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

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Leibniz rule and a heat problem with homogeneous initial and boundary data

Problem Consider the following heat equation: $$v_{xx} = v_t , v(0,t)=0, v_x(L,t)=0, v(x,0)=0.$$ Furthermore, $I(t)$ is defined by $$I(t)=\int_0^L [v(x,t)]^2 dx.$$ Complete the following: Apply ...
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39 views

Stuck while trying to simplifying this PDE identity

This is related to a fluids HW: Let $u(x)$ be incompressible vector field on n-D space, $x\in \Omega\subset R^n$ and let $\theta(x)$ be a scalar field s.t. $\int_{\Omega} \theta(x)dx=0$, $\nabla.u=0$ ...
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36 views

Wave equation with mixed boundary conditions

For the wave equation: $$ \begin{cases} u_{tt}-c^2u_{xx}=0 & 0<x<\pi, t>0 & (1)\\ u(x,0)=u_{t}(x,0)=0 & 0<x<\pi & (2) \\ u(0,t)=\sin(ct) &t>0 \\ ...
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Which one of the two solutions is right?

I tried to solve the following initial/boundary value problem of parabolic type, which can be found in book A Collection of Problems in Mathematical Physics of B. M. Budak and A. A. Samarskii. But ...
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1answer
56 views

Question on partial differential equation

Solve the following PDE. $u_t + (u_x)^2 = -u$ with $u(x,0)=x$ My attempt: After substituting $u_{x}=v$ and solving new PDE $v_t + v v_x = -v$ with $v(x,0)=1$ , I got $v(x,t)=e^{-t}$. so ...
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58 views

How do I solve a Second order differential equation that is a variation of the Sine Gordon Equation?

$$0.1 \frac{d^2 \varphi}{d\tau^2}+\frac{d\varphi}{d\tau}+\sin\varphi-1.1=0$$ Im not quite sure how to reduce this equation. The inclusion of $\sin \varphi $ throws me off some what. If it helps $\tau ...
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33 views

Finding the exact expression and asymptotic form for this Bessel equation

By setting $$f = \frac{y}{\sqrt{z}}$$, transform bessel's equation of order m, $$z^{2}\frac{d^{2}f}{dz^{2}} + z\frac{df}{dz} + (z^{2}-m^{2})f = 0 $$ into $$\frac{d^{2}y}{dz^{2}} + y(1 + ...
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1answer
69 views

Surface of constant mean curvature

From PDE Evans, 2nd edition: Chapter 8, Exercise 12: Assume $u$ is a smooth minimizer of the area integral $$I[w]=\int_U (1+|Dw|^2)^{1/2} \, dx,$$ subject to given boundary conditions $w=g$ on ...
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Is there some relationship between algebraic curves and partial differential equations that goes beyond classifying different PDE's

I ask primarily because despite not having taken that many math classes (up to two semesters of a PDE class in college), it would be very interesting if maybe we could gain intuition regarding ...
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35 views

What is required for a pde to be solvable by separation of variables?

Separation of variables is explained in my notes to solve certain types of partial differential equations. This method requires the assumption that the solution, say $u(x,t)$, is in the form ...
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1answer
21 views

Linear evolution PDE: exponential decay to equilibrium

Suppose I have a homogeneous linear evolution equation $$ \left\{\begin{array}{ll} u_t + Lu = 0,&x\in\Omega,\quad t> 0 \\ u(0,x) = f(x) & x\in\Omega \\ u(t,x) = 0, & ...
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14 views

Particular Solutions to Time Independent Nonhomogeneous PDE's in Spherical Coordinates

I find myself working with PDE's like: $$\nabla^2f = G$$ or $$\nabla^2 f + k^2f = G$$ in spherical coordinates. I know that the homogeneous form of these equations (Laplace's and Helmholtz) have ...
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30 views

Conservation of Kinetic Energy in Vlasov-Poisson System

I'm studying the very basics of kinetic theory in Vlasov Poisson Systems, and the first equation I'm studying is the free transport equation, i.e.: $$\frac{\partial f}{\partial ...
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136 views

Topology of solution to a nonlinear eigenvalue problem

Consider the elliptic PDE: $$-\Delta u= f(x) u. $$ Assume that $f,u$ are defined in some reasonable bounded domain $\Omega \subset \mathbb{R}^n$ and impose the boundary condition $u=0$ on $\partial ...
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1answer
94 views

Method of reflections for the wave equation.

