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

learn more… | top users | synonyms (2)

0
votes
0answers
10 views

Optimal relaxation parameter for the SOR method when solving Poisson equation for non-square problems

Yang and Gobbert in paper "The Optimal Relaxation Parameter for the SOR Method Applied to a Classical Model Problem" ( http://userpages.umbc.edu/~gobbert/papers/YangGobbert2007SOR.pdf ) gave proof ...
1
vote
1answer
34 views

A trivial solution of a PDE

Let $u\in C^1$ in the unit closed disk $\Omega$ be a solution of the PDE $$a(x,y)u_x+b(x,y)u_y=-u $$ Suppose that $a(x,y)x+b(x,y)y>0$ in $\partial\Omega$. Show that $u=0$. Hint: Show ...
1
vote
1answer
18 views

Quasilinear equation and their properties.

Show that the quasilinear differential equation $$u_y+a(u)u_x=0$$ with the initial condition $u(x,0)=h(x)$ has the implicit solution $$u=h(x-a(u)y)$$ Show that there exists some positive ...
0
votes
0answers
83 views

$\frac{d^2 \theta}{dt^2}=9*\frac{d^2 \theta}{dx^2}$ use method of separation of variables to find final solution $\theta(x,t)=$?

Boundary conditions: $\theta_x (0,t)=\theta_x(4\pi,t)=0$ $t>0$ Initial conditions: $\theta(x,0)=3\cos 2x$, $\theta_t(x,0)=1+6 \cos 2x$, $0<x<4\pi$ Use result: $$X''-\lambda*X=0$$ ...
1
vote
1answer
37 views

$L^1$ Estimates involving bi-Laplacian

The following inequality can be shown to be true in the cases $p>1$: If $n\ge 5$, $\frac{1}{q} = \frac{1}{p} - \frac{4}{n}$, then there exists $C_{p,q,n}>0$ such that, for every $f \in ...
0
votes
0answers
25 views

reducing this Bessel differential equation using euler's ode

$$\frac{d^{2}y}{d^{2}z} + y ( 1 + \frac{1}{4z^{2}} - \frac{m^{2}}{z^{2}}) = 0$$ I have the above DE which I managed to transform into using a scaling transformation in response to a prior question ...
0
votes
0answers
27 views

Numerically solving the diffusion-reaction equation with boundary values

I want to solve a nonlinear PDE (steady-state diffusion reaction): $\Delta u = f(u)$ That has the following boundary conditions: $u_y(x,0) = 0$ $u(x,h) = m$ I am trying to solve it via newton's ...
0
votes
1answer
12 views

Laplace-Operator of distribution-valued function (heat equation)

I'm having trouble making sense of an exercise involving this definition of the heat equation: $u'(t) = \Delta (u(t))$, $u(0) = \delta_0$ for $t > 0$ where $u : [0, \infty) \rightarrow ...
2
votes
1answer
25 views

Characteristic equation of the transport equation

(a) Write down the characteristic equations for the PDE $$u_t+b\cdot Du =f \text{ in } \mathbb{R}^n\times(0,\infty)$$ where $b\in \mathbb{R}^n, f=f(x,t)$. (b) Use the characteristic ODE to ...
1
vote
1answer
30 views

functions in $H_0^1(\Omega) \cap C(\overline{\Omega})$ are zero on the boundary

I want to solve the following problem Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set with $C^\infty$-smooth boundary. Show that for any $u \in H_0^1(\Omega) \cap C(\overline{\Omega})$ ...
1
vote
1answer
107 views

What is the weak formulation of this problem?

Find $u\in H_D^1(\Omega)$ such that $-\nabla\cdot(a\nabla u)=0$ in $\Omega$, $\dfrac{\partial u}{\partial n}=g$ on $\Gamma_N$, $u=0$ on $\Gamma_D$. The function $a(x,y)$ is ...
3
votes
0answers
31 views

Minimum Knowledges to precisely calculate PDEs (integral equations)

Basically all I want to do is to calculate (or prove) precisely equations such as $$ \frac{1}{n\alpha(n)r^{n-1}}\int_{\partial B(x,r)} u(y) dS(y) = \frac{1}{n\alpha(n)} \int_{\partial B(x,r)} ...
1
vote
1answer
37 views

When using the method of characteristics, do we ever need the equation for the gradient?

