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

learn more… | top users | synonyms (2)

1
vote
0answers
6 views

Differentiation with integration region depending on $x$ to solve for decreasing energy of wave equation

I want to show that for the general wave equation $u_{tt} - \nabla \cdot (c^2\nabla u) + qu = 0, \quad u(x, 0) = \phi(x), \quad u_t(x, 0) = \phi(x)$ we have $$ E(t) = \int_{|x-x_0| < R_0 - c_2t} ...
1
vote
0answers
12 views

Combining two results from partial integration

I have a set of two PDEs: $$\partial_{\tau}\theta+\partial_{\eta}\psi=0$$ $$\partial_{\tau}\psi=-\partial_{\eta}\theta+\alpha\partial_{\eta}^{2}\psi$$ These can be combined into a wave equation of ...
2
votes
0answers
15 views

Heat equation with mixed boundary conditions

I am trying to solve the following problem $$\left\{\begin{matrix}\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},& 0<x<1,t>0,\\ u(0,t)=\frac{\partial u}{\partial ...
0
votes
1answer
9 views

Taking derivative of energy of wave equation

Consider the variable coefficient, real valued wave equation $$ u_{tt} - \nabla \cdot (c^2 \nabla u) + qu = 0, \quad u(x,0) = \phi(x), \quad u_t(x, 0) = \phi(x), $$ where $c, q \geq 0$ depend only on ...
1
vote
1answer
15 views

Apply Periodic Boundary to PDE (Fourier Transform)

Use Fourier Transform to solve the BVP: \begin{cases} u_t + a u_x - b u_{xx} = 0, & \mbox{for } x \in [-1,1] \\ u(x,0) = f(x) \\ u(x+2,t) = u(x,t) \end{cases} I solved the problem (attached); ...
1
vote
1answer
12 views

uniqueness of heat equations and the squared integrable assumption

I am looking at the classical proof of uniqueness for the heat equation in Evans. Clearly, we differentiate under the integral sign of the square of $w$. A very basic question is, why are the ...
2
votes
0answers
19 views

Poincare Inequality in n-dimensions

I am trying to prove the Poincare Inequality on a n-dimensional box. That is a domain $ \Omega = (0,1)^n$ for $f(x) \epsilon H^{1}_{0}(\Omega) $, show there exists a constant $C$ such that ...
0
votes
0answers
18 views

Nondimensionalization of PDE with non constants coefficients

I need to try to solve, by some numerical method, a variant of the Navier-Cauchy equation of motion for an elastic, linear, isotrophic, not homogeneus body: $$ \rho_0 (\ddot{\textbf{u}} - ...
0
votes
0answers
26 views

Solving PDE's: Laplace equation on semi-infinite cylinder

Solve Laplace’s equation $\nabla^2$u = 0 inside a semi-infinite cylinder 0 < z, 0 < r < 1 with boundary conditions u(r = 1, $\theta$, z) = $e^{−z}$ and u(r, $\theta$, z = 0) = 0, where (r, ...
1
vote
0answers
4 views

Equivalence of first order quasilinear PDE to linear PDE

Given a system of nonlinear PDE of the special form: $\sum_{i=1}^n A_i(x, \phi) - \frac{\partial \phi_j}{\partial x_i} = B_j(x,\phi) $ $(1)$ with $(j=1,...,m)$ and $x \in R^n,\phi \in R^m$. If we ...
0
votes
0answers
17 views

Inequality in the proof of Weak Harnack Inequality

Let $\Omega \subset \mathbb{R}^{n}$ a bounded domain s.t $B_{1} \subset \Omega$ , $u \in H^{1}(\Omega)$ a nonnegative supersolution in the weak sense of the equation $Lu=-D_{i}(a_{ij}(x)D_{j}u)$ ...
0
votes
0answers
15 views