For the Wave Equation: $ \left\{\begin{matrix} u_{tt}-c^2u_{xx}=0 & x>0, t>0\\ u(x,0)=f(x) & x>0 & \hspace{0.5cm} (1) \\ u_{t}(x,0)=g(x) & x>0 & \hspace{0.5cm} ...
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16 views

Rarefaction waves in traffic theory with triangular fundamental diagram?

I'm studying traffic theory with the conservation law of the LWR model: $\frac{d\rho}{dt}+\frac{d\Phi(\rho)}{dx}=0$ . I would like to know which is the solution obtained with the method of ...
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1answer
44 views

PDE Cauchy problem.

Solve the following Cauchy problem. $u_t - \dfrac{1}{2} ( u_x )^2 = 0$ in $t>0$ , $x \in \mathbb{R}$. $u(x,0)= \dfrac{x^2}{2} $ on $t=0 $ , $x \in \mathbb{R}$.
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Optimal relaxation parameter for the SOR method when solving Poisson equation for non-square problems

Yang and Gobbert in paper "The Optimal Relaxation Parameter for the SOR Method Applied to a Classical Model Problem" ( http://userpages.umbc.edu/~gobbert/papers/YangGobbert2007SOR.pdf ) gave proof ...
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1answer
34 views

A trivial solution of a PDE

Let $u\in C^1$ in the unit closed disk $\Omega$ be a solution of the PDE $$a(x,y)u_x+b(x,y)u_y=-u $$ Suppose that $a(x,y)x+b(x,y)y>0$ in $\partial\Omega$. Show that $u=0$. Hint: Show ...
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1answer
18 views

Quasilinear equation and their properties.

Show that the quasilinear differential equation $$u_y+a(u)u_x=0$$ with the initial condition $u(x,0)=h(x)$ has the implicit solution $$u=h(x-a(u)y)$$ Show that there exists some positive ...
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83 views

$\frac{d^2 \theta}{dt^2}=9*\frac{d^2 \theta}{dx^2}$ use method of separation of variables to find final solution $\theta(x,t)=$?

Boundary conditions: $\theta_x (0,t)=\theta_x(4\pi,t)=0$ $t>0$ Initial conditions: $\theta(x,0)=3\cos 2x$, $\theta_t(x,0)=1+6 \cos 2x$, $0<x<4\pi$ Use result: $$X''-\lambda*X=0$$ ...
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1answer
36 views

$L^1$ Estimates involving bi-Laplacian

The following inequality can be shown to be true in the cases $p>1$: If $n\ge 5$, $\frac{1}{q} = \frac{1}{p} - \frac{4}{n}$, then there exists $C_{p,q,n}>0$ such that, for every $f \in ...
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23 views

reducing this Bessel differential equation using euler's ode

$$\frac{d^{2}y}{d^{2}z} + y ( 1 + \frac{1}{4z^{2}} - \frac{m^{2}}{z^{2}}) = 0$$ I have the above DE which I managed to transform into using a scaling transformation in response to a prior question ...
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24 views

Numerically solving the diffusion-reaction equation with boundary values

I want to solve a nonlinear PDE (steady-state diffusion reaction): $\Delta u = f(u)$ That has the following boundary conditions: $u_y(x,0) = 0$ $u(x,h) = m$ I am trying to solve it via newton's ...
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1answer
11 views

