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

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Inverse Fourier Transform of $\cos(c\omega t)$ and $\sin(c\omega t)$

I'm just needing a bit of help to understand the derivation of the inverse fourier transform of $\cos(c\omega t)$ and $\sin(c\omega t)$, in deriving D'Alembert's solution to the wave equation. I get ...
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1answer
38 views

Solving Laplace's equation

Given Laplace's equation $u_{xx} + u_{yy} = 0$ and 2 boundary conditions $u(x,y)=x^2y$ $u(cos\theta,sin\theta) = 1 + cos\theta$ in the unit disc. I want to solve laplace equation. After using ...
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1answer
23 views

Is it possible to solve pde with 2 Neumann boundary conditions (Gaussian Elimination)?

I have the following equation: $$ \nabla^2u = f $$ over $\Omega: [0,10] \times [0,10]$ where boundary conditions: $$ \left\{ \begin{array}{ll} \frac{\partial u (0,y)}{\partial x} = 0 \\ ...
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12 views

Why is the Laplace/Helmholtz equation only separable in a finite number of coordinate systems?

On MathWorld one finds that the Helmholtz equation $$(\nabla^2+k^2)\psi=0$$ is only separable in 11 coordinate systems. Similar statements can be found about the Laplace equation and maybe other ...
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24 views

Asking for help with a PDE problem.

everyone. I am relatively new to PDE and I am self-studying. I am REALLY puzzled by the following statement in a book, so I come to ask for help. If $u$ is a solution of $\Delta u = f$ in $B_1(0)$, ...
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1answer
23 views

Continuity of a composition map between Holder spaces

Let $\varphi\in C^{\infty}_0(\mathbb{R})$, $0<\alpha<1$, $\Omega\subset\mathbb{R}^d$ be a bounded domain. Is it true that the map $\Phi:C^{0,\alpha}(\bar{\Omega})\to ...
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1answer
46 views

boundary conditions for operator

if you have a Schrödinger operator on a sphere ( $\mathbb{S}^2$) $-\Delta_{\theta,\phi} \psi(\theta,\phi) + V(\theta) \psi(\theta,\phi) = E\psi(\theta,\phi),$ where the potential does not depend on ...
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1answer
33 views

Advection equation with source u/x

I am trying to solve following equation: $$ u_t + u_x + \frac{u}{x} = 0 $$ With initial condition: $$ u(x,0) = 0 $$ And with boundary condition given at x = 15: $$ u(15,t) = sin (wt) $$ I tried to ...
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1answer
26 views

The spectrum of $L:=-\Delta+V(x)$ on complex $L^2(\mathbb{R}^N)$ and real $L^2(\mathbb{R}^N)$

In general, when one talks about the spectrum of an self-adjoint operator, it is naturally considered in a complex Hilbert space (say $L^2(\mathbb{R}^N,\mathbb{C})$). Moreover, the spectral ...
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19 views

Numerical method of lines for solving PDEs

Could you please advise some literature about the numerical method of lines (MOL) for parabolic PDEs? It is a method of solving PDEs with discretizing only by space but not by time. A system of ODEs ...
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1answer
26 views

Using the convergence of Fourier Series Theorem to estimate the number of terms for Fourier Series $f(x)$

Attached are scans from my book. One of my homework problems requires me to let $f(x)=(x^2-1)^2$ for $-1 \leq x \leq 1$. I am using the book's example (Example 5) as a guideline, but it is driving me ...
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1answer
21 views

How to find the coefficients in the Fourier series solution of a 1-D heat equation?

I am trying to use Fourier's method to solve a problem. $u(x,t) = \sum \limits_{n=1}^\infty B_ne^{-(n\pi C / L)^2 t}\sin\left(\frac{n\pi x}{L}\right), B_n=\frac2L\int_0^L \sin\left(\frac{n\pi ...
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0answers
30 views

Laplacian on $\mathbb{S}^2$ has a pure point spectrum

Consider an operator $T = -\Delta + V(\theta)$ where $V(\theta)$ is $C^{\infty}$ and $T : C^{\infty}(\mathbb{S}^2) \subset L^2(\mathbb{S}^2)\rightarrow C^{\infty}(\mathbb{S}^2).$ I was wondering why ...
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25 views

This PDE is simple but I have an elemenatry problem. [duplicate]

Given this function: $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} - \omega^2u ,\quad x \in \mathbb{R}, t > 0$$ $$u(x+2\pi, t)=u(x,t) ,\quad x \in \mathbb{R}, t > 0$$ ...
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30 views

How to continue with this PDE.

