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

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Ordinary point of a Bessel DE

The Bessel DE: $$z^2\frac{\text d^2f}{\text{d}z^2}+z\frac{\text{d}f}{\text{d}z}+\left(z^2-m^2\right)f = 0.$$ The Bessel DE can be rewritten as: $$\frac{d^2f}{\text{dz}^2} + a(z)\frac{df}{ dz } + ...
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1answer
11 views

Reducing a Bessel's differential equation to a more 'useable' form

Suppose the given equation is: $$r^2\frac{\text d^2f}{\text{d}r^2}+r\frac{\text{d}f}{\text{d}r}+(\lambda r^2-m^2)f = 0$$ My text demonstrates the following: Let $$\text{z = }\sqrt{\lambda }r$$ So ...
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1answer
25 views

A domain in $\mathbb{R}^n$ with $C^2$-boundary satisfies an “outer spherical condition”

Let $\Omega\subseteq\mathbb{R}^n$ be a domain and $\partial\Omega\in C^2$, i.e. $\Omega=\overline{\Omega}^\circ$ For all $x_0\in\partial\Omega$, there exists a neighbourhood $U\subseteq\mathbb{R}^n$ ...
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2answers
51 views

How to address multiple cases in this BVP? (Laplace equation in quarter-annulus)

The original problem: $$\nabla^2 u =0 \ \ \ \ for \ \ \ 0<a<r<b\ \ \ ,\ \ \ 0<\theta <\frac \pi 2$$ $$u(r,0)=0,\ \ u(r,\frac \pi 2)=f(r),\ \ u(a,\theta)=u(b,\theta)=0$$ My ...
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2answers
64 views

Why does $D \det(I)B=\frac{d}{d t}\det(I+t B)|_{t=0}$ hold?

The following problem is from Taylor's PDE I. I do not get why $D \det(I)B=\frac{d}{d t}\det(I+t B)|_{t=0}$ hold. Since $\det(I)=1$ and $D$ is $n\times n$ matrix, the left side seems to be a matrix, ...
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How to solve the linear bi-harmonic equation using a fourier transformation?

Let $D$ be and interval in $\mathbb{R}$ or a rectangle in $\mathbb{R}^2$, e.g. $D = [0, d_1] \times [0, d_2]$. For given $f : D \to \mathbb{R}$ With $\Delta = {\partial^2 \over \partial x_1^2} + ...
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1answer
11 views

reducing a pde to a canonical form

I'm really struggling with this one and I can't seem to find what's wrong with my approach. I am given a PDE in the form $$U_{xx} + x y U_{yy} = 0,$$ and I am supposed to bring it to its canonical ...
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56 views

How to solve this boundary value problem?

I'm struggling with the following Boundary Value Problem for some time. The problem is to solve the biharmonic equation $\nabla^4\psi = 0$ with $\psi$ dependent not just on the coordinates on the ...
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Characteristics and additional conditions for differential equation

I need to solve such a DE: $$(1+x^2)u_x+u_y=0$$ And then I need to draw its characteristics. The second part of the task says: Write three additional conditions such that this equation: Has one ...
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1answer
14 views

PDE of the form $x \partial_x T - y \partial_y T = F(x,y)$ where $F$ is a given function.

Is there a known solution, or technique, for solving the following PDE? $x \partial_x T - y \partial_y T = F(x,y)$ Here, $F$ is a given smooth function $\mathbb R^2 \to \mathbb R$, and $T: \mathbb ...
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0answers
6 views

Neumann boundary conditions for Laplace equation with Raviart-Thomas elements

I am working on creating a finite element model for the Darcy equation using Raviart-Thomas elements and the mixed hybrid formulation. The problem in mixed form is this: $\mathbb{K}\nabla p = ...
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0answers
11 views

What is the definition of ``2nd-order quasilinear parabolic'' ? for partial differential systems?

I have to know why the mean curvature flows are 2nd-order quasilinear parabolic. Let $\Omega\subset\mathbb{R}^n$ be a bonded domain (or a smooth manifold of $n$ dimensional) and $N\geq 2$. When the ...
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2answers
45 views

If $\frac{\partial F^i}{\partial x^j}=0$ on a connected open set, is $F$ constant?

Let $U$ be open in $\mathbb{R}^n$ and let $$F:U\to \mathbb{R}^m$$ be a smooth map, i.e. $F\in C^\infty(U)$. It is easy to prove that if $U$ is convex and $$\frac{\partial F^i}{\partial x^j}=0\tag{1}$$ ...
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1answer
26 views

Integrating a Poisson kernel in $n$ dimensional unit sphere

Let \begin{equation*} P(x,y)=\frac{1}{\omega_n R} \frac{R^2-|x|^2}{|x-y|^n} \end{equation*} be a Poisson kernel where $x$, $y$ are in $R^n$, $|x|<R$, $|y|=R$, $\omega_n$ is area of n dimensional ...
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1answer
12 views

Numerical method with a time derivative boundary condition

I'm trying to reproduce a result from a paper I'm reading using a numerical scheme that I'm coding myself. The equation is a reaction diffusion PDE. $$\frac{\partial M}{\partial t}=\frac{\partial^2 ...
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0answers
18 views

How to construct the explicit solution of this boundary-value problem of the PDE of the first order

I want to find the explicit solutions of this boundary value problem of the first order PDE \begin{cases}\tag{1} \text{PDE: } \frac{1}{4}(u_x)^2+u u_y=u(x,y),\quad y\neq \frac{x^2}{2},\\[2ex] ...
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1answer
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Why $\int_{\partial(\Omega \backslash B_p(y))}v(x) \Delta u(x)dx=\int_{\Omega}v(x)\Delta u(x)dx$ as $p \to 0$?

