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

learn more… | top users | synonyms (2)

1
vote
1answer
316 views

question in Evans PDE book

This question might sound naive but on p.34 of his book, he is considering the Poisson-dirichlet problem with $\Delta u=-f$ on $U$ with $u=g$ on $\partial U$. He then derives a formula for the general ...
1
vote
1answer
112 views

Find solution of PDE $x \frac {\partial z} {\partial x} +y \frac {\partial z} {\partial y}=z$

Problem:Find solution of Cauchy problem for the first order PDE $x \frac {\partial z} {\partial x} +y \frac {\partial z} {\partial y}=z$,on $ D= {(x,y,z): x^2 +y^2 \neq0,z>0} $ with initial ...
1
vote
1answer
61 views

$\Delta f = 0$, $f(x) = \phi(r)$. Must $\phi'(r)>0$?

I'm reading Folland PDE and Corollary 2.3 says if $f(x)=\phi(r)$ is radial on $\mathbb{R}^n$, then $\Delta f=0$ on $\mathbb{R}^n\setminus\{0\}$ iff $\phi(r)=a+br^{2-n}$ for $n\neq 2$ and ...
1
vote
2answers
57 views

rewrite $\frac{\partial \theta}{\partial t}=k \frac{\partial^2 \theta}{\partial x^2}$ as a DE with two new variables $q_1$ and $q_2$

I am given the differential equation: $$\frac{\partial \theta}{\partial t}=k \frac{\partial^2 \theta}{\partial x^2}$$ Use the change of variables $q_1(x,t) = \frac{x^2}{kt}$ and $q_2 ...
1
vote
2answers
74 views

Comparison between Bessel's coefficients

The spatial solution is written as $$\Phi_k(r) = r^{1-\frac{d}{2}} \left(c_1 J_{1-\frac{d}{2}}(k r) + c_2 Y_{-1+\frac{d}{2}}(kr)\right).$$ In the case $d=3$, the solutions can be written as ...
1
vote
1answer
74 views

How to show that $\int_0^T (u_m(t), u_m(t))_{H} \to \int_0^T (u(t), u(t))?$ (Bochner space)

Let $p \geq 1.$ Let $$W = \{ u \in L^p(0,T;V) : u' \in L^{q}(0,T;V^*)\}$$ where $V \subset H \subset V$ is a Gelfand triple ($V$ Banach and $H$ Hilbert), and $p$ and $q$ are conjugate indices. The ...
1
vote
2answers
139 views

Laplace operator's interpretation (Laplacian)

What is your interpretation of Laplace operator? When evaluating Laplacian of some scalar field at a given point one can get a value. What does this value tell us about the field or it's behaviour in ...
1
vote
2answers
185 views

Burger's equation

I couldn't solve this problem, can you help me please? The Burger equation $$ u_y + uu_x = 0 $$ $ - \infty < x < \infty $ , $ y > 0 $ , $ u(x,0)=f(x) $ My question; is there any solution ...
1
vote
1answer
146 views

About Mean Value Property of Harmonic Function

I know the question may seem foolish to you but I am not quite sure how to show it in a decent way. My problem is to show that for bounded Borel measurable $f:\mathbb{D}^2\to\mathbb{R}$, (D1) is ...
1
vote
1answer
62 views

question about infimun

Let $\Omega \subset R^n$ a open set and $p >1$. Let $K \subset \Omega$ a compact set. Define $$A:= \{ u \in C^{\infty}_{0} (\Omega) ; \ u \geq 1 \text{ on } K\}$$ $$ B:= \{ u \in C^{\infty}_{0} ...
1
vote
1answer
31 views

How to solve for $\Gamma(X,t)$ in $\Gamma_{t,X} = S(X,t) \Gamma_X$?

Given this equation $$\frac{\partial^2 \Gamma}{\partial t \partial X} = S(X,t) \frac{\partial \Gamma}{\partial X},$$ how do you solve for $\Gamma(X,t)$? $S(X,t)$ is unknown and we impose conditions ...
1
vote
1answer
34 views

What is the definition of $H^{-k}(\mathbb{R}^n)$ and its norm?