This is given function: $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} - \omega^2u ,\quad x \in \mathbb{R}, t > 0$$ $$u(x+2\pi, t)=u(x,t) ,\quad x \in \mathbb{R}, t > 0$$ ...
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1answer
22 views

Maximum principle for heat equation with Neumann boundary conditions

Consider the initial-boundary value problem $$ \frac{\partial u}{\partial t} = a\Delta u \;\mbox{ in } \;\Omega $$ $$ a\frac{\partial u}{\partial n} = g \;\mbox{ on }\;\Gamma = \partial \Omega $$ $$ ...
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12 views

Sobolev spaces of maps between manifolds and the Palais-Smale Condition

I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts: Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean ...
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69 views

Complex Fourier Series. I Might Neeed Some Help On This Problem

The Problem: If $f(x) $ is a real funciton, rewrite the integral: $$ \frac{1}{2L} \int_{-L}^{L} |f(x)|^{2} \, dx$$ in terms of the usual Fourier Coefficients, $A_n$ and $B_n$ The attempt: Recall ...
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15 views

Find a particular solution of a nonhomogeneous modified Bessel equation

Find a particular solution of the following equation: $\frac{1}{r}\frac{d}{d r}(r\frac{d\phi}{dr})-\phi=\frac{1}{r}\delta(r-r_0)$ where $\delta(r)$ is the Dirac Delta funtion and $r_0$ is constand.
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1answer
26 views

Classification of pde

I got stuck on the following problem: Determine the subsets of $\mathbb{R}^2$ where the pde $$u_{xx}+2xu_xu_{xy}+yu_{yy}+yu_x=1$$ is elliptic, hyperbolic and parabolic respectively. Now, at first I ...
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1answer
25 views

physical meaning of heat equation

consider the heat equation $u_t=a(t)u_{xx}+f(x,t)$, $0<x<L$, $0<t<T$ subject to the initial condition $u(x,0)=g(x)$ and boundary conditions $u(1,t)=0,$ $u_x(0,t)+hu(0,t)=0$ where ...
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1answer
14 views

Dirichlet and Neumann problems uniqueness

Prove uniqueness for the Dirichlet and Neumann problems for the reduced Helmholtz equation $\triangle u − ku = 0$ in a bounded planar domain $D$, where $k$ is a positive constant. How can I ...
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1answer
20 views

Obtaining solutions to boundary value problem from general solution

Suppose $u(x,y) = \frac{y^{2}}{2} - g(ye^{-x})$ is the solution to a second-order linear PDE, where $g$ is some smooth function. If we have the additional boundary value condition that $u(0,y) = ...
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9 views

Partial differential equation question from Zachmanoglou, describing regions where the equation is hyperbolic, parabolic, or elliptic

Given an equation $2u_{xx} + 4u_{xy} + 3u_{yy} - u = 0$, how to describe the regions where the equation is hyperbolic, parabolic, or elliptic? Do I need to find the Laplacian?
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26 views

Geodesics on a perturbed submanifold of $\mathbb{R}^m$

Let us consider $M$, a Riemannian manifold of dimension $n$, isometrically embedded in $R^m$. Let us consider a geodesic $\gamma$ on $M$. Now, let us "perturb" (in other words, change slightly the ...
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76 views

Solution of parabolic PDE system

For the following parabolic PDE system, $u(x,t)$ and $v(x,t)$ are functions of independent variables $x$ and $t$, $x\in[a,b]$. \begin{equation} \begin{cases} \frac{\partial}{\partial ...
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A PDE problem related to the ratio of populations

Let $m>0$ in $\bar{\Omega}$ be a given nonconstant function, where $\Omega\subset \mathbb{R}^n$ is a bounded smooth domain. Then consider the following elliptic modeling problem: $$ \Delta ...
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18 views

Laplace equation with split boundary conditions

I am struggling with a Laplace equation with different boundary conditions: Domain: $0 < x < W, \quad 0 < y < H$, $$U_{xx} + U_{yy} = 0,$$ with these boundary conditions: $$U_x(0,y) = ...
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14 views

Three dimensional plate model

Does anyone know of a good book or paper where the natural boundary conditions for the three dimensional plate model with simply supported edges are derived? I think that the bending moments should ...
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1answer
38 views

Rewriting partial differential equation

I have some trouble rewriting a partial differential equation, more specifically the heat equation in one dimension: $ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + f(x,t)\\ $ ...
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1answer
23 views

I might need some help on this Complex Fourier Series Problem

Here is the problem: Use the Complex Fourier Series on $[-L,L] $ with complex coefficients to find a representation of $\frac{1}{2L} \int_{-L}^{L} |f(x)|^{2} dx$ Here is my attempt: The ...
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24 views

scaling 1st order quasilinear PDE and method of characteristics

Let $\alpha:\mathbb R^n\to (0,\infty)$, $b:\mathbb R^n\times\mathbb R\to\mathbb R^n$ and $c:\mathbb R^n\times\mathbb R\to\mathbb R$ be smooth functions. Then the differential equations $$ ...
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17 views

Vibrating String in air

Suppose that we have a vibrating string and we have friction in this problem.model this problem and solve Partial differential equation of that.
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1answer
31 views

Can you help me with this Complex Fourier Series Problem?

