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

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Compactly supported function.

Can anyone explain me the features of a compactly supported functions behave when they are compactly supported. I am learning PDE and I come across it very often. For example : When we define weak ...
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55 views

PDE multiplicity problem

I need help with the following question, any help is appreciated. Show that $0$ is an eigenvalues of multiplicity $1$ for the problem $-\triangle u=\lambda u$ in D $\triangledown ...
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Solution of Laplace's equation in an annulus with constant Dirichlet conditions?

What's the solution to Laplace's equation $\nabla^2V=0 $ in the annulus with centre 0, inner radius 1, and outer radius 2, with boundary conditions $V=0$ on the inner boundary and $V=1$ on the outer ...
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123 views

What is 'imbedding' with Sobolev space and $ L^2 $ space?

I want to know that the meaning of the following. $$ W^{n,1}\textrm{ is continuously imbedded into }L^2$$ Here, $W^{n,1}$ is a Sobolev space.
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396 views

Length of the zero level set of a function

Say we have a continuous function $u(x,y) : \mathbb{R}^2 \rightarrow \mathbb{R}$. I have seen several textbooks that make the following assertion: The length/area element of the zero level set of ...
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Find solutions that fit the heat function

I have an exercise as homework and I am stuck. If $k\in R$ the heat equation is $$\frac{\partial f}{\partial t} - k \left(\frac{\partial ^2f}{\partial x^2}+\frac{\partial ^2f}{\partial y^2} \right) ...
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344 views

Heat Equation Derivation and Mean Value Theorem

Farlow book PDEs for Scientists and Engineers pg. 27 shows derivation for Heat Equation. It starts by stating Net change of heat inside $[x,x+\Delta x]$ = Net flux of heat across boundaries + Total ...
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426 views

PDE with method of characteristics

Let $F(z,t)=\sum_{n=r}^k z^n p_{r,n}(t)$ with $|z|<1$ a generating function of probability. $F$ satisfies the following PDE: $$ \frac{\partial }{\partial t}F(z,t) + (z-1)(az+b)\frac{\partial ...
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614 views

Method of characteristics with constant PDE

I was just introduced to method of characteristics for solving PDE's. We solved the wave equation that is inifinitely long using this method. However I am very confused about this method. Here is a ...
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113 views

Solution for a PDE on $\Omega=[0,\pi]^3$

In Strauss's Partial Differential Equations, the eigenvalue problem $$-\triangle v=\lambda v,\qquad v|_{\partial \Omega}=0$$ is solved by separating the $x,y,z$ variables: $v=X(x)Y(y)Z(z)$, $$ ...
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346 views

On the method of characteristics

By the method of characteristics, it is possible to prove the existence of local solution for the first-order linear non homogeneous equation: $\mathcal{L}_X f+g=0$ where $X$ is a non-singular ...
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604 views

Change of variables in a partial differential equation

How to convert the following partial differential equation (pde) $$\frac{\partial V}{\partial t}=aV-as\frac{\partial V}{\partial s}-b^2s^2\frac{\partial^2 V}{\partial s^2}$$ into a pde of the form ...
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39 views

Is $C^{\infty}$ dense in $W^{k,p}$?

The $C_c^{\infty}$ are certainly not sense in an arbitrary $W^{k,p}$ space. Despite, I started wondering whether at least $C^{\infty}$ is dense? Now, this can certainly not be true for general ...
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Evans PDE, Problem 8 Chapter 2 clarification on $|x-y|$

Hi I am attempting problem 8 (Chapt2 Evans PDE). Again I found the solution on the internet. enter link description here I understood much of everything of the proof except for one line. " Since ...
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44 views

analytic solution poisson equation spherical coordinates

I'm trying to analitically solve a poisson equation $\nabla^{2}p(r,\theta,\phi)=f(r,\theta)$ in spherical coordinates; the boundary condition is $p(\infty,\theta,\phi)$=0. I'm quite used to the ...
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Help understanding variable substitutions

The concrete problem is this Let $$u(t,x)=v(\frac{x^2}t)$$ on $\mathbb R^+\times\mathbb R$. Show that $$u_t=u_{xx} \Leftrightarrow 4zv''(z)+(z+2)v'(z)=0$$ Now, while I would like to know the ...
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27 views

A solution of a differential equation of first order in the large-variable limit

The differential equation reads: $ \dfrac{\partial R (t)}{\partial t} = \dfrac{c_2}{R^2} + \dfrac{c_3}{R^3} + O(R^{-4})$, Where $c2 > 0$ and $c3 > 0$, how to get the solution of the ...
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Fourier transform of PDE on finite and infinite bound simultaneously.

