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Questions tagged [pde]

Questions on partial (as opposed to ordinary) differential equations.

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(PDE) Parabolic Equation and their BCs and ICs

I have this PDE with the BCs and ICs $ u_t=4u_{xx} \\ u(0,t)=u(2,t)=0\quad \text{For All}\quad t>0 \\u(x,0)=f(x) \\ f(x)=10x,\quad x<0,5 \\ f(x)=8,5-7x,\quad 0,5\le x<1,5 \\ f(x)=-8+4x,\...
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25 views

Solving a Partial DE

How do I solve $y^2\eta_{xx} - \eta_{x} = 0 $ for $\eta(x,y)$
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8 views

Steady State and Transient in Parabolic Equation problem

How do you distinguish steady state and transient problem from parabolic equations? I mean, how do we know the difference about "word problem" is talk about steady state or transient?
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13 views

Writing PDE in its weak form

Consider the linear PDE $$ a(x, t)v_x + b(x, t)v_t = c(x, t)v + d(x, t), $$ in $\Omega$. Derive its weak form: $$ \oint_{\partial R}{(avdt − bvdx)} = \int \int_R {((c + a_x + b_t)v + d)dxdt,} $$ ...
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1answer
39 views

Is it a valid claim that ODEs are easier to solve numerically than PDEs?

My final project in my Partial Differential Equations class involved studying one non-linear PDE in depth. In reading about my equation, I've realized that PDEs of 3 spatial variables can be re-...
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8 views

zero flux boundary condition

I need to proof horizontal divergence with the zero flux boundary conditions such that \begin{equation} \boldsymbol{\nabla} \cdot \mathbf{u} = 0 \end{equation} where $\mathbf{u}=(u,v,w)^T$ and each ...
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0answers
43 views

Convergence of harmonic functions in $L^1$ implies uniform convergence on compact sets

Resorting to an analog of what's done here, I'm trying to prove the following statement: Let $u_m: \mathbb{R}^n \to \mathbb{R}$ be a sequence of harmonic functions and suppose there exists a ...
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Are two solutions to the wave equation equal, if they have the same absolute value?

Question. Suppose that $u_1, u_2\in C^2(\mathbb R^{d+1})$ are real-valued, satisfy $\partial_t^2 u = \Delta u$, and $$|u_1(t, x)|=|u_2(t, x)|, \qquad \forall (t, x)\in \mathbb R^{d+1}.$$ Does it ...
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18 views

Verifying that $x^i e^{-a|x|^2}$ belongs to the Schwartz space

According to wikipedia, $x^i e^{-a|x|^2}$ where $i$ is a multi-index, belongs to the Schwartz space. This means that $$\sup_{x\in\mathbb{R}^n} x^{\alpha} D^{\beta} f(x) < \infty$$ where $f(x) = ...
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Reference request: Density of testfunctions in sobolev space $ W^{1}_{0}$

can someone help me out and name a good source where the following statement ist proven? $C^\infty_0(\Omega)$ is dense in $W^{1,p}_0(\Omega)$ with $\Omega \subset \mathbb{R}^n$ being an open, bounded ...
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9 views

One Boundary Condition and One Initial Condition of PDE

I have PDE (diffusion equation) on the half plane and just have one BC and one IC only. says, $u(0,t)=T_0\quad \text{Constant} \\ u(x,0)=e^{-2x^2}$ I'm try to solve this with Fourier Series and i ...
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0answers
7 views

Linear system with non-constant matrix containg entries form unknown vector

Consider a system of PDEs $$ \begin{cases} u_t = \nabla \cdot (D(u)\nabla u) + \frac{c}{K_U+c}u-ku\\ c_t = d_c\Delta c -\frac{\nu_U c}{K_U + c}u \end{cases} $$ with some boundary conditions. Here, $D(...
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1answer
17 views

Partial integro-differential equation using Laplace transform

Is it possible to solve the linear PDE analytically \begin{equation} \frac{\partial u}{\partial z} + a \frac{\partial u}{\partial t} + \int_{0}^{t} e^{-\beta (t-t')} u(z,t') dt'=f(z,t), \end{equation} ...
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11 views

Linearization of PDE with Jacobian

Suppose there is a system of non-linear PDEs. Can such a system be linearized by using a concept similar to the Jacobian for ODEs?
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1answer
31 views

PDE as a system of ODEs

It has been said that a PDE is like an infinite dimensional system of ODEs. How can one see this by a clear example? Can any PDE be transformed into an infinite dimensional system of ODEs?
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Holder regularity and PDEs

This question is out of curiousity and may be ill posed, but are there examples of evolution PDEs of the form $\partial_tu=F(u,Du,D^2u)$ which do not have parabolic regularity or smoothing, where the ...
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18 views

Fast Marching Method - How does the procedure start?

I've read about the Fast Marching Method described by Sethian in his book "Level Set Methods and Fast Marching Methods" and also in the paper "Fast Marching Methods", which can be found for free here: ...
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1answer
20 views

How to find the determinant of system of PDE given below. Also explain the character of the system.

