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

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Where to learn perturbation theory for pde (in introductory level)? [Reference Request]

Recently I've been reading the text by Falow 'PDEs for Scientist and Enginieers'. In the latter sections is contained 'Perturbation method'. This one gives only kind of computational techniques; no ...
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1answer
29 views

Help solving $\frac{\partial^3u}{\partial x^3}=\frac{\partial u}{\partial t}$ using Fourier transforms

I am trying to solve $$\frac{\partial^3u}{\partial x^3}=\frac{\partial u}{\partial t}$$ $x\in\mathbb{R},\:t >0$. Subject to the conditions $u(x,0)=f(x),\:u,\:\frac{\partial u}{\partial ...
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1answer
9 views

Regularity of semilinear PDE

I'm reading Evans' PDE book (second edition) and I tried to solve this problem but I'm a little confused Problem 7, chapter 6: Let $u\in H^1(\mathbb{R}^n)$ have compact support and be a weak ...
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12 views

Speed of Fisher Kolmogorov Wave equation in Matlab [on hold]

I have a working code for the Fisher Kolmogorov equation. Now I want to find the speed at which the wave is propagating. And that calculation would be : (x coordinate of U at time t+ delta t) - x ...
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11 views

A finite vibrating string satisfies the following initial value problem, find a solution using separation of variables

We have $u_{xx} = u_{tt}$ on $0<x<L, t\geq 0$ with boundary conditions $u(0, t) = 0$ and $u(L, t)=0$ The string is released from rest with initial displacement $u(x,0) = \left\{\begin{matrix} ...
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4 views

Exercise needed for weak solution of elliptic equations

I'm trying to find more exercise for weak theorem for 2nd order elliptic PDEs, like the exercise in chapter 6 on Evans's PDE book. Any suggestions other than Evans or Gilbarg-Trudinger? Thx!
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17 views

space of solutions of a PDE

so I have just completed part (c) and I'm now on part (d). To fill you in, I have found that $K = -\pi^2 (n^2+m^2)$ for some $n,m \in \mathbb{Z}$ and $p = n\pi, q = m\pi$ now I don't really ...
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19 views

Asymptotic bounds for the solutions of 3d wave equation

Let $u$ solve the 3-d wave equation: $u_{tt}-\Delta u =0$ such that $u=g$ and $u_t=h$ for $t=0$ and where $g$ and $h$ are both assumed to be compactly supported and smooth. I have shown that there ...
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6 views

Accessible resources to learn about bicharacteristic strips

I'm taking an introductory course in PDEs and, once seen the method of characteristics, the professor briefly talked about bicharacteristic strips and micro-local analysis. I'd hate to pass by such a ...
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17 views

Newtonian potential for ellipsoid

Is there an explicit expression of the Newtonian potential for ellipsoid? As the expression for ball is clear by its symmetry. Definition of Newtonian potential of ellipsoid $\Omega$ at x is defined ...
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15 views

A Direct Proof of Representation Theorem for Positive Harmonic Functions in the Half Plane?

Does anyone know a direct proof of this representation theorem for non-negative harmonic functions in the half-plane that doesn't appeal to a similar result in the unit disk? Also, does anyone who ...
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1answer
21 views

Strange thing about Weak Maximum\Minimum Principle?

I feel confused about this problem. I think it is obvious using Weak Maximum\Minimum principles. Since for harmonic functions. If $\Omega$ is bounded and $u\in C^2(\Omega) \cap C^0(\overline ...
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12 views

Coupled second order partial

I have a set of coupled differential equations $\frac{\partial f}{\partial t} + a_1 \frac{\partial^2 f}{\partial x^2} = -b f$ $\frac{\partial g}{\partial t} + a_2 \frac{\partial^2 g}{\partial x^2} = ...
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20 views

Proof using convolution?

there. I am a novice in graduate school. This is the first time I learn PDE in graduate level. I found it so hard. I am going to have a test next week and I am so worried about it. Since I always ...
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1answer
19 views

Solving the PDE $\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial ^2 \phi}{\partial y^2} = K \phi$ using the separation of variables

I'm trying to solve $\dfrac{\partial^2 \phi}{\partial x^2} + \dfrac{\partial ^2 \phi}{\partial y^2} = K \phi$ with $K$ constant I let $\phi = XY$ then got $X''/X + Y''/Y = K$ but I'm not sure where ...
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7 views

finding ODEs satisfied by X and Y for a PDE

for the following PDE separate the variables using $\phi(x,y) = X(x)Y(y)$ and find the ODEs satisfied by $X $ and $Y$ PDE: $$ \dfrac{\partial ^2 \phi}{\partial x^2} = \dfrac{\partial^2 \phi}{\partial ...
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1answer
6 views

Maximal principle: proof for subsolution

I have recently gone through the statement that a subsolution $v$ satisfies the maximal principle: $\sup_{\Omega T} =\sup_{\partial\Omega T}$. So if $u(x,t)>0$ is a supersolution, then how can we ...
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16 views

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
32 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
22 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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10 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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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
44 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
29 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
22 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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17 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
20 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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29 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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24 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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28 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
20 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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1answer
66 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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12 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
25 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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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
17 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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25 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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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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31 views

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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17 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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11 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
32 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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integera surface of linear PDE which contains circle [closed]

solutoin of entegeral surface of he linear PDE which contains the circle
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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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20 views

Formation of PDE [closed]

I need to find the PDE arising from the surface: $f(x^2-y^2, x^2-y^2) = 0$ How can I approach such problems when surface is given as an arbitrary function of two complex functions ?