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Questions tagged [pde]

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

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Find $P(\tau_a<\infty|X_0=x)$ given $dX_t = rdt+dW_t$ using PDE method

Let $dX_t = rdt+dW_t$ with $r$ constant, and $X_0=x$. Let $\tau_a$ be the hitting time of $X_t$ hitting $a$ for $a>0$. I want to compute $P(\tau_a<\infty|X_0=x)$ where $x<a$ using PDE method. ...
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7 views

wave equation(doppler effect)

Solve the equation of wave equation with fixed source at the origin $$\Delta p + \omega^2p=-\delta(x)$$ The question asks for solution of the form $p(r) = C\frac{e^{i\lambda r}}{r}$, where $r = |x|$. ...
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20 views

Proof If {${T_{j}}$} converge to T then {${D^k T_{j}}$} converges to $D^k T$ for every n-tuple k

Proof that Differentitaion is a continuous operation in $D'$ : if {${T_{j}}$ }converge to T then {${D^k T_{j}}$} converges to $D^k T$ for every n-tuple k. My version to proof it. : Let $\Phi \in D(\...
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Basic 2D Finite Element Question: approximate solution in terms of linear basis function

I would like to confirm the following even though it is very basic. It would be tragic to continue the topic if one does the basics wrong. Therefore, the following is asked. We have the Laplace ...
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8 views

A estimate in Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials

Picture below is from the 20th page of Oh, Yong-Geun, Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials, Commun. Math. Phys. 121, No. 1, 11-33 (1989). ZBL0693....
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elliptic problem via Mountain pass theorem

I want to study the (weak) solutions of the following problem $$ (P) \ \ \begin{cases}-\Delta u = \vert u\vert^{p-1}u \ \ \ in \ \ \ \Omega \\ u=0\ \ \ in\ \ \ \delta\Omega\end{cases} $$ where $\Omega$...
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1answer
17 views

Unboundedness of the Sobolev norm of a sequence of functions.

Let $\{f_n\}$ be a sequence of functions that are continuous and lying in $H^k(\mathbb{R}^m)$. Assuming $k>\frac{m}{2}$, and if $f_n \to f$ pointwise, where $f\in H^k(\mathbb{R}^m)$, such that $f$ ...
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20 views

Solving Reynolds Equation using Finite Difference

I am attempting to apply finite difference to Reynold's Equation (with some simplifying assumptions). For example: $$\frac{\partial}{\partial x} \left( \frac{\rho h^3}{12\mu} \frac{\partial }{\...
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1answer
32 views

A system of PDEs

Consider the following system for $u(x,y)$, $v(x,y)$. $$2x^2yu_x + 5xy^2u_y + 2x^2y^2v_y + 5xyu + x = yu_y - x^2v_x + u - 2xv = 0 $$ Prove that it is equivalent to a second order semilinear PDE. ...
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11 views

Solution lacks continuous dependence?

$u^\epsilon = u^\epsilon(x,y),$ is a scalar function. The initial data for this PDE problem is as follows $\frac{\partial^2}{\partial x^2}u^\epsilon + \frac{\partial^2}{\partial y^2}u^\epsilon = 0; ...
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17 views

Definition of $C^{m,k/2}$-capacity of a point.

I hve come across the following notation and a new term $C^{m,k/2}$-capacity of a point. I'd appreciate some reference, where I can find the definition and relevant theory.
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Characteristic Cauchy problem for 1st order pde

When the initial curve is characteristic, the Cauchy problem may have infinitly many solutions or no solution at all. In the case of existence of infinity of solutions, how to construct them? Example:...
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1answer
45 views

Show f is continously differentiable and $f' =g$.

Let T and T' be regular distributions. T=T$_{f}$ and $T'=T_{g}$ . Assume both f and g are continous. Show that f is continuously differntiable and f'=g. I define $\langle T_{f},\Phi \rangle = $$\...
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32 views

Solding pdf using Green's function.

