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

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512 views

Method of Eigenfunction Expansion

The solution of a PDE can be represented by a Fourier cosine series $$ u(x,t)=\sum_{n=1}^\infty A_n(t)\cos\frac{n\pi x}L. $$ Applying a given initial condition $$ u(x,0)=100, $$ lets us solve for ...
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3answers
502 views

Heat equation, separation of variables and Fourier transform

I have a question about the heat equation $\frac{\partial \varphi}{\partial t} = \frac{\partial^2 \varphi}{\partial x^2}$ with the conditions that $\varphi(x,t=0) = f_0(x)$ and $\lim_{x \rightarrow\pm ...
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1answer
152 views

Singular support of a tempered distribution is compact?

I am reading Introduction to the Theory of Distributions by Friedlander and Joshi. As definition 8.6.1, they define the singular support of a tempered distribution $u$ to be the complement of {$x$: ...
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61 views

The Maximum Principle and Some Notation

I am working on this question: Suppose that $\Delta u\leqslant0$ in $\Omega$ and $u=f$ on $\partial\Omega$, and $\Delta v\leqslant0$ in $\Omega$ and $v=f$ on $\partial\Omega$. Prove that ...
4
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1answer
771 views

Wave equation with variable speed coefficient

Consider the wave equation initial value problem in $\mathbb R^3$ with spatially variable wave speed, denoted by \begin{align*} \frac{\partial^2}{\partial t^2}u(x,t)-c(x)\Delta ...
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1answer
81 views

Diffusion equation with dirichlet condition by seperation of variables

The problem is this: $$\begin{cases} U_t = 3U_{xx}, \quad 0 < x < 2\pi, \\ U(0,t) = U(2\pi,t) = 0, & \\ U(x,0) = 2 \sin x + 5 \sin 3x \end{cases}$$ I want to express this as an infinite ...
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0answers
113 views

Chain rules for differential forms

I have the variable $x,y,z$ possibly depending on each other i.e. on a smooth manifold. Using the theory of differential forms I can derive $\left(\frac{\partial x}{\partial y}\right)_z ...
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1answer
329 views

Regularity of elliptic PDE with coefficients in some Sobolev space

Is there a regularity theory for elliptic equations "optimized" for coefficients in a Sobolev space $W^{k,r}$? By this, I mean a result (and the according elliptic estimates) that gives you (for $k$ ...
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1answer
96 views

How to prove this change of variables?

Consider the sets: $$Q_0=\{x\in\mathbf{R}^n:0<x_1<2, \ |x_\alpha|<1, \ for \ 1<\alpha\leq n\},$$ $$Q_l=\{x\in\mathbf{R}^n:0<x_1<2l, \ |x_\alpha|<l \ for \ 1<\alpha\leq n\}.$$ ...
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0answers
25 views

How to prove this equivalence?

Consider the general elliptic operator $$M=\sum_{i,j=1}^na_{ij}(x)\frac{\partial^2}{\partial x_i\partial x_j},$$ where $a_{ij}$ are continous functions. The function $u$ satisfies $$|Mu|\leq A(|\nabla ...
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65 views

Proving $ f \in C_b^1 ( [0,\infty) \times \Bbb R^n) $ by using the Sobolev inequality.

Let $s > 1 + n/2$ for $n \in \Bbb N$, and $s$ be an integer. If $f \in C^0 ( [0,\infty), W^{s,2} (\Bbb R^n )) \cap C^1 ( [0, \infty) , W^{s-1,2}(\Bbb R^n)) $ then how can I show that $$ f \in C_b^1 ...
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2answers
76 views

Estimate on a simple-looking integral arising from harmonic analysis/harmonic extensions

Let $z\in \mathbb{D}, t\in S^1, \beta\in \mathbb{R}$. I was dealing with the following integral arising from some other calculation regarding harmonic extension on $\mathbb{D}$: ...
2
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1answer
56 views

About the boundedness of the derivative of a function which is in special function space.

If $f \in C^1 ([0,T] , L^2) \cap C^0 ([0,T] , W^{1,2} )$, then how can I conclude that $$ \left \| \frac{\partial f}{\partial t} \right \|_{L^\infty([0,T] \times \Bbb R^n )} < \infty ?$$ Here $f$ ...
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1answer
79 views

The notation $ f \in C^0 ([0,T],X) $

In general, do the notation $$ f \in C^0 ([0,T],X) $$ imply that $\| f(t) \|_X < \infty$ for any fixed $t \in [0,T]$? Or just means $\| f (t) \|_X$ is continuous on $[0,T]$? Here $X$ is like $L^p$ ...
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1answer
67 views

Identify the distrionbutional derivative with classical derivative?

I am reading Rudin's Functional Analysis and got quite confused by his proof for theorem 7.25, which he calls Sobolev's Lemma. In proving the theorem, he defines the function $F$, and calculates its ...
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1answer
84 views

The number of initial conditions of Cauchy problem for PDE

This number, is it always equal to the order of the differential equation? in which case, one can reduce this number? thanks in advance
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0answers
56 views

A question from Hilbert and Courant's Vol II of Methods of Mathematical Physics (I might have spotted an error).

