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

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37 views

Prove domain of Dependence Inequality for the Wave Equation?

Let $(x_0,t_0)\in R^{n+1}$ with $t_0>0$, and let $\Omega$ be the conical domain in $R^{n+1}$ bounded by the backward characteristic cone with apex at $(x_0,t_0)$ and by the plane $t=0$. Suppose ...
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0answers
34 views

Range of Influence of the Wave Equation?

Suppose $u$ is a solution of the two-dimensional wave equation and that at $t=0, u=u_t=0$ outside the disc $x_1^2+x_2^2 \le 1$. Up to what time can you be sure that $u=0$ at the point ...
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2answers
66 views

Finding a solution for $x\frac{\partial u}{\partial x}+2y\frac{\partial u}{\partial y}=x^2$

To find a solution for $x\frac{\partial u}{\partial x}+2y\frac{\partial u}{\partial y}=x^2$ knowing that $u(x,y)=1$ if $xy=1.$ I thought it may be useful to do the change $v=\log x, w=\frac{1}{2}\log ...
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37 views

Neumann eigenvalue problem for the Laplacian

Let $\Omega$ be a bounded, connected open subset of $\mathbb R^n$. Consider the Neumann eigenvalue problem $$ \Delta u =\lambda u \, \text{at} \, \Omega \\ \frac{\partial u}{\partial \vec{\nu}}=0 ...
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3answers
57 views

Solving $x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y }=1$

I want to solve the differential equation $$x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y }=1$$ with the initial condition $u(1,y)=y.$ I'm very unfamiliar with possible methods to ...
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0answers
31 views

The space of distribution $H^{-1}$

Let's suppose to have a function $u$ in the space $L^\infty(0,T;H^1(\mathbb{R}^n))$ with $\partial_t u\in L^\infty(0,T;H^{-1}(\mathbb{R}^n))$. So $\partial_t u$ is a linear and continuous functional ...
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1answer
18 views

Uniqueness of solution for seperation of variables solvable PDEs

I am taking first course in PDEs and the only way i know of solving PDEs is separation of variables , and all the equations i saw had unique answers due to the ICs and BCs , but not this one : $$ ...
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1answer
8 views

General solution of a hyperbolic second order pde

How do I find the general equation to $$\frac{\partial^2 u}{\partial x \partial y} = 0$$ using characteristics. I am confused because I thought hyperbolic equations always had 2 distinct roots.
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0answers
23 views

uniformly convex domain and uniformly convex function

I want to ask a question about uniformly convex domain: Suppose $\Omega$ is unifromly convex, i.e. for each point $x_0\in\partial\Omega$, with regualrity $C^2$ smooth. Uniform convexity of domain ...
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1answer
25 views

Green's Function - PDE

I am completely stumped on Green's function regarding PDE's. There are barely any examples in my book on how to apply it. For example, if a question asks, "find the Green's function ..." to some ...
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17 views

One simple question about Fourier transformation of system of PDE's

Let's assume set of equations $$ \tag 1 \frac{\partial \mathbf A}{\partial t} = \Delta \mathbf A + a [\nabla \times \mathbf A] - d\mathbf b_{k} (\mathbf b_{k} \cdot \mathbf A), \quad \mathbf A(0) = 0 ...
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0answers
18 views

$L^{2}$ convergence and converence of distribution

Suppose that $f_{n}(x)$ are a sequence of $L^{2}$ functions which converge to a function $f(x)$ in the $L^{2}$ sense. Show that it also converges weakly in the sense of distributions, ie for any test ...
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0answers
8 views

How to solve a factorized Helmholtz equation?