Show that the function is identically Zero in certain subset

We are given a open ball D (radius = 1) in $\mathbb R^2$. and let $\{x_n\}$ be the dense sequence in the set D. Around each point $x_n$ we make a hole of radius $r_n$. The sequence $r_n$ satisfy the ...
0
votes
0answers
17 views

long time behavior of heat equation

Given the heat equation \begin{align} {{u}_{t}}-{{u}_{xx}}&=0,\quad x\in \mathbb R,\,t>0 \\ u\left( x,0 \right)&=f\left( x \right),\quad x\in \mathbb R. \end{align} If ...
1
vote
0answers
10 views

PDE: Traffic problem with initial density a piecewise function

I was studying a bit of PDE and I found very interesting traffic problems, however, I have some troubles to deal with them. I wanted to solve the following: Consider the traffic problem ...
0
votes
0answers
15 views

Weak derivative of a piecewise defined function

I am currently looking at these online notes on PDEs, page 59. How does it follow that if $f^R = \phi(x/R) f(x)$ $ \phi(x) = \left\{\def\arraystretch{1.2}% \begin{array}{@{}c@{\quad}l@{}} 1 ...
1
vote
1answer
38 views

Solve $A \partial_t w + B \partial_t\partial_x^4 w + C \partial_x^4 w + \partial_t^2 w = 0$

a non-mathematician wants me to solve a PDE. The problem is that I don't know a lot of theory to solve PDE's except the fouriertransform. This is the PDE $$A \partial_t w + B \partial_t\partial_x^4 w ...
0
votes
2answers
45 views

Method of characteristics - finding the particular solution using initial conditions

I am trying to use the method from my previous question to solve this PDE: $$ 3u_x + 2u_t = \cos x $$ with initial condition $u(x,0) = x^2$. So I need to solve these: \begin{align} \frac{dx}{ds} ...
0
votes
0answers
10 views

capillary surface problem [on hold]

Consider the capillary surface problem (⋆) )   Du  div 1 + |Du|2 = κu in Ω on∂Ω,  Dηu  1+|Du|2 =β where κ > 0, η is the outward pointing unit normal to ∂Ω and β ∈ C1(Ω) satisfies |β| ≤ 1 ...
1
vote
2answers
33 views

Method of characteristics - eliminating variables

I am trying to follow a guide for the method of characteristics; quoting the first example: We use the method of characteristics to solve the problem $ 2u_x - u_y = 0, \;\; u(x, 0) = f(x) $ ...
0
votes
0answers
59 views

About the gradient of a function in $H^{1}(\Omega)$

let $\Omega \in \mathbb{R}^{n}$ a bounded domain and $u \in H^{1}(\Omega)$ a real function. In the Leoni's Book - A First Course in Sobolev Spaces, the author define $\nabla u = (D_{1} u,\dots, ...
1
vote
1answer
18 views

checking that an initial condition holds for the heat equation

I'm trying to follow a video lecture on solving the heat equation. $I) \space u_t = ku_{xx}, x \in \mathbb{R}, t > 0$ $II) \space u(x,0)=\phi (x), $ $k$ is const, $\phi (x) $ is a ...
0
votes
1answer
33 views

More equations than unknowns for maxwell equations?

I had one curiosity regarding maxwell equations in 3-D From the curl equations, you get 6 unknowns, with 6 equations. The divergence equations add 2 additional equations. When these are combined, we ...
2
votes
1answer
52 views

How do I determine if the equation is a conservation law?

We have the PDE $\frac{\partial u}{\partial t}+a(x,y)\frac{\partial u}{\partial x}+b(x,y)\frac{\partial u}{\partial y}=0$. What would be conditions on $a$ and $b$ for the equation to constitute a ...
2
votes
0answers
20 views

Eigenvalues and Eigenvectors of an hyperbolic partial differential equations $\partial_t W + A \partial_x W = 0$

I read in a article dealing with a hyperbolic partial differential equations this statement : For any system of hyperbolic partial differential equations (pde), expressed as (1) ...
2
votes
0answers
22 views

PDE - three restrictions, wave equation (1 dimension)

I'm not very good at PDEs but this particular problem seems... Strange. It requires that the answer be "continuous (!!)" all in bold. \begin{align} u_{tt}&=9u_{xx},\quad x>0,\, t>0, \\ ...
0
votes
0answers
10 views

Question with D'Alembert formular

We have the solution of the wave equation $u_{tt}-u_{xx} = 0$ with boundary condition $u(0,t) = u(L,t) =0$ is $u(x,t) = \int_{t-x}^{t+x}u_x(0,s)ds$. My question is that can we replace the formular as ...
2
votes
0answers
14 views

Where does the name “tracking type problem” come from?

