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

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322 views

Verification Of A Solution Of A One Dimensional Wave Equation [PDE]

I'am trying to answer a question from Michael D.Greenberg's Advanced Engineering Mathematics concerning a PDE.[chapter 1:introduction to modeling Ex1.2 Q4] Verify that $u(x,t) = (Ax + B)(Ct + D) + (E ...
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2answers
788 views

Green's function

Why does the Green's function $G(r,r_0)$ of the Laplace's equation $\nabla^2 u=0$, the domain being the half plane, is equal $0$ on the boundary? How can I interpret the Laplace's equation physically? ...
2
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1answer
100 views

Why is the following a solution to the system?

I have the following in my notes, but I can't remember how it works. Please help! $\nabla^2\psi=0, \quad\psi\to 0\quad\text{as}\quad x^2+y^2\to\infty, \quad\psi (x,y,0)$ is continuous Then by using ...
2
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1answer
241 views

Deriving SDE(s) and Expectation from Given PDE

We want to solve the PDE $u_t + \left( \frac{x^2 + y^2}{2}\right)u_{xx} + (x-y^2)u_y + ryu = 0 $ where $r$ is some constant and $u(x,y,T) = V(x,y)$ is given. Write an SDE and express $u(x,y,0)$ as the ...
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2answers
128 views

Uniqueness of solution to 1st order pdes

I am given a 1st order partial differential equation $y{\partial \psi\over\partial x}+x{\partial \psi\over\partial y}=0$ subjected to boundary condition $\psi(x,0)=\exp(-x^2)$. I have found that a ...
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1answer
230 views

solving this laplacian

given the laplacian $ -y^{2}( \partial _{x}^{2} + \partial _{y}^{2} )f(x,y)=Ef(x,y) $ can we find a solution in the form $ f(x,y)= \sum_{n,m}C_{n,m} \phi (x,E) g(y,E)$ if we impose the extra ...
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2answers
298 views

Expectation of Stochastic Process Given First Hitting Time Information

Let $V_t$ satisfy the SDE $dV_t = -\gamma V_t dt + \alpha dW_t$. Let $\tau$ be the first hitting time for 0, i.e., $\tau $ = min$(t | V_t = 0)$. Let $s =$ min$(\tau, 5)$. Let $\mathcal{F}_s$ be the ...
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1answer
72 views

Changing Variables in a PDE

Consider a positive solution to $(*)\qquad du_{xx}=u(\alpha-u)$ in $(a,b)$, $u(a)=0=u(b)$, where $d$ and $\alpha$ are positive constants. Prove $u(x) \leqslant \alpha$ on $[a,b]$. i) Suppose to the ...
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1answer
219 views

Kolmogorov Backward Equation Boundary Value Problem

I need to solve the backward equation $u_t - \gamma x u_x + \frac{1}{2} b^2 u_{xx} $ subject to the final condition $ u(x,T) = (x-a)^2 $. Here a and b and $\gamma$ are constants. I am given a strong ...
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0answers
813 views

Solving the wave equation using method of characteristics

I am having a lot of trouble understanding the method of characteristics to solve the wave equation. In fact, I have a final exam tomorrow and I can't seem to get a question from a previous ...
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2answers
894 views

What does the heat kernel in the heat equation represent $u(x,t)$?

Okay so I am studying for my PDE course and I am convering Fourier transforms. In fact I am using fourier transforms to find a solution to the heat equation on an infinite length rod. After going ...
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0answers
130 views

Help with PDE proof

I have been having trouble with the following outline of a proof, any help I can get will be appreciated. Thank you in advance. Consider a positive solution to $(*)\qquad du_{xx}=u(\alpha-u)$ in ...
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1answer
103 views

Help with Partial Differential Equations

I am having problems with the following question, any and all help is appreciated. Suppose $\Delta u = 0$ in $D$ $$\displaystyle\frac{du}{d\eta} +au = 0$$ on $\partial D$ where $D$ is a bounded ...
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2answers
215 views

Parabolic time-dependent PDE notation/dual space confusion

Suppose we have the PDE $\frac{\partial u}{\partial t} - \Delta u = f(x,t)$ with some boundary conditions. I am confused about what it means to say that the weak time derivative $u_t \in ...
2
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1answer
312 views

The mean value property and local maximum

I have an exercise in P.D.E that I couldn't solve. Let $\Omega \subset \mathbb{R}^n$ be a connected open set and $u:\Omega \to \mathbb{R}$ a continuous function that satisfies the following ...
2
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1answer
230 views

Finding an analytical solution to the wave equation using method of characteristics

Okay so I am super confused on what the method of characteristics is and what it means geometrically. So my first question is if anyone could kindly explain what characteristic lines are, why its ...
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3answers
347 views

General Criteria for the Existence or Non-existence of Solutions to a PDE

What are the conditions[General Criteria] for the existence or non existence of the solutions to a PDE[Elliptic type] subject to given boundary conditions? A specific Example: Let's consider the ...
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1answer
533 views

Method of characteristics with constant PDE

I was just introduced to method of characteristics for solving PDE's. We solved the wave equation that is inifinitely long using this method. However I am very confused about this method. Here is a ...
3
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1answer
598 views

Two basic questions on PDEs (trace operator and Sobolev space)

I am a bit unsure about the role of the trace operator. I understand that if you have a PDE that is solved by a function $u$ in some Sobolev space, then it's not necessarily defined on the boundary ...
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1answer
68 views

Does the solution to $u_t=-uu_x+0.1u_{xx}$ decay in time?

