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Questions tagged [pde]

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

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2answers
28 views

Solving the Dirichlet problem$\nabla^2u=0$, $u|_{\partial \Omega}=x$ where $\Omega=\{(x,y)|x^2+y^2<2\}$

Attempt: (Seperation of variables in polar coordinates) Assume $$u=\Theta(\theta)R(r)$$ $$\nabla^2=u_{rr}+1/r .u_r+1/r^2.u_{\theta\theta}$$ so $$0=\nabla^2u=\Theta(\theta)\dfrac{\partial^2R}{\...
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0answers
13 views

Fourier Transform for heat equation on a half-plane

I'm trying to solve heat equation on a half-plane using fourier transform and I don't understand why it doesn't work (yes, I know we can use method of images or sine-transform, but I want to ...
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0answers
13 views

Partial differential Equation uniqueness

Let $\Omega\in\mathbb{R}^{n}$ be a bounded connected open set. I have the following partial differential Equation; \begin{align} \nabla\cdot\left(-D(x)\nabla \psi\right)&=F\quad \text{in}\quad \...
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1answer
13 views

Convergence of infinite series in PDE

If $$u(x,t)=\sum_{k=0}^\infty \frac{1}{(2k)!}x^{2k}\frac{d^k}{dt^k}e^{\frac{-1}{t^2}}$$ with $x\in \mathbb R$. How do is show that $u(x,0)=0$ for $x\in \mathbb R$. I know that the as $t\rightarrow 0$,...
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0answers
4 views

Greens function representation of nonlinear Poisson equation

Let $L$ be an operator and suppose the Green's function exists. That is there exist a function $G$ such that $LG=\delta$ where $\delta$ is the Dirac delta function. If $L$ is linear, one can represent ...
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0answers
11 views

Bounding the symbol of a differential operator

Consider a bundle map $P$, between the bundles $\Bbb R^m \times \Bbb R^n \rightarrow \Bbb R^m$ and $\Bbb R^m \times \Bbb R^k \rightarrow \Bbb R^m$. On each fiber above $\xi \in \Bbb R^m$, $P$ acts as ...
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1answer
32 views

Fourier transform of $\varphi_m(u)=\int |x|^mu(x)dx$

I'm stuck with the following problem Let $\varphi_m \in \mathcal{S}'(\mathbb{R}^{n})$, $n \in \mathbb{N}$, $m\in \mathbb{C}$, $0 >\text{Re}(m)>-n$ the distribution defined by $$\varphi_m(u)...
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0answers
28 views

A Poincare-like inequality

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set with continuous boundary. Prove that for each $\epsilon>0$ there is a constant $C(\epsilon)>0$ s.t. $$ \int_{\Omega}|f(x)|^pdx\leq C(\...
2
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1answer
37 views

Trying to prove an integration by parts formula

Denote pde operator $$ Lu = - div \cdot (p \nabla u ) + qu$$ where $x \in D$ and $p=p(x) > 0$ and q=q(x) are continuous on $\bar{D}$ an p has continous first partial derivatives on $\bar{D}$. I ...
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2answers
39 views

How to calculate the energy of a partial differential equation that is not parabolic or hyperbolic

I have the following partial differential equation: I'm asked to prove that if $f\equiv 0$, then the total energy (kinetic energy + potential energy) of the system decreases with time. What is ...
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1answer
25 views

Heat equation with initial boundary problem

If I have been given in heat equation the initial boundary conditions $u_x(2,t) = 1$ then how I will use this it in question as I will proceed with separation of variable method ..since in question ...
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0answers
18 views

Relation beween weak solution and classical solution .

For $ f \in L^2(\Omega) $ u is called a weak solution of \begin{cases}-\Delta u=f&\text{in $\Omega$} \\ u=0&\text{in $\partial \Omega$}\end{cases} if: 1.$u\in W_0^{1,2}(\Omega)$ 2.$\int_{\...
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0answers
13 views

Showing convexity of an operator on a Hilbert space.

