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Questions tagged [pde]

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

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1answer
17 views

how to verify a solution to the 3D wave equation

Is $E = (A\sin(k(x-ct)),0,0)$ a solution to the wave equation $c^2 \nabla^2E=\frac{\partial^2E}{\partial t^2}$? What is $\frac{\partial E}{\partial t}$?
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0answers
12 views

Understanding where shocks form

Suppose we have $$u_t + f(u) u_x = 0$$ where $t, x > 0$, and initial conditions $u(x,0) = C$, where $C>0$ is constant, and $u(0,t) = g(t)$, where $t>0$. We know the solution is $$u(x,t) = F(x-...
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0answers
5 views

Finding points in space-time where characteristics intersect

Given burgers' equation $u_t + uu_x = 0$ subject to $u(x,0) = e^{-x^2}$ for $x \in \mathbb{R}$. I want to find the points $(x_B, t_B)$ where solution breaks. Well, we know $$ t_B = - \frac{1}{\inf_{...
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2answers
30 views

Solution for $u_t+u_x=0$ using characteristics

P. Dravek and G. Holubova, Elements of Partial Differential Equations, Section 3.4 Exercise 22: Show that the initial value problem $$u_t + u_x = 0,\; u(x,t) = x \;\text{ on }\; x^2+t^2=1.$$ ...
1
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0answers
25 views

Hardy-Littlewood Inequality for Sobolev spaces

After making the mistake of applying Hardy-Littlewood-Sobolev(H-L-S) for the infinity case, I was wondering if it is possible to bound it by a Sobolev norm. Fix dimension to be $3$. H-L-S says that ...
2
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1answer
34 views

Does integral of function against all test function derivative imply function is equal to 0?

Howdy so I know that if $\int fv = 0 $ for all test functions $v$ then $f=0$ If you have however $\int fv_x = 0 $ for all test functions $v$ then does it mean that $f=0$? I guess my thought is ...
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1answer
28 views

What is the function you get when you double the arguments of sin and cos in the Fourier Series of another function?

Suppose $$f(x) =\frac{a_0}{2}+\sum_{k=1}^\infty a_k\cos(kx)+b_k\sin(kx) $$ Then what is $$S(x)=\frac{a_0}{2}+\sum_{k=1}^\infty a_k\cos(2kx)+b_k\sin(2kx)$$ in terms of $f(x)$. I tried writing down ...
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0answers
11 views

Second-order linear PDE - hyperbolic and elliptic

Consider the equation: $$u_{xx}+yu_{yy}+\frac{1}{2}u_y=0$$ For $y<0$ the equation is hyperbolic and I found its canonical form as: $$v_{\zeta\eta}=0$$, which I am not sure whether it is correct or ...
1
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1answer
26 views

Obtaining the general solution of a parabolic PDE

I am trying to obtain the canonical form of this PDE: $$(1+\sin(x))u_{xx} + 2\cos(x)u_{xy} + (1- \sin(x))u_{yy} - u_y - \cos^2(x) = 0 $$ Since the discriminant is equal to zero, the euqation is a ...
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0answers
14 views

Discrete Fourier Interpolation Proof

This question is within the context of developing the discrete Fourier interpolation. We begin with an interval $[a,b]$ with a uniform partition $a = x_0 < ··· < x_{N−1} < b$, where each $...
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0answers
24 views

The axiomatic minimum required to have unique solutions to the Schrödinger equation

Let us consider the free non-relativistic Schrödinger equation $$i\partial_t \psi =-\frac{1}{2}\partial_x^2 \psi=:H\psi.$$ Adapting Fritz John's pathological solution to the heat equation, I find that ...
2
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1answer
24 views

Weak convergence in $W^{1,p}(\Omega)$

I am a bit confused by weak convergence in Sobolev spaces. I am making this post to hopefully clarify some of my doubts. Recall that in a Banach space, we say a sequence $x_n\in X$ converges weakly ...
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0answers
14 views

