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

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Help with a nonlinear partial differential equation

let : $$\frac{\partial f}{\partial x}=f _{x}\;,\;\;\frac{\partial f}{\partial t}=f _{t}\;,\;\;\frac{\partial}{\partial t}\frac{\partial f}{\partial x}=f_{tx}\;, \;\;\ \frac{\partial}{\partial x}\frac{\...
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Poincaré's inequality proof for $u \in W_0^{1,p}(\Omega)$.

I am trying to prove Poincare's inequality for $u \in W_0^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is an open bounded set and $1 \leq p < \infty$. This is Poincare's inequality: $||...
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What are complexiton solutions of PDEs?

I have seen written in some article that the solution of PDEs containing more than one transcendental functions is called complexiton solution, is it correct ? what are properties of complexition ...
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PDEs - prove continuity of operator

Consider the following nonlinear problem $$ \begin{cases} -div(a(u)\nabla u ) =0 & \text{in $\Omega$} \\ u=0, & \text{on $\partial \Omega$ } \end{cases} $$ We can assume $\Omega$ to be a ...
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Numerical analysis of wave equation in polar coordinates:

Is there a simple solution to deal with the problem of radial symmetry when solving a pde numerically. If so can someone provide some references/resources that explain this. Any help would be greatly ...
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Almost the minimal surface equation

I came across the following quasilinear PDE: $$ \nabla \cdot \left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}} \right) = 0. $$ This is almost the minimal surface equation, except that there is a minus sign ...
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Analytical solution for a Non-linear differential equation $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+B \sin(2Cy)$

Analytical solution for a non-linear differential equation: $\frac{d^2y}{dt^2} = A \left(\frac{dy}{dt}\right)+ B \sin(2Cy)$ A,B are non-zero constants and y (position) is a scalar-value parameter ...
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24 views

Laplace Operator Times Function

I'm just going through some proofs of a PDE book and have a question about one of them. It is stated that: $$ \int_U w \Delta w \text{ d}x = -2 \int_U |Dw|^2 \text{ d}x $$ Where $w$ is a solution of ...
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Name of a Sobolev/Poincare type inequality

It is a basic analysis exercise to verify that for $u \colon [0,1] \times [0,\infty) \to \mathbb R$, if $u(0,t) = 0$ for all $t$, then $$ ||u(\cdot,t)||_\infty \leq ||u_x(\cdot,t)||_2,$$ where the ...
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Functions with constant divergence of gradient-like field $\phi\nabla \phi/|\nabla \phi|$

I would like to classify functions $\phi : \mathbb{R}^2 \rightarrow \mathbb{R}$ such that $$ \nabla \cdot \left( \phi \frac{\nabla\phi}{|\nabla\phi|} \right) = \text{const}. $$ The only examples I ...
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Enquiry regarding a step in the proof of Theorem 13 on page 35 of Evans' PDE second edition.

In the proof of symmetry of Green's function on page 36, I have: On the other hand, $v(z) = \Phi(z-x)-\phi^x(z)$, where $\phi^x$ is smooth in $U$. Thus $$\lim_{\epsilon \to 0} \int_{\partial B(x,\...
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Bounds on flux integrals

What are some handy upper bounds for surface integrals (and their proofs)? Specifically, suppose $f$ is a bounded function on a surface $S$. Do we have $$ \int_{\partial S} F \cdot n \; \mathrm{d}S \...
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Temperature/heat equation

I solved this problem $$\left\{\begin{array}{ll} u_{t}=ku_{xx}, & x\in(0,1), t>0 \\ u(0,t)=2, u(1,t)=3, & t>0 \\ u(x,0)=x^{2}+x+2, & x\in(0,1) \end{array}\right.$$ and I got this $$u(...
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28 views

Nonhomogeneous heat equation [on hold]

I really don't know how to start to solve it: $$\left\{\begin{array}{ll} u_{t}=ku_{xx}-\lambda^{2}u, & x\in(0,\ell), t>0 \\ u(0,t)=u(\ell,t)=0, & t>0 \\ u(x,0)=h(x), & x\in(0,\ell) \...
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Rewriting ODE in terms of a different variable ($z=e^x$)

Given the ODE $$x^2M''+xM'+\lambda M = 0$$ where $1<x<L$, with boundary conditions $M(1) = 0$, $M(L)=0$, we can rewrite it in the Sturm-Liouville form and get $$\left[M'\exp\left(\int\limits_0^L{...
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Affine projection of polynomial

