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

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Linear Hamiltonian System

Suppose the linear system: $\dot{z} = J \frac{\partial{H}}{\partial{z}} = J S(t) z = A(t) z$, with Hamiltonian $H=H(t,z)=\frac{1}{2} z^T S(t)z$. How can I prove that: $$\frac{d}{dt}H(t,\xi(t)) = ...
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A function to relate all its variables

I'm dealing with a function looks like $z=f(x,y_1,y_2,y_3)$; where $x$ and $y$ are independent variables. Basically what I did so far is: $z=f(x,y_1)$, where $z= \text{energy}$, $x = \text{distance}$ ...
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Problem on string vibration

Given the standard wave equation for small amplitudes, we have been asked to find the position of string $y(x,t)$, given: $y(x,0)=\sin x$, and, $y'(x,0)=\cos x$, where $y'$ depicts partial ...
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Fourier transformations and the inversion formula

I am working through the above question in preparation for an upcoming exam. I have completed part (a) and quoted the inversion formula for part (b), but I cannot see how to find a form to evaluate ...
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Backwards heat equation (stability analysis)

Problem Consider the backwards heat equation of the form $$ \left\{ \begin{aligned} u_{t} &= \lambda^2 u_{xx}, & x\in[0,L], \quad t\in[0,T]\\ u(0,t) &= u(L,t) = 0 \\ u(x,T) &= ...
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Nonlinear Partial DE

In my work I have faced with following partial differential equation $$\left(\frac{\partial u}{\partial x}\right)^2-\left(\frac{\partial u}{\partial y}\right)^2+f(x,y)\frac{\partial u}{\partial ...
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Partial derivative of $F{\equiv}0$

If you have a function $F{\equiv}0$ then is the partial derivative of $F$ with respect to any of its variables $0$? Specifically, when we have Charpit's equations for a PDE $F(x,y,u,p,q) = 0$, where ...
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An MCQ of Solving a heat equation with no boundary conditions

Let $u(x, t)$ be the solution of the equation $$\frac{∂^2 u}{∂x^2} = \frac{∂u}{∂t}$$ which tends to zero as $t → ∞$ and has the value $cos x$ when $t = 0.$ Then $u = \sum_{n = 1}^{∞} a_n sin(nx + ...
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Solution to a PDE that requires fundamental Laplace solutions

I would like to derive the spherical symmetric solutions (or in another words make use of the fundamental solutions) of the following PDE: \begin{equation} \Delta u(x)+\lambda u(x)=0, \quad ...
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Extension of bounded operators between norm spaces

Let $X$ and $Y$ be two Banach Spaces and $X_1$ be a subspace of $X$. If $T$ is a bounded linear operator from $X_1\to Y$, then, this is an extension of $T$ from $X\to Y$ such that $\|T\|_{X}\le ...
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Find the Characteristic Curve of the PDE

Let $x=x(s),y=y(s),z=z(s) ,s\epsilon\Bbb R $ , be the characteristic curve of the PDE $z_x + z_y -z = 0$ passing through the curve $x=0 , y=t , z=t^2 , t\epsilon\Bbb R$ Then what are ...
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Charpit's method - locally unique

http://www0.maths.ox.ac.uk/system/files/coursematerial/2014/3078/40/B5b2.pdf At the bottom of page 2 in these notes they give the condition for solving the equations uniquely for $p_0$ and $q_0$, but ...
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Reference for Dirichlet problem with coefficients in certain spaces.

I would like to get a reference for the following: Let $U$ be a domain with smooth boundary. We consider a parabolic differential operator $L$ acting on functions $f$ on $U$ and having the form ...
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Linear Second Order PDEs and usefulness of the canonical form.

I've been trying to solve a linear second order partial differential equation that looks relatively harmless $$ u_{xy}(1-x)(1-y)-u_x (1-x) -u_y (1-y) + u (1-d) -2w=0$$ whose domain is the unit ...
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Can't understand a notation regarding weak solution of Vlasov-Poisson system

The text is from https://cmouhot.files.wordpress.com/2010/01/chapter5.pdf . In section 1, it uses $~f_t~$ to represent a smooth solution of Vlasov-Poisson system (VPS). I think here $~t~$ in $~f_t~$ ...
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Uniqueness, symmetry, and energy behavior for the diffusion equation on an interval

Consider the boundary value problem (BVP) $u_t$ = $u_{xx}$ for $t>0$ and $x \in (0,1)$ with initial condition $$u(x,0) = \sin^4(2\pi x),\quad x \in(0,1) $$ and boundary conditions $u(0,t) = ...
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How did they solve this partial differential equation

I found this PDE in a paper I'm following, and for the life of me I can't work out how they solved it. The PDE is And they give the solution For reference the paper can be found here ...
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Fourier Transform method to solve a parabolic PDE in $\mathbb{R^n}$

Let $b\in \mathbb{R^n}$ and $c>0$. Assume $g \in C(R^n)$ has compact support and $f = f(x,t)$, $f \in C_1^2(R^n \times [0,\infty))$ has compact support. I'm trying to solve the following IBP via ...
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fourth order runge-kutta method and heavyside step function.

