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

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Showing that the map that takes $u_0$ to solution $u(t)$ is self-adjoint

Let $u$ and $v$ be the solution of the heat equation $$w'(t) - \Delta w(t) =0$$ with initial data $u_0$ and $v_0$ respectively, and with either homogeneous Dirichlet or Neumann BCs on a bounded domain ...
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1answer
22 views

Find the general solution to this PDE

I'm asked to find the general solution of $$au_{xx}+bu_{xy}=0$$ With $u=u(x,y)$ and $a$, $b$ real constants. I'm just starting with PDE's, haven't seen any resolution technique except for basic ...
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14 views

Eigenvalue equation and separable solutions

Recently, studying Quantum Mechanics I found a doubt regarding separable solutions and eigenvalue equations for differential operators. Suppose we are considering some space $\mathcal{H}$ of functions ...
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0answers
17 views

Does this inequality imply a uniform $L^\infty$ bound?

Suppose I have the estimate for $t > 0$ $$\lVert u(t) \rVert_{L^\infty(\Omega)} \leq Ct^{-1}\lVert u_0 \rVert_{L^1(\Omega)}$$ for the solution $u$ of a parabolic PDE with initial data $u_0$ on a ...
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1answer
27 views

Second order PDE with initial condition

How do I solve the equation $\frac{\partial^2}{\partial x\partial t} u(x,t)=\frac{\partial^2}{\partial x^2} u(x,t)$ with the initial condition $u(x,t=0)=\sqrt{\frac{\pi}{2}}\exp(-|x|)$ ? The solution ...
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0answers
18 views

What does “loss of regularity” mean?

I have seen a lot the phrase "loss of regularity" in references regarding PDE. (For instance, there are questions like "do solutions of 3D Navier-Stokes equations lose regularity or not?") Could ...
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3answers
41 views

How to calculate $f(x, y, z)$ given $d f = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy +\frac{\partial f}{\partial z} dz$

On a manifold with local coordinates $(x_1, \ldots, x_n)$ I have a closed 1-form $\omega$ for which $d \omega = 0$ holds. This means There must be a function $f(x_1, \ldots x_n)$ for which $d f = ...
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1answer
32 views

An equivalent theorem for Sobolev spaces in infinite dimensions

There is a proposition which states: Let $f\in W^{1}(U)$ be real valued and $h\in C^{1}(\mathbb{R})$ with $h'\in C_{b}(\mathbb{R})$. We then have $h\circ f\in W^{1}(U)$ and $$\partial_{j}(h\circ ...
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0answers
35 views

Heat equation on a torus [on hold]

Can someone please help me to solve the following problem about the existance of a unique solution of the heat equation:here is the problem
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19 views

Nonlinear second order PDE

I need to solve the following PDE (which is a maximized Hamilton-Jacobi-Bellman equation) \begin{align} rV(\theta_1,\theta_2) = \frac{(\theta_1^\rho + ...
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11 views

Quasi-linear partial differential equations. Solving them.

This is what I have as a quasi-linear partial differential equation:$$u(x_1,...,x_n), \ \ \ \ \sum_{i=1}^{n}A_i(X,u) \frac{\partial u}{\partial x_i}=A_{n+1}(X,u) \ \ \ (1)$$ Then it says let ...
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0answers
16 views

Differentiability of PDE with respect to parameters

Consider a linear partial differential equation $$ (L u)(x)=f(x) \quad \forall x \text{ in } \Omega\\ u=g \quad\text{on }\partial\Omega. $$ Assuming that $f$ and $L$ depend on a parameter ...
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11 views

Study of heat equation on a torus [on hold]

Good morning, can someone please help me to find a web site where there is the study of heat equation on a torus. thanks in advance
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0answers
19 views

Methods for first order PDEs in higher dimensions

What are the possible known methods for solving first order PDEs in higher dimensions? Is there anything else besides the method of characteristic curves? In particular, I have four first order, ...
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0answers
20 views

PDE that are unchanged under all axis-rotations

It is exactly the same question as Partial Differential Equation about Rotation question. Sadly, I gain nothing useful from the above post. Or I should say I am not familiar with the terms in the ...
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0answers
12 views

A Poincare type inequality

How to prove: $$||u||_{L^1({\Omega})}\leq c(\Omega)(||u||_{L^1(\partial\Omega)}+||Du||_{L^2(\Omega)})$$ Suppose u is smooth enough and $\Omega$ is a bounded domain with smooth boundary.
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1answer
27 views

PDE Proof that a linear combination of 2 solutions is also a solution [on hold]

Can someone please help? I've been trying to figure this for a few days now. Consider the first order PDE: $au_t + bu_x$ = 0, where a and b are constants. Show that if $u_1$ and $u_2$ are solutions ...
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16 views