Suppose $v(x)= \Gamma(x-y)=\frac{1}{n(2-n)w_n}x^{2-n}$ when $n>2$. Then $\int_{\partial(\Omega \backslash B_p(y))}u\frac{\partial v}{\partial n}=\int_{\partial\Omega}u\frac{\partial v}{\partial ...
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1answer
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If $u_k$ converges uniformly on $\partial \Omega$, does it converge uniformly on $\Omega$?

Let $u_k$ be continuous on $\overline \Omega$ and harmonic in $\Omega$. Suppose $u_k$ converges uniformly on $\partial \Omega$. Can we conclude that $u_k$ converges uniformly on $\Omega$?
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1answer
48 views

Caccipoli Inequality

I am reading the book Elliptic Partial Differential Equation by Fanghua Lin and I got stuck at the lemma 1.36 ( the Caccipoli inequality). The conditions of this lemma are: $u\in C^1(B_1)$ ($B_1$ is ...
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1answer
537 views

Numerical techniques for solving systems of first-order semi-linear hyperbolic PDEs in two variables

I need to solve such systems of PDEs numerically and I'm wondering what the "standard" method (if there is one) is. The systems I'm interested in have the form: $\frac{\partial F_i}{\partial t} + ...
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Overview of nonlinear analysis, differential equations (ODE and PDE), dynamical systems, and mathematical physics, and their relationships

The fields of (i) nonlinear analysis, (ii) ODE and PDE, (iii) dynamical systems, and (iv) mathematical physics are very huge, fertile, and, in a sense, unorganized (see Open problems in ...
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Heat Transfer FEM 2D PDE Matlab [on hold]

I'd love to know if this looks right in any way since I'm unfamiliar with Heattransfer. The Domain is correct. The heat transfer coefficient is 1. The Dirichlet BC is u(0,x) = 1 and u(y,1) = 0 ...
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18 views

Nonnegative harmonic functions

Suppose $U \in \mathbb{R}^n$ is an open domain, and $u\in C^2(U) \cap C(\bar{U})$ such that $\Delta u = 0$ in $U$. I'm working on a couple of problems pertaining to the mean value formula/harmonic ...
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1answer
29 views

Radial symmetry

This is a full theorem and proof copied from PDE Evans, 2nd edition, pages 558-559. My two questions about two parts of the proof are on the bottom of this post. THEOREM 2 (Radial symmetery). Let ...
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1answer
28 views

What are difference among natural boundary, exit boundary, regular boundary and killing boundary??

In the paper i'm reading, they used the terminologies, natural boundary, exit boundary, regular boundary and killing boundary. I can't find the difference of them and definition of them. Tell me ...
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1answer
24 views

Semilinear Poisson PDE - proving a (hopefully) simple inequality

This is from page 557 of PDE Evans, 2nd edition. My question is at the bottom of this post, but for now, here is some context for my question: LEMMA 2 (Boundary estimates). Let $u \in ...
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27 views

Exact Similarity Solutions of System of Nonlinear Partial Differential Equations [on hold]

I have been reading Self-Similarity and Beyond, by P. L. Sachdev. However, I am stuck on page 70, chapter 3, section 2. I have screen shotted the part which I am having a problem with I wonder if ...
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Help in finding a paper on nonlinear Schrodinger equations [on hold]

I'm looking for the following paper: Authors: Baillon, Jean-Bernard; Cazenave, Thierry; Figueira, Mario Title : Equation de Schrodinger non linéaire. (C. R. Acad. Sci. Paris Ser. A-B 284 ...
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1answer
62 views

Proving identity of a $C_0$ semigroup

We have a strongly continuous semigroup $\{T(t)\}_{t\ge 0}$ on a Banach space X with a generator $ A:D(A)\subset X\to X$. I need to show that $\displaystyle T(t)x = \sum\limits_{n=0}^{\infty} ...
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Is ODE essentially different from PDE or actually PDE is the generalization of ODE? [on hold]

Is ODE essentially different from PDE or actually PDE is the generalization of ODE? If so, how are they essentially different from each other?
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1answer
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The adjoint operator of the second order partial differential operator.

