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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.

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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{...
Milan's user avatar
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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 ...
Thomas Finley's user avatar
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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 ...
mfaczz's user avatar
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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 \...
Domenico Vuono's user avatar
1 vote
2 answers
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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}{\...
CBBAM's user avatar
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existence and unique for minimization problems

I am a machine learning enthusiast, and today I stumbled upon these problems, and I want to learn if we can apply the existence and unique theorems to the following two problems, $\min_{k} \space 2x^...
Papa's user avatar
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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 ...
Xemnas's user avatar
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Wave Equation general solution on 1D

I'm pretty sure the answer is simple but I need some guidance. I'm reading about the wave equation. And when we have it in the following form. $$ \frac{\partial^2 u(x, t)}{\partial t^2} = \frac{\...
silgon's user avatar
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Is ‘All differential equations have infinite solutions if there aren’t any initial conditions’ wrong? [closed]

Though the answers depend on whether we are thinking about real numbers or not. I think there are counterexamples but can’t think of any Someone on stack exchange said that $f(x)^2+f’(x)^2=0$ is a ...
zoe's user avatar
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Formulating a solution ansatz for the 1D heat equation in polar coordinates to learn the PDE in a PINN setting

Hello Math Stack Exchange Community, I am working on solving a partial differential equation (PDE) with a neural network in a PINN-like fashion, and I am seeking advice on identifying an appropriate ...
alighato's user avatar
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Question about Evans’ derivation of a Green's function

At page 34 of "Partial Differential Equations" by Evans, in order to define the Green function for the set $U$, the author defines a family of functions as the solutions of the boundary ...
Lorenzo Vanni's user avatar
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Regularity for computing the first variation

I am having a trouble understanding the regularity needed to compute the first variation for the Euler-Lagrange equation for the functional $$F(u) = \int f(u) dx$$ Suppose $u:U \to \mathbb{R}$ for ...
Morcus's user avatar
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Solving 1st order PDE including convolution

I'm studying Van Kampen's "Stochastic processes in physics and chemistry" and stuck to some exercise (p.78): That is, solving \begin{equation} \frac{\partial P(y, t)}{\partial t}=\int_{-\...
Patche's user avatar
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2 answers
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Confusion in understanding the definition of a linear and quasi-linear PDE.

I was recently studying Partial Differential Equations (PDE). While going through the basics, I stumbled across the definition of a linear PDE and quasi-linear PDE. The definition went as follows: A ...
Thomas Finley's user avatar
3 votes
2 answers
335 views

Basic Solution to the Heat Equation

As a learning example, I am trying to derive the solution to the basic Heat Equation (https://en.wikipedia.org/wiki/Heat_equation) using Fourier Transforms. As I understand, the Heat Equation can ...
konofoso's user avatar
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Proof of Paley-Wiener Theorem

I'm trying to understand the proof of the following version of Paley-Wiener theorem under the additional assumption $f \in L^2$: I understood the part $(2) \Rightarrow (1)$ but I couldn't follow a ...
heyy's user avatar
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Solution of the "reciprocal of the heat equation"?

I was playing around with the heat equation in one dimension and tried to guess what the solution to homogenous boundary conditions and a sine wave as initial condition on the interval $0<x<\pi$ ...
Alejandro's user avatar
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In this proof, how does the Fourier transform work? [closed]

I recently read the article 'E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behaviour for nonlocal diffusion equations. J. Math. Pures Appl. (9) 86 (2006), 271–291'. The Theorem 2.1. confused me....
M.A.D.M.A.N's user avatar
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How to Derive the Poisson Kernel in Higher Dimensions?

I am trying to derive the Poisson kernel $ P(x, y) $ in higher dimensions, specifically in $ \mathbb{R}^n $. I know that the result should be: $$ P(x, y) = C_{n,a} \frac{y^{1-a}}{( |x|^2 + y^2 )^{\...
Christy's user avatar
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Difference scheme for time-reversed heat conduction equation

I am working on solving the time-reversed heat conduction equation(assuming a two-dimensional space with Dirichlet boundary conditions). I have implemented the finite difference method, using first-...
focalors's user avatar
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1 answer
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Where can i find a reference with explanation of each term about Poisson equation in anisotropic media?

