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

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A general solution of a partial differential equation with $f(x,y)$

I need to find a general solution to such a PDE: $$u_x-u_y=f(x,y)$$ I am able to find a solution if $f(x,y)=0$ or $f(x,y)=u$. But I have no idea how to get the general solution. Has anybody got any ...
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Notation for vector valued integration

Suppose we have a vector field $\mathbb v=(v^1,v^2)$ on the two dimensional torus $\mathbb T^2$ and we wish to compute $\int_{\mathbb T^2} \mathbb v \cdot \Delta \mathbb v $, where the Laplacian acts ...
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2answers
27 views

Finding the particular solution of a pde.

I have solved a PDE up to the point of finding the particular solution. I am trying to find the constant $$C_n$$ I have the expression $$3x-x^2=\sum_{n=1}^{\infty} C_{n} \, \sin\left(\frac{\pi n ...
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1answer
14 views

Is $C^\infty_0(\Omega)$ complete with the norm $\|u\|_\Delta:=(\|u\|_{L^2(\Omega)}^2+\|\Delta u\|_{L^2(\Omega)}^2)^{1/2}?$

Let $\Omega$ be an open subset of $\mathbb R^n$. Is it true that $C^\infty_0(\Omega)$ is complete with the norm $$\|u\|_\Delta:=(\|u\|_{L^2(\Omega)}^2+\|\Delta u\|_{L^2(\Omega)}^2)^{1/2}?$$ Above ...
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Numerically solving a transport equation

I would like to solve this transport PDE numerically : $$ \partial_t f + v(f) \partial_x f = 0 $$ What I would want to do is "freeze" the velocity $v$ and solve a classical transport equation by ...
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19 views

wave equation on a circular domain

Consider the wave equation for the displacement $$\text{u(r,$\theta $,t)}$$ in a circular domain $$\text{0 $<$ r $<$ a, -$\pi $ $<$ $\theta $ $<$ $\pi $}$$ How do I use the separation ...
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What is the motivation for solving the Bessel equation.

My course is highly theoretical. For most part, we're taught to solve equations. But as a Physics student, I would very much like to know the motivation behind seeking the solution to the Bessel's ...
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1answer
16 views

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
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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
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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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1answer
13 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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13 views

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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1answer
24 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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1answer
12 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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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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2answers
66 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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2answers
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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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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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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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1answer
21 views

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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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
52 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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0answers
16 views

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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28 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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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
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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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1answer
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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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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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1answer
70 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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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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1answer
19 views

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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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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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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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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1answer
34 views

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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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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29 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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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
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 = ...
4
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2answers
58 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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1answer
84 views

Exponential decay estimate involving $C_{0}$-semigroup and the principle eigenvalue

Consider the heat equation defined as $$\begin{cases} \partial_{t}u(x,t)=\Delta_{x}u(x,t) &\mbox{ } t>0\mbox{, } u\in U \\ u(x,t)=0 &\mbox{ } t\ge 0\mbox{, } x\in\partial U \\ u(x,0)=g(x) ...
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1answer
17 views

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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0answers
97 views

How to find an ODE with prescribed terminal values? [closed]

Let us consider an ODE $$\frac{dx_t^y}{dt}=g(x_t^y),$$ where y is the initial condition i.e. $x_0^y=y$. Now, given a function $f$ (increasing and smooth) is it possible to find $g$ (i.e. an ODE) ...
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1answer
33 views

How to solve a particular PDE (which reminds of heat equation)

I suddenly ran into this equation: Let $u:[a,b]\times \mathbb{R} \rightarrow \mathbb{R}$ be a function satisfying: $$\partial_t u = -u' + \frac{1}{2}u''$$ with bountary conditions $u(0,x)=g(x)$ where ...
4
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1answer
48 views

Good sources to learn about Geometric Analysis

So far the topics that have captured my interest the most are analysis and geometry. I want to learn a bit more about geometric and global analysis. There seems to be no shortage of geometry books but ...
3
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0answers
23 views

Resolvent estimate of hyperbolic Laplacian

Consider the Laplacian $-\Delta$ on the hyperbolic space $\mathbb{H}^n$. For $\lambda \in \mathbb{C} \setminus [0, \infty)$, do we have resolvent estimates of the form $$\Vert (-\Delta - \lambda ...