Questions tagged [partial-differential-equations]

Questions on partial (as opposed to ordinary) differential equations - equations involving partial derivatives of one or more dependent variables with respect to more than one independent variables.

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11 views

A PDE with mixed derivative

How should one go about solving an equation of the form $$\frac{\partial^2 u}{\partial x \partial y} + x \frac{\partial u}{\partial y} = y$$ Do I need to use characteristics, or integrate first?
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An problem on weak convergence: weak convergence of a sequence implies the weak convergence of the square of this sequence?

Suppose that $\{u_m\}_m\subset H^1(\mathbb{R}^3)$ and $u_m\rightharpoonup u$ weakly in $H^1(\mathbb{R}^3)$. From the classic results, I know that there exists a subsequence such that $u_m\...
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Problem 17 Chapter 4. Evans PDE 2nd edition

Let $n=1$ and suppose that $u^\varepsilon$ solves the problem $$\left\{\begin{array}{ll} -(a(\frac{x}{\varepsilon})u_x^\varepsilon)_x=f\quad\mathrm{in}\,(0,1)\\u^\varepsilon(0)=u^\varepsilon(1)=0,\end{...
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1answer
24 views

How does one solve a 4th order PDE using Fourier transforms?

The example reads Use Fourier Transforms to solve $$ \frac{\partial u} {\partial t} + \frac{\partial^4 u}{\partial x^4} + u = g(x) $$ Where $u(t=0)= f(x)$
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20 views

Solve the problem $u_t = c^2 u_{xx} + g(x,t)$, $(x,t) \in (0,L) \times (0,\infty)$

Studying some notes on partial differential equations about the heat equation, I came across the following problem and found it interesting because of its general form: \begin{cases} u_t = c^2 u_{xx} +...
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Partial derivative involving integration to find critical values

Suppose, we have a function $X(t)=\dfrac{exp[b(t)^Tc+b(t)^TAu]}{\int_a^bexp[b(s)^Tc+b(s)^TAu]ds}$ s or t is variable belong to (a,b). Practically we take finite points in this range. c and u are p* 1 ...
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29 views

Limit of perturbation as $\epsilon \to 0$

We are looking at pde $\epsilon (u_{yy}+u_{xx}) + u_x =0$ on the unit 2 dimensional disk with $u(x,y)=f(x,y)$ on the boundary (unit circle). If $u$ is a solution then what is the limit of $u(x,y)$ as ...
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13 views

Reverse of Hausdorff Young Inequality

The Hausdorff Young Inequality gives us that $$\|\hat f\|_{L^q(\mathbb{R}^n)} \leq \| f\|_{L^p(\mathbb{R}^n)}$$ if $1 \leq p \leq 2$ and $\frac{1}{q} + \frac{1}{p} = 1$, but does the converse also ...
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Hi, What are the differences and similarities in the topics of Analytical Number Theory and Mathematical Physics. If It is wrong space, pardon i'm new

What are the differences and similarities in the topics of Analytical Number Theory and Mathematical Physics (field of theoretical physics). I want to do postgraduate studies in [...
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22 views

The heat equation with affine initial data.

Suppose that $u \in C(\mathbb{R} \times [0, \infty)) \cap C^2(\mathbb{R} \times (0,\infty))$ $$u_{t}=u_{xx}, \quad \text{in} \quad \mathbb{R} \times (0, \infty),$$ $$ u(x,0)=Ax+B, \quad x \in \...
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Boundary values for system satisfying Laplace-Beltrami equation

Consider an analytic function $F=f(x,y)+i h(x,y)$, where the real and imaginary components satisfy Laplace’s equation in $(x,y)$, i.e. $$f_{xx}+f_{yy} =h_{xx}+h_{yy} =0.$$ Now, let’s say $x=x(a,b)$ ...
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37 views

Solving a heat equation

I am trying to solve this by separating the temporary variable $t$ from the spatial variables in order to get an eigenvalue problem for $\Delta$ and an ODE in $t$ but I have not been able to achieve ...
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Determining nature of solutions for given Cauchy Data

