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

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3
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1answer
122 views

2nd Order Elliptic PDEs with functional BCs

I'm interested in studying linear second order elliptic PDEs with boundary conditions that are functionals of the solution and possibly its derivative. For example, \begin{align} \nabla^2 u(\vec{x}) ...
-2
votes
1answer
98 views

Expansion of solution to advection-diffusion equation

Advection-diffusion equation I am having trouble with the advection-diffusion equation and the proposed solution to it stated in the link above. If it were further expanded and simplified, this ...
2
votes
2answers
137 views

Lemma from PDE book

This is Lemma 6.1 from Gilbarg - Trudinger. It states "Let $\textbf{P}$ be a constant matrix which defines a nonsingular linear transformation $y=x\textbf{P}$ from $\mathbb{R}^n \rightarrow ...
3
votes
3answers
534 views

how to solve $ {\partial u \over \partial t} - k {\partial ^2 u \over \partial x^2} =0$ [duplicate]

How do I solve the following PDE for it's general solution? $$ {\partial u \over \partial t} - k {\partial ^2 u \over \partial x^2} =0$$ How do I determine the general the solution of this equation ...
3
votes
3answers
323 views

Stuck on Laplace's Equation

I am trying to solve the following: $-\theta_{yy}-\theta_{xx}=0$ $\theta(0,y)=-1$ $\theta(1,y)=1$ $\theta_y(x,0)=1$ $\theta_y(x,1)=0$ I can separate this into two different ...
3
votes
5answers
874 views

Hydrogen atom in partial differential equations

For the hydrogen atom, if $$\int |u|^2 ~dx = 1,$$ at $t = 0$, I am trying to show that this is true at all later times. What I need help is with differentiating the integral with respect to $t$, and ...
2
votes
1answer
290 views

Lemma from Gilbarg/Trudinger

This is from Gilbarg/Trudinger "Elliptic Partial Differential Equations of Second order" Lemma 4.1 (p.54 of the third edition) The lemma states that for a bounded integrable function $f$, the ...
5
votes
2answers
329 views

Intuition and applications for the p-Laplacian

Consider the p-Laplacian of a suitably nice function $u$: $\Delta_p u = \nabla \cdot (|\nabla u|^{p-2} \nabla u)$ Are there useful ways of thinking about the p-Laplace operator, or of thinking about ...
1
vote
2answers
567 views

Fourier transform of heat equation

I need to solve following partial differential equation with Fourier transform numerically. $ \frac{\partial T}{\partial t} = \nabla(c\nabla T) $ where T is temperature, c heat conductivity and t is ...
3
votes
0answers
108 views

An existence Theorem

Data: $\Omega \subset \mathbb{R}^{n}$ is an open connected (may be unbounded) set, and locally $\partial \Omega$ is aLipschitz graph. $S \subset \partial \Omega$ is measurabel and $H^{n-1}(S)>0.$ ...
2
votes
1answer
434 views

How to show that fundamental solution of Laplace equation $\in L^2 $given $ f \in L^2 $?

I need help with this homework question. The question is : Let $f:R^3\to R$ and $f\in L^2(R^3)$. $f$ is supported on a ball of radius 1/2 centred at origin. Let $u$ be the solution to $\Delta u=f$ ...
5
votes
1answer
199 views

Interior gradient bound

I would like some help with the following problem (Gilbarg/Trudinger, Ex. 2.13): Let $u$ be harmonic in $\Omega \subset \mathbb R^n$. Use the argument leading to (2.31) to prove the interior ...
0
votes
1answer
1k views

Sobolev Embedding Theorems in Dimension One

What exactly is the content of Sobolev Embedding Theorems (compact for Sobolev spaces and Hölder spaces) when we're looking at functions on the real line?
2
votes
1answer
196 views

What's a measure valued solution of a PDE?

What's a measure valued solution of a PDE? For instance the Fokker-Planck equation \begin{align} \partial_t\mu_t+\sum_i\partial_i(b_i\mu_t)-\frac{1}{2}\sum_{ij}\partial_{ij}(a_{ij}\mu_t)=0 ...
1
vote
0answers
195 views

how to find general solution from complete solution?

For an equation of the form $\displaystyle f\left(\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}\right) = 0$ the complete solution is $ z = ax + \phi (a) y + \psi(a)$--(1) and general ...
3
votes
3answers
370 views

Can a partial differential equation have two different solutions?

Consider: $$x^2p+y^2q=(x+y)z$$ where $p=\frac{\partial z}{\partial x}$ and $q=\frac{\partial z}{\partial y}$. Thus by Lagrange's Method $$\frac{dx}{x^2}=\frac{dy}{y^2}=\frac{dz}{(x+y)z}$$ ...
4
votes
2answers
242 views

Can the PDE $u_{xx} + u_{xy} + u_{yy} = 0$ be separated by variables?

