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Questions tagged [pde]

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

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Proving maximum of $u$ on $\overline\Omega$ is achieved on $\partial \Omega$ using $Lv = \sum a_{jk}\partial_{jk} + \sum b_j(x)\partial_j$

Corollary 2.14: Suppose $\overline\Omega$ is compact. If $u$ is harmonic and real-valued on $\Omega$ and continuous on $\overline\Omega$, then the maximum value of $u$ on $\overline\Omega$ is ...
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15 views

Taking the surface integral $\int_{S_1(0)}y_jy_k \ d\sigma(y)$

I'm learning how to take surface integrals on the surface of spheres in $\mathbb{R}^n$. This question is related to Calculating the surface integral $\int_{S_1(0)}y_j \ d\sigma(y)$ where I try to ...
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1answer
16 views

Rewriting the Landau equation

I read an introduction to Landau equation: $$\frac{d \vert A\rvert^2}{d t}=2\sigma \lvert A \rvert^2-l \lvert A \rvert^4.$$ And I encountered a problem on the solution of the simple model equation. ...
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0answers
22 views

Differential equation Fourier series solution, every even term should disappear but doesn't in my solution

I'm solving $$\frac{\partial u}{\partial t} - \frac{\partial^2u}{\partial x^2} = 1$$ $$u(0,t) = u(L,t) = 20, \quad L= 2.$$ $$u(x,0) = 20.$$ I start by homogenizing the equation. Let $u = v + \hat{u}, ...
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0answers
12 views

Is this an hypoelliptic operator?

$$Lu = -\Delta u + \sum\limits_{i = 1}^na_iu(x_i)$$ Is this an elliptic operator? According to definition, it doesn't seem to fit in. Is this an hypoelliptic operator?
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1answer
22 views

Differential equation of all surfaces with z axis as axis of revolution

Find the Differential equation of all surfaces with z axis as axis of revolution? How to find the generic equation form. For cone with z axis and origin as vertex, equation is $x^2+y^2=z^2 \tan^2 \...
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0answers
16 views

Calculating the surface integral $\int_{S_1(0)}y_j \ d\sigma(y)$

I'm learning how to take surface integrals on the surface of spheres in $\mathbb{R}^n$. I'm asked to compute $$\int_{S_1(0)}y_j \ d\sigma(y)$$ According to the definition above, all I have to do ...
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0answers
29 views

$u$ is harmonic in $\mathbb{R}^N$ then if $T$ is an orthogonal transformation, $u\circ T$ is harmonic in $\mathbb{R}^N$

Show that if $u$ is harmonic in $\mathbb{R}^n$ and if $T:\mathbb{R}^n\to\mathbb{R}^n$ is an orthogonal transformation then $u\circ T$ is also harmonic in $\mathbb{R}^n$. Remember that an linear ...
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0answers
18 views

Assumption on traveling wave solutions of Fisher's equation

I have a question about Fisher's equation in a biology context. For example, in Fisher's equation $u_{t} = Du_{xx} + u(1-u)$, where $u$ is a density of cell, the logistic term explains that the ...
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0answers
14 views

Find all harmonic functions in $\mathbb{R}^n-\{0\}$ of the form $u(x) = f(|x|)$, where $f:]0,\infty[\to\mathbb{R}$ is of class $C^2$

Find all harmonic functions in $\mathbb{R}^n-\{0\}$ of the form $u(x) = f(|x|)$, where $f:]0,\infty[\to\mathbb{R}$ is of class $C^2$ I followed https://math.stackexchange.com/a/732878/166180 and ...
2
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1answer
35 views

Two Different Forms of Characteristic Equations?

