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

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System of linear differential equations eigenproblem

In the Smith and Young 2001 paper on the Barotropic tide they have the governing equations... \begin{array}{rcl} u_t-f_0v+p_x & = & 0 \\v_t+f_0u+p_y & = & 0 \\p_z &= &b ...
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27 views

Conditions for solution to transport equation

Suppose we consider an operator $$ D = i \frac{d}{dx} + B(x) \colon C_U^\infty([0,\beta],\mathbb{C^n}) \to C_U^\infty([0,\beta],\mathbb{C^n}) $$ acting on smooth complex vector valued functions ...
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39 views

Is there any method that could help me solve this differential equation?

It's an order 1 linear p.d.e., but the coefficients are quite complicated. $$\begin{array}{ll}&\left(As_1^2+Bs_1s_2-(A+B+C)s_1+C\right)\frac{\partial}{\partial s_1}F(s_1,s_2,t)\\ ...
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111 views

Integral of fundamental solution to Heat Equation

On PDE Evans, 2nd edition, page 46, it says the following: LEMMA (Integral of fundamental solution). For each time $t > 0$, \begin{align} \int_{\mathbb{R}^n} \Phi(x,t) \, dx = 1 \end{align} ...
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46 views

Feynman Kac solution discontinuity at 0

In most exposition of the Feynman Kac formula $$\frac{\partial u}{\partial t}(x,t) + \tfrac{1}{2} \sigma^2(x,t) \frac{\partial^2 u}{\partial x^2}(x,t) -V(x,t) u(x,t) = 0$$ the condition of the ...
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42 views

Solution regularity of the heat equation after $t>\varepsilon$

Consider the heat equation \begin{align} \partial_t u - \Delta u &= 0 && \mbox{ in }Q=\Omega\times[0,T] \\ u &= 0 && \mbox{ on }\partial \Omega \times [0,T] \\ u(\cdot,0) ...
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50 views

To Show $ x.\nu > 0 $ on convex domain.

$\Omega\subset \mathbb{R}^n $ be a convex domain, $\nu$ be the unit outward normal of the smooth boundary, then I have to show that $ x.\nu > 0 $ for all $ x \in \partial \Omega$. Here is my try. ...
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2answers
67 views

Method of Characteristics and Initial Value Problem

$u_t + 3u_x = 2t$, $u(x,0)=\sin(x/2)$. I used the method of characteristics to get the answer, $u(x,t)=t^2 + 2\sin^{-1}(x-3t)$. Does this satisfy the initial condition? I checked for the first ...
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1answer
87 views

In the numerical solution of the Wave Equation, using finite differences, where do I obtain the spatial values from?

In trying to implement a simplistic numerical solver for wave equations, I have run into a conceptual problem that I haven't been able to solve. Consider a one-dimensional wave equation of a quantity ...
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19 views

Does the analytic solution of explicit scheme increase with time?

I would solve $U_t=U_{xx}$. To do that I do the approximation $U(x,t)\approx u_{p,q}$ and use the explicit scheme $$\frac{u_{p,q+1}-u_{p,q}}{\delta t}=\frac{u_{p-1,q}-2u_{p,q}+u_{p+1,q}}{(\delta ...
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44 views

One problem about harmonic functions

Problem. Given open, bounded set $\Omega\subset\mathbb R^d$ with smooth boundary $\partial\Omega$ and given smooth function $\varphi$ on $\partial\Omega$. As known, problem $$ \begin{cases} ...
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1answer
49 views

How to see that solution of PDE increases with time

How do I tell if the solution of $u_t=au_{xx}+bu$ on $[0,1]\times(0,T]$ and $a,b>0$ is increasing in time? $u(x,0)=g(x), u(0,t)=u(1,t)=0$.
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Partial differential equations and semigroups: explanation of an example.

The semigroup theory (as presented in Pazy's book) give us theorems that ensures existence of solutions for the abstract cauchy problem $$\left\{\begin{align*} ...
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59 views

Versions of Trace Theorems

I have a quick question about the Trace Theorem. I have been using Evans book of Partial Differential Equations to study Sobolev Spaces. The Trace Theorem is given as "If $U$ is bounded and $\partial ...
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1answer
38 views

Composition of linear differential operators is a linear differential operator

I will use multi-index notation: $$ \newcommand{\p}{\partial} P = (p_1, \dots, p_d), |P| = p_1 + \dots + p_d, \p^P u = \dfrac{\p ^{|P|} u}{\p x_1 ^{p_1} \dots \p x_d ^{p_d}} .$$ Let $A = \sum_{|P| ...
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81 views

Solution of $\frac{\partial f}{\partial t}(t,x) = 2 \frac{\partial^2f}{\partial x^2}(t,x)$

Consider the PDE $$\frac{\partial f}{\partial t}(t,x) = 2 \frac{\partial^2f}{\partial x^2}(t,x)\tag{1} $$ with $t\ge0,\ x\in\mathbb R,\ f(0,x)=e^x$. I want to find $f(t,x)$. I know that the heat ...
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50 views

Some standard elliptic estimates

Consider the problem $$\Delta_{g}u - c(n)R_{g}u + Ku^{p} = 0 \mbox{ at } M$$ where $c(n) = \frac{n-2}{4(n-1)}$, $K$ is a constant, $R_{g}$ is a scalar curvature, and $(M^{n},g)$ is a smooth ...
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51 views

Do PDEs arise in the context of Varieties?

