Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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Divergence in R^n

I have a problem from my homework that I don't have any idea how to solve it. I want to use divergence theorem... But I'm working in $\mathbb{R^n}$ that is not bounded... So, I don't think I can use ...
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1answer
20 views

Fourier transform

I find some problems about pde, I guess, concerning Distribution solution. Define $$<T,\phi> = \frac{1}{\pi}\lim_{\epsilon \rightarrow 0} \int_{|x|\geq \epsilon} \frac{\phi(x)}{x} dx.$$ Then $$ ...
1
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1answer
15 views

local integrability of function

The problem asks to show that $G(\xi,\tau)=\frac{1}{2\pi i\tau+4\pi^2|\xi|^2}$ is locally integrable on $\mathbb{R^n\times R}$. So, any compact set $K$ is inside $B_R(0)\times[-R,R]$ for some ...
3
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1answer
25 views

How to solve a PDE with a Dirac Delta and what does the PDE means?

If I have a PDE $ \Delta u= \delta(0)$ on some bounded domain in $\mathbb{R}^2$ with smooth boundary with some nice enough boundary condition. What is the solution of the PDE? And what is the PDE ...
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0answers
15 views

How to numerically solve a Green's function using mathematica? [migrated]

Suppose we have a Green's function $$LG(x)=\delta(x),$$ how to numerically solve it by mathematica? Can the mathematica read the delta function directly? For analytical calculation, we usually ...
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2answers
36 views

Approximate Sobolev function by smooth function - error estimate?

I wondered if there is, in general, a way to estimate the error (in $L^2$) between a Sobolev function and its mollified version. Let's say $\Omega\subset\mathbb{R}^n$ is bounded with smooth boundary ...
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0answers
27 views

How to solve $u_{tt}+\Delta u + x^{2}u=0$?

Let $u:\mathbb R \times \mathbb R \to \mathbb C$ be some function so the everything in the following make sense. Consider the following PDE: $\frac{\partial^{2}}{\partial t^2} u(x,t) + ...
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0answers
15 views

Explicit solutions heat equation

Is there anyway to represent explicit the solution of the following fractional heat equation: $$\partial_{t}u(x,t) + f(x,t)(-\Delta)^{\frac{1}{2}} u(x,t)=0$$ with given initial data ...
2
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0answers
24 views

Poincaré-like inequality

Let $\Omega\subset\mathbb{R}^3$ be an open bounded set. Let $\partial\Omega=\Gamma^1\cup\Gamma^2$, with $\Gamma^1\cap\Gamma^2=\emptyset$. We denote as $\Gamma^1_j$, $j=1,\dots,p_{\Gamma^1}+1$, the ...
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0answers
29 views

Asymptotics of Harmonic Functions

I am looking for some information (answers, references etc.) on existence of solutions to \begin{equation} \Delta u = f \end{equation} on the Euclidean unit ball $B \subset \mathbb{C}^n$ with ...
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1answer
14 views

Wave equation with Neumann boundary conditions

Problem def.: Solve the wave problem in a form of a Fourier series PDE: $$u_{tt}=2u_{xx}$$ BCs: $$u_x(0,t)=u_{x}(2\pi,t)=0$$ ICs: $$u(x,0)=-1,\:u_{t}(x,0)=1,\qquad0<x<2\pi$$ So far my ...
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0answers
27 views

Can anybody recognise this equation?

I just wonder if the following equation is a known special function? $$\left(u(1-u^2)\frac{d^2}{du^2}-(u^2+1)\frac{d}{du}-\frac{au}{(1-u^2)}-\frac{bu^3}{(1-u^2)}+c\right)G(u,u')=0,$$ where $a$, $b$, ...
1
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1answer
26 views

Function of first integrals pde

Show that if $F$ and $G$ are first integrals of the characteristic system of $u_{t} + c(x,t,u)u_{x} = g(x,t,u)$ then $\Psi(F(x,t,u),G(x,t,u))=0$ defines the solution $u=u(x,t)$ of this equation. It ...
3
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0answers
105 views

Complete integral of pde without independent variables

Show that the complete integral of pde $F(u,p,q)=0$ ($p=u_{x}$ and $q=u_{y}$) is $$ f(x,y,u,a,b) = x + ay + b - \int\frac{du}{g(u,a)}, $$ where the function $p=g(u,a)$ is computed from the ...
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1answer
36 views

Solve Lagrange Partial Differential Equation: $x^3p + y(3 x^2 + y)q = z(2 x^2 + y)$

I am unable to solve this Lagrange PDE: $$ x^3 p + y(3 x^2 + y) q = z(2 x^2 + y), $$ where $p = \dfrac{\partial z}{\partial x}$, $q = \dfrac{\partial z}{\partial y}$. I have found the $c_1 = ...
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0answers
12 views

