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.
22,790
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PDE on the probability that the brownian motion stays in [a,b]
This was part of an exam question.
Let $a<b$, $(X_t)$ a brownian motion and $$\forall x \in \mathbb R, t\ge 0, \quad \pi(x,t):=P(\forall s \in [0,t], x+X_t \in [a,b]).$$
Given that $\pi$ is $C^2$ (...
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15
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General solution to $\frac{\partial^n}{\partial x^n}u(x,y)=\frac{\partial^m}{\partial y^m}u(x,y)$
I have never taken a Partial Differential Equations course, so bare with me. I want to find the general solution to $$\frac{\partial^n}{\partial x^n}u(x,y)=\frac{\partial^m}{\partial y^m}u(x,y)$$out ...
- 5,052
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42
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Linearized PDE system - but why is this linear?
I am referring to this paper.
They consider the PDE system
$$
\begin{align*}
\partial_t f + v\cdot\nabla_x f & = \chi_1(v) - \rho_g f,\\
\partial _t g+v\cdot\nabla_x g & =\chi_2(v)-\rho_f g
\...
- 2,007
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22
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PDE: wave equation inhomogenous boundary problem: Where the C^2 gone?
I'm reading W. Strauss' PDE book. While studiyng wave equation inhomogenous problem I stucked at this place: 1.
When we computed $$w_{n}(t)$$ in terms of $$u_{n}(t), \pi, k(t), h(t)$$ and then ...
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2
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answers
13
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How to linearize this PDE system?
Consider the PDE system
$$
\begin{align*}
\partial_t f_1+v\cdot\nabla_x f_1&=\sigma(\rho_1\chi_1-f_1)+\chi_1-\rho_2 f_1,\\
\partial_t f_2+v\cdot\nabla_x f_2&=\sigma(\rho_2\chi_2-f_2)+\chi_2-\...
- 2,007
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16
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An isoperimetric-type inequality
I am reading some notes on de Giorgi's methods in the regularity of elliptic equations, and have come across a step which I can't make sense of. The claim is as follows (see Lemma 10 in the linked ...
- 61
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Traffic flow, if the initial density r(x,0) is decreasing, then it has a limit as t tends to infty
How to prove that if the initial density $\rho(x,0)$ of the traffic flow is decreasing, then there is $lim_{t->\infty}\rho(x,t)$
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7
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Cauchy problem for linear 2nd-order degenerate parabolic PDEs
Let's consider a PDE of the form (for a function $f$ of variables $t, x_1, \dots, x_n$)
$$\frac\partial{\partial t} f = \sum_{ij}C_{ij}(x_1, \dots, x_n)\frac{\partial^2 f}{\partial x_i\partial x_j} + \...
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The exact, formal definition of a prabolic problem/equation
While searching for information about properties of parabolic problems, I stumbled upon a publication titled "Study of Nonlinear Parabolic Problems".
This made me wonder why the writers ...
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55
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Prove that $u = |\nabla v|^2$ reaches its maximum in $\partial \Omega$, $v$ being a solution to the given Poisson pde
Let $\Omega \subset \mathbb{R}^2$ and $v: \mathbb{R}^2 \to \mathbb{R}$ be a solution to:
$$\begin{cases}
v_{xx}+v_{yy} = -2 &\text{in $\Omega$}\\
v = 0 &\text{in $\partial \Omega$}
\end{cases}$...
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35
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Every test function $\varphi \in C^{\infty}$ are bounded and its derivatives?
Would you recommend a bibliography where I can find all properties of test functions like this question, and also many examples?
- 337
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Complete, General and Singular solution of a partial differential equation [closed]
How to find the complete, singular and general solutions of a given partial differential equation?
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55
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Problem about Schrodinger equation
Consider the linear Schrodinger equation $$\partial_t u - i \Delta u = 0$$ with initial condition $u(0,x)=f(x)$. Assume $f \in L^2(\mathbb{R}^n)$.
