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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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=...
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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)=...
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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 \...
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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 $...
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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 ...
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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 ...
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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 ...
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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}&...
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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}{\...
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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}_{...
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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 } ...
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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 \...
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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.\...
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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 ...
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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, \...
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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: $$0.5u_{yy}+xu_y+x^2u=x^4$$ $u=e^{ry}$...
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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(\...
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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 ...
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$W^{2,2} \subset W^{1,p}$ 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)...
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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 ...
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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\...
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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-\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 \...
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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 ...
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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 ...
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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}&=...
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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>...
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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) \...
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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 = ...
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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\...
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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 ...
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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 $...
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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 ...
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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: $...
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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= \...
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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] \...
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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(\...
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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 $a_{ij}$ ...
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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=...
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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 ...
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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} ...
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1 answer
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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 $g(x,y)$ $(2)$:...
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$l^{p}$ 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 ...
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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{\...
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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: $$\...
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1 vote
1 answer
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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 ...
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