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

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Solution to wave equation for $u(t, 0) = \gamma(t)$

I am looking for a necessary and sufficient condition on a function $\gamma: \mathbb{R} \to \mathbb{R}$ such that there is a solution $u \in C(\mathbb{R}^2)$ for the wave equation $$\partial^2_t u(t, ...
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0answers
51 views

How to get the idea of the formula for the mean value property for the heat equation

From the mean-value property of the Laplace's equation, we have the following mean-value property: $$ u(x)=\frac{1}{a(n)r^n}\int_{B(x,r)}udy. $$ But for the mean-value property of the Heat equation, ...
2
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1answer
42 views

PDE boundary condition question regarding limits

Just as a bit of background, I'm working with the Black-Scholes PDE and I'm testing some things out by taking an initial condition for it as $\sin(S/50)$, where $S$ is the spot price (but that's ...
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0answers
20 views

Interchanging the order of integration (related to resolvent kernel)

I was reading a chapter related to resolvent kernel, and there was one step that I don't quite understand. Assuming $k(t,s)$ and $g(t)$ are continouous,$\int^t_0k(t,s)\int^s_0k(s,\tau)g(\tau)d\tau ...
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0answers
41 views

mean evolution of 1D Fokker-Planck

Given the Fokker-Planck equation on 1D with drift term $D^{(1)}(x)$, and diffusion term $D^{(2)}(x)$, and the governing equation of probability density function $$\frac{\partial f(x,t)}{\partial t} = ...
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2answers
87 views

How do I solve a PDE with multiple Dirac functions?

I am exposed to a PDE in the following form: $\frac{\partial f}{\partial t}=\alpha \frac{\partial^2 f}{\partial x^2}-\beta \frac{\partial f}{\partial x} + \mu_a P_a(t) \delta(x-1)+ \mu_b P_b(t) \...
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0answers
23 views

For $u_h(t) = \frac 1h \int_t^{t+h}u(s)\;\mathrm{d}s$, is $[\partial_t (u_h(t))]_{x_j} = \partial_t[(u_{x_j})_h(t)]?$

Let $u$ belong to $L^2(0,T;H^1(\Omega))$ and $u_t \in L^2(0,T;(H^1(\Omega))^*)$ and define the function $$u_h(t) = \frac 1h \int_t^{t+h}u(s)\;\mathrm{d}s$$ Is it true that $$[\partial_t (u_h(t))]_{...
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2answers
26 views

Numerically Solving a 3d PDE with Stochastic Terms

I'm getting a bit confused if the procedure I'm doing is correct so any feedback would be great! It's just a standard deterministic PDE for the price of a theoretic option, even if it's quite a ...
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1answer
32 views

Laplace equation on a cylinder

For the Laplace equation in 3D $$\nabla^2 u =u_{xx}+u_{yy}+u_{zz}=0$$ in a right cylinder with an arbitrarily shaped base, whose top is $z=H$, bottom is $z=0$, we assume the following boundary ...
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2answers
61 views

Can Integration constant be anything?

When the question is to solve a given ODE (without initial value), can I assign any value to the constant $C$, in order to solve it, or is there a specific constant value that must be found? In ...
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1answer
39 views

solution for heat equation

I read from Evan's PDE book, that we have the explicit solution in $\mathbb{R}^n\times [0,T]$ for the heat equation: $$ \partial_t u-\Delta u=f, $$ $$ u(t=0,x)=u_0, $$ when $f$ is $C^2$ with compact ...
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0answers
45 views

PDE $yu_{x}+xu_{y}=0, u(0,y)=e^{-y^2}$ - why the solution is determined only in the region ${x^2 ≤ y^2}$?

I am solving the PDE $$\left\{\begin{matrix} yu_{x}+xu_{y}=0\\ u(0,y)=e^{-y^2} \end{matrix}\right.$$ My results are below. In the answer to this problem there is a statement saying that "a sketch of ...
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0answers
12 views

The unique tangent of super-harmonic function on the liminf of the singularity

Let $n\ge2,B=B(0,1)\subset\mathbb{R}^n$ is the unit ball. Let $u\in C^2(B\backslash\{0\}),\Delta u\le0$ be the super-harmonic function such that $\liminf\limits_{x\to 0}u(x)=0$. Let $v,w\in C^2(B)$ ...
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0answers
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How to find out the characteristics of the following PDE?

Let $x = x(s), y= y(s), u= u(s), s \in \mathbb{R}$ be the characteristic curve of the PDE $$\bigg(\frac{\partial u}{\partial x}\bigg)\bigg(\frac{\partial u}{\partial y}\bigg) - u = 0$$ passing through ...
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22 views

Maximum principle question

Does maximum principle exist for an elliptic equation(laplace or poisson) if the domain doesn't have a boundary, for example if you take $S^1$ (the 1-torus) in 1d?
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23 views

Simple partial differentiation of d'Alembert's formula

The answer I get for $\partial_t\int_{x-ct}^{x+ct} G(s) ds$ does not agree with the answer given in a text. The answer seems to be $c(G(x+ct)+G(x-ct))$ but I am not able to arrive at this solution. ...
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3answers
44 views

Solve the initial value problem for this inhomogeneous heat equation.