Iterated convolutions w.r.t. different variables of a function

I do not understand a claim from a paper: Let $b:[0,T] \times \mathbb{R}^n \rightarrow \mathbb{R}$ be a bounded function and let $$b^{n} (t,x) = b(t,x) \ast \psi_n(t) \ast\phi_n(x), $$ where ...
3
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2answers
61 views

Bound for a certain integration

Let $\psi$ be a smooth function with compact support, and $\phi$ is smooth and $\phi'(x) \neq 0$ for any $x$ in support of $\psi$. Define $$I(\lambda) = \int_\mathbb{R} e^{i\lambda \phi(x)}\psi(x) ...
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0answers
17 views

(Variation of) The unceetainty principle

I try to look for reference to the proof of the following version of the uncertainty principle : $$||f||_{L^2} \leq \frac{4\pi}{n}\inf_{y\in \mathbb{R}^n}(\int_{\mathbb{R}^n}|x-y|^2|f(x)|^2 dx)^{1/2} ...
0
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1answer
21 views

Wave equation with $u(x,0)=\sin^{3}x$ initial condition

Problem def.: Solve the wave problem in a form of a Fourier series PDE: $u_{tt}=3u_{xx}$ BCs: $u(0,t)=u(\pi,t)=0$ ICs: $u(x,0)=\sin^{3}x,\:u_{t}(x,0)=0,\qquad0<x<\pi$ My solution: Using ...
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0answers
45 views

Method of continuity in PDEs

Theorem 5.2 (method of continuity) of Gilbarg and Trudinger's "Elliptic partial differential equations of second order" states for $\mathfrak{B}$ a Banach space and $\mathfrak{C}$ a normed linear ...
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2answers
139 views

What is the idea behind Green's function? What does it do?

I have an exam on ordinary and partial differential equations in a couple of days and there is one concept that I am really struggling with: Green's function. I have basically read every PDF-file on ...
0
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1answer
42 views

Solve the PDE $\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}=x$ using Laplace transform in $t$

Using Laplace transform in $t$, or otherwise, solve the equation for $u$: $$\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}=x$$ in the region: $x$ > 0, $t$ > 0, subject to the boundary ...
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1answer
18 views

another help with partial differential equation.

I am to solve next task. solve this PDE with boundary conditions. $\Delta u = \frac{64}{r^5}\sin\varphi, \quad 1<r<2,$ $u'_r|_{r=1} = 2\cos^2\frac{\varphi}{2}, \quad u'_r|_{r=2} = ...
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2answers
25 views

Solving with the Method of characteristics

Can anybody help me in making two characteristic functions of the equation $$xu_x+(x+y)u_y=1$$ I have found one of the Char. by equating $$\frac{dx}{x}=du$$ By which I get $$ln(x)=u$$ How to form ...
1
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2answers
203 views

How to proceed with this Second Order PDE?

I would really like to solve the following with the method you will see below. I have done some work but I have no idea how to continue. Any help would be greatly appreciated. $$ \frac{\partial ...
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0answers
24 views

Eigenvalue equation and the diffusion equation

I am running some finite-element software on Matlab that generates solutions to the diffusion equation over a punctured, rectangular domain. This is the same as the $k \times k$ matrix system ...
0
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1answer
14 views

How to reduce into canonical form

Determine the type of the following equation and reduce the PDE to its canonical form $u_{xx} + 4u_{xy} + 4u_{yy} + u = 0$. We consider pdes in the form ...
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1answer
36 views

Questions on proof that $\Vert \cdot \Vert_p$ is a norm when dealing with $L^p$ spaces

Unfortunately I never had an opportunity to take a functional analysis/PDE module so I keep running into steps I'm unsure of when I'm looking at proofs. I'm trying to fix that now with some books so ...
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0answers
14 views

Heat and Wave equation - Green's function versus Fourier series?

I am learning how to solve the heat and wave equation in bounded domains in 1 and 2D as well as in $\mathbb{R}$ and $\mathbb{R}^2$. In the latter case I have learned the representation formulas i.e. ...
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0answers
13 views

matlab code for finite volume method

I am trying to write a matlab code which is about applying Roe linearization to the 1D shallow water equation. In addition, I should include entropy fix. This is a function that I defined: function ...
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0answers
23 views

Is this finite difference approach correct? [migrated]

I am solving incompressible 2D Navier-Stokes equations with zero y-component velocity. and so, the two equations I have are: $$ \frac{\partial u}{\partial x} = 0$$ $$ \frac{\partial P}{\partial x} = ...
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0answers
13 views

Pattern Formation book

I want to initiate me into the study of pattern formation. I am motived by the work of Alan Turing (1952). What book do you recommend to star? PD : I'm studied mathematics and computation. Thanks
3
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1answer
52 views

Computing $\int_D x^3-3xy^2\,{\rm d}x\,{\rm d}y$ using the mean value property.