Let $H$ be a Hilbert space, $A(\cdot,\cdot):H\times H\to\mathbb{R}$ be a symmetric coercive bi-continuous bi-linear form on $H$ and $F:H\to\mathbb{R}$ be a continuous linear functional. Then the ...
1
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1answer
38 views

Poisson equation $-\Delta u = 1$ with Dirichlet condition

Let $R > 0$. Determine the radial solution of the problem \begin{align} - \Delta u(x) & = 1 \text{ if $|x| < R$}\\ u(x) & = 0 \text{ if $|x| = R$} \end{align} We know the fundamental ...
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0answers
9 views

Lower bound of self-adjoint operator

For a self-adjoint operator $H$, and a nonzero function $u\in H^1(\mathbb R^n)$, if I have $$ \langle H(u), u \rangle_{L^2} < 0 , ~~~~ H|_{u^\bot}\ge 0, $$ then define a new operator $L$, about $...
2
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1answer
25 views

Question about derivation in Fourier series

In the Wikipedia page for separation of variables an example is shown where the author derives $$f(x) = \sum_{n = 1}^{\infty} D_n \sin \frac{n\pi x}{L}$$ as a sine series expansion of $f(x)$. They ...
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0answers
32 views

What is the $\lim_{t\to \infty} u(1,t)$

Let $u(x,t)$ be a solution of $u_t-u_{xx}=0$ we are given the following additional information. $$u(x,0)=\frac{e^{2x}-1}{e^{2x}+1}$$ and $u(x,t)$ is bounded. I want to find the $\lim_{t\to \infty} ...
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0answers
19 views

initial boundary value problem for PDE

the problem $$ \begin{cases} u_t(x,t)=u_{xx}(x,t)+2\\ u_x(0,t)=1\\ u_x(1,t)=-1\\ u(x,0)=f(x) \end{cases} 0<x<1, 0<t $$ my approach $$ u(x,t)=A(x,t)+B(x,t)\\ A:homogeneous, B:non-homogeneous\\...
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0answers
3 views

Proving that a hyperbolic PDE admits a weak solution

I've been given a hyperbolic differential equation. The question is the following: Prove that the above problem admits a weak solution (Hint: write an equation satisfied by $v=e^{\lambda t}u$, then ...
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0answers
17 views

How to solve $-\Delta f(x,y) = 1, \ \ (x,y) \in (0,1)^2$, $f(x,y)=1, \ \ (x,y) \in \partial (0,1)^2$

Problem Solve : $-\Delta f(x,y) = 1, \ \ (x,y) \in (0,1)^2$, $f(x,y)=1, \ \ (x,y) \in \partial (0,1)^2$ Try I'm not familiar with the problem, so just spreading the formula out, $$ - \frac{\...
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0answers
8 views

Transform linear hyperbolic equation into wave equation

Suppose $\partial_{tt}u-\Delta u+ M(t)u=0$. Can we transfrom the equation into the form $\partial_{ss}\tilde{u}-\partial_{yy}\tilde{u}=f(s,y)$
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0answers
11 views

Shock and rarefaction waves as a Banach-space valued functions

A few days ago I read something on wikipedia about Bochner spaces (i.e. spaces that map some time interval into some Banach space). [Bochner space wiki]. More about these spaces could be found for ...
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0answers
37 views

$\Delta_g f = 0$ on the Riemannian Manifold $(\mathbb{R}^3 \setminus B , g)$ with conditions on boundary and at infinity

Consider the manifold $M = \mathbb{R}^3 \setminus B$ where $B$ is the ball with radius $1$ with riemannian metric $g$ (not necessarily the euclidean metric). I am looking for solutions to $\Delta_g ...
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1answer
21 views

Solve a Nonlinear PDE

I want to solve the problem $$x_1u_{x_1} + 2x_2u_{x_2} + u_{x_3}=3u,~~~~~ u(x_1,x_2,0) = g(x_1,x_2).$$ I believe I am on the right track but would really like feedback. This is what I have so far: ...
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1answer
19 views

which of following option is correct for heat equation (CSIR june 2018)

If $u(x,t)$ is the solution to $$\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^2}~, \quad 0<x<1,~t>0 \\ u(x,0) = 1 + x + \sin(\pi x) \cos(\pi x)~, \quad u(0,t)=1,~u(1,t)=2$$ ...
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2answers
35 views

Is a harmonic function in $\mathbb{R}^2$ which is $o(\ln |x|)$ a constant?

