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

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Derivation of Green's function.

Suppose $u \in C^2(\overline{U})$ is an arbitrary function. Fix $x \in U$, choose $\epsilon >0$ such that $B(x, \epsilon) \subset U$, and apply Green's formula to the region $V_{\epsilon} := U ...
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1answer
26 views

How many conditions do we need for a problem to have an unique solution?

How do we know how many initial and boundary conditions we need for a problem to have an unique solution ?? For example if we have the problem $$u_{tt}-u_{xxtt}(x,t)-u_{xx}(x,t)=f(x,t), 0<x<1, ...
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0answers
12 views

Do smooth solutions of $u_{t}(x, t) = \Delta u_{t} + u$ satisfy $\sup_{0 \leq t \leq T}\|u(\cdot, t)\|_{L^{2}_{x}} = \|u(x, 0)\|_{L^{2}}$?

Let $u(x, t) : \mathbb{R}^{d} \times [0, T) \rightarrow \mathbb{R}$ be a smooth solution to $$u_{t}(x, t) = \Delta u_{t} + u$$ with $u(x, 0) = u_{0}(x) \in L^{2}(\mathbb{R}^{d})$. Furthermore, suppose ...
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26 views

Initial&Boundary Value problem-Fourier

$$u_{t}=u_{xx}, \hspace{5mm} x>0, t>0$$ $$u(0,t)=0 \hspace{3mm} u(x,0)=f(x)$$ We want that the solutions are bounded. We are looking for solutions of the form $$u(x,t)=X(x) \cdot T(t)$$ ...
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14 views

Some questions about Krein-Rutman Theorem

I would like to figure out the Krein-Rutman Theorem. And I'm following the notes: ftp://ftp.ma.utexas.edu/pub/papers/llave/.grad/5999_chap1-1.pdf However, I got some questions. Defintion. Let X ...
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1answer
23 views

Eigenfunctions and Eigenvalues of a Linear Operator

For a math project on Schroedinger's equation I and a partner are working on we need to find eigenfunctions and eigenvalues that satisfy $L\phi_n = \lambda_n\phi_n$, where $L$ is defined as $L\psi = ...
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14 views

Hugoniot Locus/Integral curves in linearly degenerate case

Given a linearly degenerate conservation law as described in Evans' book about pdes, I am unsure about the completeness of the proof (chapter 11.2, theorem 3)that the Hugoniot locus $S_k$ of a given ...
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17 views

Solving laplace equation and decomposing

I want to solve $$ 9U_{xx}+U_{yy}=\sin (2\pi x) + \sin(2\pi y) \label{eq:1}\tag{1} $$ with $U=0$ on the boundary of the unit square. I know you would have to decompose the problem to satisfy each of ...
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19 views

Why do we take the odd extension?

When we have the initial and boundary value problem $$u_{tt}(x,t)-c^2u_{xx}(x,t)=0, x>0, t>0 \\ u(0,t)=0 \\ u(x,0)=f(x), x \geq 0 \\ u_t(x,0)=g(x), x \geq 0$$ can we apply Green's theorem or ...
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1answer
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Help solving Modified 1 dimensional heat equation

$u_t(t,x)=u_{xx}(t,x)-u(t,x)$ and $x\in(0,1), t>0$ with boundary and initial conditions $u(t,0)=0, u(t,1)=e-e^{-1}, u(0,x)=f(x)$. I tried using auxiliary function $v(t,x)=u(t,x)-(e-e^{-1})x$ but ...
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21 views

interpreting wave equation using mathematica

hi can someone please help me solving this problem: Let $u(x, t)$ be a solution of a wave equation $u_{tt} − u_{xx} = 0$, $x \in (0, 2\pi)$, $t > 0$ satisfying Neumann boundary conditions $u_x(0, ...
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2answers
35 views