In page 751 (I hope some folks have a copy of it, legal or otherwise, I have a legal one :-D), I am attaching scans of pages 750-751. http://www.mediafire.com/view/?7hu91s2t5866lqj ...
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36 views

Partial Differential Equations System with moving contour.

Can you recommend me a good site/textbook explaining how to solve and simulate Partial Differential Equations where the contour is not fixed, but can vary with time? I'm interested both to the case ...
2
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0answers
240 views

Helmholtz and Biharmonic equation examples with exact solution

I'm looking for examples of Helmholtz and Biharmonic equations in Cartesian co-ordinates with exact solutions, in order to compare results of my numerical solutions with it. I was able to find quite ...
3
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1answer
104 views

Numerical Computation of Eigenvalues

I am trying to find the first few eigenvalues of an operator defined by the following PDE: $$ \begin{cases} -\Delta u +(1-\varphi)u=\lambda u, & \text{ on }\Omega = [0,1]^2 \\ u=0 & \text{ ...
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0answers
196 views

Green's Formula as proof for harmony

I showed this to my teacher this morning, and he remarked that the argument was a bit strange...but that it might be correct; he recommended using the ball mean property (which I will do, but I'm ...
4
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1answer
152 views

Solution of Schrödinger's Equation

I was wondering what is known about the solution of the Schrödinger equation $$i h \frac{\partial}{\partial t} Ψ(x, t) =- \frac{h^2}{2m}\Delta Ψ(x,t)+V(x)Ψ(x, t)$$ for $t ∈ \mathbb{R}$. What sort of ...
3
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1answer
197 views

Free Schroedinger equation

How can one find and prove the general solution to the equation $\dfrac{\partial f(x,t)}{\partial t} =c^2i\dfrac{\partial^2f(x,t)}{\partial x^2}$ ? I can find the solutions $Ae^{ikx-E_kt}$, so I ...
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1answer
154 views

PDE on smoothness conditions and existence values

How can we show that a smooth solution of the problem $$\begin{cases} u_t +uu_x = 0 \\ u(x, 0) = \cos(\pi x) \end{cases}$$ satisfies the equation $u = \cos \pi(x − ut)$ and that $u$ ceases to exist ...
4
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2answers
346 views

Homological algebra in PDE

I have been fascinated by the power and wide applicability of homological methods in algebra and topology. Because I am also interested in PDE, there arises a natural question for me. What is ...
4
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1answer
412 views

PDE: Maximum principle + Periodic Boundary Conditions = Constant?

I'm working on a homework assignment in PDE, and I'm required to use the maximum principle to demonstrate that when $\Delta u(x)=0$ and periodic boundary conditions are applied, $u(x)$ is a constant. ...
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0answers
86 views

Looking for energy functional

I'm pretty sure that this is a stupid question, but I'm having troubles in writing down the energy functional of an elliptic pde. That is, what's the energy functional of the problem ...
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0answers
506 views

Stability of Lax-Wendroff scheme for wave equation

I'm a beginner in PDE's and numerical methods, so please go slow :-D I'm trying to show that the Lax-Wendroff scheme is stable for $|a\lambda| < 1$. The scheme is this: $v_m^{n+1} = v_m^n - ...
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2answers
103 views

Solve the dirichlet pde with the given conditions.

Solve the Dirichelt problem: $$\nabla^2u(x,y)=0$$ $$0\le x\le3,0<y<7$$ $$$u(x,0)=0, u(x,7)=sin((\pi)x/3$$ $$9\le x\le3$$ $$u(0,y)=u(3,y)=0$$ $$0\le y\le7$$ Using separation of variables I ...
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2answers
239 views

direct differentiation in PDE

How can we check by direct differentiation that the formula $u(x, t) = \varphi(z)$, where $z$ is given implicitly by $x − z = ta(\varphi(z))$, does indeed provide a solution of the PDE $u_t + a(u)u_x ...
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1answer
41 views

DPEs system which I cannot seem to solve

Consider the following DPE system: $$\left\{ \begin{array}{rcl} g_x - f_y& = &1-x^2 \\ h_x - f_z &= &3x^2 \\ h_y - g_z &=& -1 \ \end{array}\right .$$ This comes from trying ...
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1answer
749 views

shock waves characteristics

I'm trying to solve $u_t + u^2u_x = 0$ with $u(x, 0) = 2 + x$. I'm thinking to proceed by characteristics where we have above that $\frac{dx}{dt} = 1$ and $dy/dt = u^2$, but not sure if this will ...
2
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1answer
102 views

How do I solve this linear partial differential equation?

