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

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Particular integral of PDE.

The PDE $$\frac{\partial ^{2}u}{\partial x^{2}}+2\frac{\partial^{2}u}{\partial x\partial y}+\frac{\partial^{2}u}{\partial y^{2}}=x$$ Has $1.$ Only one particular integral. $2.$ a particular integral ...
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11 views

Necessary condition for local existence of solution for system of two first order PDEs

Let $f,g$ be two smooth functions on $\mathbb{R}^2$ and $U,V$ be two smooth vector fields on $\mathbb{R}^2$. What is the sufficient condition for local existence of a solution $\phi$ of equations ...
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1answer
9 views

Nonlinear first order pde with both IC and BC

I am trying to solve the first order problem, namely: $$ \begin{cases} u_{t}+uu_{x} = 0, \hspace{0.5cm} x>0, t>0 \\ u(0,t)=t, \hspace{0.5cm} t>0 \\ u(x,0)=x^2, \hspace{0.5cm} x>0 ...
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FitzHugh–Nagumo system with diffusion

I was studying the FitzHugh-Nagumo model with diffusion and I quite do not understand the meaning of it. If we consider the system without diffusion, \begin{equation}\label{FHN}\begin{cases} ...
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1answer
24 views

Solution of a sublinear elliptic problem.

In a lecture notes, the author showed the problem $\tag{$P$}$ $\begin{cases} -\Delta u = |u|^{q-2}u \textrm{ in } \Omega, \\ u(x)= 0 \textrm{ in } \partial\Omega, \end{cases}$ where $\Omega ...
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2answers
76 views

Cauchy's problem. Equation of mathematical physics

$$U_{tt} = \Delta U + x^3 - 3xy^2$$ $$U|_{t=0} = e^x \cos y$$ $$U_t|_{t=0} = e^y \sin x$$ Help me, please, with solution of this equation. Can you prompt me algorithm to find the ...
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1answer
33 views

First eigenvalue of laplacian

I know the laplacian $\Delta$ has only positive eigenvalues, but why there is a first one? Assume $\Delta$ is acting on an appropriate set of real valued functions on the bounded domain $\Omega ...
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1answer
454 views

Show $\nabla^2g=-f$ almost everywhere

let a continuous function $f(x,y,z)$ be absolutely continuous over every bounded region and let it be in $L^1$ that is $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} ...
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8 views

Is the derivative product rule true for Bochner spaces?

If $u\in L^{2}(0,T; L^{2}(\Sigma))$ with $u_{t}\in L^{2}(0,T; H^{-1}(\Sigma))$ and $v\in L^{2}(0,T; H^{1}_{0}(\Sigma))$ with $v_{t}\in L^{2}(0,T; L^{2}(\Sigma))$ is it true that $$ ...
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pde of probability density function [on hold]

I have a question about the time derivative of the probability density function, assuming a density function,$\rho(S,T|S_t,t)$, if I calculate the derivative with respect to T, that is the Fokker ...
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1answer
36 views

What does “loss of regularity” mean?

I have seen a lot the phrase "loss of regularity" in references regarding PDE. (For instance, there are questions like "do solutions of 3D Navier-Stokes equations lose regularity or not?") Could ...
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9 views

solution of the Heat equation on a closed manifold

Let $\rm(M,g_0)$ be a closed Riemannian manifold and let $(g(t),\phi(t))$ is the solution of the so-called List's flow, i.e. the following system of equations $$\left\{\begin{array}{11} ...
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1answer
52 views

Show that $\eta(z) \det P$ is a null Lagrangian

This is problem 8.7.4 from Evans' PDE book. Assume $\eta: \mathbb{R}^n \to \mathbb{R}$ is $C^1$. Show that $L(P,z,x) = \eta(z) \det P$ is a null Lagrangian. Here $P$ is a $n \times n$ matrix and $z ...
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19 views

Linearising a PDE with a general nonlinearity term?

We have the following PDE: $$ i\psi_t + \psi_{xx} + F(|\psi|^2)\psi = 0 $$ Here, $F(x)$ represents a 'general nonlinearity' term. Suppose that $F(p_0) = 0$ for some $p_0 > 0$. Then we have a ...
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1answer
22 views

Concerning the proof of regularity of the weak solution for the laplacian problem given in Brezis

I am reading Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim brezis, and I am a bit confused about the proof. The theorem is stated as: Let $\Omega \subset ...
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27 views

Cauchy Problem for PDE $\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0$

Consider the Cauchy Problem of finding $u(x,t)$ such that $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0,x\in\mathbb{R},t>0$$ Which choices of the following functions of $u_{0}$ ...
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21 views

Solving large set of PDEs and verfication of solution

Please comment on solution of following system of PDEs, I have checked solution several times but could not find any error but there are problems when I proceed with solution obtained as below. I ...
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0answers
8 views

Compact resolvent proof

I want to prove the following proposition: We consider the operator $A=A_V=-\Delta+\frac{1}{4}|\nabla V|^2-\frac{1}{2}\Delta V$(where $V$ is a polynomial) with domaine $D(A)=C_c^{\infty}(R^n)$. ...
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40 views

Differential Equation Simplification on American Put Option paper

I am currently reading An Exact and Explicit Solution for the Valuation of American Put Options by Song-Ping Zhu. It is written in the first line of equation (5) that $$-\frac{\partial V}{\partial ...
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33 views

How do I get these equations ($n$th-order deformation equation and its initial/boundary conditions)?