Solving inhomogeneous PDEs with ODEs

I am trying to understand a general method of solving an inhomogeneous PDE. I have begun with the Heat equation but am stuck with the last step. For instance, solving ...
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1answer
38 views

Separation of variables for the Laplace equation on a disk

I have the equation $$\bigtriangleup u=\frac{1}{r} \frac{\partial}{\partial r}(r\frac{\partial u}{\partial r}) + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta}=0$$ where $0<r<1$ , $-\pi< ...
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1answer
38 views

Evans PDE: Chapter 5, Problem 9 - Clarification

I've been trying to work out the solution to Question 9 in Chapter 5 of Evans, and I'm having some difficulties. I've been looking at the solution posted here: question 9 - chap 5 evans PDE And I ...
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58 views

Application of the Fundamental Theorem of Calculus

I was wondering if someone could help me with a problem I'm having. I'm reading a paper 'Spatiotemporal dynamics of continuum neural fields' and on page 13 they authors derive a model for spatially ...
5
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1answer
72 views

How to solve $f\frac{\partial f}{\partial x}+\frac{\partial f}{\partial t}=0$?

How to solve $f\frac{\partial f}{\partial x}+\frac{\partial f}{\partial t}=0$? (where $f(x,t)$ is assumed to be in $C^\infty(\mathbb{R}\times\mathbb{R^+}\rightarrow\mathbb{R})$) I can find a ...
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1answer
16 views

Proof of Mean-value for Laplace's equation.

$\textbf{ Statement of Theorem:}$ If $u \in C^2(U)$ is harmonic, then $$u(x) = \frac{1}{m(\partial B(x,r))}\int_{\partial B(x,r)} u dS = \frac{1}{m(B(x,r))}\int_{B(x,r)} u dy$$ for each $B(x,r) ...
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1answer
27 views

Do we suppose that $y$ is the variable at which $f$ is differentiable in $\mathbb{R}$ ?

In my notes there is the following: The solution of the problem $$u_{tt}-c^2u_{xx}=0, x \in \mathbb{R}, t>0 \\ u(x, 0)=f(x), x \in \mathbb{R} \\ u_t(x,0)=g(x)$$ is ...
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1answer
25 views

Canonical form of the differential equation

In my notes there is the following: Find the canonical form of the differential equation $$4u_{xx}-12u_{xy}+9u_{yy}+u_{y}=0$$ $$\Delta=(12)^2-4^2 \cdot 3^2=0$$ The canonical form will be of ...
2
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1answer
33 views

Solve an initial value problem using the directional derivative

In my notes there is the following example of solving an initial value problem using the directional derivative. The problem is the following: $$u_t(x,t)=u_x(x,t), x \in \mathbb{R}, t>0 \\ ...
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0answers
5 views

What is the parameter of the Streamline-Diffusion Method?

Recently, I want to apply streamline-diffusion(SD) method to solve Burgers Equation. \begin{align} u_t + uu_x - \varepsilon u_{xx} &= 0,\\ u(0,t) = u(X,t) &= 0,\\ u(x,0) &= u_0(x),\ \ \ ...
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1answer
30 views

Find a solution of the Laplace equation $-\Delta u=1$ with boundary condition $u=0$ on a spherical shell

Let $n\ge 2$ $B_\varepsilon$ and $\overline{B}_\varepsilon$ be the open and closed ball around $0$ with radius $\varepsilon>0$ in $\mathbb{R}^n$, respectively $R>0$, $\rho\in (0,R)$ and ...
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0answers
20 views

A physical model for pde

Do you know any mathematical model of a physical process such that it satisfies the following equations ? $$\mathrm{u_t=a(t)u_{xx}},\,\,\,0<x<1,\,\,\,t>0$$ ...
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Closed form solutions of 2D p-laplace equation.

