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

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7 views

Wave Equation with outgoing wave boundary conditions

I need some help with this problem: I have a to solve the wave equation with two initial conditions and with outgoing wave boundary conditions; i.e., $$\begin{cases} u_{tt}-u_{xx} & =0\\ u(x,0) ...
2
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0answers
18 views

Help with this eingenvalues problem

I'm trying to find the Laplacian eigenvalues for the Dirichlet problem in a right isosceles triangle $T\subset\mathbb R^2$ where its smaller side has length $c$. I saw in an article that the ...
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0answers
9 views

Parabolic PDE with diffusion matrix of zero determinant

Consider a Fokker-Planck type PDE in $\mathbb{R}^2$: \begin{equation} \partial_t\rho=\mathrm{div}(\rho\nabla V)+ D^2:\left[\sigma\rho\right] \hspace{2cm} (*) \end{equation} where we have the ...
1
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2answers
26 views

Have I obtained the proper solution to this PDE?

I'm a little stuck on this. Consider $ u_t -(1+t^2)u_x = \phi(x,t) \quad u(x,0)=u_0(x)$ Via the method of characteristics, the total derivative of $u(x,t)$ is $$\frac{du}{dt} = \dfrac{\partial ...
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0answers
57 views

Gronwall's inequality and polynomial

Given $u = u(t) \geq 0, u \in C^{1}[0,\infty)$. Suppose there is a polynomial $f$ with non-negative coefficients such that $$u'(t)\leq f(u(t)).$$ Prove that there exists $T>0,M>0$, both ...
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0answers
15 views

Finding the adjoint operator

On the interval $(0,1)$ consider the differential operator $Lu=u''''+u'$ with boundary conditions $u(0)+u'(1)=u(1)+u'(0)=0$ $2u(0)+u''(1)=2u(1)+u''(0)=0$ $(1)$ I want to find the adjoint ...
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34 views

Smoothness of solutions to the Burger's Equation

Consider $ u_t + uu_x =0 $ and an associated problem $uu_t + u^2u_x=0$ ($*$). If we let $ w= u^2$, the second PDE becomes $$ w_t + \left(\dfrac{2}{3}w^{3/2}\right)_x=0 \>. (**)$$ I am required ...
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36 views

Solve a equation involved differential operator in Matlab

I want to solve $F$ in the following equation in Matlab: $$ F + \partial_x F = Y, $$ where $F$ is a matrix (image), $\partial_x$ is the differential operator of $x$ axis, and $Y$ is a known matrix. ...
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0answers
17 views

Regularity when Dirichlet conditions are posed on the interior of a domain.

Problem: Let $\Omega_1,\Omega_2\subseteq \mathbb{R}^2$ be two domains, define $\Omega:=\Omega_1\cup\Omega_2$ as their union as well as $\Gamma:=\partial\Omega_1 \cap \partial \Omega_2$ as their common ...
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0answers
9 views

To show a given function is not the viscosity solution.

For the equation $ F(x,u,u') = -au''-1 =0$ for $ x\in (0,2)$ with $ u(0) = 0 = u(2) $ and $a(x)$ is $1$ for $x\in (0,1)$ and $2$ for $x\in [1,2)$. Need to show that the function $$ u(x) = ...
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0answers
13 views

Existence of solutions to this ODE arising from Faedo-Galerkin method?

Let $\{w_j\}$ be a basis of $H^1_0(\Omega)$ and let $\phi(x) = \frac{x}{|x|^{1-{\frac 1p}}}$ (for $2 < p < 3$). Define $$v_m(t) = \sum_{i=1}^m \zeta_i(t)w_i$$ where the coefficients ...
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0answers
11 views

boundary conditions and even/odd extensions

Why do we use an odd extension for Dirichlet boundary conditions and an even extension for Neumann boundary condition when finding solutions to the wave equation on the half line?
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23 views

Fourier Coefficients

So (r, θ) denotes polar coordinates in the plane, let a > 0 be a constant. $\nabla^2u=0$, in r < a $\frac{\partial u}{\partial r}+\gamma u=h(\theta)$ on r = a h is an arbitrary periodic ...
0
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1answer
20 views

Help with solving this linear, variable coefficient PDE

Consider the PDE (letting $u = u(t,x)$) $$ u_t + (ab - c^2 - ax)u_x - \frac{c^2}{2}u_{xx} - au = 0, $$ where $a,b,c > 0$. In a paper I'm reading, they note this PDE has been shown to be ...
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27 views

calculate divergence and infimum

Let $\Omega=\{(x,y)\in \mathbb{R}^2 \mbox{ s.t } x>0 \mbox{ and }x^2+y^2<1\}$. Define the following function $$v(x,y)=\dfrac{x}{(x^2+y^2)^{\frac14}}-x$$ and by $A(x,y)$ the $2\times 2$-matrix ...
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0answers
21 views

Completeness of solutions and the separation of variables method

The method of separation of variables is introduced in every textbook on mathematical physics. A basic question is rarely addressed: does this method exhaust all the solutions? Is there any ...
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2answers
43 views

Sobolev space $H_0^m(\Omega)$?