I am reading a paper on optics an in appendix A2 they split the Helmholtz equation into two parts and write down the solution for one of those parts (link). Helmholtz Equation where the Laplacian is ...
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0answers
33 views

Green function integration

When I'm trying to find the Green Function of Helmholtz equation for a cube $0≤x,y,z≤L$ $$\nabla^2u+k^2u=\delta(\vec{x}-\vec{x}')$$ where u=0 on the surface. I set to find the green function where ...
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0answers
17 views

Find a vector field $\mathbb{Y}$ satisfying $L_{\mathbb{X}}\mathbb{Y}=\mathbb{Z}$

Let $\mathbb{X}$ be the vector field on $\mathbb{R}^2$ given by $\mathbb{X}=(1,y)$. Let $\mathbb{Z}$ be the vector field on $\mathbb{R}^2$ given by $\displaystyle \mathbb{Z}(x,y)= \bigg( ...
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0answers
17 views

continuous dependency estimate for viscosity solutions

Let $u^i$, $i=1,2$, be viscosity solutions of \begin{align*} u_t^i + H(Du^i,x) & = 0\quad\mathrm{in}\ \mathbb R^n\times (0,\infty)\\ u^i & = g^i\quad\mathrm{on}\ \mathbb R^n\times \{t=0\} ...
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1answer
13 views

Solvability of eigenvalue problem with Schwartz data

Fix $a\in\mathbb{R}$ and define the operator $T$ acting on the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ by sending $\phi$ to $\Delta\phi-a^2\phi$. Then $T$ is clearly a bounded operator.Question is ...
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1answer
38 views

Higher order numerical PDE schemes near boundaries, implementation in MATLAB

Followup to my previous question. The first order scheme proved unstable for my pde: $$f_t + A y f_x - B x f_y =0$$ So I'm looking to implement a higher order scheme (using these tables). I was ...
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1answer
32 views

Determine adjoint operator

Let $K\in D\{(x,\xi)\in \mathbb {R}^2 : x > 0, \xi > 0\} $ and $L (\phi)(x)=\int_0^x K (x,\xi)\phi (\xi)d\xi $ for $\phi \in D (\mathbb {R})$ $D $ is the space of testfunctions I know that ...
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2answers
41 views

Show $u\in H^1(B(0;1/2))$ is holder continuous, where $u$ is a weak solution to $-\Delta u+cu=f$ for some $c\in L^q$ for some $3/2<q<2,$.

If $u\in H^1(B)$, $B=\lbrace x\in\mathbb{R}^3, |x|<1/2\rbrace$ is a weak solution to $$-\Delta u+cu=f$$ for some $c\in L^q$ for some $3/2<q<2,$ and $f\in C^\infty$, then show $u$ is holder ...
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1answer
19 views

An identity for the integral $ \int_{\partial B(0,1)}u(x_0+aw)u(x_0+cw)$ with a harmonic function $u$

This is Question 2.18 from Gilbarg and Trudinger, chapter 2. We are given that $\Omega$ is open bounded smooth boundary. Now fix $x_0\in \Omega$ and a constant $c>0$ such that ...
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2answers
39 views

Vibrating string - separation of variables

$u_{tt}=c^2u_{xx}$ where $u(x,0)=x+\sin(x)$, $u_t(x,0)=0$, $u(0,t)=u_x(\pi,t)=0$. Assume a solution $u(x,t)=X(x)T(t)\not\equiv 0$. This yielded $\lambda_n=\frac{1}{2}+2n$. For $X_n(x)$ I have ...
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0answers
37 views

Weak subsolution and composition with convex smooth function

I'm trying to solve a problem in Partial Differential Equations by Lawrence C. Evans. It goes like this Assume $u\in H^1(U)$ is a bounded weak solution of \begin{equation}-\sum_{i,j=1}^n ...
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0answers
19 views

Normal derivative property

I have seen in many papers that to obtain some results about PDEs is used the following argument: If $\phi=0$ in $\partial\Omega$ then $\bigtriangledown\phi=\dfrac{\partial\phi}{\partial n}n$, where ...
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0answers
21 views

Derivative of Smooth convex function composed with sobolev is sobolev?