Intuition of the hyperbolic, parabolic and elliptic PDEs

I am learning PDE and today I started the second-order linear PDEs. These PDEs are divided into three categories. I am really interested in knowing the intuition behind this categorization and the ...
0
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1answer
26 views

Obtaining the Fundamental Solution of $1-\partial_x^2$

How does one obtain the fundamental solution of the differential operator $(1-\partial_x^2)$? There does not seem to be any easily accesible literature specifically describing how this is done, except ...
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0answers
51 views

Unique solution for a PDE

Consider the equation $$xu_x+yu_y=\frac{1}{\cos(u)}$$ with the initial condition $$u(s^2,\sin(s))=0.$$ If we use the methods of characteristics we can obtain: $$x(s,t)=s^2e^t$$ $$y(s,t)=e^t\ \sin(s)$$...
3
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0answers
28 views

Derivation of Burgers' Equation

I'm aware that it is possible to reduce Navier-Stokes to Burgers by neglecting pressure, and that one can derive the inviscid form by considering an ideal gas and concluding that the convective ...
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0answers
15 views

Calculating the convergence order of a numerical scheme

To obtain the convergence order of a numerical scheme, the following formula is used $$ R = \frac{ \log_2 || e_{new} || - \log_2 || e_{old} || }{ \log_2 || \Delta x_{new} || - \log_2 || \Delta x_{...
1
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0answers
24 views

Finding the canonical form of a PDE

Consider the equation $$u_{xx} + yu_{yy} + \frac {1}{2} u_y = 0$$ If we calculate the discriminant of this equation we find that: $$\delta (x,y) = -y$$ So for $y<0$ it is a hyperbolic equation ...
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0answers
8 views

In $v_t-D\Delta v=f$(diffusion eq.), what can be said about $f$?

In the diffusion equation $v_t-D\Delta v=f$, there can be several different boundary conditions and properties assumed about $v$(2nd space derivative continuous, 1st time derivative continuous in the ...
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1answer
48 views

Evaluating Fourier coefficients to complete a Laplace equation solution

While solving a PDE problem involving the Laplace equation in 3D, I arrive at the following summation relation when i substitute the only non-homogeneous boundary condition available $$ \sum_{m=1}^{\...
1
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1answer
28 views

Solving $u_t + cu_x = k$ by method of characteristics

Given the 1st order linear PDE $$u_t + cu_x = k$$ with initial condition $u(x,0)=\mathrm{cosh}2x$, I am required to find a solution using the method of characteristics. Characteristic equations are ...
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0answers
6 views

Canonical form of PDE and the idea behind the classification of second-order linear PDEs

I am so sorry for asking such a question but I’m totally new to this subject and I’m confused so I’m trying to use your help and experience. Consider the equation $$(1+\ sinx)u_{xx}+ 2\ cos(x)u_{xy} +...
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0answers
22 views

Boundary condition specification

The following equation is given to me. $$ \begin{align*} \frac{\partial}{\partial t}\left(u(\mathbf{x},t)-\Delta u(\mathbf{x},t)+\Delta^{2}u(\mathbf{x},t)\right) & =f(\mathbf{x},u,u_{x},u_{y}),\...
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votes
3answers
42 views

Finding the general solution of $u u_{xy} - u_x u_y = 0,$

In the book of Berg, at page 4, at the end of the introduction, as an exercise, it is asked to find the general solution of $$u u_{xy} - u_x u_y = 0,$$ however, considering the fact that the book ...
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0answers
19 views

solutions for solving second order partial differential equations [on hold]

enter image description here enter image description here Hello, everyone, I am going to take part in an interview for applying master degree. their university provided some examples that needed to ...
1
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0answers
6 views