Consider a two dimensional fow with Cartesian velocity components u and v. Assume that flow is steady, inviscid, incompressible and irrotational. The governing equations are then mass conservation, ...
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30 views

transform second-order pde into two first-order pde

I need to transform my equation into two first-order equation. But I don't know how to do that. My equation is the Gibbons-Tsarev equation : \begin{equation} \frac{\partial u}{\partial t}\frac{\...
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Understanding proof: if $u: \Omega \subset \mathbb{R}^N \to \mathbb{R}$ is harmonic and $\nabla u \in L^2(\mathbb{R}^N)$, then $u$ is constant.

I'm trying to solve the following question: If $u$ is harmonic in $\mathbb{R}^N$ e $\nabla u \in L^2(\mathbb{R}^N)$, then $u$ is constant. I've found this solution, which I posted below, but I ...
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1answer
22 views

What are optimality conditions?

My question is rather general and in advance I appologize for not being precise enough, which is very likely. It concerns the matter: what do people understand by optimality conditions? Suppose we ...
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Diffusion Equation on the One Half Plane

How do i solve the Diffusion Equation with Fourier Series solution below? $u_t=ku_{xx}\quad,0\lt x,0\lt t \\ u(0,t)=T_0\, (constant) \\ u(x,0)=e^{-2x^2}$ It's not a homework actually. Please help ...
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Finding Green's functions in $\mathbb{R}^3$

Attempt In my book, we have the definition for Green's Function $$ G(r',r) = \frac{1}{4 \pi } \frac{1}{|r'-r|} + h(r,r')$$ is the Green's function for the Dirichlet problem on some domain of $R^3$ ...
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10 views

What is regularity of solutions?

I have been reading a publication involving numerical methods for PDEs and stumbled upon the term "regularity of solutions". My internet search hasn't given much useful information on this. Does ...
2
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1answer
32 views

Prove harmonic function inequality with the Mean-Value Property

Let $\Omega \subset \mathbb{R}^n$ open and let $u$ be a harmonic function in $\Omega.$ If $K \subset \Omega$ is compact, then prove $$ \sup\limits_{x \in K} |u(x)| \le \frac{n}{\omega_n ~dist(K,...
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16 views

Propagation of regularity for the heat equation

Let $\Omega$ be an open bounded subset of $\mathbb R^N$. Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega)$ and consider the following boundary value problem for the heat equation:...
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1answer
17 views

Derive fourier coefficient with IC dirac-delta

PDE: $$ u_t (x,t) - u_{xx} (x,t) = 0, \quad u(0,t)=0, \quad u_x(2,t)=0, \quad u(x,0)=\delta(x-x_0) $$ where $0<x_0<2$ is a given constant. I worked out my solution by using separation of ...
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2answers
35 views

limit involving harmonic function

Let $u$ harmonic function in $\mathbb{R}^3 -\{0\}$. I know that $$\lim_{x\to0} \sqrt{|x|} \cdot u(x)=k< \infty$$ I'm trying to show that $k=0$. I tried by contradiction, but I failed and I'm ...
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1answer
42 views

Finding initial conditions of 1-D Wave Equation PDE

I am trying to solve a 1-D wave equation PDE. My first step was to find all the initial conditions I could which is proving more difficult than I thought. The problem is below. The initial conditions ...
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70 views

Numerical solution to a non-linear PDE

I have this Non-linear PDE $$ \frac{\partial C}{\partial t}=\left(\frac{\partial C}{\partial x}\right)^2+C\frac{\partial^2 C}{\partial x^2} $$ Where C is a function of (x,t) It comes from the ...
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1answer
29 views

Duhamel's Principle for the wave equation

attempt a) To prove Duhamel's principle, we must show that (3) satisfies (1) and (2). One has $$ u_t = U(x,0,t)$$ and so $u_{tt} = U_t(x,0,t) = f(x,t) $. Also, is $u_x = 0$? Can someone clarify ...
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1answer
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Check my work: General solution of a PDE

I have been asked to find the "Most general solution" for $u(x,y)$ of the PDE $$\frac{\partial u}{\partial x} = \frac{x}{\sqrt{x^2+y^2}} + 3y\cos(3xy) + 3x^2y^2$$ I know you must take the integral ...
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0answers
21 views

Harmonic extension for harmonic function

Studying the mean spherical mean and the volumetric mean, this question has occurred to me. The volumetric mean is defined as follows: Let $\Omega \subset \mathbb{R}^n$ open and $f:\Omega \to \...
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1answer
54 views

Estimate for the fundamental solution of Laplace equation from Evans

After the definition of the fundamental solution of Laplace equation (page 22) $\Phi(x) = \begin{cases} -\frac{1}{2\pi} \, \log(|x|), \, & n=2, \\ \frac{1}{n \, (n-2) \, \omega_n} \, \frac{1}{|x|...
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22 views

Integration by parts, double integral and PDE

I have a function $v=v(x,y,t)$ satisfying the following PDE (degerate heat equation) $\frac{\partial v}{\partial t} - \Delta_x v - \Delta_y v - 2 \nabla_x \cdot \nabla_y v = 0$. I also have a ...
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2answers
77 views

If partial derivatives of a harmonic function are constant, is the function linear?