I have the PDE $$\frac{\partial{u}}{\partial{t}} + u \frac{\partial{u}}{\partial{x}} = −2u$$ Written explicitly, I thought the characteristic equations were $\frac{\partial{t}}{\partial{r}} = 1$, $\...
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1answer
40 views

Finite difference method for non-linear PDE

I'm trying to numerically solve the following fairly simple non-linear PDE: \begin{equation} \frac{\partial \omega}{\partial t}=-V_g(\omega)\frac{\partial \omega}{\partial z} \end{equation} I've ...
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1answer
25 views

Help solving separation of variables problem

I am trying to solve the PDE separation of variables problem for the PDE $$\dfrac{-F''(x) + 2F'(x)}{F(x)} = \lambda$$ for the cases where $\lambda$ is negative, positive, and equal to 0. I am stuck ...
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1answer
36 views

Trouble in Checking PDE Boundary Solution Problem

I'm doing this problem with some other students, but it seems that our solution doesn't work? We have the partial differential equation $\dfrac{\partial^2 u}{\partial x^2} -2 \dfrac{\partial^2 u}{\...
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0answers
15 views

Lax-Wendroff finite volume scheme derivation

I'm trying to figure out how the finite volume version of Lax-Wendroff scheme is derived. Here is the PDE and Lax-Wendfroff scheme: $$u=\text{function of x,t}\\\hat{u}=\frac{1}{\Delta x}\int_{x_{i-1/...
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1answer
31 views

Second order evolution equations from a variational problem?

I'm not an expert in calculus of variations but suppose you have $$ E = \int_{\Omega} \mathcal{L}(x,y,y')d\Omega $$ The gradient of such function is given by $$ \nabla E = \frac{\partial \mathcal L}...
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1answer
29 views

Why the 'gradient of the diffeomorphism at a point in the surface' perpendicular to the surface at that point?

This question is related to these two questions of mine: Intuition or motivation for the definition of an hypersurface. What are we actually trying to define? and Understanding this very generic ...
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0answers
21 views

Question about distributional derivative

Let $D(\Omega)$ be the set of smooth, compactly supported functions on $\Omega \subset \mathbb{R}^n$ and let $D'(\Omega)$ be the distribution. Then, what is the meaning of the following sentence? $...
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10 views

Well posedness of Heterogeneous Helmholtz weak form

I'm trying to model the Heat profile in a stationary microwave oven. In order to obtain the electromagnetic fields, I have to solve the following problem with the Finite Element Method : Find $u$ a ...
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0answers
15 views

Hyperbolic Conservation Laws

Why the name Hyperbolic Conservation law for $u_t+f(u)_x=0$ Is there any parabolic or elliptic conservation laws?
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1answer
65 views

Where did $a$ and $b$ come from in this eigenvalue problem?

I have this eigenvalue problem: $$-x^2u_{xx} + 2xu_x - 2u = \lambda x^2 u$$ for $0 < x < 1$, with boundary conditions $u_x(0) = 0, u(1) = u_x(1)$. I'm trying to find a function $M(x)$, such ...
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0answers
37 views

Stokes equations and change of variable

Let $\Omega$ be domain of $\mathbb{R}^n$ and $\Phi : \Omega \to \Phi(\Omega)$ a deformation. Consider the Stokes equations written in the deformed configuration \begin{align} - 2\mu \operatorname{div}...
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46 views

Laplace equation in unit square

I would like to solve the $\triangle u(x,y) = 0$ in the unit square, with periodic BC when $x=0,1$ and Neumann condition when $y=0,1$ $$\partial_y u(x,0) = \begin{cases} A \quad &\text{for } 0\...
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0answers
18 views

An inequality about Besov norm: transfer the index $s$ to the multiplier $\langle \xi \rangle^s$

I try to prove the following inequality about Besov norm: Let $s,\sigma > 0$ and $r \geq 1$, show that there is a constant $C = C(d,s,\sigma,r)$ such that for any $u \in \mathcal{S}(\mathbb{R}^d)$...
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2answers
53 views

Finding Solutions of Sturm-Liouville Equation Satisfying Boundary Conditions and Checking Orthogonality of Eigenfunctions

I recently asked this question about converting the DE $$y'' + 2y' + (\lambda + 1)y = 0$$ to Sturm-Liouville form: $$\frac{d}{dx}\left( e^{2x} y' \right) + (\lambda e^{2x} + e^{2x})y = 0$$ I am ...
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1answer
35 views

Converting $y'' + 2y' + (\lambda + 1)y = 0$ to Sturm-Liouville form.