Use the Green’s function obtained in problem 2 to obtain the leading term in the asymptotic expansion for $ \phi(x) $ as $ \lvert x \rvert \to \infty $, where $ \phi $ satisfies $ \triangledown^2 \phi ...
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37 views

Solve $\kappa \frac{\partial^2 u}{\partial x^2} = \frac{\partial u }{\partial t}$ using Fourier sine or cosine

Solve the problem $$\kappa \frac{\partial^2 u}{\partial x^2} = \frac{\partial u }{\partial t}$$ given that $u(0,t) = u_0$, a constant and $u(x,0) = 0.$ You may also use $$\int_{0}^{\infty}\...
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1answer
33 views

Using Fourier transform to solve for pde

Consider the initial value problem for the wave equation $\frac{\partial^2}{\partial t^2}u(x,t) = c^2 \frac{\partial^2}{\partial x^2}u(x,t)$ , $-\infty < x < \infty$, t > 0 with initial ...
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1answer
28 views

Factorising functions out of partial derivatives

I have been doing that work that requires me to use the chain rule on second order partial derivatives to replace variables (x, y) with (u, v) where u and v are functions of x and y. My question is ...
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23 views

On the growth of a sub-harmonic function [on hold]

Let us assume that a positive valued function $f\in C^2(B)$, where $B=\{(x,y);x^2+y^2<1\}$, satisfies the following inequality $$ \Delta (\ln f)\geq f^2 \quad \text{on} \,\, B\subset \mathbb{R}^2 ...
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1answer
37 views

Solve the Distribution equation $xT= 1$

My Problem is to find all solutions of the distribution equation $xT =1$. I don't know how to solve it. The solutions should be $pv\frac1x+c$ , $c \in \mathbb{R}$. My idea was to create a function ...
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2answers
37 views

Solving nonlinear inhomogeneous first-order PDE using method of characteristics

I am trying to solve the partial differential equation $$\frac{\partial \rho}{\partial t}+(A+2B\rho)\frac{\partial \rho }{\partial x}=0$$ where $A$ and $B$ are constants. This is meant to model the ...
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9 views

A statement may use regularization theorem (sobolev space)

Here is the statement: suppose $u\in W^{m,p}(U)$(sobolev space), $supp{u}\subset U$, then $u_epsilon(x)=\frac{1}{n}\int{\alpha(\frac{x_1-x_1'}{\epsilon})……}\alpha(\frac{x_n-x_n'}{\epsilon})u(x')dx'$ ...
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1answer
22 views

The definition of generalized function (pde)

The definition of generalized function in my books is as follows: generalized function is linear continuous function on fundamental spaces, such as $\mathbb{D}(R^n),\varphi(R^n)$. But in ADAMS, the ...
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1answer
35 views

To solve a PDE using Separation of Variables.

Would it be possible for someone to guide me through this problem. The PDE is, $\Delta u =0$ We have the following boundary solutions: where a and b are real numbers u(0,y)=c1, u(a,y)=c1, u(x,0)=g(...
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1answer
34 views

Solve $\frac{\partial w}{\partial t} +t\frac{\partial w}{\partial x}=1$ … [on hold]

Solve this PDE using the characteristic form $\begin{equation} \frac{\partial w}{\partial t} +t\frac{\partial w}{\partial x}=1 \\ w(x,0)=cos(x) \end{equation}$ My attempt We know $w(t)=w(x(t),t)$ ...
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1answer
57 views

Solve this PDE $\frac{\partial w}{\partial t} +x\frac{\partial w}{\partial x}=1$

Solve this PDE using the characteristic form $\begin{equation} \frac{\partial w}{\partial t} +x\frac{\partial w}{\partial x}=1 \\ w(x,0)=f(x) \end{equation}$ My attempt Let's go to rewrite the PDE. ...
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0answers
17 views