I need to show that $$\lim_{t \rightarrow 0}||u(t,x) ...
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1
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Solve the Partial Defferential Equation $z+xp-x^2 y q^2-x^3 p q=0$, where $p=\frac{\partial z}{\partial x}$ and $q=\frac{\partial z}{\partial y}$.
I use Charpit Method to solve the problem but calculation is so big. Is there any another method to solve the problem.I think I need some variable transformation which gives me a standard form then I ...
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function that is $-y$ when differentiated by $x$ and $x$ when differentiated by $y$
I need to find a real 2-variable function $f(x,y)$ that is $-y$ when differentiated by $x$ and $x$ when differentiated by $y$. Does such a function exist? If so, I would like to have a concrete ...
- 57
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Difusion - line methods - dissolution of minerals
Could someone assist me to solve this problem using the lines method? I have a 1D model that describes solute diffusion across a system with two layers: a fluid boundary layer and a porous secondary ...
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answers
44
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Change of Variables for Partial Differential Equations to remove advection term
Given a PDE
$$
\frac{\partial^2u}{\partial x^2} + g(x)\frac{\partial u}{\partial x} + \frac{\partial u}{\partial t} = 0
$$
Is there any change of variables that transforms the PDE into a PDE that ...
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18
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numerically compute eigenfunctions of $a(u,v)=\langle f(\nabla u)\nabla u,\nabla v\rangle_{L^2}$
Let $D:=(0,1)^2$ and consider the nonnegative form $$a(u,v):=\langle f(\nabla u)\nabla u,\nabla v\rangle_{L^2(D;\:\mathbb R^2)}$$ for $u,v\in L^2(D)$ where $f:\mathbb R^2\to(0,\infty)$ is a smooth ...
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inhomogeneous Helmholtz equation does not obey superposition using FEM
I am trying to solve the in-homogeneous Helmholtz equation with damping and forcing using finite element method (FEM). The equation is;
$c^2\nabla^2p - c^2\omega i d \nabla^2p + \omega^2 p = f$
where ...
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Question about a proof with moving plane method
I am reading the book Elliptic partial differential equations by Q. Han and F. Lin. In the section 3.6 "Moving Plane Methods", I come across a trouble understanding the following step:
...
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How to translate PEC boundary condition into the TE mode of Maxwell's equations?
In $2$D case, the TE mode of Maxwell's equations is
$$\nabla\cdot(\varepsilon^{-1}\nabla H_z)=\omega^2\mu_{zz}H_z,$$
where $\varepsilon$ is a $2$-by-$2$ matrix and $\mu_{zz}$ is a scalar. Suppose the ...
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answers
74
views
Strong Maximum Principle for the heat Equation - Evans - Theorem 4 - Why do we need U to be bounded.
I had a doubt in following theorem of Evans
THEOREM 4 (Strong maximum principle). Suppose $u \in C^2_1(U) \cap C(\bar{U})$ is satisfies $u_t + \Delta u = 0$ within $U_T$. (i) Then
$$
\max _{\bar{U_T}} ...
- 790
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answers
45
views
Can't get proper numerical convergence for complicated Advection-Diffusion-Reaction PDE
I have trying for quite some time to write a finite difference solver for the following advection-diffusion-reaction differential equation:
$$
\frac{\partial C}{\partial x} = \frac{1}{u(z) + u_e(x,z)}\...
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1
answer
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Reference to a Theorem (or book) about parabolic PDEs
I am studying free-boundary problems, and I see that many authors face the following standard problem from the theory of parabolic PDEs:
Given a domain $[x_1, x_2] \times [t_0, t_1)$, find a function $...
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answers
38
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Find a function that does not satisfy that IVP [closed]
Find $f\in C^{\infty}(\Bbb R,\Bbb R)$ s.t IVP has not satisfied $f(x')=0$ and $x(0)=0$.