Let $\Omega \subset \mathbb{R}^n$ be open, then $H_0^m(\Omega) := \overline{C^{\infty}_c(\Omega)}$ with respect to the Sobolev norm. The problem is that I don't really see what kind of functions are ...
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0answers
27 views

Space-Time FEM for parabolic problems

I am trying to solve a parabolic problem (an IBVP) in one spatial variable using the Galerkin method. After searching for inspiration, I find that the typical approach is to discretise the temporal ...
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0answers
15 views

Decomposition of Adomian Polynomials by MATLAB [on hold]

I just wanted to decompose/expand/ Adomian polynomials by MATLAB codes; but I am failed to do so! Here are the codes that I am failed to get the desired result, I hope some one will help. Thank you ...
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0answers
27 views

Overly Regular PDE Solution - where's my mistake?

I was looking at this question and attempting to come up with my own answer when I "proved" a statement which seems far too strong: if $f\in L^2(\mathbb R^n)$, then the unique solution to $u-\Delta ...
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0answers
22 views

Uniform convergence of the harmonic form heat flow

[${\bf NOTATIONS}$] Let $M$ be a closed Riemannian manifold of $m$ dimensional, $p\in\{1,\cdots,m\}$. $A^p:=\{\text{smooth p-forms on }M\}$. $\delta:A^{p+1}\to A^p$ denotes the formally adjoint ...
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33 views

Elliptic and Fredholm partial differential operators

As I learn from the comments to this question a non elliptic operator on a compact manifold can not be a fredholm operator. On the other hand it seems that the following operator is fredholm(At ...
2
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1answer
33 views

When does Separation of Variables yield basis of solution set?

In mathematical physics, one often employs the technique 'Separation of Variables' to find the full solution set to some linear partial differential equation. For instance, consider the differential ...
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0answers
15 views

How to solve this degenerate parabolic euqation

I want to solve the following equation \begin{align*} \label{linear-prandtl-transform-3} \begin{cases} & \partial_t \hat{w}(t, x, y) + \tilde{u(t, y)}\partial_x \hat w(t, x, y) - 2 \partial_y ...
0
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1answer
35 views

Boundedness of solutions for the Laplacian

A solution to the equation $-\Delta u+u=f$ for $f\in L^2(\mathbb R^n)$ belongs in $H^2(\mathbb R^n)$. Is it possible to obtain a solution in $H^2\cap L^\infty(\mathbb R^n)$ if $f\in L^2\cap ...
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0answers
18 views

Can $u\in W^{1,\infty}\cap H^1(\mathbb R^n)$ be approximated by a sequence $u_k\in C_0^{\infty}(\mathbb R^n)$ with $\|u_k\|_{1,\infty}$ bounded?

This problem is relevant to this but I am not really able to prove it or find a counterexample. Could anyone give a hint?
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1answer
19 views

Nonlinear partial differential equation of first order

I've got the Farlow's book, and it says that this: $u=\phi(x-g(u)t)$ is an implicit solution of the nonlinear problem: $u_t+g(u)u_x=0$ $u(x,0)=\phi(x)$ with $x,t\in\mathbb{R},t>0$. But, how ...
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0answers
25 views

Show an equation only has harmonic solution

I want to show $$\begin{cases} \Delta(\Delta u) - \nabla\cdot (\Delta u \cdot \nabla u)=0\\ \int \Delta u < \infty\\ \Delta u \ge0 \end{cases}$$ in $\mathbb{R}^2$ only has a solution such that ...
2
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0answers
25 views

Why does an infinite Neumann boundary condition become a Dirichlet condition?

Often when I read a paper I see a statement of the type: Our boundary condition at the surface is $\frac{\partial f}{\partial x} = \alpha$. In the limit of $\alpha \to \infty$ this is equivalent ...
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0answers
19 views

Extract a linear equation from a nonlinear second order PDE.

Recently, I trapped in a mathematical problem in my research, which is as following: Given a known nonlinear second second PDE, $\mathit{f}(x_1,x_2,...,x_n)=E(x_1,x_2,...,x_n)$. Its all parameter ...
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1answer
24 views

Proving Blow up of Solution by comparing with ODE

We can figure out that for the ODE $u'(t)=f(u(t))$ the solution is global if $\int_{M}^{\infty}\frac{du}{f(u)}=\infty$. $(M>0)$ I cannot find out why this condition is sufficient for blow up of ...
0
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1answer
51 views

If $u$ is a smooth solution to a PDE, show that $u \ge 0$.

Assume that $u$ is a smooth solution of the PDE from Problem 7, that $g \ge 0$, and that $c$ is bounded (but not necessarily nonnegative). Show $u \ge 0$. (Hint: What PDE does $v := e^{-\lambda ...
3
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0answers
30 views

Finite difference solution of steady-state diffusion equation with variable material properties

I'm trying to use a finite difference method to solve the steady-state neutron diffusion equation in a nuclear reactor: $$ D(x) \nabla^2 \phi(x) + \left( \frac{\nu(x)}{k} \Sigma_f(x) - \Sigma_a(x) ...
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0answers
14 views

Generalized wave equation

I am interested in understanding as much as I can about the following partial differential equation, which is a generalization of the 1D wave equation: $$\frac{\partial^2 u(x,t)}{\partial t^2} + ...
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0answers
15 views

Are 1-D line sections of 2-D point source-invoked potential distributions positive definite?