Minty-Browder theorem

Can you please give a reference where I can get the exact proof of the Minty-Browder theorem stated as follows: Observe that I below the operator $T$ is demicontinuous and need not be continuous. For ...
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31 views

Uniqueness of charge distribution

Some exercises in general physics ask for students to find specific charge distributions on the boundaries of given conductors in various of situations. Usual answers goes like follows; firstly using ...
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1answer
26 views

Solve this PDE using the characteristic form

Solve this PDE using the characteristic form $\begin{equation} \frac{\partial w}{\partial t} -\frac{\partial w}{\partial x}=-w \\ w(0,t)=4e^{-3t} \end{equation}$ My attempt We know $w(t)=w(x(t),t)$ ...
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1answer
40 views

Determine the solution of this PDE using the characteristic method (parametric form).

Determine the solution of this EDP using the characteristic method (parametric form). $\begin{equation} \frac{\partial w}{\partial t}-\frac{\partial{w}}{\partial x}=0 \\ w(x,0)=e^{2x} \,\,\,\, \text{...
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1answer
26 views

a theorem about regularization of function

Theorem: Let $u(x)$ is locally integrable function on $R^n$, $\phi= e^{\frac{1}{|x|^2-1}}, when |x|<1;\phi= 0, when |x|\geq 1 $ $\alpha(x)=\frac{1}{C}\phi$ where $C=\int_{R^n}\phi(x)dx$ and $\...
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1answer
35 views

What is the definition of embedding? (pde)

I see the definition in book which tells that: If we say linear space $A$ is embedding in $B$, it means $A$ is contained in $B$ and the mapping $A\to B$ is continuous. But I also see the definition ...
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0answers
32 views

How to get $u\in C^0([0,T];L^2(\Omega))$?

For a bounded smooth domain $\Omega$ in $\mathbb{R}^N$, We know that if $u\in L^2(0,T;H_0^1(\Omega)$ and $\partial_tu\in L^2(0,T;H^{-1}(\Omega))$ then $u\in C^0([0,T];L^2(\Omega))$. However, now we ...
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1answer
32 views

DGL: variational problem in $H^1$

I've given $f \in L^2(\Omega), g \in H^1(\Omega)$. I want to find $u \in H^1(\Omega)$ such that $$ -div (A \nabla u ) + <b, \nabla u> + cu = f$$ in $\Omega$ $$u = g$$ on $\Gamma$. In order to ...
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2answers
42 views

Forward Euler PDE (grid method) misunderstanding - Is the question missing a detail

We are interested in solving the advection equation $u_t = u_x$ where $0\leq x < 1$, $t \geq 0$ with periodic boundary conditions and $u(x,0) = f(x),f(x) = f(x+1)$ In the grid $x_0,x_1,\dots x_N$, ...
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0answers
55 views

existence and uniqueness for semilinear heat equation

Consider a semilinear heat equation on $[0,1]$ $$ \frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}+b(t,x,u(t,x)). $$ We assume here the periodic boundary conditions, that is $u(0,t)=u(1,...
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0answers
9 views

Truncation function

Define the truncation function $ T_k(s)=s$, if $|s|\leq k$ and $T_k(s)=k\frac{s}{|s|}$, if $|s|\geq k$. Let $w\in A_p$ (the class of Muckenhoupt weight, https://en.wikipedia.org/wiki/...
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0answers
22 views

Solve the following wave initial value problem by using the D'Alembert's formula

Solve the following wave initial value problem by using the D'Alembert's formula: $$\frac{\partial^2 u}{\partial t^2}−4\frac{\partial^2 u}{\partial x^2}= 0 \text{ with } u(0, x) = 1, \frac{\partial u}...
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1answer
34 views

Game Theory Reccomendation, Mean Field Theory

I'm about to do a sort of reading course with a mathematics professor wherein I read and teach him about Game Theory. He claims not to know Game Theory. After that, we aim to read about Mean Field ...
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26 views