I just started reading Gerald Folland's book Fourier Analysis and Its Applications. I have a question about problem 1 from section 1.3. The problem is the following. Derive pairs of ordinary ...
2
votes
0answers
2k views

A question on Lagrange's Method for solving partial differential equation

The question is to solve $(y-z)p+(z-x)q=(x-y)$ where $p=\frac{\partial z}{\partial x}$ and $q=\frac{\partial z}{\partial y}$ The solution I am referring to has this following line: ...
1
vote
2answers
173 views

nonlinearity of PDE's

(1) $u_x + u_y = 0$ (2) $u_x + yu_y = 0$ (3) $u_x + uu_y = 0$ (4) $u_{xx} + u_{yy} = 0$ (5) $u_{tt} − u_{xx} + u^3 = 0$ (6) $u_t + uu_x + u_{xxx} = 0$ (7) $u_{tt} + u_{xxxx} = 0$ (8) $u_t − iu_{xx} = ...
0
votes
1answer
502 views

linearity and nonlinearity of the PDEs below

I just want to verify and check my understanding to see if I can do this right. a)$ u_x + u_y = 0$ This is linear since: $(u+v)_x + (u+v)_y = u_x +u_y + v_x+v_y $ and $c(u_x+u_y) = cu_x + cu_y$ b) ...
3
votes
1answer
267 views

getting the fundamental solution of Laplace's equation from the heat kernel

Since Laplace's equation is related to the steady state of heat flow problems, I'm guessing that there is a way to get from the heat kernel to the fundamental solution of Laplace's equation by letting ...
2
votes
1answer
241 views

Too many maximum and comparison principles [closed]

Why are there so many maximum and comparison principles in the study of partial differential equations? It is scary to try and learn them because there are hundreds of them. Does every type of domain ...
1
vote
1answer
45 views

Differential system on the torus 2

I've got the following question related to my previous post and I was suggested to write a new question, so there you are. The question is the following: prove that if $f$ is smooth, periodic and ...
3
votes
1answer
429 views

Weak convergence in Sobolev spaces involving time

Let $$Lu:=-\sum_{i,j=1}^n(a^{i,j}(t,x)u_{x_i}(t,x))_{x_j}+\sum_{i=1}^nb^i(t,x)u_{x_i}(t,x)+c(t,x)u(t,x)$$ and the associated bilinear form ...
1
vote
2answers
69 views

Lebesgue integration for $u \in C^{\infty}_c$

Let $u \in C^{\infty}_c(\Bbb{R}^d)$, where $C^{\infty}_c(\Bbb{R}^d)$ is the family of infintly differentiable functions with a compact support. Is $u$ in $L^2(\Bbb{R}^d)$? I think that $u$ is in ...
0
votes
1answer
306 views

Numerical Solution of Systems of PDE

Could someone give me some reference to the Numerical solution of a System of PDEs of following type.. (which also encompasses strongly elliptic system of PDEs) in 2D or 3D. $$\left\{ ...
0
votes
1answer
237 views

Advection-diffusion equation

The advection-diffusion equation I am working with has the form ...
1
vote
2answers
252 views

Wave equation with initial and boundary conditions - is this function right?

If $y(x,t)$ satisfies the 1-dimensional wave equation $$\frac{\partial^2y}{\partial t^2}=c^2\frac{\partial^2y}{\partial x^2}\quad\text{for }0\leq x \leq l$$ with boundary conditions ...
1
vote
1answer
190 views

Implicit function theorem and PDE; do we get uniqueness?

Please see this page: The implicit function theorem: A PDE example. In the implicit function theorem they quote, uniqueness is not mentioned. But the inverse function theorem (which is equivalent to ...
1
vote
1answer
556 views

Does solve PDE by combination of variables always cannot find the general solutions?

Combination of variables is the technique that reducing the PDE to one independent variables (i.e. become ODE) by introducing a suitable change of variables. But the general solution of a PDE should ...
10
votes
1answer
197 views

A type of local minimum (2)

Data: $\Omega \subset \mathbb{R}^{n}$ is an open connected (may be unbounded) set, and locally $\partial \Omega$ is a Lipschitz graph. $S \subset \partial \Omega$ is measurable and $H^{n-1}(S)>0$. ...
2
votes
1answer
429 views

Weak existence and uniqueness for linear PDE system

I am interested in proving existence/uniqueness to: find $u(x,t)$, $v(x,t)$ such that $$u_t - a_1u_{xx} - a_2u_x - a_3u -a_4v = f$$ $$v_t - a_5u_{xx} - a_6u_x - a_7u - a_8v_{xx} - a_9v_x - a_{10}v = ...
6
votes
3answers
701 views

Elliptic Regularity for solutions in distributional sense

I know that there are a lot of (great) books treating regularity of weak solutions of elliptic pdes (such as Gilbarg-Trudinger), but what about regularity of very weak solutions, that is, solutions in ...
3
votes
2answers
262 views