What is the definition of $H^{-k}(\mathbb{R}^n)$ and its norm? How can I understand the fact $$\|f\|_{H^{-k}(\mathbb{R}^n)}=\|(I-\triangle)^{-k}f\|_{H^{k}(\mathbb{R}^n)}.$$
1
vote
1answer
101 views

Why this functional isn't differentiable?

I saw in the article Alt, H. M. and Caffarelli, L. A. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math., 325, (1981), 105–144. That the functional ...
1
vote
1answer
165 views

Numerical methods for a second order PDE boundary value problem

Is there any numerical method (like the method of splitting of variables for the equation of 2D-diffusion) for solving the following boundary value problem $$\left\{\begin{aligned}&\frac{\partial ...
1
vote
1answer
70 views

Over-constrained general solution to wave equation

d'Alembert's formula states that the general solution to the one-dimensional wave equation is $$ u(x,t) = f(x+ct) + g(x-ct).$$ for any well-behaved functions $f$ and $g$. This is a well-known and ...
1
vote
1answer
65 views

how to solve this partial differential equation

while in its equilibrium position a uniform string stretched between the points (0,0) and (ℓ,0) (hint cn=0 since equilibruim)
1
vote
1answer
44 views

Is there a function $u$ with $g(u)$ regular whose “truncation” $g(T(u))$ is not regular?

I hope the title is not to misleading. Assume you have a continuous function $g:R\to R$ and a Lebesgue measurable function $u:\Omega\to R$ for some bounded domain $\Omega$ such that $g(u)\in ...
1
vote
1answer
171 views

Smooth solutions of the Laplace equation with Neumann data

Let $D$ a connected domain, $\Delta u = 0$ on $D$ and $\partial_n u = 0$ on $\partial D$. Using energy methods I can show, that two solutions of this problem are unique up to a constant. Now, how can ...
1
vote
1answer
117 views

Solving the equation $\nabla u=f$.

Let $\Omega\subset\mathbb{R}^N$ be a open set, $\mathcal{D}(\Omega)=C_0^\infty(\Omega)$ and $D'(\Omega)$ the set of distributions. Suppose that $f_i\in \mathcal{D}'(\Omega)$ for $i=1,\ldots,N$. Define ...
1
vote
1answer
56 views

$\bigcup_{n}V_n$ is dense in $V$ implies $\bigcup_{n}L^2(0,T;V_n)$ is dense in $L^2(0,T;V)$?

Let $V$ be a separable Hilbert space with basis $w_j$ and let $V_n$ denote the linear span of $w_j$ for $j=1,...,n$. Clearly $V_n$ are Hilbert spaces and $V_n \subset V_{n+1}$ for all $n$. We have ...
1
vote
1answer
33 views

transforming it into a heat equation (how I write down the solution)

$u_t + a\cdot\bigtriangledown u + bu = \Delta u $ $ u(x,0) = f(x) $. $a$ and $b$ are constants... I wish to write down explicit formula for a solution. I tried to make some change of variables to get ...
1
vote
1answer
1k views

Solving Laplace's equation using finite differences

I having coding in MatLab to approximate solutions to Laplace's equation in 2D using finite differences. I was able to do it without much problem. I learnt about how to implement this using this: ...
1
vote
1answer
167 views

Mean Value Property of Harmonic Functions

I can't prove this theorem: "Let $\Omega$ is a bounded domain, $u\in C^2(\Omega)$ satisfy $\Delta u=0(\geq0,\leq0)$, then for any ball $B=B_R(y)\subset \subset \Omega$, we have ...
1
vote
1answer
509 views

Relationship between Turing bifurcation, saddle-node bifurcation, and Hopf bifurcation?