While solving a Quasi-Linear PDE, I found that the general solution is given by $$u(x,y)=x+g(x^3-y^3)$$, where $g$ is an arbitrary function. Now, how can we determine the nature of solution given a ...
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Determine the support of $x\mapsto\int_{\partial B_t(x)}(y-x)\cdot\nabla f(y)\:{\rm d}y$, when $f$ is compactly supported

Let $f\in C^2(\mathbb R^3)$ with $\operatorname{supp}f\subseteq\overline B_{r_0}(0)$ for some $r_0>0$ and$^1$ $$g(t,x):=\int_{\partial B_t(x)}(y-x)\cdot\nabla f(y)\:\sigma_{\partial B_t(x)}({\rm d}...
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36 views

Solving parabolic pde with piecewise-constant coefficients with Feynman-Kac

Consider the partial differential equation: $$\frac{\partial u}{\partial t}(x,t) + \mu(x,t) \frac{\partial u}{\partial x}(x,t) + \frac{1}{2} \sigma^2(x,t) \frac{\partial^2 u}{\partial x^2}(x,t) - V(x,...
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39 views

How 2nd derivative is determined? Question is from string theory (A string model).

[enter image description here][1] This question is related to Mathematical modeling. I have many times to get second derivative but failed to drive. I would remain thankful for resolving my issue. [1]:...
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17 views

Identity in surface integral over $S_R(0)$

I'm currently taking a course in PDE and one of the questions from the book we are following asks us to prove the following identity: Let $R>0$ and $f: S_{R}(0) \rightarrow \mathbb{R}$ a continuous ...
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22 views

derive a ode from minimal surface eqaution

I want to derive a ode from minimal surface equation, so I define $u(x)=f(r)$ where $r=|x|$ then we have $\frac{\partial r}{\partial x_i}=\frac{x_i}{r}$ $\frac{\partial u}{\partial x_i}=f^\prime(r)\...
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Proving that solution one and 2 are equivalent of given PDE

I found the solution of the given PDE by two methods but I want to check whehter both the solutions are the same or not. can some help me?
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Homogeneous semi-infinite wave equation with inhomogeneous boundary and initial conditions

On page 115 of the third edition of Logan's Applied Partial Differential Equations, the reader is asked to solve the following: $$u_{tt} = u_{xx}, \,\,\: x,t>0$$ $$ u(0,t) = \sin t ,\,\,\: t\ge 0 $$...
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Parabolic linear PDE with non-homogenous Neumann boundary condition.

Let $\Omega \subset \mathbb{R}^n$ be an open bounded set with lipschitz boundary, $T>0$, $f \in L^2(0,T;L^2(\Omega))$, $g \in L^2(0,T;H^{\frac{1}{2}}(\Omega))$ and $h \in H^1(\Omega).$ I'm trying ...
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26 views

Hamiltonian-Jacobi-Bellman Equation for Mayer Problem

All papers I have read define the Hamilton-Jacobi-Bellman equation starting from the Bolza Problem like so: $$\min_u J=\phi(x(t_f))+\int_{t_0}^{t_f} L(x(t),u(t),t)dt$$ subjected to $$ \frac{dx(t)}{dt} ...
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1answer
11 views

How to handle boundary conditions that are piece wise functions

Consider the PDE $$u_{t} = k u_{xx}$$ subject to the conditions $$u(0, t) = 0, u(L, t) = 0$$ and $$u(x, 0) = \begin{cases} 1, 0 < x < \frac{L}{2} \\ 0, \frac{L}{2} < x < L \end{cases}$$ ...
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1answer
35 views

Problem on partial derivative

I have some confusion here. The question is $u=x^n\:f\left(\frac{x}{y}\right)$ . I need to find the $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$ and shows that $x\frac{\partial ...
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23 views