I studied about Poisson equation on anisotropic media with equation as such $$ k_{11}\dfrac{\partial ^2 u}{\partial x^2} + (k_{12} + k_{21})\dfrac{\partial ^2 u}{\partial x \partial y} + k_{22}\dfrac{\...
Cedric Mohammad A.C.'s user avatar
2 votes
0 answers
59 views

Bochner-Sobolev spaces with second time derivative and embeddings

In the weak theory of evolution PDEs, the Bochner-Sobolev spaces are frequently used. For $a,b \in \mathbb{R}$ and $X,Y$ banach spaces, we define these spaces as $$W^{1,p,q}(a,b,X,Y) = \left\{v, \ \...
Maths_GEES 's user avatar
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Invariant Solutions of PDEs-Linear fokker-planck equation with an odd drift

I am currently writing my master thesis on Lie group analysis and recently I came across this infinitesimal generator: I am trying to obtain group invariant solutions in their implicit form and so ...
George Beliyiannis's user avatar
1 vote
1 answer
44 views

Solving ODE system with less equations

In sensitivity analysis, there is a set of equations called sensitivity equations. They're obtained by differentiating your initial IVP with respect to the parameters. For example: If your IVP is: $\...
nileebolt's user avatar
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7-point North-East ILU decomposition

I am using a 7-point difference operator (actually a 5-point difference operator with $b, f = 0$) to discretize the 2D model anisotropic problem. $ L_h $ in stencil notation is given. $ L_h = \begin{...
Sonny Jordan's user avatar
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24 views

Onsager conjecture and properties of Besov Spaces

I am currently studying the result in Peter Constantin 1, Weinan E, and E. S. Titi, Onsager’s Conjecture on the Energy Conservation for Solutions of Euler’s Equation link:https://web.math.princeton....
Radoslav Habarda's user avatar
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1 answer
57 views

Well-posedness of $\partial_t^3 u = \Delta u$

I am interested in the well-posedness of the following PDE: $$\partial_t^3 u = \Delta u.$$ It resembles both the heat equation and the wave equation, but with a third-order time derivative. Although ...
Zhang Yuhan's user avatar
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1 answer
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Doubt in Hopf's Lemma

This is a problem from Evans' PDE book: Assume $u$ is connected. Use (a) energy methods and (b) the maximum principle to show that the only smooth solutions of the Neumann boundary-value problem: $$\...
Kadmos's user avatar
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2 votes
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Does this equation imply "non-linear" waves?

For one-dimensional bounded domain $x \in [0, L]$ consider $\partial_{t} v = -\frac{1}{\rho}\partial_{x}\rho - v $ $\partial_{t} \rho = -\partial_{x}(\rho v)$ with initial and boundary data as $v(0, ...
YoussefMabrouk's user avatar
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1 answer
51 views

Radius of balls in estimates for De Giorgi method

I wanted to ask a question regarding the radius of balls used to get the different estimates to establish both the jump from $L^2$ to $L^{\infty}$ and the Holder continuity later on in the proof of De ...
Thomas Petit's user avatar
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28 views

Elliptic Equation on sphere

The literature I read recently says that the function $$\phi_1(\theta)=(\theta\cdot e_d)_+$$ defined on the sphere solves the equation $$ -\Delta _{\mathbb{S}}\phi _1=\left( d-1 \right) \phi _1\qquad \...
zik2019's user avatar
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42 views
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Regularity of continuous approximation on time discretization

I have been trying to understand how to deal with continuous approximations when dealing with time discretizations, consider the abstract PDE $$ \partial_t u + A(u) = f(u) $$ and then instead ...
Daniel Moraes's user avatar
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0 answers
52 views

Numerically solved PDE of Ornstein–Uhlenbeck process on 2-Simplex violates conservation of probability

Thanks for your consideration. I'm working to create a solution of an Ornstein-Uhlenbeck process with a force that takes mass towards the centre of a Simplex. I'm assuming absorbing boundaries. The ...
CRTmonitor's user avatar
-1 votes
0 answers
30 views

PDEs inequality solution [closed]

i have a PDE inequality that i tried a lot to get a general analytical solution or at least a particular family of solutions. The PDE inequality is: $$ \frac{\partial }{\partial r}\left( y\frac{\...
Soufiane Fares's user avatar
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0 answers
23 views

Calculating functional derivative for a Physics-Informed Neural Network (PINN) using Automatic Differentiation

I'm working with a Physics-Informed Neural Network (PINN) to approximate the solution of a 1D Poisson equation: $\frac{d^2u}{dx^2} = f$ Here, I have an MLP with weight parameters $\theta$ that takes a ...
Yanyan Wang's user avatar
1 vote
1 answer
37 views

Local existence NLS

I am reading the book of Tao on dispersive PDE. In a proof of local existence of the solution of NLS, he wrote that the following inequality holds: $$||u|^{\alpha -1}u - |v|^{\alpha -1}v| \leq c (|u|^{...
N230899's user avatar
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0 answers
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Changing coordinates of $2$nd order partial operators

Let's work in $\mathbf R^n$. If we want to change coordinates $\mathbf x\to\mathbf r$, with them related like $$ \mathbf x=\mathbf x(\mathbf r) $$ Then, the second order generic partial operator in ...
Conreu's user avatar
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-2 votes
0 answers
20 views

regularity of the weak solution on the cube [closed]