I haven't learned how to do this yet, but a friend gave me this question to do. He said it was on an exam he did and it was a fun puzzle. $xu_x-yu_y=2u$ That's pretty much all he gave me. Is this ...
4
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1answer
60 views

One way transform of a pair of differential equations

This question concerns the Korteweg-de Vries equation. It is known that the transform $F=f^2+f_x$ transforms $$F_t-6FF_x+F_{xxx}=0$$ into $$f_t-6f^2f_x+f_{xxx}=0$$ where $F=F(x,t), f=f(x,t)$ ...
3
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1answer
139 views

What is the name of this equation?

I have found this picture but I don't know the name of the equation in it. Another thing, what kind of plots are those in the picture? I have also tried to re copy it: $$ ...
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2answers
88 views

corollary of maximum principle

Let $ \displaystyle{ U \subset \mathbb R ^n }$ open, bounded and connented. If $ u \in C(U) \cap C(\bar U)$ such that: $$ \Delta u =0 \quad \text { in U}$$ $$u=g \geq 0 \quad \text {in} \quad ...
2
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0answers
78 views

nonlinear partial differential equation

I'm considering the following type of PDE: $u_t=\frac{u_{xx}+u_{x}}{u_t^2}$. Are there any currents methods for studying the well posedness of such an equation at zero. (I don't have much of a ...
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1answer
581 views

An inequality in Evans' PDE

In Section $9.2$ Theorem $5$ of Lawrence Evans' Partial Differential Equations, First Edition the author proves that for a large enough $\lambda$, the equation $$\begin{array}-\Delta u+b(\nabla ...
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0answers
77 views

Solving a non-linear differential operator equation

Sorry if the title is misleading. I am trying to obtain $V(r)$ from the following equation: $\frac{\partial^2}{\partial t^2} C(r,t) = (-\frac{\partial^2}{\partial r^2} + V(r))^2 C(r,t)$ I can do ...
2
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1answer
261 views

General Sobolev Embedding

Is it possible to deduce the general Sobolev embedding into Holder spaces, $$W^{k,p}(\Omega) \hookrightarrow C^{\ell, k-\frac{n}{p} - \ell}(\Omega)$$ for $\ell<k \in \mathbb N$, $p > ...
2
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2answers
2k views

Heat equation in cylindrical coordinates with Neumann boundary condition

Given a cylinder of internal radius $r_0$ and external radius $r_1$, the heat equation in cylindrical coordinates that represents the behaviour of the temperature inside the cylinder, can be written ...
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1answer
120 views

teaching a little nonlinear PDE to an undergraduate

I would like to teach a little nonlinear PDE to an undergraduate who is taking a course in second-order linear boundary value problems. I have never taught nonlinear PDE before, although it is my ...
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1answer
81 views

A question about the integral of partial derivatives.

Let $A$ be $n \times n$ Hermitian matrix with its component $a_{ij}$. Let $v$ be a $n$ dimensional column matrix with its component $v_i$. Let $a_{ij} \in C^{\infty} ( \Bbb R^n)$ and $v_i \in W^{1,2}( ...
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0answers
272 views

Can infinitesimal generator be defined by the time-inhomogeneous stochastic process?

The following is the definition of infinitesimal generator from Oksendal. Let $\{X_t,t\in[0,T]\}$ be a time-homogeneous It\^o diffusion in $\mathbb{R}^d$. The $\textit{infinitesimal generator}$ ...
4
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2answers
782 views

What exactly are partial differential equations?

What exactly are partial differential equations? I know what differential equations are but I want to know what a PDE is since the Schrödinger equation for example is a PDE too. Also, is there a good ...
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1answer
439 views

Mean Value Property of Harmonic Functions Proof Step

I'll only include the step that throws me off unless more info is requested, but this is from LC Evans PDEs book: $$ \displaystyle \lim_{t \to \, 0^+ } \left[ \frac{1}{n\,\alpha(n) \, t^{n-1}} \int_{ ...
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1answer
497 views

A question on energy equipartition of the wave equation

I am solving a initial value problem for the wave equation $$ u_{tt}=u_{xx}\ \ in \ \ \mathbb{R}\times (0,\infty), \ \ \ u=g, \ u_{t}=h \ \ on \ \ \mathbb{R}\times \{0\} $$ for some com[actly ...
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1answer
134 views

Theoretical Understanding of Fourier Series

I understand mathematically how it is derived, and how it works (ie how it is applied and the significance of Fourier Series in solving PDEs). What I want to know is why it works, theoretically and ...
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1answer
155 views

Solving a PDE possibly with method of characteristics or other methods

For a PDE $$(x-y^{2}) u_{x} + u_{y} = 0$$ I've tried to use method of characteristics. But I've failed to do so. It was because of the term $x-y^{2}$; I don't know how to integrate this on the ...
3
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1answer
161 views

Poisson integral on $\mathbb{H}$ for boundary data which is orientation-preserving homeomorphism of $\mathbb{R}$

Let $f$ be a real-valued function (in my case, an orientation-preserving homeomorphims of $\mathbb{R}$) on the real line $\mathbb{R}$ which is not in any $L^p$ -space. Let us take the simplest example ...