I'm trying to solve this IVP for heat equation, $$u_t-\frac{1}{4}u_{xx}=e^{-t}~~\text{ in }-\infty<x<\infty,~t>0,$$ $$u(x,0)=x^2.$$ By the superposition principle, the solution should equal ...
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1answer
61 views

Method for solving 2nd order linear PDE of three variables

For the 2nd order linear PDE below, please give method(s) to solve it, working, a solution, and what conditions the solution can exist? $$\sin(t)\frac{\partial^2y}{\partial t^2}+\cos(t)\frac{\partial ...
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1answer
51 views

Questions about the proof of Poisson's formula for half-space in Page 38 of Evans' book

I have questions about the proof of Theorem 14 (poisson's formula for half-space) in Page 38. Let $K(x,y)$ be the Poisson's kernel for $\mathbb R^n_+$: $$K(x,y)=\frac{2x_n}{na(n)}\frac{1}{|x-y|^n}dy\...
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0answers
23 views

Distribution Convergence in PDE theory

I'm trying to follow an Example (5.1) from a PDE book (Vasy). I was having trouble following the proof of the following question (modified): Define a bump function: ${\delta_j}^{-n}\phi(\frac{x}{\...
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4answers
54 views

What is the geometric interpretation of the solution to PDE $xu_x+yu_y=0$

I have the following PDE $$xu_x+yu_y=0$$ for which I get the characteristic function $$y=cx$$ along which the u(x,y) is constant. The general solution is $$u(x,y)=f(\frac{y}{x})$$. I understand ...
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1answer
31 views

Proof verification for the Friedrich's inequality in $\mathbb{R}$

I'm trying to prove the Friedrich's inequality in $\mathbb{R}^n$, but I first want to prove it in $\mathbb{R}$ because using iterated integrals I can then use the same idea in $\mathbb{R}^n$. So let $\...
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0answers
37 views

Does anyone recognise this non-linear diffusion equation?

I'm doing some work on modelling cell migration, I've derived this particular form of a non-linear diffusion equation to describe the mean behaviour of a stochastic model I'm studying. I was wondering ...
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0answers
33 views

problem regarding exponential solution of heat equation?

Let the heat equation $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x_1^2}+ \frac{\partial^2 u}{\partial x_2^2}+ \frac{\partial^2 u}{\partial x_3^2}, \ t \geq 0 , \ x= (x_1, x_2, x_3)$...
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0answers
9 views

Crandall-Ishii lemma on unbounded domains

In the User's Guide to Viscosity Solutions, there is an example (Section 5.D; page 30) using the Crandall-Ishii lemma on unbounded domains. Equation (5.17) claims $$ -3\alpha\left(\begin{array}{cc} I\...
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6answers
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Why is it useful to show the existence and uniqueness of solution for a PDE?

Don't get me wrong, I understand that it is important in mathematics to qualitatively study the problems given. But I would like to know to what extent this helps for example, to actually solve the ...
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2answers
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Intro to PDEs: $\Delta$ use in the maximum principle proof for Laplace's Equation

I am in a class called Intro to Partial Differential Equations (I am an undergrad student and all of this is very new to me, so please be patient with me and what may appear to be a silly question). I ...
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0answers
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The maximum principle of positive super-harmonic function

Let $0<u\in C^3(\mathbb R^n),\Delta u\le0$, Show that $\forall r>0,\forall|x|\ge r,|x|^{n-2}u(x)\ge\min\limits_{|y|=r}u(y)r^{n-2}$ Using Kelvin transform $\displaystyle v(x)=\left(\frac {r}{|x|}...
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2answers
71 views

heat equation from half space to the whole space

I understand that the solution to the heat equation can be analytically written down if the equation is defined on the entire real like. However, even if the equation is defined on the half space(...
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1answer
47 views

Analytic Solutions to differential (heat) equation.

I've searched around, but I couldn't find anything too helpful on the subject, so here goes. I am trying to find (if possible) an analytical solution to the differential equation of the form: $$ \pm\...
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0answers
15 views

Yet another question about "characterization of $H^{-1}$ in Evans

I'm referring to Evans' "Characterization of $H^{-1}$ in his book Partial Differential Equations. It seems like all of this applies just as well if we take the dual of $H^{1}(U)$ instead of $H_{0}^{...
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1answer
27 views

specific solution to pde $u_x + yu_y =0, u(0,y)=y^3$

I have $$\left\{\begin{matrix} 4u_{x}-3u_{y}=0\\ u(0,y)=y^3 \end{matrix}\right.$$ with general solution $$u(x, y) = f(ye^{−x})$$ so using the boundary condition I get $ \ \ u(0,y)=y^3 \ $ so $\ \ ...
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1answer
37 views

geometric interpretation of the solution to PDE $u_x + yu_y =0$

The equation $$u_x + yu_y =0$$ has the general solution $$u(x,y)=f(ye^{-x})$$ The characteristic curves should look like I tried to plot the solution using google. Is this plot correct? ...
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1answer
18 views

clarification to specific solution of the first order linear PDE problem $4u_{x}-3u_{y}=0, u(0,y)=y^3$