Laplace-Operator of distribution-valued function (heat equation)

I'm having trouble making sense of an exercise involving this definition of the heat equation: $u'(t) = \Delta (u(t))$, $u(0) = \delta_0$ for $t > 0$ where $u : [0, \infty) \rightarrow ...
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1answer
25 views

Characteristic equation of the transport equation

(a) Write down the characteristic equations for the PDE $$u_t+b\cdot Du =f \text{ in } \mathbb{R}^n\times(0,\infty)$$ where $b\in \mathbb{R}^n, f=f(x,t)$. (b) Use the characteristic ODE to ...
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1answer
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functions in $H_0^1(\Omega) \cap C(\overline{\Omega})$ are zero on the boundary

I want to solve the following problem Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set with $C^\infty$-smooth boundary. Show that for any $u \in H_0^1(\Omega) \cap C(\overline{\Omega})$ ...
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1answer
85 views

What is the weak formulation of this problem?

Find $u\in H_D^1(\Omega)$ such that $-\nabla\cdot(a\nabla u)=0$ in $\Omega$, $\dfrac{\partial u}{\partial n}=g$ on $\Gamma_N$, $u=0$ on $\Gamma_D$. The function $a(x,y)$ is ...
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31 views

Minimum Knowledges to precisely calculate PDEs (integral equations)

Basically all I want to do is to calculate (or prove) precisely equations such as $$ \frac{1}{n\alpha(n)r^{n-1}}\int_{\partial B(x,r)} u(y) dS(y) = \frac{1}{n\alpha(n)} \int_{\partial B(x,r)} ...
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1answer
37 views

When using the method of characteristics, do we ever need the equation for the gradient?

I have a question about the method of characteristics (as defined in the PDE book by Evans). He summarizes the characteristic equations as (see eq. 11 on p. 98): (using $p=Du,z=u$ and the variable ...
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1answer
28 views

The initial condition for a heat equation with stationary solution subtracted

I am presented with the following question for exam revision: Heat is supplied at a prescribed rate $Q(x) > 0$ (per unit volume) to an isotropic conducting rod that occupies the region ...
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Is there any pde whose solution evolves as a partial Fourier integral?

Is there any partial differential equation such that the its solution evolves as partial Fourier integral (continuous version of partial sum) of a function $f(x)$ which might be an condition or ...
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31 views

If $\forall v \in V, \ a(Tu,v)=(u,v)$ is $T$ a bounded an regular operator?

Let $V, H$ two Hilbert spaces infinite dimensional. If the bilinear form $a(.,.)$ satisfies There exists a constant $\alpha>0$ such that $\forall v \in V, \ a(v,v)\geq \alpha \|v\|^2.$ There ...
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1answer
33 views

Exam questions on sobolev spaces, sobolev embedding [closed]

Hi I am doing some preparation for research in nonlinear PDE. I have almost finished reading the chapter of sobolev spaces and want some questions to test my understanding on various important ...
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1answer
25 views

Nonexistence for small data

For context of this question, please refer to page 689 of PDE Evans, 2nd edition. My question is at the bottom. As an example, consider this initial-value problem in three space dimensions: ...
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separation of variables 3D cylindrical

Given the wave equation for the displacement $$u(r,\theta,t)$$ in a circular domain $$0<r<a \text,-\pi<\theta<\pi$$ Use the separation of variables to reduce the problem to an ODE. ...
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1answer
34 views

Cauchy Problem and continuity of derivatives

I've been asked to solve the Cauchy problem: $ \left\{\begin{matrix} 2u_{x}+3u_{y}=0 & \hspace{0.1cm} (1)\\ u(x,0)=|x| & \hspace{0.2cm} (2)& \end{matrix}\right. $ Using the method ...
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2answers
36 views

Help Solving this 1D Linear Parabolic PDE

Let $u = u(t,x)$ satisfy the PDE $$ \frac{\partial u}{\partial t} = \frac{1}{2}c^2\frac{\partial^2 u}{\partial x^2} + (a + bx)\frac{\partial u}{\partial x} + f u, $$ where $a,b,c,f \in \mathbb{R}$ are ...
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Finding an approximate solution to a differential equation using finite difference method.

