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

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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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8 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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using method of lines to solve a pde heat equation

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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$L^{2}$ 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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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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19 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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8 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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9 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
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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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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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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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6 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
28 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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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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33 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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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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1answer
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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
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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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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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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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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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27 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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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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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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12 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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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
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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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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
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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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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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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1answer
67 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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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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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
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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
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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1answer
14 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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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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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
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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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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 ...
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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
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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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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14 views

What conditions must an operator meet, to have only real eigenvalues?

Given the problem $Lu = \lambda u$, what properties must $L$ have, for all its eigenvalues to be real? An answer in the context of (partial) differential equations would be appreciated.
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Notational question on vector field/differential operator

I am looking at something related to Hormander condition. The paper is 'On local smoothness of generalised and hypotellipticity of second order differential equations' by Oleinik and Radkevich. It ...
3
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2answers
33 views

EigenFunction for $\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x} =\frac{2f^2}{x}$

When studying a computer vision problem I end up with a function $f(x,t)$ that satisfying $\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x} =\frac{2f^2}{x}$. My question includes two ...
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1answer
34 views

Change of variables - PDE

I was just wondering how do I use change of variables to obtain a more suitable equation to solve for the following PDE? If I know how to do that then I am sure I can solve the rest. ...