I have a question about the method of characteristics (as defined in the PDE book by Evans). He summarizes the characteristic equations as (see eq. 11 on p. 98): (using $p=Du,z=u$ and the variable ...
0
votes
1answer
29 views

The initial condition for a heat equation with stationary solution subtracted

I am presented with the following question for exam revision: Heat is supplied at a prescribed rate $Q(x) > 0$ (per unit volume) to an isotropic conducting rod that occupies the region ...
2
votes
0answers
41 views

Is there any pde whose solution evolves as a partial Fourier integral?

Is there any partial differential equation such that the its solution evolves as partial Fourier integral (continuous version of partial sum) of a function $f(x)$ which might be an condition or ...
0
votes
0answers
32 views

If $\forall v \in V, \ a(Tu,v)=(u,v)$ is $T$ a bounded an regular operator?

Let $V, H$ two Hilbert spaces infinite dimensional. If the bilinear form $a(.,.)$ satisfies There exists a constant $\alpha>0$ such that $\forall v \in V, \ a(v,v)\geq \alpha \|v\|^2.$ There ...
2
votes
1answer
35 views

Exam questions on sobolev spaces, sobolev embedding [closed]

Hi I am doing some preparation for research in nonlinear PDE. I have almost finished reading the chapter of sobolev spaces and want some questions to test my understanding on various important ...
0
votes
1answer
25 views

Nonexistence for small data

For context of this question, please refer to page 689 of PDE Evans, 2nd edition. My question is at the bottom. As an example, consider this initial-value problem in three space dimensions: ...
0
votes
1answer
16 views

separation of variables 3D cylindrical

Given the wave equation for the displacement $$u(r,\theta,t)$$ in a circular domain $$0<r<a \text,-\pi<\theta<\pi$$ Use the separation of variables to reduce the problem to an ODE. ...
1
vote
1answer
34 views

Cauchy Problem and continuity of derivatives

I've been asked to solve the Cauchy problem: $ \left\{\begin{matrix} 2u_{x}+3u_{y}=0 & \hspace{0.1cm} (1)\\ u(x,0)=|x| & \hspace{0.2cm} (2)& \end{matrix}\right. $ Using the method ...
4
votes
2answers
36 views

Help Solving this 1D Linear Parabolic PDE

Let $u = u(t,x)$ satisfy the PDE $$ \frac{\partial u}{\partial t} = \frac{1}{2}c^2\frac{\partial^2 u}{\partial x^2} + (a + bx)\frac{\partial u}{\partial x} + f u, $$ where $a,b,c,f \in \mathbb{R}$ are ...
1
vote
2answers
24 views

Finding an approximate solution to a differential equation using finite difference method.

I have a differential equation $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=-2$$ on the square $$0 \leq x,y \leq 1$$ subject to the boundary conditions $u=0$ along $x=0$ ...
1
vote
1answer
46 views

Laplace equation on a disk

I have the Laplace equation $$\Delta u=\frac{1}{r} \frac{\partial}{\partial r } \left(r \frac{\partial u}{\partial r} \right)+\frac{1}{r^2} \frac{\partial u }{\partial \theta^2}=0$$ on a unit disk ...
4
votes
2answers
57 views

Finding a characteristic equation of second order PDE?

How to find the characteristic equation of the following PDE $$PDE: (\sin^2 {x} ) u_{xx}+ (\sin {2x})u_{xy}+(\cos^2x)u_{yy}=x$$
3
votes
2answers
36 views

Meaning of the following, partial derivatives..

What is the meaning of $${\partial^kG \over \partial t^k} \in C$$ how is this function explained $G(t,s)$, does it mean that the k-th derivative of $G$ is continuous. I've done some studying on this ...
1
vote
0answers
77 views

Total differential proof , need help understanding. Integration factor.