Try:
I found a paper that talks about that and gives an example but from $\Bbb R^2$ to $\Bbb R$, on page 347, I ...
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answers
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Why are solutions to PDEs eigenfunctions of the symmetry generators?
Consider the case of Laplace's equation
$$\nabla^2 f = 0$$
After the standard procedure of separation of variables, the general solution to this equation is found to be a linear combination of ...
- 207
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votes
2
answers
32
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Substitution on a Temperature Problem
Given real parameters $A,B,C$ consider the temperature problem with non-homogeneous boundary conditions:
$$u_t=ku_{xx}, \;\;\; u=u(x,t), \;\;\;0\leq x\leq \pi,\;\;\; t,k> 0$$
$$u_x(0,t)=u(0,t)+...
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Evolving quadratic form with heat equation on sphere (to arrive at Trace)
In this answer to MO question "Geometric interpretation of Trace" (the 9th highest upvoted question on the site!), the following interpretation of the trace is given:
$$\operatorname{Tr}(A) ...
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Ricci Flow: The existence of potential of Curvature
For a compact Riemannian Manifold $(M,g)$ without boundary. $R$ as the scalar curvature. And $d\mu$ is the Riemannian volume form.
So we can define the average of the scalar curvature $r:= \frac{\...
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When can you substitute in a differential equation?
I'm working on a differential equation, but I am not allowed to substitute itself back in at a later step, could someone tell me why?
Here's my work. Consider:
$$\frac{dx(t)}{dt} = W(x(t)) + Q(x(t),y(...
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answers
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How can I solve the given PDE equation [closed]
Solve the following PDE:
$$
U_t+(U−1)U_x=2
$$
with initial value:
$$
U(x,0)=\begin{cases}
1 &\text{for $x<0$}; \\
1−x &\...
2
votes
1
answer
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views
Convolution of Mollifier and function $f$ converges to $f$ on all compact subsets?
I am studying the book by Evans L.C., Gariepy R.F. - Measure theory and fine properties of functions. On the chapter about local approximation of Sobolev functions, he introduces the function
$\eta:\...
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answers
25
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Compactness criterion on subsets of fractional Sobolev spaces
Are there sufficient conditions for a family of functions $F \subset H^s_0(\Omega)$ $(s>0)$ to be relatively compact in $H^s_0(\Omega)$, where $\Omega \subseteq \mathbb{R}^n$ is compact. $n$ is ...
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Prove the lower semi-continuity of inner products [duplicate]
Let $\mathcal{H}$ be a Hilbert space equipped with inner product $(\cdot,\cdot)$. Suppose that a sequence $u_n\rightarrow u$ weakly in $(\mathcal{H},(\cdot,\cdot))$, I want to prove that $$(u,u)\le \...
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Is there a less time consuming way of resolving the factors of the given equation?
I considered an equation of the form ax2 + by2 + cxy + dx + ey + f = 0 for solving non-homogeneous partial differential equations. I could not find any way to resolve the equation apart from trial and ...
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How to use an interpolation argument to prove $A\hookrightarrow L^p(\mathbb R)$ for $p\in [2, +\infty]?$
Assume you have that
$$A\hookrightarrow L^2(\mathbb R) \quad\text{ and }\quad B\hookrightarrow L^\infty(\mathbb R).$$
Suppose also that $A\hookrightarrow B$.
By using the above information, is it ...
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The name and intuition as well for this type of equation?
I am not a person who has expertise in partial differential equations, but I wish to gain an abstract-level intuition of a potentially very well-known equation written below at (3). For a starter, I ...
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When $f/\frac{\partial f}{\partial x}$ does not depend upon $x$?
Suppose that $f(x,y,\ldots)$ is a multivariable non-constant rational function. When $f/\frac{\partial f}{\partial x}$ does not depend upon $x$?
My attempt: The question equivalent to when
$$
\frac{\...