So I'm trying to model a hydrodynamic system that introduces a sudden "jump" in the value of a function at a specific time. The system is solved with a Runge-Kutta fourth order method. I have a ...
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What does this symbol mean here in this context?

The colon ("$:$" sign) - I am seeing this a lot in Chapter 8 of PDE Evans, like this (page 497 of the 2nd edition): THEOREM 6 (Pressure as Lagrange multiplier). There exists a scalar function $p ...
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If $f\in C^0(\overline{\Omega})$ is subharmonic and $-f$ is subharmonic, too, then $f\in C^2(\Omega)$ and $\Delta f=0$

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain and $f\in C^0(\overline{\Omega})$ be subharmonic, i.e. for each closed ball $\overline{B}\subseteq\Omega$ it holds: $$u\in C^2(B)\text{ is ...
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Laplacian Smoothing Irregular Initial Data

Apparently for many parabolic and elliptic PDEs the (ir-)regularity of initial data does not have any significant impact on the regularity of (weak) solutions. Very often people when people talk ...
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If $f\in C^0(\mathbb{R}^n)$ is subharmonic and $\limsup_{|x|\to\infty}f(x)\le 0$, then $f$ must be non-positive in $\mathbb{R}^n$

Let $f\in C^0(\mathbb{R}^)$ be subharmonic, i.e. for each closed ball $\overline{B}\subseteq\mathbb{R}^n$ it holds: $$u\in C^2(B)\text{ is harmonic in }B\text{ and }f\le u\text{ on }\partial ...
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$\Delta u = 0 \Rightarrow \Delta (u\circ \phi)=0$?

Let $U,V \subset \mathbb{R}^n$ be open sets and $u \in C^2(U)$ satisfies $\Delta u =0.$ Let $\phi: V \rightarrow U$ be a surjective $C^2(V;U)$ function, then I am looking for sufficient conditions ...
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Solving inverse differential equation

How to solve $$ \frac{1}{D_{x}^{2}+D_{x}D_{y}-12D_{y}^{2}}(ye^{3x+y}+x^{3})? $$ I understand that I need to use $$ ...
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1answer
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The separation of variables in a non-homogenous equation (theory clarification)

I know "copying and paste" method from resources aren't permitted but the text is fairly long and given the amount of time I have to learn PDE (as an exchange student beside having to adapt to a ...
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using Rayleigh's Quotient to determine the interval along which lamda lies to avoid considering all possible cases of lamda

I have heard in my lectures that When solving for a PDE using the separation of variables, one checks for all possible cases of $$\text{$\lambda >$0,$\lambda $=0,$\lambda <$0}$$, but this ...
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PDE — Wave Equation with extra term

How would I go about solving the following PDE: $u_{tt} = u_{xx} + A\sin(\omega t)$ with the boundary conditions: $u(0,t) = 0, u(1,t) = 0, u(x,0) = 0, u_t(x,0) = 0$. I have tried separation of ...
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Boundary conditions that yields no solutions to the coefficient?

I want to make sure I'm not overlooking certain steps as I've already spent an hour looking through. The heat equation is given as: $$ \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} ...
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1answer
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When the false transient method could make an elliptic PDE easier to solve numerically?

I think that I do not fully understand the false transient method. This method consists on introducing a time derivative to an elliptic PDE to convert it to a parabolic PDE. However, it does not make ...
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Laplace Operator in $3D$

I am looking to find the radial part of Laplace's operator in three dimensions. I looked up Laplace's operator in spherical coordinates and from there I guess the radial part is: ...
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Find wave equation initial condition such that solution consists of right-going waves only

Let $u(x,t)$ solve the wave equation $u_{tt}=c^2u_{xx}$ and let $u(x,0)=A(x)$ for some function $A(x)$. Find the function $B(x)=u_t(x,0)$ such that the solution consists of right going waves only. My ...
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An imbedding inequality in PDE.

u is a function of 3-dimension, I'm trying to prove this: $\|u\|_4^4 \leq C \|u\|^2_{H^1} \|u\|_{L^2}^2$ Anyone can shed light on this? Thanks very much.
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Inhomogeneous heat equation with source term orthogonality