How to integrate to solve a PDE with mixed partials in the integrand

Problem Statement: Determine the equlibrium temperature distribution inside a circular annulus $r_1\leq r \leq r_2$. If the outer radius is at temperature $T_2$ and inner radius at temp $T_1$. So ...
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1answer
26 views

Odd and Even Fourier Series Extension of $f(x)=x$ on $[0,\pi]$

I'm confused on finding the odd and even extensions of $f(x) = x$ on $[0,\pi]$. I know the general forms and how to find the co-efficients, but for the sin series, $f(0)$ =/= $f(\pi)$, so then I only ...
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1answer
21 views

Online course for numerical methods/analysis of PDEs

Could anybody recommend an online course for implementing numerical methods to solve PDEs which can supplement reading? This is with a view to writing an implementation to solve the Monge-Ampere ...
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1answer
12 views

Non-homogeneous First Order PDE Method

Just to be upfront, this is a homework question, but I'm just stuck on one particular part and I want to see what I'm doing wrong. The PDE in question is the following: ...
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0answers
18 views

How to find the general solution of this PDE

I've been assigned the following partial differential equation as an introductory exercise: $$u_{xy}+au_x+bu_y+abu=0$$ Where $a,b$ are constants and $u=u(x,y)$. Having seen barely anything except ...
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1answer
30 views

Partial differential equation 5 [on hold]

Which change of variable should I do to solve this PDE? $u_{xy}(x, y) + au_x(x, y) + bu_y(x, y) + abu(x, y) = 0$
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1answer
50 views

Compare analytic model with numerical, mass spring system.

So I'm trying to solve a problem here and I have been working on it all day, clearly i'm in need of some guidance. I have a rod of length $L$ and cross section area $A$, Young's modulus $E$ and ...
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0answers
26 views

The eigenfunction of $-\Delta$.

Let $u_k$ be eigenfunctions for operator $-\Delta$ over domain $\Omega\subset \mathbb R^2$, open bounded, smooth boundary. Then, we know that $u_k$ forms a basis for $L^2$. Let $u\in H_0^1(\Omega)$ be ...
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26 views

method of characteristics for non-linear PDE

I'm trying to solve the PDE $u_x^2-u_y^2=8u$ with initial conditions $u(x,x)=f(x)$. I have that $F(x,y,u,p,q)=p^2-q^2-8u$, with $p=u_x, q=u_y$, and then \begin{equation*} \begin{array}{ll} ...
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1answer
28 views

System of differential equation (Matrix form)

I'm trying to solve this system $$ M\ddot{X}(t) = KX(t) $$ where M is a known diagonal matrix and K is a symmetrical known matrix. I'm asked to do the ansatz $Y(t) = M^{1/2}X(t)$ where $M^{1/2} = ...
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0answers
17 views

Step 2 of Strichartz's Estimate Proof.

I am stuck with Step 2 of the Strichartz's estimate. My question is actually a continuation of a topic which has been raised some time ago and it could be seen here Technical question about Strichartz ...
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0answers
16 views

Differentiation with integration region depending on $x$ to solve for decreasing energy of wave equation

I want to show that for the general wave equation $u_{tt} - \nabla \cdot (c^2\nabla u) + qu = 0, \quad u(x, 0) = \phi(x), \quad u_t(x, 0) = \phi(x)$ we have $$ E(t) = \int_{|x-x_0| < R_0 - c_2t} ...
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1answer
32 views

Combining two results from partial integration

I have a set of two PDEs: $$\partial_{\tau}\theta+\partial_{\eta}\psi=0$$ $$\partial_{\tau}\psi=-\partial_{\eta}\theta+\alpha\partial_{\eta}^{2}\psi$$ These can be combined into a wave equation of ...
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0answers
18 views

Heat equation with mixed boundary conditions

I am trying to solve the following problem $$\left\{\begin{matrix}\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},& 0<x<1,t>0,\\ u(0,t)=\frac{\partial u}{\partial ...
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1answer
13 views

Taking derivative of energy of wave equation

Consider the variable coefficient, real valued wave equation $$ u_{tt} - \nabla \cdot (c^2 \nabla u) + qu = 0, \quad u(x,0) = \phi(x), \quad u_t(x, 0) = \phi(x), $$ where $c, q \geq 0$ depend only on ...
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1answer
23 views

Apply Periodic Boundary to PDE (Fourier Transform)

Use Fourier Transform to solve the BVP: \begin{cases} u_t + a u_x - b u_{xx} = 0, & \mbox{for } x \in [-1,1] \\ u(x,0) = f(x) \\ u(x+2,t) = u(x,t) \end{cases} I solved the problem (attached); ...
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1answer
14 views

uniqueness of heat equations and the squared integrable assumption

I am looking at the classical proof of uniqueness for the heat equation in Evans. Clearly, we differentiate under the integral sign of the square of $w$. A very basic question is, why are the ...
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27 views

