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

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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 ...
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0answers
205 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 ...
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1answer
97 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
185 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 ...
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1answer
139 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 ...
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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
153 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 ...
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2answers
281 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 ...
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1answer
355 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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74 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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431 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
101 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
186 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
39 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
633 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 ...
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1answer
101 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 ...
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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 $F=F(x,t), f=f(x,t)$ ...
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1answer
136 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
85 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 ...
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70 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
447 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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74 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 ...
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1answer
243 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 > ...
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2answers
1k 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
110 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
239 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 $\textit{infinitesimal generator}$ ...
3
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2answers
566 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
288 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
375 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
129 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
144 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
147 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 ...
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1answer
110 views

Why do we use this class of functions?

This is mainly concerned with studies coming from Gilbarg and Trudinger. Elliptic Partial differential equations are of the form $Lu =f$, where $L = a^{ij}D_{ij} + b^iD_i + cu$, and most of the ...
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2answers
2k views

Heat Equation with One Non-Homogeneous Boundary Condition

I posed myself the following PDE because it would be interesting to graph: $$ u_t=u_{xx},\qquad0<x<L,\qquad t>0,\\ \begin{align} u(0,t)&=\sin^2\frac t2,\\ u_x(L,t)&=0,\\ ...
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2answers
226 views

Mean-value formula for inhomogeneous harmonic functions

I ma working on Evans' PDE textbook problems, but I am stuck with the following problem about modification of the proof of the mean-value formula for harmonic functions. I cannot really see how to ...
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1answer
315 views

A question on a solution of an inhomogeneous heat equation.

I am now working on the following PDE equation (Evan's PDE textbook Section 2.5 No.14) \begin{align} u_{t}-\Delta u + cu=f \ \ & on \ \ \mathbb{R}^n\times (0,\infty) \\ u=g \ \ & on \ \ ...
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2answers
652 views

Solution of Wave Equation using Reflection Principle

A sample problem for an exam is as follows: Consider the wave equation $U_{tt} = 4U_{xx}, 0 < x < 1$ with $U(0,t)= U(l,t)= 0$ and $U(x,0)= x(1-x)$, $U_t(x,0)= \pi$. Find $U(1/4,1/4)$ ...
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2answers
101 views

Mathematical explanation of problems behind time and space derivatives being second order

$\left(\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)\phi = \frac{m^2c^2}{\hbar^2}\phi$ with the wave function $\phi$ being a relativistic scalar: a complex number which has ...
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1answer
169 views

Mean Value property for harmonic functions on regions other than balls/spheres

Let $u$ be a harmonic function on $\mathbb R^n$. We know that if $B$ is a ball centered at $x$, then $$u(x) = \frac{1}{|B|} \int_B u(y) dy = \frac{1}{|\partial B|} \int_{\partial B} u(z) dS(z).$$ I ...
3
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1answer
197 views

On the regularity of the Laplace equations and tensor products and such

To start with, let me apologize for my ignorance as I know next to nothing about partial differential equations. My question is about the tensor product of Banach spaces but actually I do not ...
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53 views

which subarea of math text book study about the theory of smooth function?

In another word, which subarea does the theory of smooth function have? I would like to know the list of book on analysis that i could learn more about smooth function.
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78 views

I need to find a specific maximum principle

I need a maximum principle that says: If $L$ is an elliptic operator and $u$ is a positive function ($u\in C^2(\Omega)$, with $\Omega\subset\mathbb{R}^n$ an unbounded domain) such that $Lu\geq0$ ...
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2answers
302 views

Heat equation with initial values $U(0,t)=U_1$, $U(L,t)=U_2$,$\forall t$.

My problem is given as Arbitrary temperatures at ends . If the ends $x=0$ and $x=L$ of the bar in the text are kept at constant temperatures $U_1$ and $U_2$ respectively, what is the temperature ...
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1answer
79 views

About fourier transform in the PDE

In the PDE as below$$ \partial_t u - \frac{i}{\rho} ( - \partial_x^2 )^{\rho /2} u = 0 \;\;\;(t,x) \in \Bbb R^2 $$ How can I prove that $$ (- \partial_x ^2 )^{\rho/ 2} = \scr F ^{-1} | \xi |^\rho ...
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2answers
485 views

Ways to solve a 2D Laplace equation

I'm looking for a survey of methods to solve Laplace equation in two dimensions. Is there a book describing them with hints regarding their applicability for various cases? I mean analytical ...
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1answer
115 views

Sobolev Approximation Theorem Example

The approximation theorem states that if $U$ is bounded, $u \in W^{1,p}(U)$ for some $1 \leq p < \infty$ then there are functions $u_m \in C^\infty(U) \cap W^{k,p}(U)$, such that $u_m \rightarrow ...
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38 views

characteristics proyections of a PDE

Let $u(x,y)$ by the integral surface of the equation: $a(x,y)u_x+b(x,y)u_y+u=0$ Where $a,b$ are positives differential function in the hole plane. Let $D=\{(x,y)||x|<1 ,|y|<1\}$ How do I ...
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1answer
54 views

Stability in PDE with the $L^2$ norm

$u_t +au_x = f(x,t)$ for $ 0<x<R$ and $t>0$ $u(0,t)=0$ for $t>0$ $u(x,0)=0$ for $0<x<R$ I have manage to get: $\int_0^R \! u^2(x,t) \, \mathrm{d} x \leq e^t \int_0^t \int_0^R \! ...
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1answer
113 views

Magnus expansion for linear operators

I want to learn about Magnus expansion for the time dependent pdes of the form $u_t(t,x)=A(t)u(t,x)$. According to the wikipedia explanation http://en.wikipedia.org/wiki/Magnus_expansion $A$ has to be ...