An Sobolev-type inequality of $1-d$ torus

In Kishimoto-Tsutsumi's paper published by Math Research Letter 2018, I see the following inequality in the last line of page 10: $\| f g \partial_x h \|_{H^{-s}(T)} \lesssim \| f \|_{H^s(T)} \| g \...
1
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2answers
18 views

non-trivial solution boundary problems

Given the boundary problem $$X''=\mu X,\;X(0)=0,\;X'(L)=0$$ We need to find the non trivial solution $X(x)$ that satisfies the above equations. Let $\mu<0$, $\mu=-k^2$ for some $k>0$. The ...
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0answers
22 views

Domain for which there exists a unique solution for PDE

I have solved the PDE: \begin{cases} xu_x+yu_y=\sec u \\ u(s^2,\sin s)=0 \end{cases} using the method of characteristics, and if I’m correct we have: \begin{align} x(s,t)&=s^2e^t \\ y(s,t)&=e^...
1
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2answers
48 views

General solution of a PDE- Lagrange or Characteristics method

I am trying to find the general solution of the PDE: $$xu_x + (1+y)u_y= x(1+y)+xu$$ If the initial condition is $$u(x,6x-1)=\phi(x)$$ then what is the necessary condition for $\phi$ that guarantees ...
2
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2answers
60 views

Solving PDE using characteristic method

I am trying to solve the partial differential equation $x\ u_ x - u\ u_y = y$ with the initial condition $u(1,y) = y$ , using the mathod of characteristics. My problem is with y and z , I mean $$\...
1
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0answers
9 views

Dual local smoothing and retarded local smoothing for Schrodinger equation

This exercise is from Tao's Nonlinear Dispersive Equations: Local and Global Analysis, Exercise 2.54. Let $u$ be a solution to the inhomogeneous Schrodinger equation $i\partial_t u+\Delta u=F$, ...
4
votes
2answers
69 views

Burgers equation with sinusoidal bump initial data

Suppose we have $u_t + uu_x = 0 $ with $$ \phi(x) = u(x,0) = \begin{cases} 0, && x \leq 0, x > 1 \\ \sin \pi x, && 0 < x \leq 1 \end{cases} $$ If we parametrize our curve with ...
3
votes
1answer
31 views

$-\Delta u=f$ in $L^1$ but $u_{x_ix_j}$ not in $L^1$ ($i\neq j$)

I want to show the following function is a counterexample for Poisson equation with $L^1$ RHS but its solution is not in $W^{2,1}$: Let $n\geq3$ and $x\in\mathbb{R}^n$. Let $f(x)=|x|^{-n}(\log|x|)^{-...
2
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0answers
45 views

What is a neat way to solve $\nabla\mathbf{u}+\nabla\mathbf{u}^T=\mathbf{\mathbf{C}}$?

Let $\mathbf{u}:\mathbb{R}^3\to\mathbb{R}^3$ be a smooth enough vector field that satisfies the following equation $$\nabla\mathbf{u}+\nabla\mathbf{u}^T=\mathbf{C},\tag{1}$$ where $\nabla\mathbf{u}$ ...
0
votes
1answer
26 views

Obtaining the general solution of $a u_x + b u_y + c u = 0$ by using the solution on the characteristic curve

Consider the following first order linear PDE $$a u_x + b u_y + c u = 0.$$ I have solved this PDE by changing my coordinates to $x' = ax + by $ and $y' = bx - ay$, and the general solution was $\...
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0answers
15 views

Interpolation of Discrete Dynamical System [on hold]

Let $ A:\mathbb{N}\times \mathbb{R}^d\rightarrow \mathbb{R}^d, $ be a flow on $X$, such that there exists a function $f:\mathbb{R}^d\rightarrow \mathbb{R}^d$ satisfying $$ \lim\limits_{n\mapsto \infty}...
4
votes
0answers
49 views

Derivation of 2D Korteweg-de-Vries equation

Coming from engineering rather than mathematics, I am recently dealing with non-linear partial differential equations e.g. like the well known Korteweg-de-Vries equation: $$u_{t} + uu_x + u_{xxx} = 0$$...
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votes
0answers
14 views