Quoting from http://jxshix.people.wm.edu/2009-harbin-course/mississippi-bifurcation-2.pdf a Turing bifurcation occurs when for an ODE and related PDE $u' = f(u,v), v' = g(u,v)$ $u_t = d_1 \nabla ...
1
vote
1answer
372 views

A Semi-infinite right circular cylinder problem

A semi-infinite right circular cylinder whose axis lie along the $z$ axis, has its base on the $x$-$y$ plane.The base is maintained at a constant potential $V_0$ and the side of the cylinder is ...
1
vote
1answer
68 views

explain $df(tx).x = \sum_{i=1}^n {\partial f\over \partial x_i}(tx)x_i \hspace{1cm} x\in \mathbb R^n$

the question is : let $U$ be a Neighbourhood of the origine of $R^n$ and : $x\in U \Rightarrow tx \in U , \forall t\in U $ let f be a numeric function defined in U , and $f(0)= 0$ if we have ...
1
vote
2answers
50 views

Inequality for solution of boundary-value problem

Let $\varphi(x,t)$ be a sufficiently smooth solution of the problem $$\left\{\begin{array}{rcll} \frac{\partial\varphi}{\partial t}(x,t) - \frac{\partial^2\varphi}{\partial x^2}(x,t) & = & ...
1
vote
1answer
258 views

Parabolic PDEs: Boundary conditions

I'm working through Pinchover and Rubinstein's "Introduction to Partial Differential Equations" and am trying to understand the motivation for studying Sturm Liouville problems. To this end, I am ...
1
vote
1answer
185 views

the first eigenfunction of Dirichlet problem

Let $\Omega$ be a bounded planar domain which has a axis of symmetry and $T:\Bbb{R}^2\longrightarrow\Bbb{R}^2$ symmetry with respect to this axis. Let $u_{1}(x)$ be the first eigenfunction of ...
1
vote
1answer
97 views

solution to heat equation in a particular case

$$\frac{\partial u}{\partial t}(t,x)-\frac{1}{2}\frac{\partial^2 u}{\partial x^2}(t,x)=0,\ \ t>0, x\in\mathbb{R}$$ $$u(0,x)=\max(x,0)$$ $$\frac{\partial v}{\partial ...
1
vote
1answer
146 views

laplace equation in a rectangle with boundary condition

$u_{xx}+u_{yy}=0 \quad in \quad the \quad rectangle \quad 0<x<a \quad 0<y<1$ $u=0 \quad on \quad y=1$ $u=j(y) \quad on \quad x=0 $ $u_y +u=0 \quad on \quad y=0$ $u_x=0 \quad on ...
1
vote
2answers
519 views

Diffusion process. Distribution vs transition probability.

I need confirmation on the following problem: Take a SDE of the form: \begin{equation} dX_t=a(X_t,t)dt+b(X_t,t)dW_t \end{equation} where all the conditions, such that the solution $X_t$ is defined ...
1
vote
1answer
170 views

sobolev space-equivalence of scalar product

Let $f \in L^2(\mathbb{R}^n).$ Why the equation $\Delta u - u = \dfrac{\partial f}{\partial x_i}$ admits a unique solution $u \in H^1(\mathbb{R}^n)$? 2- Prove that there exist a constant $C \geq 0$ ...
1
vote
1answer
65 views

Differentiation of a 2-variable function

Let $$u(x,t)=\frac{1}{2c}\int_0^t \int_{x-ct+cs}^{x+ct-cs}f(y,s)\,dy\,ds$$ then, this satisfies the PDE : $$u_{tt}=c^2u_{xx}+f \quad \text{and} \quad u(x,0)=u_t(x,0)=0$$ Can you help me get ...
1
vote
1answer
329 views

Von Neumann Stability Analysis

I came across the following task recently: Use the von-Neumann stability analysis to investigate the stability of the discrete form of $\frac{\partial c}{\partial x} = \frac{\partial^2 c}{\partial ...
1
vote
1answer
181 views

Poincare inequality on $H^1_0(M)$

Is it possible to deduce the Poincare inequality for functions in $H^1_0(M)$ from the Poincare inequality for functions in $H^1(M)$ with mean value 0? $M$ is a hypersurface with non-empty boundary.
1
vote
1answer
131 views

The problem of concentration (Clarification of statement in Evans: Weak Convergence Methods for Nonlinear PDE)

I am working through Evan's book on Weak Convergence Methods for Nonlinear PDE. He assumes that $U$ is an open bounded smooth subset of $\mathbb{R}^n$ and that $1<q<n$. In ($\S$D) concerning ...
1
vote
1answer
253 views

Weak Differentiability of Holder functions

Is it true that every Holder function is weakly differentiable? If not please give counterexample. Thanks
1
vote
1answer
62 views

How to establish the estimate?