I am trying to convert the DE $$y'' + 2y' + (\lambda + 1)y = 0$$ to Sturm-Liouville form. I know that Sturm-Liouville form is $$\frac{\partial}{\partial{x}}\left( p(x) \frac{\partial{\phi}}{\...
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1answer
32 views

Understanding this proof about the principle of the maximum

I'm reading a proof from my teacher about the weak principle of the maximum that is, $$\max_{\overline{\Omega}} u = \max_{\partial\Omega} u$$ when $u$ is harmonic ($\Delta u = 0$) in $\Omega$ and $u\...
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2answers
61 views

Stuck with boundary value PDE problem

The last time I posted this I got many down votes and I don't know why. Maybe because I forgot to include my work? I got the PDE $\dfrac{\partial^2 u}{\partial x^2} -2 \dfrac{\partial^2 u}{\partial x ...
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0answers
26 views

Solve this pde $(D^{2}-3DD'+2D'^{2})z = (2+4x) e ^{\left(x+2y\right)}$ [on hold]

Being weak in math, I've tried and failed to solve this question. Any help is much appreciated.
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2answers
45 views

How to solve $(f_x)^2+(f_y)^2=4(1-f(x,y))(f(x,y))^2$?

Let $f:\Bbb R^2\to \Bbb R$ such that $$(f_x)^2+(f_y)^2=4\Big(1-f(x,y)\Big)\Big(f(x,y)\Big)^2,\qquad 0<f(x,y)<1.$$ then which functions satisfy the above property?
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Proof of existence for the $a$ coefficient in pdes like $u_t + a\left( t,x \right) u_x = 0$

Let $u_0 \in C^1$ and let $X \left( t,x \right) \in C^2$ s.t $X$ has an inverse w.r.t $x$ : there exists $Y\in C^1$ s.t $x = X \left( t, Y \left( t,x \right) \right) = Y \left( t, X \left( t,x \right)...
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47 views

Non linear differential equation in $\mathbb{R}^2$

I am looking for $F: \mathbb{R}^2 \to \mathbb{R}^2$ such that $$\nabla F\cdot F = (1+ xy) \binom{x}{y},$$ where for $F = \binom{f_1}{f_2},$ we denote $\nabla F = \binom{\nabla f_1^T}{\nabla f_2^T}.$ ...
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1answer
18 views

method of separation of variables for heat transfer with mixed conditions

$u_t=c^2 u_{xx}, \quad \forall \quad 0<x<L \quad t>0$ $u_x(L,t)=0, \quad u(0,t)=0$ $u(x,0)=f(x)$ Can anyone please tell me how to apply the boundary conditions to this problem and ...
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1answer
28 views

PDE - heat equaltion with \cos(x) [on hold]

$$u_t=u_{xx}+\cos(x)$$ $$u(x,0)=e^{2x} \forall x$$ My idea: First of all if we can remove $\cos(x)$ we will get heat equation. And the heat equation we know how to solve(separation of variables, ...
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0answers
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How to derive the solution of the wave equation in a finite interval with inhomogenous boundary conditions $u(0,t)=h(t),v(l,t)=k(t)$

We have the following problem: $$u_{tt}=c^2u_{xx}$$ $$u(x,0)=u_{t}(x,0)=0$$ $$u(0,t)=h(t),u(l,t)=k(t).$$ From the first two conditions one can deduce that $u(x,t)=0$ for $x>ct$ since the solution ...
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0answers
27 views

Finding Green's function of the PDE

I am writing as I would like to know method to obtain Green's function of the following PDE, which appears in the paper "ON THE FIRST PASSAGE TIME DENSITY OF A CONTINUOUS MARTINGALE OVER A MOVING ...
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1answer
14 views

Biharmonic function is the real part of $\bar{z}f(z) + g(z)$

First I showed that $\triangle u(x,y) = \partial_{xx} u+ \partial_{yy} u$ under the change of variable $z=x+iy, \bar z=x-iy$ we have $\triangle \tilde u(z,\bar {z}) = 4\partial_{\bar z}\partial_{{z}} \...
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0answers
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Solution of Laplace equation