I'm studying the second order elliptic partial differential equations in the 'Partial Differential Equations, EVANS'. The section 6.2.3 begins with defining the adjoint operator $L^*$ of the operator ...
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prove or disprove an inequality on bounds of derivatives for radial functions

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any ...
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2answers
33 views

Linear Hamiltonian System

Suppose the linear system: $\dot{z} = J \frac{\partial{H}}{\partial{z}} = J S(t) z = A(t) z$, with Hamiltonian $H=H(t,z)=\frac{1}{2} z^T S(t)z$. How can I prove that: $$\frac{d}{dt}H(t,\xi(t)) = ...
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Applying different boundary conditions in a quasi Helmholtz problem.

I have tried to solve this exercise from Applied Partial Differential Equations-Richard Haberman, Consider the two-dimensional eigenvalue problem $$ \nabla^2\phi+\lambda\sigma(x,y)\phi=0 $$ ...
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Behavior of a Solution to Heat equation Compactly Supported in Time

We say that a function $g\in C^{\infty}(\mathbb{R}^{n})$ is of Gevrey class $G^{\sigma}(\mathbb{R}^{n})$ if for each compact $K\subset\mathbb{R}^{n}$, there exist $C,R>0$ such that $$\sup_{x\in ...
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1answer
48 views

Maximum principle-estimation

Let $S=\{x \in \mathbb{R}^2 \mid |x| <1\}$. Using the maximum principle I have to show that the solution of the problem $$-\Delta u(x)=f(x), x \in S \\ u(x)=0, x \in \partial{S}$$ satisfies the ...
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Exercise to Heat equation

I am dealing with the following exercise on a generalized heat equation Let $U$ be a open bounded subset of $\mathbb{R}^n$ and $T \in (0, \infty).$ Now, let $u$ be a solution to $$(\partial_t - ...
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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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Charpit's Method

Find the complete integral of partial differential equation $\displaystyle z^2 = pqxy $ ? I have solved this equation till auxiliary equation: $\displaystyle ...
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Fisher's equation

In question 2)c)i) here https://www.maths.ox.ac.uk/system/files/legacy/3333/b08_10_0.pdf then $$\frac{\partial u}{\partial t} = u(1-u-\beta v) + \frac{\partial^2 u}{\partial \xi^2}$$ and ...
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2answers
39 views

Heat equation-unicity

We have the folllwing problem: $\begin{cases} & \dfrac{\partial u}{\partial t} = k \dfrac{\partial^2 u}{\partial x^2}, 0 < x < l, t > 0\\ & u(0,t)=0,\\ & \dfrac{\partial ...
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1answer
65 views

wave equation with neumann boundary and initial condition hat function

I have the following boundary and initial value problem on $0\leq x\leq 1$: $u_{xx}=u_{tt}$, $ f(x)=u(x,0) = \begin{cases} 0& \textrm{ if $0\leq x\leq 1/4$} \\ x-1/4& \textrm{ ...
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1answer
33 views

Boundary conditions that yields no solutions to the coefficient?

I want to make sure I'm not overlooking certain steps as I've already spent an hour looking through. The heat equation is given as: $$ \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} ...
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1answer
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Solving the heat equation

I have the heat equation $$\frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}$$ where $0\leq t<\infty$ and $0\leq x\leq1$ The Boundary conditions are $u(0,t)=0$ and $u(1,t)=1$ with ...
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1answer
26 views

Solve the following PDE: $(1+\sqrt{z-y-x})z'_x+z'_y=2$

Solve the following PDE: $(1+\sqrt{z-y-x})z'_x+z'_y=2$ given that $z(x,2x)=2x$. I want to explain to you how we were taught to solve these at class, and this method seemed to work with other ...
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19 views

Mollifiers and smooth path connetction

Generaly it is about smooth path connection. I have a two smooth paths f,g and a points x,y,z of open subset of $R^n$ such that f is smooth path from a to y and g is smooth path from y to z. I define ...
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20 views

Bounds for the solution of heat equation using convolutions [on hold]

We know that the solution of the heat equation $$u_t=u_{xx}$$ with initial condition $ u(0,x)=u_0(x)$ is given by $$u(t,x)=(u_o * H_t)(x)$$ where the heat kernel is given by ...
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1answer
60 views

heat conduction problem

Find the solution of the heat conduction problem $U_{xx} =4U_t , 0 < x < 2, t>0;$ $U(0,t)=0, U(2,t)=0, t>0$; $U(x,0)=2\sin(\frac\pi2x)-\sin(\pi x) + 4\sin(3\pi x), 0 \le x \le 2 $ ok ...
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1answer
32 views

Equivalence of Dirichlet problems. Gilbarg & Trudinger

I do not understand the proof of theorem 11.4 in the book "Elliptic Partial Differential Equations of Second Order" by Gilbarg & Trudinger. The reason is that I do not understand the text right ...
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1answer
36 views

If $u \in H^1(0,\infty)$, does $u_x(\infty) = 0$?

Let $u \in H^1(0,\infty)$. Then is it true that $u'(x) \to 0$ as $x \to \infty$? I am wondering by Green's formula/IBP in this setting; do I get something like $$\int_0^\infty u_{xx}v = ...