This is homework so no answers please Here is what I mean specifically The full problem is $f:\mathbb{R}\to \mathbb{R}$ smooth and convex, bounded $u\in H^{1}(U)$ and $v\in H_{0}^{1}(U)$, is it ...
4
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1answer
54 views

Solving PDE by Laplace Transform

Use Laplace transforms to solve the boundary value problem $$Y_{xx}(x,t)-2Y_{tx}(x,t)+Y_{tt}(x,t)=0, \quad 0<x<1, t>0$$ $$Y(x,0)=Y_t(x,0)=0, \quad 0<x<1$$ $$Y(0,t)=0, \ Y(t,1)=F(t), ...
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1answer
29 views

Basis of $W^{1,p}_0\cap L^2$ using $(\lambda_i, v)_{H^s_0} = \mu_i(\lambda_i, v)_{L^2}$

Let $p > 1$. Define $\lambda_i$ by the eigenfunctions of the problem $$(\lambda_i, v)_{H^s_0} = \mu_i(\lambda_i, v)_{L^2}\quad\text{for all $v \in H^s_0(\Omega)$},$$ where $s$ is chosen ...
2
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1answer
58 views

Stability of nonlinear system of PDE's

Let's assume system $$ \tag 1 \frac{\partial \mu}{\partial t} = \gamma (\mathbf B \cdot \mathbf E), $$ $$ \tag 2 [\nabla \times \mathbf E] = -\frac{\partial \mathbf B}{\partial t}, $$ $$ \tag 3 ...
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0answers
20 views

What does instability mean and examples, boundary condition

The Upwind-Scheme for the numerical solution of first order PDE's (homogenous case) of the form $u_t + cu_x = 0$ is given by $$ u_j^{n+1} = \left\{ \begin{array}{ll} u_j^n - \frac{c\Delta t}{\Delta ...
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0answers
33 views

Heat equation, boundary gradient singularity

Consider the Cauchy-Dirichlet problem for the heat equation in a non-cylindrical region $\Omega \subset \mathbf{R}^+ \times \mathbf{R}$: $\Omega = \{ (t,x): \; 0 \leq t \leq 1, \; x \leq ...
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1answer
16 views

Simple problem about Laplace Equation in a domain

Suppose that "$u$", is solution of the problem $$\triangle u=0, r<R $$ $$u_{r}(R,\phi)=f(\phi), 0<\phi\ < 2 \pi$$ Show that $$\int_{0}^{2 \pi}{f(\phi)d\phi}=0$$ I know what this question ...
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0answers
29 views

Unbounded Entropy Solution to Burger's Equation

I need to deduce that $ u(x) = \left\{ \begin{array}{lr} \frac{-2}{3}(t+\sqrt{3x+t^2}) & t^2+4x>0\\ 0 & t^2+4x<0 \end{array} \right. $ is an unbounded ...
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16 views

Helmholtz equation in 3D

I understand solving the Helmholtz equation, $\nabla^2u + \lambda^2u = 0$, when $\nabla^2 = \partial_{xx} + \partial_{yy}$. However, I am not sure of the form I should obtain when $\nabla^2 = ...
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1answer
28 views

Separation of variables in the PDE $u_{tt}=c^2 u_{xx}$.

I'm stuck trying to work with the constants to the solution to the SOV problem Given the following equation: $u_{tt}=c^2 u_{xx}$ and the following conditions: $u(0,t)=0=u(\pi,t)$, $u(x,0)=0$, ...
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1answer
43 views

How to use D'Alembert formula for Neumann boundary conditions on a finite interval?

I have a PDE to solve that I am not sure how to do. I know how to solve this using D'Alembert's formula for Dirichlet boundary conditions but I do not know how to solve it for Neumann boundary ...
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0answers
26 views

Traces of $W^{1,\infty}$ functions

Let $\Omega$ be a Lipschitz domain and $p \in (1,\infty)$. It's known that if $u \in W^{1,p}(\Omega)$ then $u_{|\partial \Omega} \in W^{1-\frac{1}{p},p}(\partial \Omega)$. I'm wondering if the ...
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1answer
33 views

Voltage Distribution Inside a Cylinder [closed]

I was assigned this problem, and quite honestly I do not know where to begin. If I could get some help and an explanation of the Bessel function, also? Thank you. I know my conditions are: ...
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1answer
73 views

