Questions tagged [partial-differential-equations]
Questions on partial (as opposed to ordinary) differential equations - equations involving partial derivatives of one or more dependent variables with respect to more than one independent variables.
23,572
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Can any one teach me about proof of Theorem 7 in page 282 of the Evans's PDE book more friendly?
I think that I am begginer of PDE. I am reading the Evan's PDE, p.282, Theorem 7 and some question arises :
Q.1. First underlined statement : Where did $C$ come from? In previous sentence Evans wrote ...
- 2,656
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Oven temperature
This question is a crosspost from this one on another StackExchange subsite due to the recommendation given there. The question arose in the kitchen, not while doing homework.
The motivation
Consider ...
- 717
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Solutions to this odd Diff Eq with an integral in it?
I’ve come to this unusual Diff Eq in a statistical physics problem. A real positive density function $\rho\left(r,v\right)$ is a function of two real positive variables $0\leq r<\infty$ and $0\leq ...
- 2,702
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General question: Is there theory/literature for the solution of symmetric PDEs?
I'm currently struggling with $x^2 \frac{\partial^2f}{ \partial y^2} = y^2 \frac{\partial^2f}{\partial x^2}.$ I managed to get some of the solutions to the first order PDE $x \frac{\partial f}{\...
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Saying solution of a PDE is continuous to the boundary
Suppose $\Omega \subset \mathbb{R}^n$ is a domain and $u$ is a function on it such that $\Delta u$ converges to $0$ at every boundary point of $\Omega$, can we say from here that $u$ extends ...
- 342
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Well-posedness result for a linear parabolic equation on torus
Consider the following linear parabolic equation in one spatial dimension for $u=u(x,t)$ on the one-dimensional torus $\mathbb{T}^1,$ meaning $x \in \mathbb{T}^1$ and $t \in (0, T]:$
$$ \partial_t u- ...
- 169
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Method of images for a sphere
I know that for a circle, the process for modifying Green's function to account for the boundary condition $u(a,\theta) = 0$ is to (from geometric arguments) create an image at the point $\textbf{x}_0*...
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Questions about the Pohozaev identity for toda system.
Consider the following Toda system:
$$
\left\{\begin{array}{l}
-\Delta u_1=2 e^{u_1}-e^{u_2} \\
-\Delta u_2=-e^{u_1}+2 e^{u_2}
\end{array}\right.
$$
in $\mathbb{R}^2$.
I'm reading Analytic aspects of ...
- 625
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How to Solve the Differential Equation Involving Pauli Matrices and Time-Dependent Terms?
I am trying to solve the following differential equation analytically`:
$$
{\rm i}\,\partial_{t}
\begin{pmatrix}
u^{+}
\\
u^{-}
\end{pmatrix} =
\left[\rule{0pt}{5mm}\,2\alpha
\left(n - vt\right)\sigma^...
- 41
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Confusion in Partial Derivation of an Equation containing Quaternion
I found a way to rotate a 3D vector using a given unit quaternion. Thanks to this answer.
Now, let's say I want to rotate a gravity vector: $\overrightarrow{g} = \begin{bmatrix} g_x\\ g_y\\ g_z\\ \end{...
- 107
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1
answer
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Why is the term $H(x,y)u_{yx}$ omitted in every definition of a linear 2nd order PDE in two independent variables?
I was studying about PDEs when I came across the following definition of the general form of a second order linear PDE in $n$ independent variables:
Definition 1: The most general second-order linear ...
- 3,065
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What is an Affine PDE?
Is $\begin{align}
\frac{\partial^4u}{\partial x^3 \partial y}\,&+x\,\frac{\partial^3u}{\partial y^3}+7=0
\end{align}$ a linear, affine or quasilinear PDE?
I understand what a linear and a ...
- 3
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Calderon-Zygmund inequality for Neumann problems
The question is simple:
Let $\Omega$ be a bounded smooth domain. Then for any function $w\in W^{2,p}(\Omega)\cap C^1(\overline \Omega)$, such that $\frac{\partial w}{\partial \eta}=0$ on $\partial \...
- 2,682
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Why are kernels often singular on the diagonal?
Many kernels/integral operators are given in terms of a function that is singular near the origin:
For example, the heat kernel on $\mathbb{R}^d$:
$$
\operatorname{K}\left(t,x,y\right) =
\frac{1}{\...
- 6,275
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Stuck on last step of reducing to canonical form of $u_{xx} + x^2u_{yy} = 0$
I have most of the question done but I've no idea how to get the last step. The correct final answer is supposed to be
$$ u_{\lambda\lambda} + u_{\sigma\sigma} = -\frac{u_\lambda}{2\lambda} $$
I saw ...
- 31