- 4,701
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votes
1
answer
53
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Help Solving and Understanding a Temperature Problem
Consider the following temperature problem:
$$u_t(x,t)=ku_{xx}(x,t), \;0\leq x \leq \pi,\;\; t,k >0$$
with boundary conditions:
$$u_x(0,t)=u(0,t)$$
$$u_x(\pi,t)=u(\pi,t)$$
$$u(x,0)=f(x)$$
I know ...
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1
answer
65
views
Poincaré inequality on a Riemannian manifold
On a compact subset $\Omega \subset \mathbb{R}^n$, the Poincaré inequality states
$$\|u\|_{L^p(\Omega)} \leq C \|\nabla u \|_{L^p(\Omega)}. \tag{1}$$
When we generalize to a compact Riemannian ...
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101
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How can I continue to solve this PDE?
I´m having trouble finding the solution of this PDE
$$\frac{\partial^2 u}{\partial x\, \partial t}=\frac{\partial^2 u}{\partial x^2}$$
on $-\infty < x < \infty$, $t>0$, and initial condition $...
0
votes
0
answers
47
views
Mathematical theory of plasma
I am working on a heavily mathematically project about plasma. In particular, I want to find references that treat the problem from microscopic models that include relativistic and magnetic effects (...
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1
answer
33
views
Simple notation question regarding orthogonality of sines used in solving PDEs
Just to preface by providing the context for the problem leading up to the question, below is a brief paraphrase of setting up the problem.
$$u_{xx} + u_{yy} = x^2-x + y^2 - y; \ \ 0 \le x,y \le 1 \\ ...
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0
answers
30
views
Exponential decay for a smooth solution to a parabolic PDE
Consider the following problems.
Problem 1. Let $\Omega \subset \mathbb{R}^n$ be open and bounded with smooth boundary, L an elliptic operator defined by $$Lu=-D_j(a^{ij}D_iu)$$ where $a^{ij}=a^{ji}$ ...
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votes
0
answers
18
views
Finding continuous subsequence in Lebesgue Bochner space
Assume you have a sequence $x_n(t)$ in $L^p(0,T;X)$ $1<p<\infty$, where $X$ is Banach.
Under which conditions or in which special case can one extract a subsequence that is continuous in $t$ wrt....
- 718
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0
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45
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Perturbation of an infinite-time horizon evolution system
I'm dealing with an evolutionary problem described by the equation:
$$\dot y(t)=F(y(t)) +f(t), \quad y(0)=y_0$$
If I suppose that this equation has a solution in a variational sense over an infinite-...
- 99
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votes
0
answers
63
views
English translation of a book of J.L. Lions?
Anyone knows if there is an english translation of the book
J.L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires, Dunod,
Paris, 1969
?
Or do someone at least know an ...
- 718
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0
answers
24
views
The real CFL condition for cylindrical laplacian
I've been exploring the CFL (Courant-Friedrichs-Lewy) condition in polar coordinates and have observed that previous inquiries haven't yielded a satisfactory answer. I've come across this paper which ...
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0
answers
67
views
How to calculate the limits of the integral and the outward vector? How to make change of variables in this proof?
I would like to see the calculations behind this proof I found here:
Proof of polar coordinates theorem in Evans' PDE Book
That is, I am struggling to understand the following:
How do we ...
- 337
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0
answers
44
views
If $u\in W^{1,p}(\mathbb{R}^n) \cap C^1{(\mathbb{R}^n)}$ may I suppose that $|u(x)|=\frac{1}{\omega_n} \int_{B_{1}(x)}|u(x)|dy?$ How to prove it?
If $u\in W^{1,p}(\mathbb{R}^n) \cap C^1{(\mathbb{R}^n)}$ and $\omega_n$ is the volume of a ball, may I always suppose that
$$|u(x)|=\frac{1}{\omega_n} \int_{B_{1}(x)}|u(x)|dy?$$
How to prove it?
I ...
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