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

learn more… | top users | synonyms (2)

0
votes
0answers
14 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 ...
0
votes
0answers
16 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
27 views

ODEs arising from harder method of characteristics questions

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
26 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
48 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
17 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
26 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
41 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
15 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) ...
1
vote
0answers
14 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. $u_{tt}=9u_{xx}$, $x>0$ , $t>0$ $u(x,t)=e^{-x}$ ...
0
votes
0answers
9 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
13 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
32 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) ...
0
votes
1answer
8 views

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
33 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
8 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
27 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
28 views

Reduce the PDE to Canonical form. [on hold]

$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 ...
1
vote
1answer
8 views

How to calculate the transition density for a multivariate jump process

I have the following stochastic process: $dX = (A-I)XdN$, where $X$ is a $2\times1$ vector of random variables, $A$ is a constant, real, symmetric, $2\times2$ matrix, $I$ is the identity matrix and ...
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) = ...
1
vote
0answers
19 views

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

Let $X$ be a separable Banach space endowed with a centered nondegenerate Gaussian measure $\gamma$. Then consider $f\in\mathcal{F}C^{1}_{b}(X)$. I want to prove that $f$ is in ...
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
35 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
16 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 ...
0
votes
1answer
20 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
32 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 ...
1
vote
1answer
13 views

PDE Method of characteristics with initial condition

I wanted to solve the following PDE with initial condition $$ u_t+tu_x=0, $$ $$ u(x,1)=f(x),$$ where $f(x)$ is a given function, using the method of characteristics. I explain what I have done. First ...
1
vote
1answer
18 views

How to show contradiction in the Hardy inequality when the singularity power is greater that 2.

Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have $$ \Lambda \int_{\Omega} ...
2
votes
0answers
21 views

fourier transform for pde equation

I was solving the pde using fourier transform: $u_{tt}-u_{xx}+m^2u=0$ with initial values $u(0,x)=f(x)$ and $u_t(0,x)=g(x)$. I have received the answer $$U(t,k)=Ae^{-it \sqrt {k^2+m^2}}+Be^{it \sqrt ...
0
votes
0answers
30 views

Is this partial differential equation solvable?

Ok so I am asked to set up a partial differential equation and then motivate why it is solvable. I'm only 2 weeks into my course so we are not asked to solve anything yet. However, if someone would ...
0
votes
0answers
15 views

About PDE solution: H^1 norm bounded = bounded in L^2?

Let $\Omega \subset \mathbb{R}^d~(d=2,3)$ be an open bounded set with Lipschitz continuous boundary $\Gamma$. We assume that $\Gamma$ consists of two disjointed parts, i.e, $\Gamma = \Gamma_{c} \cup ...
0
votes
2answers
42 views

how come $C^{1}_{0}$ is not complete under norm $||.||_{1,2}$

why the space $C^{1}_{0}$ is not complete under the norm $||.||_{1,2}$? by some counter example $||u||1,2=(\int_{\Omega} (|\nabla u |^2+|u|^2))^{1/2}$
1
vote
0answers
28 views

Are the set of probability functions with compact support in a fixed closed ball complete under the Wasserstein norm?

Let $B_R$ be a closed ball of radius $R$ in the space $\mathbb{R}^d$. As the title suggests I have this feeling that the set of functions $$S:= \left\lbrace f:\mathbb{R}^d \to \mathbb{R} ...
0
votes
1answer
32 views

Fourier cosine series giving nonsense answer

I'm currently trying to find the cosine Fourier series of $f(x) = \left | \sin \frac{\pi n }{L} x\right |$ on the interval $0 < x < L$. I first started by calculating the first term of the ...
-1
votes
0answers
15 views

Sketching partial differential equation

I have found a solution to my pde however I want to try and sketch it however I don't know where to start. pde
0
votes
0answers
23 views

Find the value of the partial derivative

Find the value of $\dfrac{1}{D_x^2-D_y^2}{\sin(x-y)};D_x=\dfrac{\partial }{\partial x};D_y=\dfrac{\partial }{\partial y}$ I am used to finding the value of $\dfrac{1}{F(D)}\sin ax$ where $D^2$ ...
0
votes
0answers
38 views

Solution of boundary value problem using Fourier series

I want to solve the following PDE using Fourier series. $u(x,y): \Omega \to \mathbb{R}$, $\Omega=(0,\pi)\times (0,2\pi)$ $u-3u_{xx}-u_{yy}= 3\sin(2x)-\sin(5x)$ $u_{xx}$ and $u_{yy}$ are second ...
0
votes
0answers
45 views

What is a very good book about Green's Functions?

I would really like to learn in a good way about Green's Functions. Their use, their derivations, how to use them to solve partial differential equations and so on. I'm a theoretical physicist so my ...
1
vote
0answers
24 views

Spherical Bessel expansion of Green function

Any reference/advice would be good. I can use eigenfunction to solve the Green function for $$\Delta u(x) + k^2 u(x) = \delta(x - y)$$ boundary condition given as $u = 0$ on $\partial B(1)$, unit ...