Consider $$u_{xx} + u_{yy} = 0 $$ on the bounds: $$o < x < L$$ and $$-\infty<y<\infty$$ The initial condition is: $$u(0,y) = f(y)$$ and $$u(L,y)=g(y)$$ I've tried performing fourier ...
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28 views

Solution to a pde

I have a PDE system that I am trying to solve at steady state. When I make the appropriate substitutions, I get an equation of the following form: $$\frac{1+M}{M}\frac{d^2 M}{d x^2}=1$$ Is there a ...
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37 views

Is there a name for this type of linear partial differential equation?

I came across the following form of first order linear partial differential equation and I was wondering if there is a name for it? $$ \frac{\partial g}{\partial x} \frac{\partial f}{\partial y} - ...
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$L^1$ Estimates involving bi-Laplacian

The following inequality can be shown to be true in the cases $p>1$: If $n\ge 5$, $\frac{1}{q} = \frac{1}{p} - \frac{4}{n}$, then there exists $C_{p,q,n}>0$ such that, for every $f \in ...
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functions in $H_0^1(\Omega) \cap C(\overline{\Omega})$ are zero on the boundary

I want to solve the following problem Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set with $C^\infty$-smooth boundary. Show that for any $u \in H_0^1(\Omega) \cap C(\overline{\Omega})$ ...
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106 views

What is the weak formulation of this problem?

Find $u\in H_D^1(\Omega)$ such that $-\nabla\cdot(a\nabla u)=0$ in $\Omega$, $\dfrac{\partial u}{\partial n}=g$ on $\Gamma_N$, $u=0$ on $\Gamma_D$. The function $a(x,y)$ is ...
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Finding an approximate solution to a differential equation using finite difference method.

I have a differential equation $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=-2$$ on the square $$0 \leq x,y \leq 1$$ subject to the boundary conditions $u=0$ along $x=0$ ...
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Characterization of $H^k$ by Fourier transform

Let $u\in H^k(\mathbb{R}^n)$ a function with complex-valued. Question 1: If $u\in H^k(\mathbb{R}^n)$ and $D^\alpha u\in L_2(\mathbb{R}^n)$, by definition for each $|\alpha|\le k,$ then we have ...
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55 views

Trouble understanding method of characteristics-PDE for solving the Cauchy Problem

So I am trying to understand the non-algorithmic part of the method of characteristics for solving a first order quasilinear PDE: $ a(x,y,u)u_{x}+b(x,y,u)u_{y}=c(x,y,u) \hspace{1cm } $ (1) I ...
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37 views

Solve the following PDE using Fourier transform

Solve the following 3-D wave equation using Fourier transform $$PDE: u_{tt}=C^2[u_{xx}+u_{yy}+u_{zz}],\qquad-\infty<x,y,z<\infty,\qquad t>0$$ $$BC: u(x,y,z,t)\rightarrow 0\qquad as \qquad ...
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34 views

Quasi-linear PDE with Cauchy conditions:

Am attempting to solve the partial differential equation given by $$u_t + uu_x = 0$$ for some wave function $u(x, t)$, subject to the conditions $$u(x, 0) = \phi(x) = 1-x^2$$ I started by using the ...
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Counter example in Sobolev spaces: Is there a standard example for showing that $H^1(\mathbb{R}^2)\not\subset L^\infty(\mathbb R^2)$?

One way to define the Sobolev space $H^s(\mathbb{R}^n)$ is as the completion of the $C^\infty$ functions on $\mathbb{R}^n$ under the norm $$\lVert f \rVert_{H^s(\mathbb{R}^n)}:= \left( ...
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Solving PDE using normal mode

Given a linearised PDE $u_t=u_{xx}+\mu u$ where $x\in[0,1]$. A hint given is $u=V(x)\exp(st+ikx)$, where $s$ can be complex and $k$ is real. When I substituted into the PDE, I get ...
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Deciding when to use $\mu^2$ or $-\mu^2$ in separation of variable (PDE'S)

I have the following question Consider the two-dimensional PDE on $u = u(x,y)$ $$u_{xx}-u_{yy}=0$$ $$u(x,0)=\phi(x)$$ $$u_y(x,0)=0$$ where $\phi:\mathbb{R}\rightarrow \mathbb{R}$ is a function ...
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Does the Laplace operator include the second derivative with respect to time variable?