Consider the 2-D potential distribution induced on the plane $y=0$ by a point source positioned at $(0, -y_0, 0)$ in the open halfspace below that plane. The material below the plane is assumed ...
0
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1answer
27 views

Finite and infinite speed of propagation for wave and heat equation

What is the formal definition of Finite and infinite speed of propagation? I have searched for it, is the finite one means the solution is only determined by a bounded region? Also I do not ...
0
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1answer
38 views

Exponential decay estimate for $(x,t) \in U_T$

Suppose $u$ is a smooth solution of $$\begin{cases}u_t - \Delta u + cu = 0 & \text{in }U \times (0,\infty) \\ \qquad \qquad \quad \, \,u=0 & \text{on } \partial U \times [0,\infty) ...
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1answer
36 views

Exponential decay estimate

Assume $u$ is a smooth solution of $$\begin{cases} u_t - \Delta u = 0 & \text{in }U \times (0,\infty) \\ \qquad \quad u=0 & \text{on }\partial U \times [0,\infty) \\ \qquad \quad u = g ...
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0answers
14 views

nonincreasing energy [closed]

Hi can someone please help me to answer the following question: Consider a non-homogeneous heat equation on a real line: ut−uxx=f Where x∈R, and t>0, u=0,x∈R, and t=0. Assume that f is a continuous ...
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1answer
21 views

Solution to inhomogeneous first order linear PDE

The homogeneous part of the solution is easy I have a idea but not sure if it is correct. non-homogeneous part of the solution of $a(x,y)u_x+b(x,y)u_y=f(x,y)$ is:$\int_C f(x,y/(a^2+b^2)ds$ where ...
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0answers
26 views

behavior of the solution of the heat equation [closed]

Hi can someone please help me to answer the following question: Consider a non-homogeneous heat equation on a real line: $$u_t − u_{xx} = f(x) \in \Bbb R,$$ where $t > 0, u = 0, x \in \Bbb R$, and ...
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0answers
29 views

Heat equation with inital value [closed]

Let $u(x, t)$ be solution of heat equation $$u_t - u_{xx} = 0$$ for $x \in \mathbb{R}$, $t > 0$, while for $x \in \Bbb R, t = 0$, $u = g$, where $g$ is a twice differentiable function with compact ...
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0answers
28 views

ENERGY OF A telegraph equation

Let u(x, t) be solution of telegraph equation u_tt + αu_t − u_xx = 0 x ∈ R, t > 0, u = g u_t = h x ∈ R t = 0, where g and h are twice differentiable functions with compact support. Assume that u(x, ...
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0answers
31 views

Sturm-Liouville problem with vibrations - probably easy for most.

Trying to do this one... A model for the transverse vibrations of a stretched string with variable density ρ and tension τ (both continuous and strictly positive on the closed interval [0,l]): PDE: ...
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0answers
21 views

Solving a Partial Differential Equation [closed]

Solving Partial Differential Equation $f_{x}-r^{3} (f_{y})^{2}+3xf^{2}=0$. $f=f(x,y)$ is real-valued function of two variables $x,y$ and $f(x,ry)=rf(x,y)$ for $r>0$. Very thanks for your help.
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2answers
46 views

Wave with a source on the half-line

Problem: $u_{tt} - c^2u_{xx}=f(x,t) \quad 0<x<\infty, t>0$ $u(x,0) = 0, u_t (x,0) = 0 \quad (for \quad all\quad 0<x<\infty)$ $u(0,t)=0 \quad (for \quad all\quad t>0)$ Ok so on ...
1
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1answer
58 views

General solution to wave equation of half-line with nonhomogeneous Neumann boundary

Problem Use the general solution to solve the signalling problem with homogeneous wave equation on the half line, homogeneous IC and nonhomogeneous Neumann boundary conditions. Where c>0 is a ...
-1
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1answer
27 views

simple boundary value problem [closed]

The boundary conditions are both for the first derivative. What are the procedures to get this solution?
2
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1answer
58 views

Compactness Sobolev embedding for radial functions on $\mathbb{R}^N$

I want to show: Let $N\geq 2$ and $2< q <2^\ast$. Then the embedding \begin{align} H^1_{\text{rad}}(\mathbb{R}^N)\hookrightarrow L^q(\mathbb{R}^N) \end{align} is compact. I was able to show ...
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0answers
22 views

Question on calculus of functions that take values in Banach Spaces

I've recently started a course on PDEs, and one of the first constructions we made was the following. Say we have a PDE: $$F(t,x_1,\dots,x_n,u,u_t,u_{x_1},\dots,u_{x_n},\dots) = 0$$ with $u(t,x): ...