Find the minimum and maximum value of u(x, t) in the region for a solution of the initial value problem [on hold]

Suppose that $u(x, t)$ is a $C^2$ solution of the initial boundary value problem (PDE) $u_t − ku_{xx} = 0$, $0 < x < 1, t > 0$ (IC) $u(x, 0) = 4x(1 − x)$, $0 ≤ x ≤ 1$ (BC) $u(0, t) = 0$, $...
1
vote
1answer
36 views

Solving a linear PDE: $x^2 u_x - 2u_y - xu = x^2$

For a homework problem in my course on PDEs, I'm to find the general solution to the following PDE: $$x^2 u_x - 2u_y - xu = x^2$$ In addition, I'm to find the solution corresponding to the initial ...
3
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0answers
20 views

Is the solution to the Poisson equation an analytic function in general?

I am wondering this because I just came from my PDE's 2 class and we talked about the regularity of the Laplace equation and elliptic functions DE's. My professor the stated the following which we ...
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0answers
20 views

Invariance of Solution of Laplace equation on $\mathcal{B}_{1}(0)$ under funny composition

Suppose $v:\mathbb{R}^{2}\rightarrow\mathbb{R}$ solves $\Delta v=0$ on $\mathcal{B}_{1}(0)$. Let $\theta\in(0,2\pi)$ and define $M:\mathcal{B}_{1}(0)\rightarrow\mathcal{B}_{1}(0)$ by $M(x,y)=(x\cos\...
1
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1answer
33 views

plot solution that becomes multivalued

Consider burgers equations $u_t + u u_x =0 $ with initial data $u(x,0) = e^{-x^2}$. The characteristics are given by $$ dx/ds = u, dt/ds = 1, du/ds = 0 $$ with initial curve $\Gamma = (r,0,e^{-r^2})$...
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1answer
15 views

convergence order of numerical PDE scheme

Suppose we have a pie $Lv=0$ and we approximate it using the scheme $L_k^n u_k^n = 0$ Suppose that this scheme convergence to the solutions so we have that $|| Lv - L_k^n u_k^n || \to 0 $ as $\Delta ...
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votes
0answers
55 views

How can I solve the following linear first-order PDE with exponential coefficients? [on hold]

I tried to solve the following PDE by the method of characteristics, but it doesn't help. I wonder if any one can help or provide guidance to solve it. \begin{align} &\frac{\partial T}{\partial t}...
3
votes
0answers
27 views

PDE: Find an explicit expression for the solution of the IVP by using Fourier Transform

Find an explicit expression for the solution of the IVP: $u_{t}+cu_{x}+u=0$ $u(0,x)=f(x)$ by using Fourier Transform That's how I started but don't quite understand what to do after! Any help ...
2
votes
1answer
35 views

What does constant along characteristic mean

When we have linear or quasilinear first order pde $$ a(x,y,u) u_x + b(x,y,u) u_y = c(x,y,u) $$ And suppose we have found characteristics with prescribed Cauchy data $u |_{\text{curve}} = \phi$ I ...
2
votes
2answers
57 views

Explicit solution to IVP of PDE $\rho_t = [\rho (1-\rho)]_x$

When trying to determine the density profile $\rho(t,x)$ of a system of particles I came across the PDE: $$\frac{\partial \rho}{\partial t}=\frac{\partial}{\partial x}\big(\rho (1-\rho)\big), \qquad\...
-1
votes
0answers
10 views

Weakly star convergence on $L^{\infty}$.

i would like hints for the convergence in the yellow LINE, please.
2
votes
1answer
115 views

Laplacian with Integral BC(s)

I want to solve the three-dimensional laplacian $$\nabla^{2} T = 0$$ where $\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$ defined on $...
-2
votes
1answer
24 views

Completeness of unit ball [closed]

Suppose $X$ is a probability measure space. Then show that unit ball of $L^{\infty}(X)$ is complete with respect to $L^2$ norm.