While investigating a physics problem, I found the following PDE: $\vec\nabla. (|\vec\nabla P|^p \vec \nabla P) = 0 $ Where $\vec\nabla =(\dfrac{\partial }{\partial r},\dfrac {1}{r} ...
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0answers
21 views

Solving $du/dt=a \Delta u + b (\Delta u)^2$

Consider the function $u(\boldsymbol{x},t)$, where $u:\mathbb{R}^n \times \mathbb{R}_+ \rightarrow \mathbb{R}$ ($\mathbb{R}_+$ denotes nonnegative reals). My question is related to the PDE ...
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continuity in $H_0^{1,2}$

Asuume that $u \in H_0^{1,2}(\Omega)$ and $f \in L^2(\Omega)$, and $\int_\Omega \nabla u \nabla v \, dx=-\int_\Omega fv \, dx$ holds for all $v\in C_0^1(\Omega)$. Show that $\int_\Omega \nabla u ...
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How do we deduce the condition for the solution?

Suppose that we have the differential equation $$u_t(x,t)=k^2u_{xx}(x,t), x \in (0,l), t>0$$ $$u(x,t): \text{ heat of rod at the position } x \ (0 \leq x \leq l )$$ If we have Dirichlet ...
1
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1answer
27 views

Sobolev Embedding and Uniform $C^1$ bound

I am currently reading a paper and am a little confused about the following, which for clarity, I distill into the following question: Suppose $\{w_i\} \subset C^2(\mathbb{R}^n)$ is a sequence of ...
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0answers
24 views

Energy inequalities with negative sobolev number.

Let $\phi\in H^{s}$ such that the following energy inequality is true: $$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\cdot)\|_s \, dt $$ where $P$ is a strictly hyperbolic linear operator. For ...
2
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1answer
23 views

Functions belonging to $L^p(\mathbb{R}^n ; \mathbb{R}^m)$ if and only of their norm belongs to $L^p(\mathbb{R}^n)$

The definition I have of $f \in L^p(\mathbb{R}^n ; \mathbb{R}^m)$ is that we require each component function to be in $L^p(\mathbb{R}^n)$. Is is true that $f \in L^p(\mathbb{R}^n ; \mathbb{R}^m) ...
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0answers
19 views

Partial Differential Equations Black Scholes Problem

Part 1) Consider the Black-Scholes problem $$\frac{\partial A}{\partial t}+\frac{\sigma^2B^2}{2}\frac{\partial^2A}{\partial B^2}+rB\frac{\partial A}{\partial B}-rA=0 ...
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1answer
38 views

Black-Scholes Problem

Consider the Black-Scholes problem $$\frac{\partial A}{\partial t}+\frac{\sigma^2B^2}{2}\frac{\partial^2A}{\partial B^2}+rB\frac{\partial A}{\partial B}-rA=0 \hspace{3mm}\textrm{and}\hspace{3mm} ...
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0answers
16 views

First order elliptic pseudodifferential operator and Sobolev space

Let $P$ be an elliptic pseudodifferential operator of order 1 on a compact manifold $M$. Also let the spectrum of $P$ lie inside $(0, \infty)$. I am trying to see how one could prove that $P : H^1(M) ...
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48 views

How do I apply this PDE as an image filter?

I'm trying to preprocess a height map image with a helmholtz-type equation as described in this paper. The equation is: $$ddx(h') + ddy(h') + y(h'-h) = 0$$ I solved for h and got: ...
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40 views

PDE first order

I'm heading the book Elements Of Partial Differential Equations -Sneddon 1957. At chapter two exists this exercise "Eliminate the arbitrary function $f$ fron the equation $$ z= ...
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1answer
25 views

Separation of variables and basis of solutions?