Method of characteristics - traffic flow

Solve subject to the initial conditions $$ \rho(x,0) = f(x) \tag{1} $$ $$ \frac{\partial \rho}{\partial t} + 3x\frac{\partial \rho}{\partial x} = 4 \tag{2} $$ then I believe I should have $$ \...
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0answers
52 views

Fourier transform of equation $f_t + \nabla \cdot (gf)=0$

I'm trying to take the Fourier transform of this equation $f_t + \nabla \cdot (gf)=0$, because I want to know whether I can get kernel of this equation. Here we assume $\nabla \cdot g=0$. For example, ...
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0answers
43 views

Prove a Gagliardo-Nirenberg inequality in $W^{2,p}$

$\textbf{Problem}$ Integrate by parts to prove: \begin{align*} \int_{\Omega} \vert Du \vert^p dx \leq C\Vert u \Vert_{L^p(\Omega)}^{1/2}\Vert D^2u\Vert_{L^p(\Omega)}^{1/2} \end{align*} for $2\leq ...
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1answer
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A specific problem on : Does bounding of the Sobolev norm can cause bounding of a higher derivative?

Let $f \in H^k(\mathbb{R}^m)$, $k>\frac{m}{2}$. Given any $f$, such that $\|f\|_{H^k(\mathbb{R}^m)}<K$ , and any $\phi \in C^{\infty}(\mathbb{R}^m)\cap H^k(\mathbb{R}^m)$, such that $\|\phi\|_{...
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1answer
12 views

Weak lower semicontinuity property of a bounded, coercive and linear operator

Let $A\colon V \to V^*$ be a bounded linear coercive operator on a Hilbert space $V$. Does it follow that for if $u_n \rightharpoonup u$ in $V$ (weak convergence) then $$\langle Au, u \rangle \leq \...
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24 views

Integral of Laplacian on complete non-compact manifold

Let $(M,g)$ be a complete non-compact manifold, $\Delta$ denote the Laplacian operator on smooth functions. Q Is this true that $\int_M(\Delta f\cdot f)dvol_M\geq0$ for any function $f\in L^2(M)$ ...
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0answers
12 views

Minimizer of certain functional is strong $\Lambda$-minimizer

I'm interested in the quantitative isoperimetric inequality. I am currently reading https://arxiv.org/pdf/1007.3899.pdf and I have some questions regarding lemma 3.5 which states a certain situation ...
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0answers
23 views

Solutions of $\partial_t f =-\Delta^2 f$

I'm looking for references (or direct solutions :) ) of how to solve the following partial differential equation : $\partial_t f =-\Delta^2 f$ , with $\Delta f$ the Laplacian of $f$. I'm interested ...
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34 views

How to find analytical solution for $E(t)$ for the heat equation

For the given PDE system how do I find the total energy without having to calculate the complete solution? I've tried equating $E(0)$ and $E(\infty)$ using the steady-state solution, but to no avail.
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1answer
27 views

How to reduce a second-order differential equation given a substitution

Given the general partial differential equation $$a \frac{\partial^2u}{\partial x^2} + 2b \frac{\partial^2 u}{\partial x \partial y} + c\frac{\partial^2u}{\partial y^2} = 0,$$ if $ac - b^2 > 0$, ...
3
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0answers
42 views

Differential invariants of PDEs

I have been reading 'Symmetry and Integration Methods for Differential Equations' by Bluman and Anco. I'm trying to make sense of differential invariants and PDEs... It is confusing. There is a ...
3
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1answer
29 views

A inequality between $||u||_{p}$ and $||\gamma (u)||_{p, \partial \Omega}$, where $\gamma$ is the Trace Operator?

Does someone know any inequality between $||u||_{p}$ and $||\gamma (u)||_{p, \partial \Omega}$, where $\gamma$ is the Trace Operator? I need to find something like $||u||_{p}\leq C||\gamma (u)||_{p, \...