Please i need your help to solve the following problem: Let $V(Y)$ belong to $\mathbb{R}[Y_1,...,Y_d]$. Prove that one can find an affine change of coordinate $Y=AX+B, (X_1,..,X_d)$ on $\mathbb{R}^d$...
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24 views

PDE with analytical solution in some cases

I studied PDE, if a classification of them means studying. I haven't studied solving methods for PDE so I'd like to have an elementary answer if it is possible. I got this one: $$ \frac{\partial \...
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24 views

Textbook/monograph for microlocal analysis

I want to grasp the theory of microlocal analysis and apply this theory to some PDEs in $R^n$. But most textbooks I found put much priority on manifolds. Sadly, I know little about them and don't ...
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Partial differential equation transformation

Consider the partial differential equation $$i \frac{\partial \psi}{\partial t} - \left(i\nabla + \mathbf{A} \right)^2 \psi = 0 \tag{1}$$ for the scalar function $\psi(x,y,z)$ and the vector ...
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PDE with Robin like boundary conditions

Solve for $\phi:[0,L]\times [0,\infty) \to \mathbb R$: \begin{align*} \frac{\partial^2}{\partial t^2} \phi(x,t) &= c^2\frac{\partial^2}{\partial x^2} \phi(x,t) \\ \frac{\partial^2}{\partial t^2} \...
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23 views

Energy inequality heat equation

Consider $u \in C_1^2(\Omega \times [0,T]), \Omega\subset\mathbb{R}^n$ as a solution of the problem $ u_t - \Delta u = f, \text{ in } \Omega \times (0, T]$, $u = 0, \text{ on } \partial\Omega \...
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uniqueness of weak solutions for parabolic pde Evans

Hi I am trying to understand Evan's proof on uniqueness of weak solution in chapter 7. For the proof of theorem 4 below, I can see (35) and (36) make sense. But I have difficulty to see how Gronwall's ...
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Periodic boundary conditions Poisson

Given the Poisson eq. $\Delta u = -f$ , $u \in H^{1}(\Omega)$ , where in 1d we have $\Omega = [0,1]= S^{1}$ that has periodic boundaries and given we have mean value $\int_{S^{1}} u dy= u(x)$, what ...
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Hardys Inequality on Integrals

My PDE script lists a form of Hardy´s inequality which is not the common one: Let $u\in C^\infty([0,\infty))$ and $\delta < -\frac{1}{2}$. Then: $(\int_0^\infty |r^\delta u(r)|^2 dr)^{\frac{1}{...
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The 3rd term of the energy estimates in chapter 7 Evans PDE

Hi I wonder someone could help me check my understanding of getting an inequality of the estimate for the 3rd term $\|u'_m\|_{L^2(0,T;H^{-1}(U))}$ correctly. This inequality need to be checked is ...
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67 views

Explanation of spaces of functions in PDE

Let's consider following equation: The problem $$ \begin{cases} -\operatorname{div}\left( p\left(x\right) \nabla{u} \right) + q(x)u = f \quad\text{... on } \Omega \\ u = h(x) \quad\text{... on } \...
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Origin of Non-physical Singularities in Linear Elastostatics

The theory of linear elastostatics produces non-physical singularities in the stress field, typically in locations of stress-concentrations, e.g., corners. The corresponding partial differential ...
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Fractional powers of Markov generators

Let $H$ be the generator of a symmetric Markov semigroup on $L^2(\mathbb{R}^n).$ Why the fractional power $H^\alpha$ (defined on a proper domain) with $0 < \alpha < 1$ turn out to be the ...
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Test function with bounded gradient

How to construct a test function (radial) which is zero outside a ball of radius $2r$ and $1$ on the ball of radius $r$ but the gradient is bounded by $\dfrac{1}{r}$.
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energy estimates Evans PDE chapter 7

Hi I am looking at the proof of theorem 2 energy estimates in Evans PDE. I have some difficulties regarding the estimate for each term. First for the first term. Q1 I am a little vague how (23) is ...
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How to approximate the probleme non-linear by finies elements $\mathbb{P}_1$

I have to approximate $u^1$ by finies elements $\mathbb{P}_1$ (such that $h=\frac{1}{4}$ and $V_h=\{v\in C(I),\quad v(0)=v(1)=0 \}$) \begin{cases} \dfrac{\partial u}{\partial t}-\dfrac{\partial}{\...
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Can “eigenvalues” in eigenfunction expansion be non-scalar?