I am analyzing the following first order PDE problem and have difficulties with understanding the solution $$\left\{\begin{matrix} 4u_{x}-3u_{y}=0\\ u(0,y)=y^3 \end{matrix}\right.$$ I understand ...
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0answers
22 views

Question on the solution to obtaining the truncation error for the Crank-Nicholson finite-difference scheme

I'm working on an exercise in a textbook that asks to derive the local truncation error for the Crank-Nicholson finite-difference scheme at the point $(ih, jk)$ for the partial differential equation $\...
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0answers
28 views

The $L^2$ convergence of semi-$p$-lapace equation

This question is similar to the one I post early here. But this one might be more reasonable I think... Let $g\in L^\infty(\Omega)$ be given, where $\Omega\subset \mathbb R^2$ is open bounded with ...
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0answers
28 views

The convergence of $p$-laplace equation

Let $g\in L^\infty(\Omega)$ be given, where $\Omega\subset \mathbb R^2$ is open bounded with smooth boundary. Define, for $1<p\leq 2$, $$ u_p:=\operatorname{argmin}\left\{\int_\Omega|u-g|^pdx+\...
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1answer
66 views

A general version of Gronwall's inequality

For the following $$|u(t)|^p\le C_1 \int_0^t |u(s)|^p\,ds+C_2$$ using Gronwall inequality, we have $$|u(t)|^p\le C_2(1+C_1 te^{C_1 t})$$ Now, my question is, for $$|u(t)|^p\le K_1 \int_0^t(1+|u(s)|^2)...
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2answers
69 views

Why is $u(x,y)$ “independent” of $x$?

From Wikipedia: A relatively simple PDE is: $$\frac{\partial u}{\partial x}(x,y)=0$$ This relation implies that the function $u(x,y)$ is independent of $x$. Why is that last line true? ...
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contraction mapping proof

I was reading a paper, and there is a proof I don't understand. How did they get from eq(6.4) to eq(6.5) using the norm they defined? Any help would be appreciated. Thanks in advance! Here is the ...
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1answer
39 views

Existence and Smoothness of Vector Calculus Identities [closed]

How to proof there exist smooth and globally defined solutions to all Vector calculus indentities ? For example: proof there exist smooth and globally defined solutions to the divergence of the curl ...
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21 views

Coupling Boundary Condition of one PDE with source term of another PDE

We have a system of equations, wherein the BC of one PDE is coupled with the source term of another PDE. We have a regular 2D unit grid in x and y. There are two PDEs to be solved The first PDE (...
3
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1answer
77 views

Behavior of fundamental solution to heat equation after projection

I am considering the behavior of $$\frac{1}{h}\|(1-P_h)S(h)v\|,\tag{1}$$ and $$\frac{1}{h}\|(1-P_h)S(h)P_hv\|,\tag{2}$$ as $h\to 0^+$ for a fixed good enough $v$. I hope to show one of them converges ...
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Bound of mollified in $H^{-2}$

Let $f\in L^2((0,T); H^2) $ with $ \partial_t f \in L^{2}((0,T);H^{-2}) $ and let $ \eta_{\varepsilon} $ a standard mollifier sequence in $ (t,x) $, then there exists a constant $ C $ independent of $ ...
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0answers
79 views

Solutions to Laplace's equation between surfaces of a constant coordinate

In Physics (specifically electrostatics) it is often encountered that one must solve the Dirichlet problem for Laplace's equation \begin{equation} \nabla^2 \varphi(x_1,x_2,x_3) = 0 \end{equation} ...
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1answer
47 views

Math needed to study Navier-Stokes existence and smoothness problem [closed]

This is Navier-Stokes existence and smoothness problem. I think the main problem is that I am not familiar with the mathematics of the Navier-Stokes existence smoothness problem. What kind of math ...
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1answer
52 views

comparison of two first order pde problems $u_x + cu_y =0$ vs. $u_x + cu_y =1$

I am analyzing two similar first order PDE problems. 1) $u_x + cu_y =0 \ \ $ and $ \ \ u(0,y)= sin(y)$ and 2) $u_t + cu_x =1 \ \ $ and $ \ \ u(0,y)= sin(y)$ As I understand 1) The solution ...
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1answer
11 views

Dimension of boundary of a bounded domain; what to use for Sobolev inequalities

Let $\Omega$ be a bounded (at least) Lipschitz domain in $\mathbb{R}^{N}$. Its boundary $\partial\Omega$, if I'm right, is $(N-1)$-dimensional object embedded in $\mathbb{R}^N$. We can define Sobolev ...
0
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2answers
60 views

what does the triangle in PDE notation mean?

what does the triangle sign used in PDE equations represent? example: $$u_{tt}- \Delta u =0$$ I took the below example from the Evan's book PDE p.4 (example 9). The equation is presented as the ...
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0answers
12 views

elliptic and parabolic theories for unbounded domain in R^n

I'm self-reading Evan's PDE book. It discusses elliptic and parabolic theories under the condition that the domain $U$ is open and bounded, and the functions vanishes on $\partial U.$ I'm wondering if ...