Now we're trying to find a solution for: $$ \mu(t,x):\qquad(*) \frac{\partial \mu}{\partial x}P- \frac{\partial \mu}{\partial t}Q + \mu\left(\frac{\partial P}{\partial t} - \frac{\partial Q}{\partial ...
1
vote
0answers
16 views

Classification of quasi-linear PDEs

According to my textbook, to classify a quasi-linear PDE into elliptic, parabolic or hyperbolic, one should solve the following set of equations (As an example of course): $$a_{1}\frac{\partial ...
0
votes
1answer
36 views

Extension of Fourier Transform and Plancherel Theorem

I'm very confused with the ideia of extension Fourier transform of $L^1(\mathbb{R}^n)$ to $L^2 (\mathbb{R}^n)$. I start with a $u\in L^1(\mathbb{R}^n)$ and I use the limit and the Banach property to ...
3
votes
2answers
32 views

dirichlet principle: why $u-g\in W_0^{1,2}(\Omega)$?

Let $D\subset\mathbb R^n$ be open and bounded. Consider $\Delta u=f$ in $\Omega$ and $u=g$ on $\partial\Omega$. Let $g\in W^{1,2}(D)$ and $f\in L^\infty(D)$. Then the minimizer of $$ I(u)=\int_\Omega ...
1
vote
2answers
34 views

Characterization of $H^k$ by Fourier transform

Let $u\in H^k(\mathbb{R}^n)$ a function with complex-valued. Question 1: If $u\in H^k(\mathbb{R}^n)$ and $D^\alpha u\in L_2(\mathbb{R}^n)$, by definition for each $|\alpha|\le k,$ then we have ...
0
votes
1answer
30 views

$u_{tt} + au_t = c^2u_{xx}$ for some $a>0$ implies that the energy is not increasing

Could someone please help me with the following question? I got stuck somewhere. Given a function $u(t,x)$ satisfying the relationship: $$ u_{tt} + au_t \ = \ c^2u_{xx} \qquad \text{ for some } ...
1
vote
0answers
25 views

Calculus of variations - unilateral constraints [duplicate]

Question about Evans states, chapter 8.4.2! We have $I[w] := \int_U \frac{1}{2}|Dw|^2 - fw\, dx$, among all functions $w$ belonging to the set $$\mathcal{A} : = \{w \in H_0^1(U) : w \geq h \, ...
0
votes
0answers
26 views

Connection between parabolic and elliptic PDEs

I am trying to verify the well-posedness of some PDEs which I solve numerically. There is a paper which proves the well-posedness of a similar problem using Ladyzhenskaya's book on hyperbolic ...
1
vote
1answer
56 views

Trouble understanding method of characteristics-PDE for solving the Cauchy Problem

So I am trying to understand the non-algorithmic part of the method of characteristics for solving a first order quasilinear PDE: $ a(x,y,u)u_{x}+b(x,y,u)u_{y}=c(x,y,u) \hspace{1cm } $ (1) I ...
0
votes
0answers
21 views

Spherical equation with Laplace Transform

I have a question about my approach to another question asked here: $$u_{tt}=C^2[u_{xx}+u_{yy}+u_{zz}],\qquad-\infty<x,y,z<\infty,\qquad t>0$$ $$BC: u(x,y,z,t)\rightarrow 0\qquad as \qquad ...
1
vote
0answers
71 views

Cauchy problem for nolinear PDE

Problem: Find the solution of the following Cauchy problem $$ u_{ttxx}(x,t) = (u_{tt}(x,t))^2 \quad \quad (1) $$ $$ u(x, 0) = \phi_1(x) \quad u_t(x, 0) = \phi_2(x) $$ $$ u_{tt}(x, 0) = \phi_3(x) ...
1
vote
2answers
38 views

Solve the following PDE using Fourier transform

Solve the following 3-D wave equation using Fourier transform $$PDE: u_{tt}=C^2[u_{xx}+u_{yy}+u_{zz}],\qquad-\infty<x,y,z<\infty,\qquad t>0$$ $$BC: u(x,y,z,t)\rightarrow 0\qquad as \qquad ...
1
vote
1answer
26 views

