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

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Kirchoff Formula

I just learned Kirchoff's formula for the wave equation in $\mathbb{R}^3$, but the book doesn't provide an example on how to apply the formula. An example I found, Solve the wave equation ...
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11 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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Parallelogram Rule - PDE

Use the parallelogram rule to evaluate $u(1,0.3)$ and $u(1.3,0.3)$ if $u$ satisfies $$u_{tt}=4u_{xx}, \quad 0<x<1,t>0$$ $$u(0,x)=x(x-1),u_t(0,x)=0,\quad 0<x<1$$ $$u(t,0)=4t^2, \ ...
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0answers
7 views

Boundary value problem for general elliptic equations

I have showed that if $U$ is connected then the only smooth solutions of \begin{align} -\Delta u=0 \;\;\;\text{in } U && \frac{\partial u}{\partial \nu}=0 \;\;\; \text{on } \partial U ...
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1answer
41 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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15 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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1answer
12 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
40 views

heat equation with perfectly insulated end

In one of my tutorial question about $1$-dim heat equation,a question about heat equation with pefectly insulated end at $x=0 $ and $ x=l$ with $u(x,t)$ as temperature function,TAs used as perfectly ...
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1answer
15 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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2answers
29 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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21 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
11 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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13 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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using method of lines to solve a pde heat equation [on hold]

the heat equation to be solved is du/dt=du2/dx2 domain is 0 the RHS is dicretised into sets of odes and solved with any stiff ode solver in matlab thanks
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6 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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64 views

Uniqueness theorem in differential equation

Let $g(t) , F(r)$ be measurable functions on $(0,a)$ and assume $g(t)$ be nonnegative and $\int_{0}^{a} (g(t)) dt < \infty $ and $F(r)>0$ , and for every $\delta >0$ we have ...
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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
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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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Find a function to satisfy a necessary condition on a system of pdes

Consider the following set of PDE's $\displaystyle \frac{\partial u}{\partial x}(x,y)=f(x,y,u(x,y))$ $\displaystyle \frac{\partial u}{\partial y}(x,y)=1$ $u(x_0,y_0)=u_0$ Show ...
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25 views

Inverse non homogeneous heat problem unicity

Consider the inverse problem: Determine the pair $u = u(x, t)$ and $f = f(x)$ satisfying $$u_{t}(t, x) - u_{xx}(t, x) = f(x) \hspace{0.5cm} t > 0, x \in ]0, L[;$$ $$u(t, 0) = 0 \hspace{0.5cm} t ...
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0answers
45 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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1answer
23 views
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MATLAB, 1st order 2d hyperbolic equation, problem with convergence.

Follow up to my previous question: MATLAB: solving 1st order hyperbolic equation in 2 spacial dimensions The equation I'm solving has the form: $$f_t + A y f_x - B x f_y =0$$ I wrote the following ...
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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
68 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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1answer
29 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
35 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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Approximating $v \in W^{1,p}_0(\Omega)$

Let $p > 1$. Let $V=W^{1,p}_0(\Omega)$ and $H=L^2(\Omega)$. Suppose $\{\lambda_i\}$ is a basis in $V \cap H$ which is smooth, and orthonormal in $L^2$. Given $v \in V$, is it possible to find a ...
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1answer
15 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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How to find multipliers in charpit's method [on hold]

After getting to the subsidiary equations in charpit's method i am unable to see further. It will be a great help if any one can explain how to choose the multipliers i mean is there a criteria for ...
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0answers
17 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 ...
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0answers
14 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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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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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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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

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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1answer
32 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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0answers
13 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
35 views

Holder continuity using Sobolev imbedding

We assume for any $V\subset \subset U$ and $1<p<\infty$ $||u||_{W^{2,p}(V)}\le C(||\Delta u||_{L^p(U)}+||u||_{L^1(U)})$ for some $C=C(V,U,p)$. Given, $B=\{x∈R^3,|x|<1/2\}$ and we suppose ...
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1answer
18 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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General form of a connection with zero curvature

I am looking for proofs of the following two theorems: Theorem 1. On a connected and simply-connected open set $\Omega\subset\mathbb{R}^3$, functions $L^p_{ij}\in C^1(\Omega)$ are given that satisfy ...
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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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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
24 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
28 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 ...
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1answer
24 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 ...
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1answer
31 views

Voltage Distribution Inside a Cylinder [on hold]

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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22 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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10 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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2answers
32 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 ...
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0answers
5 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-􀀀 ...