Compact embedding of sobolev space in Lp

I'm working through an exercise that asks if the embeddings $$\dot{H}(\mathbb{R}^3) \subset L^6(\mathbb{R}^3)$$ and $$L^6(B(0,R)) \subset \dot{H}(B(0,R))$$ are compact using the function $\phi(x) = \...
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1answer
35 views

Quasi Linear PDE solution using characteristics

The question in the textbook reads: Consider the PDE: $$\sin(t)\frac{\partial u}{\partial x} + x \frac{\partial u}{\partial t} = \frac{x}{t}$$ subject to $u = x$ on $t = 4$. My characteristics can't ...
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1answer
38 views

PDE for a damped oscillator

I'm studying a PDE that describes a damped oscillator, given by: $$u_t + uu_x = -\gamma u $$ with initial conditions $u(x,0)=f(x)$. I derived the characteristics as: $$\frac{dx}{dt} = u$$ and $$\frac{...
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15 views

Momentum integral in axisymmetric turbulent flow

Context: I am trying to derive an equation given in a Journal of Fluid Mechanics paper (2.2). It deals with the analysis of an axisymmetric turbulent wake where cylindrical coordinate system has been ...
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1answer
25 views

Solve differential equation by separation of variables

Using separation of variables find solutions to the equation $3u_{xy}=u$. My attempt: $u(x,y)=X(x)Y(y) \Rightarrow u_{xy}=X'(x)Y'(y) \Rightarrow 3X'Y'=XY$ From what I've seen and read, it's necessary ...
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1answer
45 views

Circularly Symmetric Heat Equation

The circularly symmetric heat equation is $\frac{\partial{u}}{\partial{t}} = k\frac{1}{r}\frac{\partial}{\partial{r}}(r\frac{\partial{u}}{\partial{r}})$ When we have the boundary conditions being $u(a,...
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1answer
47 views

ODE solution is monotonic with respect to initial value?

Consider the ODE $$\frac{dy}{dt}=f(y,t), \quad y(0)=a,$$ where we may assume that $f$ is Lipschitz continuous and there is a unique solution $y(t)$ on $[0,2]$. My questions is following: Is $y(1)$ a ...
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1answer
38 views

Complex integral over surface of sphere

How do we go about computing the integral $$\int_{|x|=t} \frac{e^{ikx}}{|x|} d\sigma$$ where $d\sigma$ is the measure of the sphere of radius $t$ in $\mathbb{R}^3$? My approach thus far has been to ...
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25 views

how to find the range of fractional Laplace operator in fractional Sobolev space?

Let fractional Sobolev space $W^{s,p}(I)=\{f\in L^p(I):(x,y)\to\frac{f(x)-f(y)}{|x-y|^{\frac1{p}+s}}\in L^p(I)\times L^p(I)\}, s,\alpha\in \mathbb{R}, s\in (0,1),$ with norm $$\|f\|_{W^{s,p}}=\|f\|_{L^...
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31 views

Physical intuition for the representation formula for harmonic functions

Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ with $C^{1}$ boundary, $u$ is harmonic in $\Omega$ and belongs to $C^{1}(\bar{\Omega})$. Then by Green's formula we have for every $x \in \Omega$...
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1answer
45 views

Two time derivatives of kinetic energy of fluid

Suppose $D$ is a smooth domain, $\rho > 0$ is fluid density (constant) and $u \in C^1([0,1];D)$ is the fluid velocity. Let $K(t) = \frac{1}{2}\int_D \rho \vert u \vert^2 dV,~0\le t \le 1,$ be the ...
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22 views

Wave Function of a particle in a cylindrical tube

I'm trying to figure out how to write the wave function and how to determine the time-independent states of a particle in an open-ended cylindrical tube with a set radius and length. I am currently ...
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1answer
55 views

Telegraph Equation separation of variables

I've been trying to solve the telegraph equation by the method of separation of variables. The equation is given by: \begin{align*} u_{tt}+au_t+bu&=c^2u_{xx}, \quad 0<x<l, \quad t>0\\ u(x,...
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Weighted Poincare ineqality