In PDE-constrained optimization problems, the distributed constrol problem $$ \begin{array}{ll} \displaystyle \min_{y,u} & J(y,u) = \frac{1}{2}\|y-y_d\|_{L^2(\Omega)}^2 + ...
0
votes
2answers
33 views

Separation of Variables for second order PDE

I have a PDE that I have attempted to solve using the method of 'separation of variables' $$u_t = (1+2t)u_{xx} \,\,\,\, 0 \leq x < \pi, t \geq 0 $$ With initial and boundary conditions: $$u(0,t) ...
1
vote
1answer
16 views
+50

Where can i find references to proofs of 1D,2D (partially 3D) Navier Stokes Equation?

I'm currently trying to get into PDE's and as part of a course i'm focusing on proofs on existence of solutions to the Navier-Stokes Equations. Although existence of solutions has been proved for 1D ...
1
vote
0answers
27 views

Solve this recurrence relation via a first order partial differential equation?

Find a general formula for $a_{n,k}$ , for $n,k\geq1$. We have initial values $a_{1,1}=1$, and $a_{1,k}=0$ for $k>1$. The recurrence relation is: $a_{n+1,1}=-a_{n,1}$ , for $n\geq1$ and ...
0
votes
0answers
36 views

solution of linear elliptic equation

Can you please help me to show that if $\Omega\subset\mathbb R^n$ is a $C^2$ domain and $f$ is an application which belongs to $L^2(Ω)$ and $u$ is a weak solution of the linear elliptic equation: ...
0
votes
0answers
9 views

transform to autonomous linear equation

I would like to ask that which are the equations could be write as the form of autonomous linear equation $u_t = Au_{xx}$ I just known the heat equation, (we take $A = Lapacian$) or the wave ...
1
vote
0answers
29 views

rayleigh quotient of eigenvalue problem (sturm liouville theory and partial differential equations)

I am reading "A First Course in Partial Differential Equations with Complex Variables and Transform Methods" (Weinberger, p. 168). if we have the eigenvalue problem $$ (pu')'- qu + \lambda \rho u = 0 ...
1
vote
0answers
10 views

PDE Von Neumann Problem- Physical Interpretation

The Von Neumann Problem is as such: $\Delta u = f(x,y,z)$ in $\ D$ $\frac {\partial u} {\partial n} = 0$ on bdy $\ D$. Using this you can prove that for there to be a solution to this Von Neumann ...
-2
votes
0answers
29 views

Reduce the PDE to Canonical form. [closed]

$u_{xx} + 5u_{xy} + 6u_{yy} = 0 $ Find the fundamental solution if possible. I think what needs to happen is you need to find dy/dx using the quadratic formula, then simplify the equation using a ...
0
votes
1answer
12 views

Solving the heat equation with piecewise IC

I have the solution to the heat equation, with the BC's and everything but the IC applied. So I am just trying to solve for the coefficients, the solution without the coefficients is $$u(x,t) = ...
2
votes
0answers
50 views
+100

Proving that if $f\in\mathcal{F}C^{1}_{b}(X)$ then $f\in W^{1,p}(X,\gamma$) for $p>1$

Let $X$ be a separable Banach space endowed with a centered nondegenerate Gaussian measure $\gamma$ and $H$ the Cameron-Martin space. Then consider $f\in\mathcal{F}C^{1}_{b}(X)$. I want to prove that ...
1
vote
0answers
22 views

Solve IVP with the method of characteristics (quasilinear PDE) (+shockwaves)

The question i just can't figure out one bit is: Solve for the continious function $u(x,t)$: $$u_t+(1-2u)u_x=0 $$ $$-\infty < x < \infty, t>0$$ $$u(x,0)= \left\{ \begin{matrix} \frac{1}{4} ...
1
vote
1answer
20 views

Prove Continuity of a multivarible function.

