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

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Diffrental Equation Solution

How can I solve this equation? ∂f/∂x=(a-x/y)∂f/∂y a=constant f? Thank you.
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0answers
17 views

Find a solution that satisfies Laplace's equation in polar coordinates

How may I find a solution that solves Laplace's equation in polar coordinates, subject to the boundary conditions? In particular, I need to find one solution that satisfies $$\Delta u = 0,$$ subject ...
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0answers
9 views

General solution of boundary value problem

I have to find the general solution of the following boundary value problem with the use of Fourier method. $$u_t(x,t)-u_{xxt}(x,t)-u_{xx}(x,t)=0, 0<x< \pi, t>0\\u(0,t=0),t>0$$ ...
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0answers
9 views

Prove that there is at most one solution with Green's identity

Prove with the use of Green's identity that the boundary value problem $$\frac{\partial}{\partial{x}} \left( (1+x^2) ...
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1answer
21 views

Why do these equalities stand ?

In my notes there is the following theorem: Let $X_k : [a,b] \rightarrow \mathbb{R}$, $k=1, \dots , n$ an orthogonal system of functions and $X: [a,b] \rightarrow \mathbb{R}$, then $\forall c_1, ...
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0answers
12 views

Numeric Evaluation of Double Surface Integral over Greens Function with Singular Points

I'm currently using python to numerically evaluate the follow expression ...
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0answers
23 views

Why is the problem in polar coordinates in that form ?

We have the initial and boundary value problem $$u_{xx}(x,y)+u_{yy}(x,y)=0 , x^2+y^2<1 \\ u(x,y)=0 \\ u(1, \theta)=\sin{\theta}, 0< \theta< \pi$$ $$U_{\rho \rho}(\rho, \theta)+ ...
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1answer
20 views

How have we found the conditions of the problem from the graph?

In my notes there is the following : $$u_{xx}(x,y)+u_{yy}(x,y)=0$$ $$u(x,0)=f(x), 0 \leq x \leq l \\ u(0,y)=0, u(x,\pi)=0 \\ u(l,y)=0$$ How have we found these conditions from the graph?? ...
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2answers
46 views

Why are the eigenfunctions linear independent?

At a Sturm-Liouville problem how do we know that the two eigenfunctions that we have found are linear independent?? For example we have the following problem : $$X''+\lambda X=0 \\ X(0)=X(2\pi) \\ ...
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2answers
13 views

PDE Solving: Difference between Similarity Solution and Characteristics?

As far as I understand, both the method of characteristics and similarity solutions allow us to reduce certain partial differential equations to ordinary differential equations which can then be ...
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0answers
33 views

A question in PDE/Sobolev spaces [on hold]

I'll be grateful if you help me or give me hints to answer the following: Suppose $\Omega$ is a bounded domain in $\mathbb R^N$ with $C^1$ boundary $f\in L^2(\Omega)$. Prove that for every ...
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1answer
21 views

Solution of the Laplace equation in polar coordinate.

Solve the following PDE: $$\phi(r,\theta) = \begin{cases} \Delta \phi=0 & \quad \text{for $a \le r\le b$ }\\[8pt] \phi=V & \quad \text{for $r=b$} \\[8pt] \phi+ C \sin(n\theta)=0 & ...
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1answer
18 views
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0answers
25 views

Can't replicate solution to a non-linear PDE ($- \square \varphi + \lambda \varphi^3 = 0$)

I am trying to replicate the solution of this paper : http://arxiv.org/pdf/0807.2179.pdf Which is, roughly, for the quartic scalar field theory, $- \square \varphi + \lambda \varphi^3 = 0$ a set ...
3
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1answer
23 views

Find a harmonic function in the cylindrical shell between $r=a$ and $r=b$

Calculate $\phi$, satisfying $\nabla^2 \phi=0$ between the two cylinders $r=a$, on which $\phi=0$, and $r=b>a$, on which $\phi=V$. I calculate it and found the solution is ...
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1answer
17 views

Solving the Laplace equation in a rectangle, using the separation of variables

Suppose I have $f_{xx}+f_{yy}=0$ on a region $R=\{(x,y):0\leq x\leq\alpha,0\leq y\leq\beta\}$ with boundary conditions $f(0,y)=f(\alpha,y)=0$, $f(x,0)=g(x)$, and $f(x,\beta)=h(x)$. I considered a ...
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1answer
31 views

System of ODEs obtained by using the method of characteristics for $u_x + 2u_t - 4u = e^{x+t}$

I have a question which requires me to use the method of characteristics in order to solve the PDE $u_x + 2u_t - 4u = e^{x+t}$. This results in the system of ODE's $\frac{dx}{dr} = 1 , \frac{dt}{dr} ...
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0answers
21 views

Nonlinear heat equation $u_{t} = \Delta(u^{4})$

Consider the nonlinear heat equation $u_{t} = \Delta(u^{4})$ in $\{x \in \mathbb{R}^{3}: |x| < 1\}$ with $u = 0$ on $\{x \in \mathbb{R}^{3}: |x| = 1\}$. The problem I am working on is to show that ...
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1answer
20 views

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
32 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
17 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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0answers
29 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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0answers
69 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
26 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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0answers
20 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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0answers
21 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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0answers
21 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
13 views

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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0answers
23 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
40 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
39 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
45 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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0answers
59 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 ...
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1answer
73 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
28 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
34 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
22 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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0answers
16 views

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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0answers
16 views

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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24 views

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 ...
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1answer
29 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
34 views
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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) ...
2
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0answers
21 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
40 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} ...