Inclusion relation of Hölder space

Here is Bruce K. Driver's lecture notes on Hölder space. Theorem 24.14 (left as an exercise there) reads: Let $\Omega$ be a precompact open subset of $\mathbb{R}^d$,$\alpha,\beta\in [0,1]$ and $k, ...
0
votes
1answer
182 views

Green's function for fractional operators

I am studying some papers about the fractional laplacian, and I am stuck on a formula that I do not understand. I would like to ask if anybody can give me some help. In this paper, on page 12, there ...
1
vote
1answer
67 views

Finite element method - Dual functional Error estimate

I have an equation -$u''=1,~~x\in(0,1)$. I solved it numerically by Finite element method. and find an approximate solution. $u_{h}$. As you know to do this I defined bilinear and linear functionals ...
2
votes
0answers
228 views

Drum membrane wave equation general solution (non-symmetrical)

I think I am stuck in solving this problem. It involves a wave equation in a circular membrane, so polar coordinates must be used: $u_{tt}=c²(u_{rr}+{1\over r}u_r+{1\over r²}u_{\theta\theta})$, ...
1
vote
1answer
94 views

Evaluating partial derivatives to apply finite differences

I need to solve the following partial differential equation (Cahn-Hilliard) using finite differences: $$\frac{\partial c}{\partial t} = \nabla^2h + \cdots$$ where $h = c(1-c)(1-2c)$. The question I ...
0
votes
1answer
316 views

Separation of variables to solve PDE with variable coefficients

If the coefficients of the PDE $$\partial_t u = a(x,t)\partial_{xx}u(x,t) + b(x,t)\partial_x u(x,t) + c(x,t)u(x,t) + d(x,t)$$ are in some Hölder space, apparently we can solve this via separation of ...
1
vote
1answer
279 views

Determine characteristics for the method of characteristics

I'm trying to understand the method of characteristics to solve first-order PDEs. As an example in his course, my professor solve this PDE for $u(x,y)$: $$x\frac{\partial u}{\partial ...
0
votes
1answer
305 views

How to solve linear partial differential equation?

Given that $P, Q, R$ is a function of $x, y, z$, we have $$ P {\partial z \over \partial x} + Q{\partial z \over \partial y} = R \hspace{2 cm} (1)$$ The book says that $ u = \phi(v) $ is the solution ...
2
votes
1answer
112 views

Some questions about distribution theorem

Given an equation $P(D)u=0$, where $P$ is a polynomial (not equal to a constant). Here are some basic information about the distributional solution $u$: If $P$ has at least one real root, then there ...
0
votes
1answer
96 views

Please help to solve the pde problem

Prove that $$u=e^{-4t}\cos\omega x$$ is a solution of the one-dimensional wave equation $$\frac{\partial u}{\partial t}=c^2\frac {\partial^2 u }{\partial x^2}.$$ I found $$\frac{\partial u ...
2
votes
0answers
99 views

Solved pde, please help to check the solution

Show that the function $u=\ln(x^2+y^2)$ is a solution of the two dimensional Laplace equation $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0.$$ I have found an answer ...
0
votes
1answer
401 views

Existence of a weak solution due to Fredholm alternative in “Evans-Partial differential Equation”

I have a question about a proof in Evans "Partial Differential Equation, Theorem 5 (Third Existence Theorem for weak solution) in Chapter 6.2. They have shown that the equation $$Lu=\lambda u+f ...
4
votes
1answer
240 views

A type of local minimum

Data: $\Omega \subset \mathbb{R}^{n}$ is an open connected (may be unbounded) set, and locally $\partial \Omega$ is aLipschitz graph. $S \subset \partial \Omega$ is measurabel and $H^{n-1}(S)>0.$ ...
9
votes
2answers
230 views

Are there n-th roots of differential operators?

In analogy to a Dirac operator, it seems to me that formally, the equation $$\frac{\partial^n}{\partial x^n}f(x,y)=D_yf(x,y)$$ is solved by $$f(x,y)=\exp{(x \sqrt[n]{D_y})}\ g(y).$$ Is there a ...
1
vote
0answers
212 views

Decay of the fundamental solution of fractional laplacian equations

It is well known that the fundamental solution $\Gamma_1$ in $\mathbb{R}^n$ of the Schrödinger operator $-\Delta + 1$ decays exponentially fast, viz. $|\Gamma_1(x)| \leq C_1 \mathrm{e}^{-C_2|x|}$ as ...
1
vote
2answers
264 views

Solving Poisson Equation

I have the following equation: $$ \begin{cases} H_{xx} + H_{yy} = xy \\ H(x,0) = 0 \\ H(x,1) = x \\ H(0,y) = 0 \\ H(1,y) = 0 \end{cases} $$ We want to solve this, so from inspection of ...
1
vote
1answer
225 views

A Question about the strong maximum principle in Evans Partial differential equation

Evans stated the strong maximum principle as follows: $U\subset\mathbb{R}^n$ a bounded and open set. If $u\in C^2(U)\cap C(\overline{U})$ is harmonic within $U$. Then, ...