If a PDE can be solved by separation of variables. Then the superposition of the solutions found via this method can form all other solutions to the PDE. Is this statement correct? If it is ...
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12 views

Probability density function for a PDE with random inputs

I am looking for a general method or alternatively few textbook examples of deriving a probability density function for a solution of partial differential equation with random inputs in the equation ...
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1answer
20 views

Variant of Ladyzhenskaya’s inequality

I am trying to show that if $\Omega \subset\subset \mathbb{R}^2$ with $C^1$ boundary and $ u \in W^{1,2}(\Omega)$ then \begin{equation*} \int u^4 < C \left(\int u^2 \right)^2 + C \left(\int u^2 ...
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1answer
22 views

Show that this function is weakly differentiable

I need to show that the function \begin{equation} u(x,y) = 1-x_1^2 \quad x_1>0 \\ u(x,y)=1+x_1^2 \quad x_1 \leq 0 \end{equation} is weakly differentiable on the unit ball. It is clear what the ...
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1answer
20 views

Deflection of String

I am trying to determine u(x,t) for a string of length L=1 and c^2=1 when the initial velocity is 0 and initial deflection with small k(.01) is as follows: ...
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27 views

Hyperbolic PDE with strange condition.

The 1D wave equation part is not tricky, but I am having trouble dealing with the max condition. I was thinking of using d'Alembert's formula somehow but I am not sure how to use it in this case. ...
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19 views

Elliptic regulartiy for nonlinear elliptic equations

Here is the question: Let us consider the Schrodinger type equation $$ \left\{ \begin{aligned} &-\Delta u + u = |u|^{p-2}u \quad \text{in } \mathbb{R}^N \\ &u\in H^1(\mathbb{R}^N) \qquad ...
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0answers
66 views

Elliptic regularity in an unbounded domain

Let $\Omega \subset \mathbb{R}^N$ be an unbounded domain with nonempty boundary. Let $$\mathcal{D}^{1,2}(\mathbb{R}^N) = \{ u \in L^{2^*}(\mathbb{R}^N) : \nabla u \in L^2(\mathbb{R}^N ; \mathbb{R}^N) ...
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1answer
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Neumann boundary conditions for PDE

I have a question about Neumann boundary condition for PDE. Suppose $\Omega$ is an open bounded set in $R^n$ with a smooth boundary $\partial \Omega $. Then, a homoegenous Neumann boundary condition ...
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Does $\operatorname{div}\left(\nabla G +xG\right)=0\Longleftrightarrow \nabla G +xG=0$?

Let $G$ be a function of $\mathcal{C}^2(\textbf{R}^d$;$\textbf{R}^*_+)$ such that $G \in \operatorname{L}^1(\textbf{R}^d)$. I read on the Internet that one has the following equivalence ...
2
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1answer
35 views

Existence and uniqueness for the Cauchy problem for the Laplace equation

Given a Laplace equation $u_{xx}+u_{yy}=0$ in $\mathbb{R}^2$ with initial conditions $u(x,0)=u_0(x)$, $u_y(x,0)=u_1(x)$. Is this problem uniquely solvable in general? Does someone have a ...
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0answers
22 views

What are the main methods to solve an evolution PDE and how are they applied?

When one sees an evolution PDE, what are the reflexes that he should have in order to tackle it. What I mean is what are the main methods that have been developed so far and to which kind of PDEs ...
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1answer
27 views

Second order PDE

Kindly help me with this... $$U_{xy} + yU_{yy} + \sin(x+y)=0$$ Here $A =0$, so how to calculate the characteristic equations ? as $$ {dy\over dx} = {B^2 \pm \sqrt D\over2A} $$
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0answers
17 views

Scaling of the Dirichlet Laplacian eigenvalue [on hold]

Let Ω be a smooth domain and let $λ_1(\Omega)$ be the first (or lowest) eigenvalue of the Dirichlet Laplacian in $\Omega$. How can I show that for $\beta>0$, $λ_1(\beta\Omega)=\frac{1}{\beta^2} ...
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15 views

What does the Sturm-Liouville theory say about separation of variables?

In this question Completeness of solutions and the separation of variables method one of the comments says that the condition for the solutions formed by separation of variables to be a basis for all ...