My work involves designing numerical methods to solve PDEs on manifolds. This situation often arises in many applications in biology, chemistry, physics etc. I've recently been reading about ...
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58 views

When is $W^{m,p}(\Omega)$ dense in $L^p(\Omega)$?

Let $\Omega\subset\mathbb{R}$, $m\in\mathbb{N}$ and $1\leq p<\infty$. What are the (most common) sufficient conditions (if such conditions exists) that we can impose on $\Omega$, $m$ and $p$ to ...
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27 views

About Cauchy Kovalevskaia Theorem

I'm reading an article that says we can garantee a solution u=u(x,t) with $(x,t)$ in a neighboor of origin of $(0,0) \in \mathbb R^{m+n}$ for the problem $$ L_j u = 0, j=1, \ldots n, $$ where $$L_j = ...
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41 views

a question about generalized derivatives in Sobolev space

Let $\delta>-\frac{1}{2}$, $s\in (0,\delta+\frac{1}{2})$, $X_+(x)=x , x>0$ or $X_+(x)=0, x\le 0$. How to prove that$(1-\lambda^2)_+^{\delta}\in W^{s,2}(R)$? Where $W^{s,2}(R)$ is a Sobolev ...
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47 views

Green's function of an operator

Given a differential operator $L$, how is its Green's function defined? I know that for a an initial condition problem it is the function so that the solution is defined by $u = G*f$, but I couldn't ...
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General integral of an PDE

Consider the PDE $$ \frac{\partial u}{\partial t}+y\frac{\partial u}{\partial x}-a^2x\frac{\partial u}{\partial y}=0 $$ To find the general integral by the method of characteristics, I construct the ...
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2answers
66 views

Problem with solving PDE

I'm trying to solve this equation: $u_{tt} = u_{x_1x_1} + u_{x_2x_2} + u_{x_3x_3}$ $u(x,0) = x_1^2\sin(x_2+x_3)$ $u_t(x,0) = 0$ In what form to find a solution? I tried in form $u = ...
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79 views

property of local Sobolev space

The local Sobolev space,defined as $W^{k,p}_{loc}(\Omega)$, is the space such that for any $u \in W^{k,p}_{loc}(\Omega)$ and any compact $V\subset \Omega$, $u \in W^{k,p}(V)$. I am just wondering if ...
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29 views

Counterexamples for p-Laplacian equations showing that $C^{1,\alpha}$ is the best regularity.

It is well known that the weak solutions of p-Laplacian equations are $C^{1,\alpha}$. Do they have more regularity? I have heard that no more regularity can be obtained. But I can't find the ...
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70 views

Is the classical derivative the weak derivative on any domain?

I was looking at the following definition of the weak derivative: Let $\Omega$ be a domain (ie an open connected subset of $\mathbb{R}^n$). Suppose $u,v \in L_{1,loc}(\Omega)$ and \begin{equation} ...
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comformal surface parameterization of simply-connected Riemann surface with boundary

$M$ and $S$ are two simply-connected surfaces with boundaries $\partial M =\partial S \neq \emptyset$, respectively. Both $M$ and $S$ are disk-like and smooth enough. Here, we assume $M$ is planar ...
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1answer
21 views

About the proof of regularity of second order elliptic equation

In the proof of interior regularity of elliptic equation, it uses the difference quotient: $D^h_k u := \frac{u(x+he_k)-u(x)}{h}$, $e_k$ is the coordinate vector in the $k$ direction, $k=1,\ldots, n$. ...
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Hyperbolic PDE with Flux Singularity

Consider the system of balance laws $$\partial_t u+\partial_x F_{\epsilon}(u) = S (u),$$ where $u = u(x,t) \in R^n$ denotes the state vector, $F: R^n \to R^n$ is the flux and $S:R^n \to R^n$ some ...
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1answer
61 views

Poisson equation on half space

On the closed upper half space of R3 i.e. all $x, y,$ and $z\ge0$ find functions $u, v$ satisfying: $$\Delta u = 1\text{ and }u(x, 0)=0$$ and $$\Delta v = \delta(0, 0, 1).$$
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Problem that is not well-posed

I'm currently self-studying partial differential equations using the book: An Introduction to Partial Differential Equations with MATLAB. On one of the chapter exercises, I'm faced with the question ...
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The space $C^1([0,T]\times \Omega)$ for $\Omega$ open and bounded

Let $\Omega$ be open and bounded. Is there anything nice I can say about the space $C^1([0,T]\times \Omega)$ and its inclusion in some Bochner like spaces? If $f \in C^1([0,T]\times \Omega)$ then ...
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Solutions to Dirichlet problem on the half space with $L^{\infty}$ boundary data.

