Tagged Questions

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

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2
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0answers
26 views

How can two PDE's have the same classical solution, but different weak solutions?

For example: The implicit solution for the inviscid Burgers' Equation, in two forms: $u_y+(\frac{1}{2} u^2)_x = 0$ and $(u^2)_y+(\frac{2}{3} u^3)_x = 0$ share the same smooth solutions, but they ...
1
vote
0answers
60 views

Derivatives of Differential operators

Let $V=H^1(\Omega)$ and $U = L^{\infty}(\Omega)^2$. Take for example, a differential operator, $L:U\times V \rightarrow V^* $ such that: \begin{equation} L(u,y) = \Delta y + \vec{u}(x)\nabla y ...
2
votes
1answer
22 views

How to identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$? (measurability of function)

How can we identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$? For my understanding: $Q_T:=(0, T)\times \Omega$; DEF1: $L^2(Q_T)=\{u: (0, T)\times \Omega \to \mathbb{R}, \mbox{measurable and} ...
3
votes
1answer
52 views
+50

Inviscid Burger's equation solution

I managed to understand the method of characteristics to get the solution of the transport equation. However, I am getting stucked finding the final solution for the Burger's equation given the ...
1
vote
1answer
37 views

My attempt to prove an inequality get stuck——————where do I go wrong?

Hi, there. Bellow is my attempt. I don't know if I have gone in the wrong way and I am stuck. My attempt: Using Green's representation formula, $u(y)=\int_{\partial \Omega}u \frac{\partial ...
1
vote
0answers
25 views

Fundamental solution of Laplacian on manifold

I'm looking for a reference for the result that there exists a fundamental solution for the Laplacian on a flat torus $$\Delta \Gamma(x-y) = \delta(x-y), \quad x,y \in \mathbb T^2.$$ and that, ...
1
vote
1answer
36 views

Fundamental solution of heat equation on a compact Riemannian manifold

Let $(M,g)$ be a compact Riemannian manifold of $m$ dimensional. Then there exists a sequence $(\phi_i, \lambda_i)_{i\in\mathbb{N}}\subset C^\infty(M)\times\mathbb{R}_{\geq0}$ such that ...
1
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0answers
26 views

How to solve fourth order pde similar to biharmonic equation.

I'm trying to solve a fourth order pde similar to the biharmonic equation $ 0=\frac{\partial ^4}{\partial x^4}u(x,y)+Q\frac{\partial ^4}{\partial x^2 \partial y^2}u(x,y)+\frac{\partial ^4}{\partial ...
2
votes
1answer
28 views

Weak form of $-\frac{d}{dx} \left[(1+x) \frac{du}{dx}\right]$

I am doing some self studying on J.N. Reddy's book, An Introduction to the finite element method, 1st edition. (This was the only one available at the local library). In exercise 2.12, one reads ...
0
votes
0answers
21 views

Crank-Nicolson method for solving nonlinear parabolic PDEs

Is the Crank-Nicolson method appropriate for solving a system of nonlinear parabolic PDEs like $\partial u/\partial t - a\Delta u + u^4 = 0$ ? I tried to apply this method for solving such system but ...
1
vote
0answers
39 views

Question regarding the dual space of $H_0^1(\Omega)$

Given $\Omega\in R^N$ open bounded with smooth boundary. We define $H^{-1}$ to be the dual space of $H_0^1(\Omega)$ and from Evan's PDE book, chapter $5$, we know that for any $f\in H^{-1}$, there ...
0
votes
0answers
36 views

Prove $U$ is subharmonic?

My attempt: Integration by parts says $\int u \triangle \varphi=\int\triangle u \varphi$. We know the left hand side is always $\ge 0$, and hence $\int\triangle u \varphi \ge 0$, since $\varphi \ge ...
1
vote
0answers
13 views

Existence of fundamental solutions of Heston's PDE

The Heston's PDE is a variable coefficient linear PDE. I know the existence of fundamental solution of constant coefficient linear PDE has been proofed by Bernard Malgrange and Leon Ehrenpreis in 1954 ...
1
vote
1answer
27 views

Define a “rotation of u” by $R_{A}u\doteq u\circ A $ with A an orthogonal $n\times n$ matrix and “$\circ$” means composition.

Show that $\Delta (u\circ A )=(\Delta u)\circ A$. (Note: $u(x)\in C^{m\geqslant 2} (\mathbb{R}^{n}) $) I tried to treat u as merely a number in $\mathbb{R}^{n}$ and then take the Laplacian to each ...
1
vote
0answers
6 views

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 ...
2
votes
1answer
31 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 ...
1
vote
1answer
23 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 ...
0
votes
0answers
26 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} ...
0
votes
0answers
6 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!
0
votes
0answers
31 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 ...
0
votes
1answer
22 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 ...
0
votes
0answers
8 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 ...
2
votes
0answers
20 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 ...
0
votes
0answers
19 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 ...
0
votes
1answer
28 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 ...
0
votes
0answers
14 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} = ...
0
votes
0answers
34 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 ...
0
votes
1answer
21 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 ...
0
votes
0answers
12 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 ...
0
votes
1answer
8 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 ...
0
votes
0answers
21 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 ...
3
votes
1answer
42 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 ...
0
votes
1answer
24 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 \\ ...
0
votes
0answers
13 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 ...
1
vote
0answers
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 $B_1(0)$, ...
0
votes
1answer
25 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 ...
0
votes
1answer
47 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 ...
0
votes
1answer
34 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 ...
2
votes
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 ...
0
votes
0answers
20 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 ...
1
vote
1answer
27 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 ...
0
votes
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 ...
2
votes
0answers
33 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 ...
0
votes
0answers
25 views

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

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

How to continue with this PDE.

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

Maximum principle for heat equation with Neumann boundary conditions

Consider the initial-boundary value problem $$ \frac{\partial u}{\partial t} = a\Delta u \;\mbox{ in } \;\Omega $$ $$ a\frac{\partial u}{\partial n} = g \;\mbox{ on }\;\Gamma = \partial \Omega $$ $$ ...
0
votes
0answers
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 ...
1
vote
0answers
70 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 ...
0
votes
0answers
17 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.
2
votes
1answer
28 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 ...