Earlier, I asked a question What does 'equating the like-power of $q$ ' mean? and I already got the answer about the meaning of a particular phrase in a certain context. However, my real question ...
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1answer
8 views

How to find spacial periodicity

I am struggling with the following question. Consider $$U(x,t)=2\cos\left(kx-\sqrt{\frac{a}{b}}k^2+(\Delta k)^2)t\right)\cos\left((\Delta k)(x-2\sqrt{\frac{a}{b}}kt\right)$$ and suppose that $\Delta ...
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Solving or knowing something about a non-linear PDE which is “almost” linear?

Let $a>0$ be fixed. I have the following PDE: $u=u(t,x)$, $t\in [0,1]$, $x\in \mathbb{R}$, $$-\partial_t u = |\partial_x u| + \frac{1}{2}\partial_x^2 u, \quad ...
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How can I actually solve this kind of partial differential equations?

$$ x \frac{\partial z}{\partial x}+t \frac{\partial z}{\partial t}+y \frac{\partial z}{\partial y}=xyt$$ I can see that one soultion for this equation is $$z=(1/3)xyt+ C$$ however how can one solve ...
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2answers
107 views

In what conditions a weak solution is a classical solution?

I'm studying elliptic equations in divergence form $$-D_{j}(a_{ij}(x)D_{i}u) + c(x)u = f(x) \text { in a domain } \Omega \subset \mathbb{R}^{n}$$ I call a function $u \in H^{1}(\Omega)$ a weak ...
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1answer
30 views

Subsequence and diagonal process

We consider a sequence of functions défined on $\mathbb R^n$ by $f_m(x)=f(\frac{x}{m}),\ \forall m\in \mathbb{N}$ such that : 1) $f=1 $ in $B(0,1)$ 2) $\mathrm{supp\,} f\subset B(0,2)$ 3) $f ...
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18 views

Resource of learning Fokas Method for linear PDEs

I want to study Fokas Method for solving linear PDEs, but I don't seem to find a good resource for that topic (Found some papers by Deconinck, which are difficult to follow). Is there maybe a resource ...
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1answer
23 views

Showing a Fourier Series for $\sin (x)$ on $(0,\pi)$

I'm not sure how to type math symbols on here so I'll try to be as clear as possible. My homework problem wants me to show that the Fourier cosine series for $\sin\left(x\right)$ on ...
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1answer
14 views

Inequalities with negative Sobolev

In this paper I am read, it say that $||\triangle u||_{H^{-2}} \leq c||u||_{L^2}$, where $u$ solves the heat equation with zero boundary conditions on the boundary. I am still getting use to negative ...
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Applying Boundary Condition to Finite Element Matrix

Several times now I have seen the following done without justification and I cannot figure out why it can be done: Consider the 1 dimensional "pde" $-u'' = f, u(0) = a, u(1) = b$ over $[0,1]$. We ...
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2answers
600 views

Relationship between integral equations and partial differential equations

In my functional analysis class, we did a lot of problems involving integral equations such as proving existence & uniqueness using spectral theory and the Banach fixed point theorem. I've never ...
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computing the heat kernel for small times

The heat kernel on a two-dimensional manifold $M$ has the well-known expression $$H(p,q,t) = \sum_{i=1}^\infty e^{\lambda_i t}\phi_i(p)\phi_i(q)$$ where $\phi_i, \lambda_i$ are the eigenfunctions and ...
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On Atiyah-Singer Index theorem and Atiyah-Bott fixed point theorem.

I would like to understand the statement of Atiyah-Singer and Atiyah-Bott theorems enough to see lots and lots of applications they have. My background is pretty thin with some basic algebraic ...
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1answer
76 views

Green's Function for a Semi-Circle

Find Green's function for $\Omega = \{(x,y) \in \mathbb{R}^2\mid x^2+y^2 < r^2, y>0 \} $. I tried to find it and I know that $G(x,y)=\frac{1}{2}\pi\cdot \ln\left(\frac{1}{|y-x|}\right) + ...
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1answer
20 views

Existence and Uniqueness of Minimization problem in Sobolev space

"Consider the functional $$ F(u) = \int_{\Omega}[ \frac{1}{2} \nabla u(x) |^2 + g(x) u(x)] dx $$ and the set $$K = \{ v \in H^1 (\Omega): v = 0\ \text{on} \ \partial \Omega\ in\ the \ sense\ ...
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1answer
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Question regarding Evan's proof of Global Approximation by $C^∞(\overline{U})$ functions

The page where the proof is is on Google Books. A picture of the statement and the proof. I reproduce the statement of the result: Suppose $U$ is bounded with $C^1$ boundary, and $u∈ W^{k,p}(U)$. ...
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18 views