The effect of the CFL number in the numerical solution in this conservation law

I've been studying the very basics of numerical methods applied to conservation laws, and I'm having trouble understanding the role of the CFL number in the upwind scheme. I want to understand it (if ...
1
vote
1answer
51 views

The functional $I[w] = \int_U \frac{1}{2} |Dw|^2 - fw \, dx$ is weakly lower semicontinuous

I am studying calculus of variation, and I need to prove that $I[w] = \int_U \frac{1}{2} |Dw|^2 - fw \, dx$ with $f \in L^2(U)$ is weakly lower semicontinuous on $H_0^1(U)$. In classes, I only ...
2
votes
0answers
42 views

Is my change of variables of a PDE correct?

I've calculated a lot and checked the first derivatives with wolframalpha. Still I'm not sure if I have done everything correctly, could someone have a look please? Original PDE: \begin{align*} u_t- ...
3
votes
1answer
30 views

Laplace equation in a circle - where is my mistake

We want to solve $r^2u''_{rr}+ru'_r+u''_{\theta \theta}=0$ where $0\leq \theta <2\pi$ and $0\leq r \leq 2$, given that $u(2,\theta)=\cos(2\theta)$. I managed to work out a simple solution, but ...
2
votes
1answer
39 views

Solve the lst order pde using the method of characteristics

Solve the following lst order $pde$ using the method of characteristics: $ u_t + 3t^2 u_x = -u$ With $ u(x,0) = \begin{cases} e^{-1} & \quad x<0\\ e^{x-1} & \quad ...
1
vote
0answers
26 views

Flux estimate for semilinear wave equation

This is from §12.1.2 (pages 660-661) of PDE Evans, 2nd edition. My question is at the bottom of this post. DEFINITION. Fix $x_0 \in \mathbb{R}^n$, $t_0 > 0$ and define the backwards wave cone ...
2
votes
2answers
29 views

Solving a PDE with the Poisson equation

New to PDEs. This one is apparently solved with the Poisson equation but I have no idea how. I would appreciate any help. $$u_t=u_{xx}+e^{-t}\sin t$$ and $$u|_{t=0}=e^{-x^2}$$
3
votes
1answer
16 views

$ \left( \ u_{tt} \ = \ c^2u_{xx}, \quad u(0,x) \ = \ 0, \quad u_t(0,x) \ = \ 0 \,\text{ decay }\right) \quad \Longrightarrow \quad u \equiv 0 $ ?

I have to show the following: Let $u$ be a classical solution of the following system: $$ u_{tt} \ = \ c^2u_{xx}, \quad u(0,x) \ = \ 0, \quad u_t(0,x) \ = \ 0 $$ Satisfyying the decay requirement: ...
3
votes
2answers
48 views

Problem involving divergence theorem and laplacian squared

Let $B=\{x\in\mathbb{R}^m:|x|<1\}$ and $u\in C^3(\bar{B})$. Suppose that $u=0$ on the boundary $\partial B$ of $B$. Show that $$ \int_B(|\Delta ...
2
votes
1answer
17 views

Positivity of solution to Laplace equation

I'm studying PDE and at the moment I'm reading L. Evans' book. The strong maximum principle states that; if $u\in C^2(U)\cup C(\bar U)$ is harmonic in $U$, where $U$ is connected and if there exists ...
4
votes
0answers
85 views

Heat equation proving smoothness

I have a question regarding a PDE course: Let $T$ be the strongly continuous semigroup which belongs to the heat equation, thus with generator $A$ is the Laplacian. Suppose we have $g \in ...
0
votes
1answer
30 views

Dirichlet energy

From PDE Evans, 2nd edition: Chapter 8, Exercise 17: Let $u,\hat{u} \in H_0^1(U)$ both be positive minimizers of the Dirichlet energy $$I[w] := \int_U |Dw|^2 \, dx,$$ subject to the constraint ...
0
votes
1answer
29 views

1D green function

I want to solve the problem $$ \frac {\partial^2u}{\partial t^2} + \alpha \frac {\partial u} {\partial t} = f(t)$$ $$u(0)=u'(0)=0$$ $$f(t)=exp(-\beta t)$$ Using distributions (Green functions). ...