Poincare Inequality in n-dimensions

I am trying to prove the Poincare Inequality on a n-dimensional box. That is a domain $ \Omega = (0,1)^n$ for $f(x) \epsilon H^{1}_{0}(\Omega) $, show there exists a constant $C$ such that ...
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0answers
21 views

Nondimensionalization of PDE with non constants coefficients

I need to try to solve, by some numerical method, a variant of the Navier-Cauchy equation of motion for an elastic, linear, isotrophic, not homogeneus body: $$ \rho_0 (\ddot{\textbf{u}} - ...
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40 views

Solving PDE's: Laplace equation on semi-infinite cylinder

Solve Laplace’s equation $\nabla^2$u = 0 inside a semi-infinite cylinder 0 < z, 0 < r < 1 with boundary conditions u(r = 1, $\theta$, z) = $e^{−z}$ and u(r, $\theta$, z = 0) = 0, where (r, ...
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0answers
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Equivalence of first order quasilinear PDE to linear PDE

Given a system of nonlinear PDE of the special form: $\sum_{i=1}^n A_i(x, \phi) - \frac{\partial \phi_j}{\partial x_i} = B_j(x,\phi) $ $(1)$ with $(j=1,...,m)$ and $x \in R^n,\phi \in R^m$. If we ...
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0answers
20 views

Inequality in the proof of Weak Harnack Inequality

Let $\Omega \subset \mathbb{R}^{n}$ a bounded domain s.t $B_{1} \subset \Omega$ , $u \in H^{1}(\Omega)$ a nonnegative supersolution in the weak sense of the equation $Lu=-D_{i}(a_{ij}(x)D_{j}u)$ ...
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17 views

Show that the function is identically Zero in certain subset

We are given a open ball D (radius = 1) in $\mathbb R^2$. and let $\{x_n\}$ be the dense sequence in the set D. Around each point $x_n$ we make a hole of radius $r_n$. The sequence $r_n$ satisfy the ...
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long time behavior of heat equation

Given the heat equation \begin{align} {{u}_{t}}-{{u}_{xx}}&=0,\quad x\in \mathbb R,\,t>0 \\ u\left( x,0 \right)&=f\left( x \right),\quad x\in \mathbb R. \end{align} If ...
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PDE: Traffic problem with initial density a piecewise function

I was studying a bit of PDE and I found very interesting traffic problems, however, I have some troubles to deal with them. I wanted to solve the following: Consider the traffic problem ...
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17 views

Weak derivative of a piecewise defined function

I am currently looking at these online notes on PDEs, page 59. How does it follow that if $f^R = \phi(x/R) f(x)$ $ \phi(x) = \left\{\def\arraystretch{1.2}% \begin{array}{@{}c@{\quad}l@{}} 1 ...
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1answer
42 views

Solve $A \partial_t w + B \partial_t\partial_x^4 w + C \partial_x^4 w + \partial_t^2 w = 0$

a non-mathematician wants me to solve a PDE. The problem is that I don't know a lot of theory to solve PDE's except the fouriertransform. This is the PDE $$A \partial_t w + B \partial_t\partial_x^4 w ...
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2answers
45 views

Method of characteristics - finding the particular solution using initial conditions

I am trying to use the method from my previous question to solve this PDE: $$ 3u_x + 2u_t = \cos x $$ with initial condition $u(x,0) = x^2$. So I need to solve these: \begin{align} \frac{dx}{ds} ...
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capillary surface problem [closed]

Consider the capillary surface problem (⋆) )   Du  div 1 + |Du|2 = κu in Ω on∂Ω,  Dηu  1+|Du|2 =β where κ > 0, η is the outward pointing unit normal to ∂Ω and β ∈ C1(Ω) satisfies |β| ≤ 1 ...
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2answers
33 views

Method of characteristics - eliminating variables

I am trying to follow a guide for the method of characteristics; quoting the first example: We use the method of characteristics to solve the problem $ 2u_x - u_y = 0, \;\; u(x, 0) = f(x) $ ...
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61 views

About the gradient of a function in $H^{1}(\Omega)$

let $\Omega \in \mathbb{R}^{n}$ a bounded domain and $u \in H^{1}(\Omega)$ a real function. In the Leoni's Book - A First Course in Sobolev Spaces, the author define $\nabla u = (D_{1} u,\dots, ...
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1answer
18 views

checking that an initial condition holds for the heat equation

I'm trying to follow a video lecture on solving the heat equation. $I) \space u_t = ku_{xx}, x \in \mathbb{R}, t > 0$ $II) \space u(x,0)=\phi (x), $ $k$ is const, $\phi (x) $ is a ...
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1answer
33 views

More equations than unknowns for maxwell equations?

I had one curiosity regarding maxwell equations in 3-D From the curl equations, you get 6 unknowns, with 6 equations. The divergence equations add 2 additional equations. When these are combined, we ...