Let $f : \mathbb{R}^{ 2 } \rightarrow \mathbb{R} $ be a harmonic function. Suppose $$\lim _ { | x | \rightarrow \infty } \frac { | f ( x ) | } { \ln | x | } = 0$$ Prove or disprove that $f$ is a ...
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0answers
29 views

2D Heat equation with Initial Data

I have the following 2D heat equation: $$u_t - \Delta u = e^t$$ where $(x_1, x_2) \in \mathbb{R}^2, t > 0, u(x_1, x_2, 0) = cos(x_1) sin(x_2)$ I am looking to find the general solution $u(x_1, ...
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1answer
102 views
+150

Understanding Optimal Transport in One Dimension.

I'm trying to understand these lecture notes. https://sites.ualberta.ca/~mathirl/IUSEP/IUSEP_2018/lecture_notes/Pass1.pdf I understand the formulation of the Monge Problem. However, I'm having trouble ...
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1answer
44 views

Solving the PDE $xu_{x} + 2yu_{y} = 3u, u(x,y,0) = g(x,y)$

I was trying to solve the PDE: $xu_{x} + 2yu_{y} = 3u$ with $u(x,y,0) = g(x,y)$ I thought of using method of characteristics So the initial curve looks like $x = a, y=b , z = g(a,b)$ with $\frac{...
2
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1answer
46 views

Heat equation initial value problem (General Solution)

I have the following equation: $u_t - u_{xx} = 0$ with initial data $u(x, 0) = e^{kx}$ for some constant $k$ and $x \in \mathbb{R}, t > 0$ I'm looking for the general solution $u(x, t)$. So far, I ...
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0answers
11 views

Conditions of entropy - Lax and Oleinik geometric

I have to prove that for a flux function $f$ convex the two entropy conditions are equivalent: (Lax) Discontinuities at form $f'(u_l)>s>f'(u_r)$ (Oleinik) Discontinuities at form $\...
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1answer
20 views

Sobolev space is complete - confusion about one step of the proof

In the proof of why a Sobolev space $W^{k, p} (U)$ is complete (for example, in Evans), I understand why we have $||D^\alpha u|| \leq ||u||_{W^{k, p} (U)}$. What I am confused about is how that ...
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2answers
25 views

how can i solve this PDE (IVP) by using Fourier transform?

$$ u_t=\alpha^2u_{xx}\\ u(x,0)=e^{-x^2} $$ where t>0 i tried to solve it, but not certain and i got hint $$ \int_{-\infty}^{\infty} e^{-\xi^2+i\xi x}d\xi=\sqrt\pi e^{-x^2/4} $$ thanks for your ...
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0answers
14 views

Is there a solution that is not a traveling wave solution for Fisher KPP equation?

If there is one, what do they look like? It is better someone explain the following equation $u_t-\Delta u=u(1-u)$.
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0answers
14 views

Proving that two variational problems are equivalent

Let $\Omega$ be an open set of finite measure. Let $\lambda_1(\Omega)$ be the first Dirichlet eigenvalue for the Laplace operator, i.e. $$- \Delta u = \lambda_1(\Omega) u, \ \ \ \ \ \text{in} \ \...
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0answers
10 views

Deriving the residual error for finite elements.