I'm trying to prove the following: Let $f:\mathbb{R^n}\times\mathbb{R} \to \mathbb{R}$ be a continuous function. We define $$F(x,t) = \int_{0}^{t}f(x,s)ds $$ Prove that F is also continuous. I ...
1
vote
1answer
12 views

Non autonomous system?

Let us consider the wave equation $u_{tt}-u_{xx}=0$. I have the two following questions: a) If we have the boundary condition $u(0,t) = u(\pi+t,t) = 0$, for all $0 < t < \infty $ the given ...
1
vote
0answers
12 views

Potential theory solution for Variable coefficient Poisson with Dirichlet Boundary conditions

I am looking for a potential theory representation for the following equation in $2$D: $$\vec{\nabla} \cdot \left(a(x) \vec{\nabla}u\right) = 0 \,\, \forall x \in \Omega \,\, (\spadesuit)$$ $$u = g ...
0
votes
0answers
10 views

Poisson integral formula the following Harnack inequality and Liouville theorem

Suppose $u$ is a nonnegative harmonic function in $B_R(x_0)\subset \mathbb{R}^n$. Prove by the Poisson integral formula the following Harnack inequality: $$ ...
0
votes
0answers
15 views

Find the Green's function for the Laplace operator in the upper half space

Find the Green's function for the Laplace operator in the upper half space ($x_n>0$) and then derive a formal integral representation for a solution of the Dirichlet problem $$ \Delta u=0 \text{ ...
1
vote
1answer
36 views

Inequality for proof of Density Theorem

Someone could help me white this question or indicate some reference? Lemma:For any $\epsilon>0$ There exits a $C=C(n,\epsilon)$ such that for $u \in H^{1}(B_{1})$ with $|\{ x \in B_{1} ; ...
0
votes
1answer
17 views

What does $u_0(x)$ represent?

I am looking at the heat equation and in my notes it says the initial temperature distribution $u(0,x)=u_0(x)$. what does this mean? What does $u_0(x)$ represent?
0
votes
1answer
9 views

Where does $u(t,x) \to u(t,x)-a-(b-a)x$ come from?

I know the heat equation is $$\frac{\partial}{\partial t} u(t,x)=\frac{\partial^2}{\partial x^2} u(t,x)$$ I know that $u(t,x)$ is the temperature distribution at time $t$ at the point $x$. We assume ...
0
votes
0answers
11 views

Which numerical method gives the most accurate solutions of Helmholtz equation for arbitrary domains?

There are many numerical methods for the solutions of PDE's such as FDM, FEM, SEM, Meshfree methods etc. I'm wondering which method gives the most accurate Dirichlet eigenvalues (and corresponding ...
1
vote
1answer
22 views

Reducing a PDE to a dimensionless form with change of variables

I am working through the following example to refresh my memory on how to use the chain rule when changing variables: Change of variables (PDE) \begin{equation} \begin{split} ...
2
votes
1answer
34 views

Analyze : $u_t-u^2u_x +cu =0, u(x,0)=g(x)$

Analyze : $u_t-u^2u_x +cu =0 $, $ u(x,0)=g(x)$. From This we have following $$\begin{align} \frac{dt}{ds} &=1 \\ \frac{du}{ds} &=c \\ \frac{dx}{ds} &=-u^2 \end{align}$$ then how to ...
0
votes
0answers
24 views

heat equation on a surface

Probably I have not well understood the heat equation: please, can you confirm or correct the followings ? (The question raised in this post is similar to Heat Equation on Manifold but they don't ...