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

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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
37 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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34 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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23 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 domain ...
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22 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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1answer
24 views

Factoring $R(ry')'-y(rR')'=[r(Ry'-R'y)]'$

In a problem this formula was used and I'm not seeing how this factor using the chain rule was derived. Other than calculating the derivative of the two that someone else already solved and showing ...
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106 views
+200

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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2answers
47 views

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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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
15 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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35 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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14 views

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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1answer
27 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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19 views

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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30 views

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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19 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
33 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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24 views

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
26 views

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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1answer
24 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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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
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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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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34 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
25 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
24 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
34 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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1answer
25 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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29 views

Young inequality question [duplicate]

I am looking for $\alpha_1,\alpha_2,\alpha_3$ such that the following inequality is true:$a^\frac{1}{80}\le \alpha_1\frac{a^{\frac{1}{4}}}{b^{\frac{3}{4}}c^6}+\alpha_2b^{\frac{1}{16}}+\alpha_3 ...
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23 views

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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Finite difference method and division by zero problem with no flux boundary condition

I am trying to implement an angionesis model described by Anderson and Chaplin in 1998. The model is based on a set of PDEs defined on an unit square with the following no-flux boundary condition ...
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1answer
42 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 ...
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58 views

Question using Young inequality

In fact i am looking for $\alpha_1,\alpha_2,\alpha_3$ such that the following inequality is true:$a^\frac{1}{80}\le \alpha_1\frac{a^{\frac{1}{4}}}{b^{\frac{3}{4}}c^6}+\alpha_2b^{\frac{1}{16}}+\alpha_3 ...
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32 views

Problem in a computation: am I doing something wrong?

This pdf gives me the following formula for the Laplacian in polar coordinates: $$\Delta f=D_{r_n}^2f+\frac{n-1}{r_n}D_{r_n}f+\frac{1}{r_n^2}\Lambda_nf,$$ where $n$ is the dimension of the space we ...
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7 views

How to balance highest order derivative term with that of nonlinear term is system of PDEs?

Please see following PDEs in $F(\xi, \eta)$ and $G(\xi, \eta)$: $\left( F \right) F_{{\xi,\xi}}+{F_{{\xi}}}^{2}+F_{{\xi,\eta}}G_{{\xi} }-F_{{\xi,\xi}}G_{{\eta}}+F_{{\eta,\eta}} =0$ $\left( F \right) ...
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26 views

Formal solution vs Classical solution

What's the difference between a formal and classical solution of a PDE. An explanation via an example is much appreciated.
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39 views

Can $f\in L^2(\Omega)$ imply $\nabla f\in [(H^1(\Omega))^*]^n$?

This is closely related to a previous question of mine. The only difference is the definition of $H^{-1}(\Omega).$ Suppose $f\in L^2(G_R)$ where $$ G_R=\{x\in\mathbb{R}^n\mid ...
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2answers
59 views

Fourier Transform Dirac Delta

I have recently learnt about tempered distributions, and how one can define the Fourier transform of a tempered distribution $v$ to be $\hat v$ so that $$\langle\hat v,\varphi\rangle=\langle v,\hat ...
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2answers
40 views

Solve $\frac {\partial^2 v}{\partial r^2}+\frac 1r \frac {\partial v}{\partial r}-\frac v{r^2} =0$

$$\frac {\partial^2 v}{\partial r^2}+\frac 1r \frac {\partial v}{\partial r}-\frac v{r^2} =0$$ i did $$\frac {1}{r}\frac {\partial }{\partial r}\bigg( r \frac {\partial v}{\partial r} \bigg) =\frac ...
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1answer
39 views

Best source to study partial differential equations (PDE) [duplicate]

Want to understand partial differential equations (linear and non-linear) more deeply. I am not a mathematician and I am more interessted in a more practical source that is teaching this topic from a ...
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1answer
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Can I write $H^1$ as $H^1_0 \oplus H^1_{\perp}$?

Let $\Omega\subset \mathbb{R}^d$, with $d\in \{1,2,3\}$ be an open bounded, simply connected domain. Define $H_0^1$ as the subspace of $H^1$ whose member functions have vanishing trace on the ...
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3rd order linear PDE with constant coefficients “general solution”

I am trying to solve the following PDE, where $F=F(z,w)$ and indices indicate partial derivatives withoug given boundary conditions (If the general case is to hard, lets say we have Dirichlet boundary ...
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1answer
37 views

Confusion with changing variables in second order DE

So in my physics assignment, we're given Schrodinger's equation: $$\frac{-\hbar^2}{2m}\frac{d^2\psi}{dx^2}+e\xi x\psi=E\psi$$ We're asked to substitute a function of the form $w(x)=Ax+B$ to arrive at ...
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2answers
47 views

Scale invariant definition of the Sobolev norm $\|\|_{m,\Omega}$ for $H^m(\Omega)$

I learned the following from Constantin and Foias's Navier-Stokes Equations (Chapter 4): We say that a function of a bounded open set $\Omega\subset\mathbb{R}^n$, $c(\Omega)$, is scale invariant ...
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Linear stability analysis of PDE for porous media model

I'm reading the article of Chuang et al. (2006), "The porous media model for the hydraulic system of a conifer tree: linking sap flux data to transpiration rate". They determine the follow PDE ...
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1answer
56 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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31 views

Question that involves an inequality

Can someone help me to answer the following question: I we have $\forall \epsilon >0$ there exist a constant $C_{\epsilon }>0$ such that: $\forall u\in C_0^{\infty}(R^{2n})$ ...
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1answer
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How to apply Picard–Lindelöf theorem to the whole domain.

Quick question about Picard–Lindelöf theorem. We know the existence and uniqueness of a local solution around $[t_0-\epsilon,t_0+\epsilon ]$. But what if we have Lipschitz continuity everywhere, does ...
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15 views

Verify the Green's function for Helmholtz equations

It is well known that $$ G(x)=\frac{1}{4\pi}\frac{\exp(ik|x|)}{|x|} $$ is the Green's function for Helmholtz equation $$ (\Delta+k^2)f=0 $$ in $\mathbb{R}^3$. My question is, given $v\in ...
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1answer
22 views

Find the solution $u=u(x,y,z,t)$

Find the solution $u=u(x,y,z,t)$ of the problem: $$\frac{\partial^2u }{\partial t^2}=\Delta u=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2}$$ ...