I am using the following set of notes for adaptive finite elements (https://www.ruhr-uni-bochum.de/num1/files/lectures/AdaptiveFEM.pdf) and am trying to go through the error calculations on page 29&...
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0answers
47 views

Studying Differential Equations without Physics Experience [on hold]

As I get closer to becoming a graduate student, I'm trying to figure out what my interests in math are and which one I would like to pursue and do research in. Recently, I've been thinking about going ...
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0answers
28 views

Help me Solve this Steady State 2-D Heat equation

So I tried to solve this PDE by separation of variables which was not applicable because of non-homogeneous nature of this PDE. Then I assumed ( after reading from a book ) : $$ T(x,y)=v(x,y) + \phi(...
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2answers
77 views

Solution of second ordinary equation

i have the following question. Let $\phi_1$ and $\phi_2$ fundamental system solutions on an interval $I$ for the second order equation $$ y''+a(x)y= 0. $$ Prove that there exists fundamental system ...
2
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0answers
34 views

Expressing $\frac{\partial u}{\partial t}=k\frac{\partial^2 u}{\partial x^2}+S(x,t)$ as an integral involving $S(x,t)$

I am trying to express the solution of$$\frac{\partial u}{\partial t}=k\frac{\partial^2 u}{\partial x^2}+S(x,t) \tag{1}$$ for $-\infty< x<\infty$ with initial condition $u(x,0)=0$ as an integral ...
1
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1answer
46 views

Mellin transform of $e^{iat}$

When doing the change of variables $v=-iat$, shouldn't the limits be reversed? Or is it because its the same as $v=\frac{at}{i}$ I cant see why this is not the case
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0answers
14 views

Relation between adjoint of trace operator and Dirac delta

Assume $u \in \mathscr{D}(\mathbb{R}^n)$ is a distribution of order $k$, with compact support on a smooth manifold $\Gamma \subset \mathbb{R}^n$. We know that we can write this distribution (Thm 2.3.5,...
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0answers
29 views

Homogeneous heat equation with time dependent boundary condition in sphere (General case)

I have been going through a derivation and am looking for some guidance. The problem is as follows: Flow of heat in a sphere given by: $$\frac{\partial v}{\partial t} = k\left(\frac{\partial ^2v}{...
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0answers
20 views

Numerical solution PDEs system

I have to solve this system of PDE: $$ \left\{ \begin{array}{c} \frac{\partial^2u_c(x,y)}{\partial x^2}+a\frac{\partial^2v_c(x,y)}{\partial x\partial y}+b\frac{\partial^2u_c(x,y)}{\partial y^2} =0\\ ...
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0answers
8 views

Prove neumann stable if condition met on operator

Please tell me if this is correct : Prove : A finite difference method realization is neumann stable provided for each $n$ and $i$ $|c_i^{n+1}|\leq |c_i^n|$. I think Neumann stable means $u^{n+1}\...
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0answers
11 views

Green's function for homogeneous PDE

I was looking for some Green's function method to solve a homogeneous PDE with nonhomogeneous boundary conditions (i.e., $Lu=0$ in $D$ with $u=f(\mathbf{x})$ in $\partial D$), but most of the ...
1
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2answers
26 views

Separation of Variables with functions which are not eigenfunctions

I'm studying a book named 'Computational Techniques for fluid dynamics' by Fletcher , and it solves a heat conduction problem with mixed boundary conditions. $$\frac{\partial T}{\partial t}-\alpha \...
0
votes
1answer
23 views

Why is this eigenvalue problem solved by $\phi(r) = J_v(\alpha r)$

I can't seem to see how the bessel function $J_v(ar)$ solves the problem. The eigenvalue problem has an $\alpha^ 2\phi(r)$ term Ive tried writing $x = \alpha r$ in the expression for $J$ but cant ...
3
votes
0answers
28 views

Regularity of harmonic functions

I have a question on a fundamental property of harmonic functions. Let $\Omega \subset \mathbb{R}^d$ be a domain. We define $H^{1}(\Omega)$ by \begin{align*} H^{1}(\Omega)=\{u \in L^{2}(\Omega,m) \...
2
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0answers
41 views

Obtaining linear tridiagonal system from PDE in hydraulic fracturing

I'm trying to re-solve the governing equations in hydraulic fracturing modeling $$ \frac{\partial q}{\partial x} + \frac{2hC}{\sqrt{t-\tau(x)}} + \frac{\partial A}{\partial t} = 0 , \qquad 0<x&...