I have a differential equation $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=-2$$ on the square $$0 \leq x,y \leq 1$$ subject to the boundary conditions $u=0$ along $x=0$ ...
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1answer
46 views

Laplace equation on a disk

I have the Laplace equation $$\Delta u=\frac{1}{r} \frac{\partial}{\partial r } \left(r \frac{\partial u}{\partial r} \right)+\frac{1}{r^2} \frac{\partial u }{\partial \theta^2}=0$$ on a unit disk ...
4
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2answers
56 views

Finding a characteristic equation of second order PDE?

How to find the characteristic equation of the following PDE $$PDE: (\sin^2 {x} ) u_{xx}+ (\sin {2x})u_{xy}+(\cos^2x)u_{yy}=x$$
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Meaning of the following, partial derivatives..

What is the meaning of $${\partial^kG \over \partial t^k} \in C$$ how is this function explained $G(t,s)$, does it mean that the k-th derivative of $G$ is continuous. I've done some studying on this ...
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Total differential proof , need help understanding. Integration factor.

Now we're trying to find a solution for: $$ \mu(t,x):\qquad(*) \frac{\partial \mu}{\partial x}P- \frac{\partial \mu}{\partial t}Q + \mu\left(\frac{\partial P}{\partial t} - \frac{\partial Q}{\partial ...
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Classification of quasi-linear PDEs

According to my textbook, to classify a quasi-linear PDE into elliptic, parabolic or hyperbolic, one should solve the following set of equations (As an example of course): $$a_{1}\frac{\partial ...
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1answer
35 views

Extension of Fourier Transform and Plancherel Theorem

I'm very confused with the ideia of extension Fourier transform of $L^1(\mathbb{R}^n)$ to $L^2 (\mathbb{R}^n)$. I start with a $u\in L^1(\mathbb{R}^n)$ and I use the limit and the Banach property to ...
3
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2answers
31 views

dirichlet principle: why $u-g\in W_0^{1,2}(\Omega)$?

Let $D\subset\mathbb R^n$ be open and bounded. Consider $\Delta u=f$ in $\Omega$ and $u=g$ on $\partial\Omega$. Let $g\in W^{1,2}(D)$ and $f\in L^\infty(D)$. Then the minimizer of $$ I(u)=\int_\Omega ...
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1answer
26 views

Characterization of $H^k$ by Fourier transform

Let $u\in H^k(\mathbb{R}^n)$ a function with complex-valued. Question 1: If $u\in H^k(\mathbb{R}^n)$ and $D^\alpha u\in L_2(\mathbb{R}^n)$, by definition for each $|\alpha|\le k,$ then we have ...
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1answer
30 views

$u_{tt} + au_t = c^2u_{xx}$ for some $a>0$ implies that the energy is not increasing

Could someone please help me with the following question? I got stuck somewhere. Given a function $u(t,x)$ satisfying the relationship: $$ u_{tt} + au_t \ = \ c^2u_{xx} \qquad \text{ for some } ...
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0answers
25 views

Calculus of variations - unilateral constraints [duplicate]

Question about Evans states, chapter 8.4.2! We have $I[w] := \int_U \frac{1}{2}|Dw|^2 - fw\, dx$, among all functions $w$ belonging to the set $$\mathcal{A} : = \{w \in H_0^1(U) : w \geq h \, ...
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26 views

Connection between parabolic and elliptic PDEs

I am trying to verify the well-posedness of some PDEs which I solve numerically. There is a paper which proves the well-posedness of a similar problem using Ladyzhenskaya's book on hyperbolic ...