I am trying to find a weighted Poincare inequality in the following form: $$\int_D w|\nabla u|^p dx \ge C \int_D w |u|^p dx$$ where $1<p<\infty $, $u\in C_0^{\infty}(D)$ and the weight $w\in ...
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1answer
25 views

Reference and questions about evolution equations

I want to learn about evolution equations, that is equations: $$ u'(t)=A(t)u(t)+f(t)$$ where $A(t)$ are unbounded operators on a Banach Space $X$. I'm interesting also in the case of $A(t)$ being ...
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23 views

Question on the explicit solution of Laplace equation

I am just learning about the fundamental and the general solution of the Laplace equation $$\triangle u = 0.$$ The main Theorem seems to be, that with the fundamental solution $$\Phi(x) = \frac{1}{d(d-...
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55 views

Is the following wave equation problem well posed?

Let $\Phi(x,y,z,t)$ be the velocity potential and $\eta(x,y,t)$ be the free surface elevation induced by an impulsive surface source $\delta_x\delta_y\delta'_t$ with support on the boundary $z=0$ at ...
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1answer
49 views

Fourier sine transform problem

I would just like to check that the following is correct. I am given the Fourier equation: $$\frac{\partial\phi}{\partial t}=\alpha\frac{\partial^2\phi}{\partial x^2}\,\,\,\,\,\,\,x,t\ge0$$ subject to:...
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16 views

if $f(s)$ is derivative twice and $u(x,y)=f(ax+by)$ prove $u$ is correct in $u_{xx}u_{yy}-(u_{xy})^2=0$ [closed]

I don't know if it needs any actual PDE knowledge or not but I can't understand what to do here.
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33 views

$u(x, t)$ is a solution for $u_{t} = u_{xx} - ru$ if $E(t) = 1/2 \int_0^L u^2(x,t) \ dx$. Prove $\lim\limits_{t \rightarrow \infty } E(t) = 0$ [closed]

$u(x,t)$ is a solution for $$u_{t} = u_{xx} - ru, \ \ \ \text{for} \ \ 0 < x < L , \ \ \ t > 0,$$ and $$u(0,t)=0 \ \ \ \ \ \text{and} \ \ \ \ \ u(L, t) = 0 .$$ If $$E(t)= 1/2 \int_0^L u^2(x, ...
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18 views

Explicit formula of the solution of $u_{tt}=au_{xx}$ and for which values of $a$ is this a “wave equation”?

Let $a\in\mathbb R\setminus{0}$ and $u\in C^2((0,\infty)\times\mathbb R)$ be a solution of $$u_{tt}=au_{xx}\tag1.$$ I'm trying to find an explicit formula of $u$ using the ansatz $$u(t,x)=v(t)w(x);\...
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1answer
20 views

Question on approximation by smooth functions in Sobolev Space

Quite a small question. In Evan's PDEs he states the following theorem Assume $U$ is bounded and $\partial U$ is $C^1$. Suppose $u\in W^{k,p}(U)$ for some $1\leq p<\infty$. Then there exist ...
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1answer
33 views

Physical intuition behind no extremum of a function

During many of the courses (my background is fluid dynamics), I have seen that if a function $\phi(x,y)$ is smooth and continuous and satisfies a diffusion/Laplace equation of the form: $$\frac{\...
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14 views

Strong convergence and pointwisly convergent [closed]

I stumbled upon this in my course. Question **(fn) is a sequence in L2 space, such that fn converges pointwisly to f, where f belongs to L2. Study the strong convergence of (fn) in L2.?? ** Answer **...
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1answer
62 views

Weak solution of nonlinear PDE

Let $\Omega \subset \mathbb{R}^n$ is a bounded domain with smooth boundary.Prove there exists a positive constant $\epsilon_0$ so that for all real numbers $\epsilon<\epsilon_0$ and $f\in L^2(\...

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