This is a question on the lecture notes. Basically we have the usual heat equation: $$\frac{\partial y}{\partial t}(x,t)=k^2\frac{\partial^2 y}{\partial^2 x}(x,t)+F(x,t)$$ We also have the usual ...
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Numerical scheme for system of PDEs

I'm trying to solve the following coupled PDE system for my master thesis: \begin{align} \kappa_0\frac{\partial p}{\partial t}&=- \nabla \cdot v \\ \rho_0\frac{\partial v }{\partial t} &= ...
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Important ODE Solutions for Solving PDEs

Which ODEs pop up most often in the study of Partial Differential Equations such as the Heat Eq, Laplace Eq, Wave Eq, etc. At least in the homogeneous case. What are their solutions? I'm going to take ...
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Sturm-louville heat problem

Could I request for an example to the above question? I've read thru the regular sturm-liouville theory but have no idea how should the theory be applied to this problem. I understand that the ...
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Clarification on matters relating the the differential operator in Sturm Liouville

My set of two notes, one of which might contain a printing error. In Sturm-Liouville problem, is the differential operator expressed as $$L(y) = \frac{d}{\text{dx}}\text{((P(x) ...
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Why are diffeomorphism-invariant PDE not elliptic?

In reading geometric analysis papers, I frequently encounter a statement of the form, "The PDE in question is diffeomorphism-invariant, and therefore cannot be elliptic." My vague understanding is ...
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Sine Fourier series and smooth function.

I was reading through my text on PDEs and came across a theorem (or perhaps Lemma) that states: "For any smooth function $g_1(y)$ with $g_1(0) = g_1(h) = 0$, it can be expressed as a Fourier sine ...
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Converting solutions to separation constant to Cosh and Sinh

The Laplace's equation inside a rectangle is: $$u_{\text{xx}}+u_{\text{yy}}\text{=0}$$ The IC's are: $${u(0,y)=g(y)}$$ $${u(L,y)=0}$$ $${u(x,0)=0}$$ $${u(x,H)=0}$$ Via method of separation we have ...
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Growth of Tychonov's Counterexample for Heat Equation Uniqueness

Define a function $\varphi$ on $\mathbb{R}_{+}$ by $$\varphi(t):=\begin{cases}e^{-1/t^{2}}, & {t>0}\\ 0, & {t\leq 0}\end{cases}\tag{1}$$ It is well-known that $\varphi$ is ...
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Why are porous medium equations posed on connected domains? Shouldn't it be done on a domain with holes (or pores)?

The porous medium equation is supposed to model gas flow through porous media (i.e. some object with holes in it). Why then, in theory of weak solutions, do people study the equation on a sufficiently ...
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Charcteristic directions

Well in my assignment I need to find the characterstic directions of a PDE. A characteristic direction is a line along which the function behaves as an ODE. The problem PDE is: $$\frac{\partial^2 ...
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Motivated by Level Sets, how can I show that minimizing this functional is equivalent to this PDE?

I would like to show, that minimizing the functional $$F(g)=\alpha\int_\Omega |\nabla g(x)|^2dx+\mu \int_\Omega (g(x)-f(x))^2dx $$ is equivalent to solving the differential euqation $$-\alpha\nabla ...
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Schwarz Reflection Principle for Harmonic Functions

Given $\Omega \subset \mathbb{R}^n$ define $\Omega^+ = \Omega \cap \{x_n>0\}$ and $\Omega^0$, $\Omega^-$ analogously let $u \in C^2(\bar{\Omega}^+)$ be harmonic and such that $\frac{\partial ...
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Rewriting multivariate second order diffrential equation as system of first order

I hope someone can shed some light on the steps taken in between, as I have the answer and the problem, but no idea how to get there: Given the second order differential equation $$\frac{\partial^2 ...
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Regularity theorem for PDE: $-\Delta u \in C(\overline{\Omega})$ implies $u\in C^1(\overline{\Omega})$?

I stumbled over this question in the context of PDE theory: Let $U$ be connected,open and bounded in $\mathbb{R}^n$ and $u \in C^0(\overline{U}) \cap C^2(U)$ and $\Delta u \in C^0(\overline{U})$ with ...
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Solving equations of the form $\phi(D_{x},D_{y})u=0$

I am taking a course called partial differential equations. We are mostly learning about solving equations of the form $\phi(D_{x},D_{y})u=0$, for example $(D_{x}^{2}+D_{x}D_{y}+D_{y}^{2})u=0$ or ...
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Dirichlet problem for Laplace equation in $\Omega=\left(\overline{B\left(0,a\right)}\right)^{c}$

I'm having troubles with the proof of the following theorem: let $\Omega=\left(\overline{B\left(0,a\right)}\right)^{c}$ and $U\in \mathbb{R},U>0$, then there exists a unique function $\psi\in ...