Let $Q:=[0,1]^d$ and $g\in L^2(\Omega)$ consider the PDE : $$ \left\{ \begin{array}{ll} -\Delta f=g & \text{in $Q$} \\ f\equiv 0 & \mbox{on $\partial Q$}, \end{array} \...
Alucard-o Ming's user avatar
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0 answers
26 views

Diffusion equation with complex coefficient

This is the solution for diffusion equation(3 dimensional): Now, I want to solve $u_t=\frac{\epsilon+i}{2}\Delta u$ (in $R^3$), $u(X,0)=\phi(X)$; where $\epsilon>0$ and $i$ is complex number. Use ...
郭冠廷's user avatar
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0 answers
30 views

Extension of a Hölder function on the product space

Let $k\geq0, \alpha>0$ and let $\Omega\subset \mathbb{R}^d$ be a $C^{k,\alpha}$ convex domain. Suppose $f:[0,1]\times \overline{\Omega} \to \mathbb{R}$ satisfies the following: For each $t\in[0,1]$...
Stephen_lamb's user avatar
1 vote
0 answers
48 views

Numerical computation of the basic reproduction number (R0) for reaction-diffusion-advection epidemiological models

I am working on a "reaction-advection-diffusion" type epidemiological model using a system of partial differential equations (PDEs). From this PDE model, I would like to numerically compute ...
Nell's user avatar
  • 43
1 vote
1 answer
120 views

Solving Laplace equation

I have the Boundary value problem below using the method of Separation of Variables $$u_{xx}+u_{yy}=0, \ x\in[0,2\pi], \ y\in[0,2\pi]$$ $$u(x,0)=\cos x, \ x\in[0,2\pi]$$ $$u(x,2\pi)=1,\ x\in[0,2\pi]$$ ...
HarrisModel's user avatar
3 votes
0 answers
67 views

Doubt on general solution of PDE

I have the PDE $$ \alpha^2 \frac{\partial^2 z}{\partial x^2} + \beta^2 \frac{\partial^2 z}{\partial y^2}=0 $$ I want to find the general solution using the change of variables $$ \begin{cases} &...
baristocrona's user avatar
1 vote
1 answer
38 views

Doubt in proof of Second Existence Theorem for Weak Solutions

I have the same doubt as in here, but none of the answers there seem to make things clearer to me. When Evan proves the Second Existence Theorem for Weak Solutions, he asserts on Step 4 that: $v-K^*v=...
Kadmos's user avatar
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1 vote
1 answer
60 views

Green function of a differential equation

Let $$ Lu(x)=a(x)\frac{d^2u(x)}{dx^2}+b(x)\frac{du(x)}{dx}+c(x)u(x), $$ We define its Green function $G_0(x,y)$ by $$ LG_0(x,y)=\delta_x(y) $$ in the sense of distribution. It's esay to get this Green ...
Rayyyyy's user avatar
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1 vote
0 answers
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A question in Strauss PDE exercise 9.4.2

In chapter 9.4 of Strauss PDE, we try to find the solution of 3D Diffusion Equation. And there is a exercise which is closely relating to the proof, as following: $$\lim_{t\to0} \iiint_{R^3} S_3(X-X',...
郭冠廷's user avatar
2 votes
0 answers
72 views

Mean curvature flow: Second time derivative?

Suppose I have a 2D surface in $\mathbb R^3$ undergoing mean curvature flow, i.e., the motion of a point on the surface instantaneously can be described as $$\frac{d\mathbf x}{dt}=-H\mathbf n,$$ where ...
Justin Solomon's user avatar
1 vote
2 answers
113 views

Finding all $ f: \mathbb{R}^2 \to \mathbb{R} $ s.t.: $ \frac{\partial f}{\partial x} - 3 \frac{\partial f}{\partial y} = 0 $

I need to find all continuously differentiable functions $ f: \mathbb{R}^2 \to \mathbb{R} $ such that $ \frac{\partial f}{\partial x} - 3 \frac{\partial f}{\partial y} = 0. $ I was given the hint ...
NivGeva's user avatar
  • 29
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0 answers
58 views

Finding a paper: Feynman-like diagrams, but for differential equations [closed]

About 15 or 20 years ago I saw an interesting article about a new way to solve differential equations using Feynman-like diagrams. I vaguely remember it has some strings, rings, strings passing inside ...
LucasBr's user avatar
  • 109
1 vote
0 answers
43 views

Scaling and Helmholtz equation

Solutions to Laplace equation and powers thereof have some convenient invariance properties that solutions to Helmholtz equation $\Delta u-u=f$ apparently lacks, especially in regards to scaling, ...
undefined's user avatar
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