Let $u: \mathbb{R}^2 \to \mathbb{R}$ be a harmonic function. If $\frac{\partial u(x,y)}{\partial x} = k_1$ and $\frac{\partial u(x,y)}{\partial y} = k_2,\forall (x,y) \in \mathbb{R}^2, k_1, k_2 \in \...
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0answers
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$\Gamma$ convergence of $W^{1,p_n}$ norm as $p_n\rightarrow p$

Given $\Omega$ a bounded and open subset of $\mathbb R^N$. And a sequence $p_n$ such that $p_n>p_0>1$ and $p_n\rightarrow p$. Define the functional $$ F_n(u)=||u||_{W_{\varphi}^{1,p_n}(\Omega)}, ...
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2answers
36 views

Convergence of sequence of function for a bounded sequence in the Sobolev space

Let $u_n$ be a bounded sequence in $W_{0}^{1,p}(\Omega)$. Then upto a subsequence one has $$ u_n\to u \mbox{ weakly in}\,W_{0}^{1,p}(\Omega). $$ How the following statement is true? $$ \int_{\Omega}|\...
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1answer
38 views

Showing that a harmonic function is constant

Let $u:\mathbb{R}^n \to \mathbb{R}$ be harmonic function and suppose that exist $C>0$ such that $|u(x)| \leq C(1+\sqrt{\|x\|})$. I want show that $u$ is constant. My first idea: Show that $C(1+\...
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0answers
29 views

Derivative of a special double integrale in the calculus of variation

Let $u$ be a real function defined on $[0, T]$ and the functional $$V(u) = \iint\limits_{[0,T]^2}f(x,y)u(x)u(y)\mathrm dx\mathrm dy$$ and $f(x,y)$ is symmetric , continuous and can be writen on this ...
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1answer
19 views

Green's formula for the Laplacian defined in a neighborhood of the surface

Source: https://arxiv.org/pdf/1705.00069.pdf On page 4, it says that the surface Laplacian of a function $u$ (I will use different letters here) defined on a neighborhood of the boundary $\partial M$...
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0answers
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Lax-Wendroff method for nonlinear hyperbolic systems of conservation laws

In page 127 of R.J. LeVeque's "Numerical Methods for conservation laws" (Birkhäuser, 1992), There are various ways that [the Lax-Wendroff method for constant-coefficient linear hyperbolic systems] ...
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27 views

Proving the existence and uniqueness of partial integro-differential equation

I am working on a type (shown below) of nonlinear partial integro-differential equation with conformable fractional derivative, $$Τ_t^α u(x,t)=g(x,t)+u(x,t)+\int_0^t \int_0^x (f(y,s))^p/y^{1-α} s^{1-...
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1answer
44 views

Why does reducing a PDE with non-homogeneous boundary conditions work?

From my lecture notes: $$\frac{\partial u}{\partial t}(x,t)-K\frac{\partial^2 u}{\partial x^2}(x,t)=f(x,t), \qquad 0<x<L, \qquad 0<t<T,$$ $$u(0,t)=\mu_1(t), \qquad u(L,t)=\mu_2(t), \...
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0answers
24 views

Solve a PDE, modified Heat $u_t+u_{xx}/2 +xu = 0$; Where $u(T,x)=1$.

$u$ is a function of space $x$ and time $t$. Assume that it has smooth derivatives and all of the requirements to have a solution. Also initial/final time condition $u(T,x)=1$. Where $T$ is fixed. ...
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A specific function on a semicircle [duplicate]

Let's define $\Omega \subset \mathbb{R}^2$ as $\Omega = \{(x, y) \in \mathbb{R}^2 | x^2 + y^2 < 1, y > 0 \}$. Now, suppose $\Delta$ is a linear operator on from $C^2(\Omega)\cap C(\overline{\...
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1answer
44 views

Show that given function is identically zero

Let $D \subset \mathbb{R}^{2}$ be and open and bounded set and $u \in C^{2}(D)\cup C^{0}(\overline{D})$ be a solution of $$ -\bigtriangleup u + u^{3} + uu_{x}^{3} + u_{y}^{2} = 0 $$ in D and $$u \...
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0answers
38 views

Showing that a harmonic function is 0

Let u(x) a harmonic function in $\mathbb{R}^n$ such as: \begin{equation} \int_{\mathbb{R}^n}|u(x)|dx =K< \infty \end{equation} Show thtat $u(x)=0$, $\forall x \in \mathbb{R}^n$. Using the ...
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0answers
20 views

Verification regarding Neumann conditions at time derivative (1 Viewer)

just a question regarding neumann conditions, I seem to have forgotten these things already. I think this question is answerable by a yes or a no. So given the 2D heat equation, $$ \frac{\partial T}...
3
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2answers
65 views

Dirichlet problem with odd function.

Let $\Omega \subset \mathbb{R}^2$ be open, bounded and symmetric with respect to the origin. Let$ f:\partial \Omega \to \mathbb{R} $ be odd and continuous function. Show that if $u$ is solution of: $$...