1}{(4\pi (t-s))^{N/2}} e^{-\frac{|x-y|^2}{4(t-s)}} f(y,s) dy ds $$ is a solution of $$ \begin{cases} u_t - \Delta ...
ThiagoGM's user avatar
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0 answers
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Why is optimal transport theory so relevant?

I see plenty of papers published with "optimal transport" in their title and I know that at least 2 Fields medal in the last 10 years were assigned for something related to optimal transport ...
qervert's user avatar
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Solving for pressure in NS equation

Let us consider the Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational forces provided by the formula: \begin{align} \dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\...
MrPie 's user avatar
  • 59
4 votes
2 answers
279 views

Uniqueness for linear elliptic PDE given existence

I am having an issue showing that zero is the only solution for the following PDE. Let $M=[-1,1]^2$ be a two dimensional square. And let $f\in W^{k,p}(M)$ for $k$ and $p$ arbitrarily high (or $C^\...
Shashi's user avatar
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1 vote
1 answer
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The meaning of the "mass matrix" on a PDE

Caveat Note, I have reviewed the question below with a similar name, and this is not a duplicate. I am asking about the Mass matrix on a PDE while the reference below is asking about the Mass matrix ...
krishnab's user avatar
  • 2,341
2 votes
1 answer
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Characteristics Method PDE - Solution Verification

$$\begin{cases}-x\partial_xu +y\partial_yu =-x^2u \\ u(x,1) = e^{-x} \end{cases}$$ I recently ask a question on PDE and I hope I understand how it works. Let $z(t)=u(x(t),y(t))$ $$\begin{cases}\dot{x}...
Turquoise Tilt's user avatar
0 votes
2 answers
258 views

Charasteristic Method for PDE

Hi i'm struggling a little with solutions of PDE. I have to solve the following $$\begin{cases}\partial_xu +y^2\partial_yu = 2yu+y^2\\ u(0,y)=y \end{cases}$$ I want to use the method of ...
Turquoise Tilt's user avatar
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1 answer
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Physical significance of repetitive roots of a PDE

Consider, the following Linear PDEs with homogenous coefficients, $$(D^4 + D'^4 -2D^2D'^2)z =0; \qquad......{\rm eqn}.1$$ $$(D^3D'^2 + D^2D'^3)z =0; \qquad......{\rm eqn}.2$$ Here z is a ...
Banoffee π's user avatar
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1 answer
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A continuous function in $W^{1,p}(I)$ function with continuous weak derivative is a $C^1(\overline{I})$ function

In Theorem 8.2. of Brezis's book of Functional Analysis, says that a function $u \in W^{1,p}(I)$ has a continuous representante $\tilde{u}$ such that $$ \int_y^x u'(t) dt = \tilde{u}(x) - \tilde{u}(y)....
ThiagoGM's user avatar
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1 vote
1 answer
114 views

Solving the 1D Heat Equation on [a,b] rather than [0,L]

Solve the 1D Heat Equation on $x \in [a,b]$ $$ \frac{\partial ^2 T}{\partial x^2} = \frac{1}{\alpha} \frac{\partial T}{\partial t}$$ $$ T(a,t) = T(b,t) = 0, T(x,0) = T_0(x) $$ Now, I know that ...
STL's user avatar
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2 votes
0 answers
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domain of definition of a semilinear PDE $x∂_xz + y∂_yz = −z^2$

I'm reading page 55 of a differential equation text. §3.5 Domain of definition gives an example As a simple example which illustrates this behaviour you can consider the equation $$ x∂_xz + y∂_yz = −...
hbghlyj's user avatar
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2 votes
1 answer
194 views

Solving a PDE with non-zero IC

Given the function $f(x,t)$, solve the following PDE $$\partial_{tt}f+ 2\partial_{t}f- \partial_{xx}f+f=0$$ BC: $$f(x=\pm \infty, t)=0$$ IC: $$f(x,t=0)=g(x), \quad f_t(x,t=0)=0$$ I tried solving it ...
ochem1's user avatar
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1 vote
0 answers
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Characteristics for a transport equation with discontinuous coefficient

Consider the transport equation on $[0,T] \times \mathbb{R}$: $$\partial_{t}u+ q(t,x)\partial_{x}u=0$$ where the initial data $u(0, \cdot) = u_{0}(\cdot)$ is Lipschitz continuous in space and time and ...
duelspace's user avatar
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1 vote
1 answer
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A physical interpretation of a very known elliptical PDE

I am looking for some reference that deals with applications; in real life, of the very known elliptical equation $$ \begin{cases} \begin{aligned} -\Delta u(x) + u(x) &= f(x), x \in \Omega \\ u(x)...
ThiagoGM's user avatar
  • 915
3 votes
1 answer
131 views

Solving a particulary tricky PDE

I'm currently working on a problem that involves PDEs. I rarely work with them and have never formally learned any solution strategies other than the one I'm going to describe to you. However I ...
Zedssad's user avatar
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1 vote
0 answers
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Solving a systems of linear PDE

I've come across a relatively simple system PDE, now I don't want to go into the specifics but rather ask some more general questions about solving PDEs. Consider three functions $f = f(x,y), g = g(x,...
Nitaa a's user avatar
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1 vote
1 answer
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On the Fredholm Alternative for PDE's in Evan's book

I have been studying Fredholm Alternative for PDE's in the book Evans - Partial Differential Equations. The result is: Theorem 4 (page 321) Precisely one of the following statements holds: either $(\...
ThiagoGM's user avatar
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0 answers
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For the solvability of the Poisson equation $\Delta u = f$ on manifold with boundary.

For Poisson equation $\Delta u = f$ in bounded domain in $\mathbb{R}^n$, we can directly get the solution by Green function. For Poisson equation $\Delta u = f$ on closed Riemannian manifold, the ...
Elio Li's user avatar
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0 answers
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General solution of the wave equation in 3D

In the context of a physics calculation, I have a wave equation with the following form, where $v$ is a constant and $f$ depends on $\vec{x}=(x,y,z)$ and on the time $t$: $$\bigg(\dfrac{\partial^2}{\...
Wild Feather's user avatar
5 votes
0 answers
85 views

PDE with a non-classical boundary condition

Assume that one has a classical PDE, say: $u_t(t,x)-u_{xx}(t,x)=0$, $t\in (0,1)$, $x\in (0,2)$, and $u(0,x)=0$. Then we can prove existence (and uniqueness) of solution when boundary conditions: $u(t,...
S. Euler's user avatar
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