I consider the inequality as follows: let $a>0,b>0$, and satisfy $$2a^2-b^2\leq C(1+a).\tag{1}$$ If we assume that $b\leq a$, then there exist a constant $C_1>0$ such that $$a\leq ...
1
vote
1answer
110 views

Which one is the right definition of eigenvalues for a differential operator?

This question might be trivial, but I have problems understanding the definition of eigenvalues for the Laplacian \begin{equation} \Delta : C^2(U) \to C(U). \end{equation} on some open, bounded domain ...
1
vote
1answer
281 views

Linear PDE and shock waves

Using the methods of characteristics for a linear partial differential equation, e.g. $$u_t + au_x = f,$$ can there be a noncontinuous solution, i.e. is there any example where the characteristics do ...
1
vote
1answer
470 views

Solution for 3D wave equation for vector fields

I've been trying to find the solution to the 3-D wave equation for vector fields, that is: $$\nabla^2 \vec{u}(\vec{x}, t) = \frac{1}{c^2} \frac{\partial^2 \vec{u}(\vec{x}, t)}{\partial t^2}$$ Given ...
1
vote
2answers
252 views

An easy partial differential equation

I have just entered the study of ODEs. However, the professor, without having talked at all about it in class, asked us to solve the following partial differential equation: $\displaystyle ...
1
vote
1answer
140 views

Question on proof of deformation lemma

This question pertains to Rabinowitz : Minimax methods in critical point theory. This is kind of a shot in the dark, since it's unlikely anyone actually has the book on hand and it's not on the web. ...
1
vote
1answer
291 views

How to Solve PDE using techniques of Separation Variables in this question [SOLVED]

Hi guys my name is Maxwell. This is my first time I asking question in this forum. I hope someone can help me this problem out : Question says : $ U_{xx} + U_{yy} = U $. Solve this PDE product ...
1
vote
2answers
227 views

solution of diffusion equation

The diffusion equation $$\frac{\partial ^2 u}{\partial x^2} = \frac{\partial u}{\partial t},\;\;u = u(x,t),\;\;u(0,t) =0 = u( \pi ,t),\;\;u(x,0) = \cos x \sin 5x $$ admits the solution $( ...
1
vote
1answer
907 views

weak subsolution

Assume $u\in H^1(U)$ is a bounded weak solution of $$-\sum_{i,j=1}^n(a^{ij}u_{x_i})_{x_j}=0 ~~~in~ U$$ Let $\phi:R\rightarrow R$ be convex and smooth,and set $w=\phi(u)$ Show $w$ is a weak ...
1
vote
1answer
244 views

The number of characteristic curves of the PDE [closed]

The number of characteristic curves of the PDE $(x^2+2y)u_{xx}+(y^3-y+x)u_{yy}+x^2(y-1)u_{xy}+3u_x+u=0$ passing through the point $x =1$, $y =1$ is 1. $0$ 2. $1$ 3. $2$ 4. $3$ how can i solve this ...
1
vote
1answer
88 views

Spherical mean of a solution to $\Delta f = |x|^\alpha$

Let $f \in C^2(\mathbb{R}^n)$ be a solution of $\Delta f = |x|^\alpha$, for some $\alpha > 0$. Let $M_f(r) = \frac{1}{\sigma_{n-1}r^{n-1}}\int_{S(r)}f(x)d\sigma(x)$ be the spherical mean of $f$ ...
1
vote
1answer
117 views

Find all functions of class $C^2$, $f:\mathbb R^2\to\mathbb R$ such that $\frac{\partial^2f}{\partial x \partial y} = 0$

Please can you help me to find all functions of class $C^2$, $f:\mathbb R^2\to\mathbb R$ such that $\frac{\partial^2f}{\partial x\partial y} = 0$. Thank you so much!