Is it possible to solve $\qquad Uxx + Uyy=0$ and $\qquad U[0, y]=0, U[1,y]=0,U[x,0]=0,U[x<0.3,1]=10,U[x>0.7,1]=10$ and Neumann condition in $(0.3, 0.7)$?
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2answers
41 views

Transform the differential equation $yu_x(x, y) - xu_y(x, y) = xyu(x, y)$ by introducing new variables $s = x^2+y^2$ and $t = e^{-x^2/2}$

Transform the differential equation $$yu_x(x, y) - xu_y(x, y) = xyu(x, y)$$ by introducing new variables $s = x^2+y^2$ and $t = e^{-x^2/2}$. Then determine the general solution to the equation. ...
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0answers
45 views

Unclear inequality of L2 norms (Poisson equation for modeling flow)

I encountered a problem working through a paper about modeling flow with the use of the Poisson equation (source given below). There appears an inequality of L2 norms I don't understand so far. Your ...
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0answers
14 views

Green's function for heat equation with source

Suppose I have a PDE: $$\frac{\partial T}{\partial t}=\frac{\partial}{\partial x}\left(k(x)\frac{\partial T}{\partial x}\right)+Q(t)$$ with the following initial/boundary conditions: $T(0,x)=0$ and ...
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1answer
32 views

Show that problem is well defined for each time

We have the Cauchy problem of the equation $u_t+xu_x=xu, x \in \mathbb{R}, 0<t<\infty$ with some given smooth ($C^1$) function $g$ as initial value. I want to check if the problem is well ...
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0answers
25 views

Constructing Green's function for PDE from homogeneous solution [closed]

I have some complicated PDE and I know its homogeneous solution. Is there a way to construct Green's function from this homogeneous solution, akin to the Sturm-Liouville case for ODE (e.g. http://www....
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0answers
12 views

Optimal condition for viscosity solutions

My question is related to exercise 11 chapter 10 Evans PDE book second edition "(Continuation) Prove that the value function...". I tried to prove it by contradiction (without any success yet), ...
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0answers
29 views

Uniqueness of pde - How do we get $E(0)=0$ ?

Let $\phi \in C^1(\mathbb{R})$ and periodic. We consider the problem $$u_t=u_{xx}, \ x\in \mathbb{R}, 0<t<\infty$$ with initial data $\phi$. I have computed a formula for a solution $u$. I want ...
0
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0answers
30 views

How can the following Initial-boundary value problem be solved?

I have used Laplace Transformation and got the solution in Laplace domain, but it seems very complicated to be transformed back to the real-time domain. Is it possible to solve the problem with ...
2
votes
1answer
27 views

Existence and uniqueness of the solution of a PDE equation

How can we prove the uniqueness in $[0, +\infty)\times (0,1)$ of the solution of a PDE as the following: $\frac{\partial}{\partial t}v(t,x)=(\frac{1}{2}x-\frac{1}{4})\frac{\partial}{\partial x}v(t,x)+...
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0answers
21 views

Finding the Green's function for a 2D domain $D \not\equiv \mathbb{R^2}$

Consider the BVP $$\nabla^2 u=F \quad\text{in} \quad D,$$ $$u=f \quad \text{on} \quad \partial D.$$ Let $(x,y)$ be a fixed point in $D$ and let $(\xi,\eta)$ be a variable point. Let $r$ be the ...
1
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1answer
22 views

Infinite speed of propagation of the heat equation

Consider the heat equation $u_t = \Delta u$ on $\mathbb{R}^n$ with initial data $u(0, x) = f(x)$. Suppose $f$ is smooth and compactly supported. Do we necessarily have that $u$ has non-compact support ...
2
votes
2answers
53 views

Clarification of Qualitative Behaviour of BVP Solutions Example

I have the following example from my PDE textbook (Essential Partial Differential Equations by Griffiths, Dold, and Silvester): I don't understand how the authors found the solution $u(x)$. The ...