Solving a PDE equation

Could you please help me to solve this equation: $$\frac{\partial^2}{\partial x^2}E(x,t)-LC_1 \frac{\partial^2}{\partial t^2}E(x,t)+LC_2 \frac{\partial^4}{\partial x^2 \partial t^2}E(x,t)=0 \qquad ...
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0answers
15 views

Second order PDE with initial and boundary conditions

I'm trying to solve the following PDE: $u=u(x,y)$ $\left\{ \begin{array}{1 1} \partial_x ^2 u-6\cdot\partial_x \partial_yu+9\cdot\partial_x^2u=x^2+y^2\ \\ \ u(0,y)=0 \\ \partial_x u(0,y)=y ...
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0answers
11 views

Dirichlet problem with Specific Boundary Condition

How to solve the Dirichlet Equation $D = \{(x, y) \in \mathbb R^2 : 0 < x < a; 0 < y < b\}$ for Laplace Equation for following boundary condition? $$u(x, 0) = x; \,u(x, \pi) = x( \pi-􀀀 ...
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1answer
38 views

d'Alembert's Solution: Should anything more be added to it?

I have this initial value problem: $$ u_{tt}- c^2u_{xx} = 0; \,0 < x < 1; \,t > 0; $$ $$ u(x, 0) = 0; \,u_t(x, 0) = 1; 0 <= x <= 1; $$ $$ u(0, t) = u(1, t) = 0; t >= 0: $$ ...
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1answer
40 views

Charge distribution on an arbitrarily shaped conductor

From physics we know that given a charged conductor put in vacuum (no external electric fields), the charge distribution on its surface is approximately proportional to the curvature of the surface on ...
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1answer
29 views

Dirichlet problem on rectangle with nonhomogeneous boundary conditions

I want to solve the following problem in Dirichlet Problem: Let $D = \{(x, y) \in \mathbb R^2 : 0 < x < a; 0 < y < b\}$ be a rectangle with boundary $\partial D$. I want to solve ...
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0answers
37 views

How to prove this property?

I have a function $S(x,y)$ which satisfies the following PDE $$\frac{\partial S(x,y)}{\partial y}=-H\left(x,\frac{\partial S(x,y)}{\partial x}\right)$$ where the known function ...
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0answers
21 views

Physical interpretation of the integral formula for the solution of Laplace equation with Dirichlet/Neumann boundary condition

Suppose we have a bounded domain $D$ with smooth boundary, with $G(x,y)$ being the Green's function for the Poisson equation on $D$, i.e. $G(x,\cdot)=0$ on $\partial D$ and $\Delta_y ...
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1answer
20 views

Positivity of Poisson kernel

Let $K(x,y)$ be the Poisson kernel for the Dirichlet problem of Laplace equation on a bounded domain $D$ with smooth boundary, i.e. for a harmonic function $u$ on $\bar D$ with $u|_{\partial D}=g$, we ...
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0answers
23 views

PDE: Green's function and the method of images

I am stuck on a problem with the method of images. The formulation is rather simple; Solve the for green's function given by $\nabla^2 G = \delta( \underline{x} - \underline{x}_0)$ in the wedge ...
2
votes
1answer
31 views

Semi-infinite plate problem when 2 edges are equal temperature and one edge is a function of position

Here is the problem posted: Now here is my solution for a) I knew to discard the $e^{ky}$ and $\sin(kx)$ due to $T\to 20$ when $y\to\infty$ and $T= 20$ at $x = 0$. Which leaves me with ...
0
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1answer
32 views

D'Alembert Formula where PDE has only one boundary condition

Solve the initial boundary value problem $$u_{tt}=4u_{xx}, \ x>0,t>0$$ $$u(x,0)=\frac{x^2}{8}, \ u_t(x,0)=x, \ x\ge0$$ $$u(0,t)=t^2, \ t\ge0.$$ I used D'Alembert forula and got ...
1
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2answers
43 views

Characteristics method applied to the PDE $u_x^2 + u_y^2=u$

I am trying to solve: $u_x^2 + u_y^2=u$ with boundary conditions: $u(x,0)=x^2$. Unfortunately it leads to equations that makes no sense (sum of squares is $0$ and all constants are $0$). I would be ...