Find the Complex Fourier Series of $F(x) = \cos(2x) + \sin(x)$ on the interval $[-\pi, \pi]$ Here is my attempt: The complex Fourier Series is in the form $\cos(2x) +\sin(x) = \sum_{n= ...
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16 views

Solving a first order linear PDE with three unknowns in characteristic equation

Consider the following PDE: $$x(x^2+u)u_x-y(y^2+u)u_y=(x^2-y^2)u.$$ The characteristic equation is as follows: $$\frac{dx}{x(x^2+u)}=\frac{dy}{y(y^2+u)}=\frac{du}{(x^2-y^2)u}.$$ If we try to solve ...
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22 views

Hints on solution to $u_t-\Delta u+cu=f$

Consider the problem (Evans, Ch 2, 14) $$ u_t-\Delta u+cu=f ,x \in \mathbb R^n\times (0,\infty)$$ $$ u=g , \mathbb R^n\times {t=0} $$ If $u$ solves $ u_t-\Delta u=f$, $u=0$ on and $v$ solves ...
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148 views

Find a scalar potential of $v(x, y)= g(y/x)(-1/x, 1/y)$

Let $v(x, y)= g(y/x)(-1/x, 1/y)$ be a vector field on $Ω$ where $Ω := [(x, y) : x > 0, y > 0]$. $g:\mathbb{R}→\mathbb{R}$ is continuous. Find a scalar potential of $v$ in terms of an ...
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31 views

Finding a solution to $xu_x+(y+1)u_y=u-1$ given an initial condition

Consider the following PDE: $$xu_y+(y+1)u_y=u-1.$$ Using this formula: $$\frac{dx}{x}=\frac{dy}{y+1}=\frac{du}{u-1}.$$ This yields $c_1=\frac{y+1}{x}$ and $c_2=\frac{u-1}{x}.$ We have: ...
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1answer
27 views

Help with biharmonic functions

I tried to use using Green's formula to prove $\triangle u=0$. But it turned out to be wrong because $u=0$ doesn't imply $\triangle u=0$ on the boundary. I am a novice in graduate pde, so I didn't ...
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PDE using $\theta$ method in Matlab

I'm trying to solve this problem numerically in Matlab: $ \left\{ \begin{array}{rl} \frac{\partial P}{\partial t} &= \frac{\partial^2 P}{\partial x^2} \ \ \ (\star) \\ P(x,0) &= 1 \\ ...
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1answer
32 views

Local barrier implies barrier?

there. This is part of the textbook of Gibarg's PDE: My question is that how to verify the part in red? How to know $\overline w$ is continous in $\overline \Omega$? Thanks so much! Your help ...
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Prove a function is subharmonic?__Gilbarg Exercise

My attempt: By a long time google search I found in a book about how to prove it in the case of harmonic function. Just construct another function $v$ which is harmonic in the $\Omega$, then prove ...
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38 views

Why the integral of a function with compact support is equal to zero?

In the text below (from PDE by Evans) the author says that the integral zero because the function has compact support. Why is that? I am referring to the last two lines and equation (4).
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2answers
63 views

Separation of variables PDEs

In this answer, he has three cases $(\lambda = 0, \lambda \lt 0, \lambda \gt 0)$. I understand the first does imply it is linear, hence it isn't consistent with the initial conditions, and looking at ...
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25 views

Is solving Poisson's equation in Polars different from Cartesian?

I'm having trouble figuring out how to separate variables in polar coordinates in 2D. In cartesian coordinates it is fairly simple to use eigenfunction ideas because I can group together the x, y ...
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1answer
52 views

PDEs for string deflection.

Okay, I have to find $u(x,t)$ for the string of length $L=\pi$ when $c^2=1$. I know: $$\text{wave equation}: \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$$ $$u(x,0)=\frac ...
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17 views

Trouble Understanding Derivation in Example 1.2.2 in Amaranath PDE Book

I'm having a bit of trouble understanding how my book derived one of its equations. The example goes: Consider the surfaces of the form $F(u,v)=0$ where $u=u(x,y,z)$ and $v=v(x,y,z)$ are known ...
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1answer
63 views

Uniqueness in boundary value problem for the biharmonic functions

My attempt: I tried to use the Green's representation formula twice. The Green's reprensentation formula:$u(y)=\int_{\partial \Omega}(u(x)\frac{\partial G(x-y)}{\partial v}-G(x-y)\frac {\partial ...
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21 views

Why this operator is not fredholm?

Define $f:S^{2}\to \mathbb{R}$ by $f(x,y,z)=z$. Let $D:=D_{\nabla f}$. As I learned from the following post this operator is not counted as a fredholm operator.( I did not underestand, ...
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23 views

Conceptual difficulty about eigenfunction expansions

I'm having a fairly large conceptual block on my understanding of eigenfunctions - I've tried out two different methods that yield embarrassingly different answers, and I would appreciate if someone ...