Showing the continuity of $\partial_{x_ix_j}v$ and $\partial^2_tv$

How to show that if $\begin{cases}\partial^2_tv(v,t;s)-\Delta v(x,t;s)=0\quad \text{for}\ (x,t)\in\mathbb R^2\times\mathbb R_{>s}\\v(x,s;s)=0,\quad\partial_tv(v,s;s)=f(x,s)\quad \text{for}\ ...
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1answer
419 views

PDE Evans Chapter 7 problem 16

Problem 16 of chapter 7 states Use problem 15 to prove that if $u$ is the semigroup solution in $X=L^2(U)$ of $$ \left\{ \begin{array}{rl} u_t - \Delta u =0 & \text{in } U_T \\ u=0 & ...
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3answers
126 views

Find $\lim_{t\to\infty}u(1,t)$, where $u(x,t)$ is a solution of $\frac{\partial u}{\partial t}-\frac{\partial^2u}{\partial x^2}=0$

Let $u(x,t)$ be a solution of $$\frac{\partial u}{\partial t}-\dfrac{\partial^{2}u}{\partial x^{2}}=0\text{ with}\\ u(x,0)=\frac{e^{2x}-1}{e^{2x}+1}.$$Then $\lim\limits_{t\to\infty} u(1,t)$ is equal ...
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Derive the partial differential equation

"Consider the function $$ F(u) = \int_{\Omega}[ \frac{1}{2} \nabla u(x) |^2 + g(x) u(x)] dx $$ and the set $$K = \{ v \in H^1 (\Omega): v = 0\ \text{on} \ \partial \Omega\ in\ the \ sense\ of\ ...
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1answer
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First Order PDE, How to Deal With This Boundary Condition?

1. The problem statement Find a solution of $$\frac{1}{x^2}\frac{\partial u(x,y)}{\partial x}+\frac{1}{y^3}\frac{\partial u(x,y)}{\partial y}=0$$ Which satisfies the condition $\frac{\partial ...
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PDE realated to heat equation with exponential additive term

I want to solve a PDE realated to heat equation with exponential additive term $${\partial u\over\partial t}={1\over x^2}{\partial\over\partial x}(x^2{\partial u\over\partial x})+e^u$$ I dervived ...
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1answer
446 views

Heat Kernel Property

Let $\phi$ be the Heat Kernel in $\mathbb{R}^n$, i. e. $$\phi (x,t)={(4\pi t)}^{-n/2}\exp\left( - \frac{\mid x \mid ^2}{4t}\right)$$ and let $u$ satisfy Heat equation. Show that: ...
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1answer
24 views

Explicit Finite Difference Scheme For Approxating a p.d.e

$\frac{du}{dt} = \frac{d}{dx}[\frac{1}{x^2+1}\frac{du}{dx}]$ I am trying to approximate this pde with a finite difference scheme but I am confused with the d/dx. Do I just take the derivative of ...
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What is the discretization matrix of 2D Poisson equation of finite diffence with checkerboard (black and red) pattern?

Given the problem$-\Delta u(x,y)=f(x,y)$ on unit rectangle $\Omega=[0,1]^{2}$ and $u(x,y)=g(x,y)$ on $\partial\Omega$, what is the finite difference matrix associated with step size $h=1/(2N+1)$ where ...
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33 views

Young inequality application

Can someone please help me to correct the following inequality: Let $a,b,c$ be three strictely positif numbers we have : $a^{\frac{1}{108}}b^{\frac{3}{4}}c^6\le ...
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1answer
24 views

Partial Differential Equations- General solution with different separation constants

So I have the wave equation $$\frac{\partial^2u}{\partial t^2} = c^2 \frac{\partial^2u}{\partial x^2} $$ and I know the process to split it into the two ODE's which are $F''(x)-nF(X) = 0$ (Used $n$ ...
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1answer
33 views

The particularity of $k$ being an integer in the solution of a DE

Related to the Thomas's comment in the question : Eigenvalues of the circle over the Laplacian operator, is there anyone could tell me why the periodic function $g$ has a fundamental set of solutions ...
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15 views

First order linear pde

Let $xyu=C_{1}$ and $ x^{2}+ y^{2}-2u= C_{2}$, where $c_{1}$ and $c_{2}$ are arbitary constants be the first integrals of the pde $$ x(u+y^2)\frac{\partial u}{\partial x}- y (u+x^{2})\frac{\partial ...
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1answer
39 views

Fourier series coefficients in PDEs

I have a problem that involves solving a PDE using separation of variables. For context, here is the question: $u(x,t)$ is the displacement of a string at position $x$ and time $t$, which is ...
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1answer
26 views

Geometric View of First-Order Quasilinear PDEs

Theorem 1 in page 4 of the book Numerical Solution of Partial Differential Equations in Science and Engineering by L. Lapidus: The general solution of the quasilinear PDE $$a(x,y,u)u_x + b(x, y,u)u_y ...