This question is a bit nebulous but I don't have a particular example in mind... In general under certain assumptions, one can use eigenfunction expansion to represent an operator. For instance, for ...
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31 views

Half strip neighbourhoods for regular surfaces

Let $S$ be a regular connected and compact surface in $\mathbb{R}^3$. It is well known that such surfaces admit a global tubular neighborhood, of thickness $\epsilon>0$. In particular, by ...
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Green's function for the Laplace or Helmholtz equations for an open rectangle in 2d or cylinder 3d?

I have only ever worked with free space Green's functions, or Green's functions for for the upper half space in 2d. So is it possible to determine a Green's function for the Helmholtz equation or ...
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52 views

A differentiation with first principles question for two variables

I know this question is probably quite easy but it's been some time since I've done any sort of calculus and since a google search failed to turn up anything relevant to this specific question I ...
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Finding time scales of a PDE using scaling [closed]

When we take the 1D Heat Equation $$\cfrac{\partial T}{\partial t}=\kappa\cfrac{\partial^2 T}{\partial x^2}$$ in the case of a long thin rod and apply the scalings: $x = L\hat x$$t=\tau\...
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Proof of the Schwarz Lemma (simplified Cauchy/Clairaut Theorem)

My lecture notes states the following lemma (sometimes called Cauchy's Theorem or Clairaut's Theorem) without proof: Lemma. (Schwarz) Assume that $v_\xi$, $v_\eta$, and $v_{\xi\eta}$ exist and are ...
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Compactness of the resolvant of a Schrodinger operator

I want to prove that the closure of an operator $A$ with domain $D(A)= \mathcal{C}^{\infty}_c(\mathbb{R}^n)$ given by: $A=\Delta+\frac{1}{4}||\nabla V(x)||^2-\frac{1}{2}\Delta V(x)$ has a compact ...
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Solving n-dimensional first order linear pde

While working on a problem in game theory, I'm stuck at a problem which requires me to solve the following linear first order PDE on $K$ independent variable: $\sum_{k=1}^K(\frac{\partial u}{\partial ...
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Operator commutes with spectral projection

Let $E$ be the spectral measure to an (unbounded) self-adjoint operator $A$. Is there a sufficient and necessary condition so that for a bounded interval $I$ we have $E_I A= AE_I$?
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Pdes -theoretical answer

Question.Let $Ω$ be a bounded Connected on $R^3$ with smooth boundary $\partial{Ω}$.Let $u$ be a harmonic function on $Ω$ with continuous derivatives on $Ω\cup \partial{Ω}$ prove that. $$\iint_V \ {\...
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How to find other equations of lagrange for the Initial Value Problem

Find the solution of the Initial Value Problem $(x-y)\dfrac{\partial u}{\partial x}+(y-x-u)\dfrac{\partial u}{\partial y}=u$ where $u(x,0)=1$ . My try: $\dfrac{dx}{x-y}=\dfrac{dy}{y-x-u}=\dfrac{...
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Solving $\sin(\sqrt{\lambda}L) + \beta \cos(\sqrt{\lambda}L)\sqrt{\lambda} = 0$

I'm working with the ODE $$-\frac{d^2u}{dx^2}=\lambda u$$ and trying to find eigenvalues and eigenfunctions corresponding the boundary conditions $$u(0)=0, u(L)+\beta \frac{du}{dx}(L)=0$$ Assuming ...
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Differential Equation Invariant under Isometric Mapping

I started reading a book on Finite Elements ("Finite Elements", Braess) and one section describes elliptic PDE's of the form $Lu = f$. The author goes on to say "If a differential equation is ...
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“Inverse” Helmholtz Decomposition

So I am trying to write a report on the Helmholtz decomposition theorem on $\mathbb{R}^3$. The theorem states that under certain conditions, every vector field $\textbf{F}:U \subseteq \mathbb{R}^3 \to ...
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Tensor product space dense in $H_0^1$?

Given $C_c^{\infty}((a_i,b_i))$ for $i \in \{1,2\}.$ Is it then true that $$C_c^{\infty}(a_1,b_1) \otimes C_c^{\infty} (a_2,b_2)$$ is dense in $H_0^1((a_1,b_1) \times (a_2,b_2))$? Clearly, by ...
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Ineqality involving operators

I need your help to solve the following problem: We define the operatorss $a$ and $b$ with domain $\mathcal{C}_0^{\infty}(\mathbb{R}^n)$ by: $a=\partial_x+\frac{1}{2}\partial_xV(x)$ (where $V$ in ...
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Is it possible to learn partial differential equations without an analysis course?

I need to learn partial differential equations and it seems that I need to take a course in analysis beforehand. Is it possible to learn PDE without analyais and only having learned ODE? Are there any ...