Consider the Dirichlet problem with boundary data $f(x)\in L^{\infty}(\mathbb{R}^{d-1})$ on the halfspace $\mathbb{R}^{d}_{+}$, where $y>0$ and $x\in\mathbb{R}^{d-1}.$ One can prove that ...
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73 views

Solving $u_t-u_{xx}=u(1-u)$ with initial/boundary conditions

How would one go about solving, or describing the solutions to this non-linear PDE, the heat equation with an extra non-linear term. $$u_t-u_{xx}=u(1-u)$$ Suppose, $$u=u(x,t),\quad x\in[0,L],\quad ...
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1answer
54 views

A diffusion partial differential equation, or Sturm-Liouville eigenvalue ODE

What is the analytical solution for the following diffusion partial differential equation (initial value problem)? $$\frac{\partial f}{\partial t} = (ax^2+b)\frac{\partial f}{\partial ...
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47 views

Partial Differential Equation With a Boundary Condition

Consider $$ -y\frac{\partial F}{\partial x} +x\frac{\partial F}{\partial y} = G(x,y) $$ with the condition $F(x,0) = 0$ for all $x > 0$. How does one show that this initial-value problem has a ...
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179 views

How to use Fourier Transform to solve the Airy's equation?

Definition: If $f\in L^1(\mathbb{R}^n)$, the Fourier Transform of $f$ is the function $\hat{f}$ given by $$\hat{f}(y)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^n}e^{-ix\cdot ...
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Uniqueness of Neumann conditions for Laplace equations

I know that generally Neumann problem only has unique solutions up to constants. But what about this case (which was brought to me by my friend): If $\Omega$ is a bounded region and its boundary is ...
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61 views

Energey Method about the uniqueness of the initial boundary problem

I am still working on this,however, I don't know how to use energy method to prove the uniqueness.Any hint or suggestion from you would be appreciated. Let U $\subset R^{n}$ be open, bounded, with ...
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Difference between the fundamental solution and the Green function

A lot of sources (books, internet courses, articles etc.) deal with just one of the two: Green function, and the fundamental solution. I wasn't able to find a distinction, but I suppose there is one. ...
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Help solving a differential equation using Green function

By using the appropriate green function, I need to find the function ${\rm u}\left(x,t\right)$ that solves the equation $$ {\rm u}_{xx}\left(x,t\right) + {\rm u}_{t}\left(x,t\right) = {\rm ...
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Regularity theory for $H^k$ space

Lat $\Omega$ be a bounded domain in $\mathbf{R}^n$ with smooth boundary. Let $a(u,v)=\int_\Omega \Sigma a_{ij}\partial_iu\partial_jv+cuv$ where $a_{ij}$ and $c$ are smooth functions on $\bar{\Omega}$ ...
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53 views

About Heat Equation and Maximum Principle

I can not prove the following problem. In area $\Omega$, we have $v\in C^2(\Omega\times[T,\infty))\bigcup C^1(\partial\Omega\times[T,\infty))$ satisfy the following equation: $$\frac{\partial ...
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Change of variables FPE

Given the partial differential equation: $$\partial_tP(z,t)=-\partial_z[(-z^2+A)P(z,t)]+D\partial_{zz}P(z,t)$$ where $A$ and $D$ are constant parameters. how to remove $z^2$ term by substitution?
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49 views

Question from a conservation law example in Evans' PDE book

I'm trying to fill in some details in an example given in Evans' PDE book, chapter 3.4, example 1 on page 139. Starting with an initial-value problem for Burgers' equation: \begin{equation} ...
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43 views

Homework: Solve the poisson equation in the outer sphere

Our teacher asked us to solve the poisson equation: \begin{eqnarray}\left\{\begin{array}{ccc}\Delta u &= &0 \\ u|_{\partial \overline{B(0,R)}} & = & g \\ \lim_{|\vec{x}|\to\infty} ...
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77 views

Laplace boundary value problem

I came across the following Laplace bvp: $u_{xx}+u_{yy}=0\ $ for $\ 0<x<1,\ 0<y<2$ $u(x,0)=u(x,2)=0$ $u(0,y)=0$ $u(1,y)=y(2-y)$ I didn't have any problems solving it. It was quite ...
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PDE oscillation problem

How do I make a start to this question? I am unsure how the given system of equations relate to the pde. Once I know that it would be a trivial to find the eigenvalues and use the given criteria. ...
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97 views

Solution to heat equation, differentiation under integral sign

For $f\in L^1(\mathbb{R}^n)$ a solution to the heat equation $\frac{1}{2}\Delta u=\frac{\partial}{\partial t} u$ is given by $$u(x,t)=(2\pi t)^{-n/2} \int\exp{\left(-\frac{\lVert ...