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

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3
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0answers
51 views

Estimating/approximating a very high dimensional unbounded poisson's equation

Consider the poisson equation on an unbounded domain. Suppose that the solution is known to exist. $$ \Delta u=f $$ I would like to estimate the solution of the this equation at a given point $x_0$. ...
1
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0answers
42 views

Solving $\nabla^2U(x,y)=0$ on a donut with two inhomogeneous boundary conditions

I am given $\nabla^2U(x,y)=0$ on a donut-shaped region, with the inner circle being of radius $r_1$, and the outer circle $r_2$. In polar coordinates, the relation is $$U_{rr}+\frac1rU_r+\frac1{r^2}...
1
vote
2answers
188 views

book for numerical methods for solving pde

I need to find some masters-level exercises about numerical methods for solving pde. Are there any good references?
1
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1answer
81 views

Long time behavior heat equation on infinite line

We know that a solution to the Cauchy problem on $\mathbb{R}$ : $u_{xx}=u_t$ with condition $u|_{t=0}=\varphi(x)$ is of the form $$u(x,t)=\dfrac{1}{2\sqrt{\pi t}}\int_{-\infty}^{\infty}\exp\left({\...
0
votes
1answer
12 views

Curve that lies on a solution surface

Suppose the solution surface is given as $$f(x,y,u)=0$$. A curve $$C$$ lies on the solution surface. What does this means?
0
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1answer
23 views

Why is $\Delta u$ bounded, if $u\in C^2(\overline{\Omega})$ and $\Omega\subseteq\mathbb{R}^n$ is a bounded domain?

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain and $u\in C^2(\overline{\Omega})$. Why must $\Delta u$ be bounded?
2
votes
1answer
48 views

Large and small time PDE solution

I have the following solution for a PDE $$ u(x,t)=(2x+4t-10)+2e^{\frac{-1}{2}t}+\sum_{n=1}^{\infty} \frac{(-1)^n cos(\frac{n\pi}{2}x)e^{\frac{-1}{4}n^2\pi^2t}}{n^3\pi^3(n^2\pi^2-2)} $$ I want to ...
0
votes
1answer
50 views

How is an ODE a consistency condition?

I was reading a text on Optimal Control Theory by E. Todorov, when I came accross this passage (on page 10): An ODE is a consistency condition which singles out specific trajectories without ...
1
vote
2answers
53 views

Solving simultaneous PDEs

Given the equations (1):$$\frac{\partial u}{\partial t}+g\frac{\partial \eta}{\partial x}=0$$ and (2):$$\frac{\partial\eta}{\partial t}+H\frac{\partial u}{\partial x}=0$$ can we combine the two ...
0
votes
1answer
53 views

Convergence of $\partial_{x_j} u(x,t)$ when $u$ converges in $L^2$ norm.

I hope you can help me with this question. We take $u(x,t)\in L^\infty_{loc}(\mathbb{R},H^1(M))\cap Lip_{loc}(\mathbb{R},L^2(M))$, the derivatives $\partial_{x_j} u $ exist and are continuous, i.e $\...
2
votes
2answers
94 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 ...
1
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0answers
43 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. $$u_t(x,t)-u_{xxt}(x,t)-u_{xx}(x,t)=0, 0<x< \pi, t>0\\u(0,t=0),t>0$$ $u(x,t)=...
0
votes
1answer
36 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, \...
1
vote
0answers
33 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)+ \frac{1}{\rho^...
1
vote
1answer
27 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?? ...
4
votes
1answer
143 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) \\ ...
0
votes
2answers
244 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 ...
0
votes
1answer
66 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 & \quad ...
1
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1answer
32 views

Boundedness of a sequence in $L^\infty(I,H^1(M))\cap\mbox{Lip}(I,L^2(M))$ implies that its temporal derivative is bounded as well

I asked my question in mathoverflow, but it seems to be inappropriate there, so I try my luck here. http://mathoverflow.net/questions/203258/boundedness-of-a-sequence-in-l-inftyi-h1m-cap-lipi-l2m-...
4
votes
2answers
52 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 $$\phi=\frac{V}{\log b-...
2
votes
1answer
127 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 ...
1
vote
1answer
48 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} =...
9
votes
1answer
229 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 ...
1
vote
1answer
52 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 \...
1
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1answer
158 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, ...
1
vote
0answers
23 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 ...
2
votes
1answer
50 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 $$u(x,t)=X(x) \cdot T(t)$$ $X(...
1
vote
1answer
59 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 = \...
2
votes
1answer
57 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 ...
6
votes
1answer
137 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 ...
0
votes
1answer
182 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 ...
1
vote
2answers
84 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 $$\begin{cases}u_t=u_{xx}+g(x)\\...
0
votes
1answer
126 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< ...
1
vote
1answer
715 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 ...
1
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0answers
71 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
votes
1answer
81 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 ...
1
vote
1answer
37 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) \...
1
vote
1answer
32 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 $$u(x,t)=\frac{1}{2}[f(x+ct)...
1
vote
1answer
42 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
votes
1answer
53 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 \\ u(x,...
0
votes
1answer
104 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
30 views

A physical model for pde

Do you know any mathematical model of a physical process such that it satisfies the following equations ? $$\mathrm{u_t=a(t)u_{xx}},\,\,\,0<x<1,\,\,\,t>0$$ $$\mathrm{-ku_x(0,t)=hu(0,t)},\,\,\...
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vote
0answers
45 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} \dfrac{\...
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vote
0answers
51 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 $\frac{du}{...
1
vote
1answer
48 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 ...
9
votes
0answers
196 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
votes
1answer
27 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
votes
0answers
54 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 \hspace{3mm}\textrm{and}\hspace{...
1
vote
1answer
83 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} A(B,T)...
3
votes
0answers
83 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: $$\dfrac{ddx(h')}{...