I am asked to compute $$\int_D x^3-3xy^2\,{\rm d}x\,{\rm d}y,$$where $D = \{ (x,y) \mid (x+1)^2+y^2 \leq 9, \text{and }(x-1)^2+y^2 \geq 1 \}$. Granted, $u(x,y) = x^3-3xy^2$ is harmonic (it is the real ...
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0answers
17 views

Is there any connection between these two definitions of coercivity (ellipticity) in PDE and bilinear form?

In the field of partial differential equation, we say that the following operator \begin{equation} Lu=-\sum\limits_{i,j=1}^n (a^{ij}u_{x_i})_{x_j}+\sum\limits_{i=1}^n b^i u_{x_i}+cu \end{equation} is ...
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0answers
31 views

What is the solution to this PDE?

I have been trying to solve this PDE, but until now I have not got anything. I tried using Mathematica, but Mathematica didn't give me a solution. I know that some PDE's don't have a general solution, ...
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0answers
27 views

solving two dimensional diffusion advection equation.

I know that the solution to one dimensional diffusion advection equation is easy to obtain. However, was wondering if the same is true for two dimensional linear diffusion advection equation, i.e., ...
0
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1answer
20 views

How to solve second order ODE?

Let $u: \mathbb R^{d} \times \mathbb R \to \mathbb C$ be a function, and $f, g \in \mathcal{S}(\mathbb R^d)$ (Schwartz Space). We consider, the following initial value problem (IVP): $ ...
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1answer
20 views

Proof of property of unique nearest point equal to convex.

In the picture below, I really don't know how to use the compactness to get there will be a ball $C$ with maximal radius. I report this book for my classmate and teacher. I tell they the $r$ has a ...
0
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0answers
26 views

Finding eigenvectors of the Laplacian operator

I am running some finite-element software on Matlab that generates solutions to the diffusion equation over a punctured, rectangular domain. This is the same as the $k \times k$ matrix system ...
0
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1answer
30 views

Integration by parts in solving Poisson equation

In Partial Differential Equations by Evans, page 24, from (11) to (13), he changes Laplacian for x to Laplacian for y, but I don't know why this can be done. Can you tell me why? Thanks.
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1answer
42 views

An example of a sequence of functions that is not pointwise convergent

I've been searching for an example of a sequence $f_n(x)$ of functions that is not pointwise convergent, i.e.: $$\lim_{n\rightarrow \infty}\left | f_n(x) - f(x) \right | = 0$$ but I cannot find one. ...
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0answers
20 views

Can we apply an Itō formula to the solution of a SPDE?

Let $V\subset H\subset V^\ast$ be a Gelfand triple $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(W_t)_{t\ge 0}$ be a ...
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1answer
32 views

Convergence of a Sequence of functions Uniformly Mean Pointwise [duplicate]

For each of the following, give an example of a sequence of functions $f_n(x)$ that converges to f A. uniformly but not in the mean square sense. B. in the mean square sense but pointwise nowhere. ...
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0answers
54 views

Rephrase a multiparameter SDE indexed by time and space as an infinite dimensional SDE indexed by time

Let $\mathcal V_t\subseteq\mathbb R^3$ be the bounded space occupied by a closed particle system at $t\ge 0$ and $$[0,\infty)\ni t\mapsto X_t\in\mathcal V_t\tag 1$$ be the movement of a fixed particle ...
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0answers
14 views

What does it mean a solution of navier-stokes equations around another solution? How does he obtain that last equation?

Im reading an article where the autor uses some kind of galerkin approximation method. The method works like this. He finds the solution of the nonlinear equations on some space $H_m$ and then the ...
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8 views

mellin transform

Could you help me how (π) in the boundary condition where w(x,0)=2π transfer into 2T/s when we solve the equation ? It is really important.
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10 views

Separation of Variables PDE on Klein Gordon Equ

I have to use separation of variables on the 3-D Klein-Gordon equation: $ c^2 \bigtriangledown^2 \Psi - \frac{\partial ^2}{\partial t^2}\Psi + (\frac{mc^2}{\hbar})\Psi = 0 $ where $ \Psi (r,t) = ...
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0answers
29 views

How many boundary/intial conditions are needed in order to solve this set of PDEs?

i will split the question to 3 parts, where each part is a question in itself, and each part is more general than the previous one. 1.) for the following set of 5 1st order 1D PDEs, how many ...
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31 views

solution to a fourth order PDE

I need to solve this fourth order PDE. the domain is $\Omega = D_1 \otimes D_2$ where $D_i$ are unit discs $D_i = \{(x_i,y_i)| x_i^2 + y_i^2 < 1\}$. The variable $R(x_1,y_1,x_2,y_2)$ satisfies - ...
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0answers
11 views

Numerical scheme for first order partial differential equation.

Does anyone know some numerical methods for solving such equations? $$ \begin{cases} V_t + \langle V_x,\, F(t, x) \rangle = 0\\ V(0, x) = h(x) \end{cases}, \quad 0 \leqslant t \leqslant T, $$ where ...