Does the Laplacian of a function $f(x,z,t)$ equal $f_{xx} + f_{yy} + f_{tt}$? We aren't sure whether or not time is included in it or not.
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Fourier transformations and the inversion formula

I am working through the above question in preparation for an upcoming exam. I have completed part (a) and quoted the inversion formula for part (b), but I cannot see how to find a form to evaluate ...
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Find the Characteristic Curve of the PDE

Let $x=x(s),y=y(s),z=z(s) ,s\epsilon\Bbb R $ , be the characteristic curve of the PDE $z_x + z_y -z = 0$ passing through the curve $x=0 , y=t , z=t^2 , t\epsilon\Bbb R$ Then what are ...
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29 views

Laplace Operator in $3D$

I am looking to find the radial part of Laplace's operator in three dimensions. I looked up Laplace's operator in spherical coordinates and from there I guess the radial part is: ...
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45 views

Orthogonality lemma sine and cosine

I want to know how much is the integral $\int_{0}^{L}\sin(nx)\cos(mx)dx$ when $m=n$ and in the case when $m\neq n$. I know the orthogonality lemma for the other cases, but not for this one.
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How do I solve this PDE (diffussion equation) using the sepration of variables method?

$$\frac{\partial u}{\partial t} =\nu\large(\frac{\partial^2u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} \large), 0 < r < a, t >0.$$ Subject to the conditions $$\frac{\partial ...
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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 ...
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61 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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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 ...
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Finding a solution to a PDE

EDIT: This is for a production scheduling problem with quadratic production and linear inventory costs. The goal is to \begin{equation*} \max_{u} \int_{0}^{T} -(c_{1}u^{2} + c_{2}y) dt ...
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What does the solution of a PDE represent?

So I took a course in PDE's this semester and now the semester is over and I'm still having issue with what exactly we solved for. I mean it in this sense, in your usual first or second calculus ...
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How can we get a contradiction?

How could we show that the problem $$u_{tt}(x, t)-u_{xt}(x, t)=0, x \in \mathbb{R}, t \in \mathbb{R}, \\ u(x, -x)=0, x \in \mathbb{R}, \\ u_t(x, -x)=x, x \in \mathbb{R}$$ doesn't have any smooth ...
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Historical Development of PDEs

I would be really thankful if someone could tell me good references giving development of techniques of Solving PDEs why such equations are important. Regards, Harish
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Method of characteristics- $u=$ constant or $u = f(y)$?

Say $u(x,y)$ is a function of $x$ and $y$ and suppose we have the following pde - $u_x - u_y = 0$ This equation has the following characteristics - $\frac{dx}{1} = \frac{dy}{-1}$ $\frac{du}{dx} = ...
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Solve $\frac{\partial^2z}{\partial x \partial y}= x^2y$

Question: Find the particular solution of the following PDE using separation: $$\frac{\partial^2z}{\partial x \partial y}= x^2y$$ such that \begin{align} z(x,0)&= x^2 \\ z(1,y) &= ...
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Have I obtained the proper solution to this PDE?

I'm a little stuck on this. Consider $ u_t -(1+t^2)u_x = \phi(x,t) \quad u(x,0)=u_0(x)$ Via the method of characteristics, the total derivative of $u(x,t)$ is $$\frac{du}{dt} = \dfrac{\partial ...
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From $f(x,y)=g(3x-4y)$ and $f(x,0)=\sin x$ for all $x$, how can I find $f(x,y)$?

I found $f(x,y)=g(3x-4y)$ and $f(x,0)=\sin x$ for all $x$. How can I find $f(x,y)$? It seems like $g(3x)=\sin x$ for all $x$. But how can I find $g(3x-4y)$?
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Fourier sine series of cosine?

In the middle of a PDE I'm trying to solve, I've gotten $$\sum_{n=1}^\infty T_n(0) \sin(nx) = \cos(3x)$$ Is this even possible? How can you expand a cosine (even) in terms of sines (odd)? Did I ...
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61 views

Does the wave equation require an initial function for one of its derivative?

Is it possible to find an explicit solution to the wave equation: $$ \partial_t^2u-c^2 \partial_x^2 u=0 \\ u(x,0)=f(x), \ u(cx,x)=g(x) $$ or do we need information about a derivative of $u$ as well?