Consider the following PDE: $$ \begin{align} &u_t+uu_x=0.1u_{xx},\qquad 0<x<1,t>0\\ &u(x+1,t)=u(x,t),\qquad t\geq 0 \\ &u(x,0) = \sin 2\pi x,\qquad 0\leq x\leq 1 \end{align} ...
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1answer
244 views

An (elementary?) Question on Weak Derivatives

Let $U\subset \mathbb{R}^{n}$ be an open set and $f:U \to \mathbb{R}$ a continuous function which is piecewise $C^{1}$. This is: there is a partition of $U$ by (say, a finite number of) open sets ...
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1answer
1k views

Deriving expression for steady state flux and concentration: questions relating to diffusion

Can anyone tell me where to begin? How do I find the expression for steady state flux and steady state concentration for example? What assumed knowledge is implicit in the question? What common ...
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2answers
541 views

Non-uniqueness of solutions of the heat equation

For the heat equation $(\partial_t-\partial_x^2)f(t,x)=0$ defined on $[0,T)\times(-\infty,\infty)$, to obtain uniqueness of the initial value problem, usually it is required to limit the growth of the ...
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1answer
378 views

How to use FEM to solve a PDE

I come from a Computer Science background and I've been around searching for a concise tutorial that tells me how Finite Element Method is used to solve a certain partial differential equation. ...
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2answers
453 views

Explaining the method of characteristics

I am learning about solving p.d.e.s by the method of characteristics at the moment. I was given an "algorithm" to solve these problems but I want to know also what is going on, how it works and what ...
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0answers
1k views

Finding coefficients of a double Fourier series

This is the end of a PDE (heat equation in 2D) I am trying to solve with bounds from $0 < x < L$ and $0 < y < H$. It is a Newmann condition problem (i.e. all derivatives of $x$ and $y$ at ...
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0answers
118 views

Energy functional for the 1-dimensional wave equation

Let $u_{tt}=u_{xx}$ in the strip $\{(x,t):0\leq x \leq \pi, t \geq 0\}$ with boundary conditions $u_x(0,t)=k_0u(0,t)$ and $u_x(\pi,t)=k_1u(\pi,t)$. I am asked to prove that $E=\frac{1}{2}\int_0^\pi ...
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1answer
318 views

Deriving Fick's Principle from the Equation of Conservation of Matter

I don't know where to start with the following problem: Can anyone give me any pointers? (For maximum assistance, please adapt your responses and solutions to be understood by a beginner, ...
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2answers
317 views

Solving a simple second order ODE with initial condition

Okay i've been at it for far too long now. It comes from a bigger question from working with a PDE. I did seperation of variables and now I am stuck near the end of the problem. Here is the ODE in ...
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1answer
316 views

Well-posedness of the Poisson problem with mixed boundary conditions

let $\Omega \subset \mathbb R^n$ be a subdomain whit Lipschitz boundary, i.e. locally any part of the boundary looks like the graph of a Lipschitz continuous function, after some affine coordinate ...
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1answer
60 views

Nontrivial u such that $\Delta u(x) = u(x)$ on a compact domain with zero Dirichlet condition?

Let $u(x)$ be a solution to the problem $\Delta u(x) = u(x)$ on a compact domain with smooth boundary. Furthermore demand that $u(x)=0$ on the boundary. Is there an easy argument why $u(x)$ has to be ...
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1answer
332 views

Maximum Principles in Laplace Equation

I am currently practising some past year exams questions on intro to PDE in my school and I have problem doing the following: Let $\Omega$ be a bounded domain and let $u$ satisfies: $-\Delta u+ ...
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1answer
95 views

how to apply maximum principle in this PDE?

I have a PDE in the bounded domain $\Omega$: $-\Delta u+ a(x)u=0$ with $u=0$ on $\partial \Omega$, and $a(x)>0$. How do I show that $u\equiv 0$ in $\Omega$? I think I should use maximum principle ...
2
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1answer
154 views

Solving $yu_{xx}+(x+y)u_{xy}+xu_{yy}=0$

Let $yu_{xx}+(x+y)u_{xy}+xu_{yy}=0$, how would you go about solving this? So far, I have show it is hyperbolic everywhere except for the line $y=x$ and have been attempting to find the characteristic ...
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1answer
671 views

Wave Equation Solution by Factoring Operators

To solve the Wave Equation $$ u_{tt} - c^2 u_{xx} = 0$$ One method is to start with operator factorization $$ u_{tt} - c^2 u_{xx} = \bigg( \frac{\partial }{\partial t} - c \frac{\partial ...
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1answer
1k views

“Show by direct substitution…”

Show by direct substitution that $$ P(x,t) = \frac{1}{\sqrt{4 \pi D t}} \exp \left( - \frac{x^2}{4Dt} \right) $$ is a solution to the equation $$ \frac{\partial P}{\partial t} = D \frac{\partial^2 ...
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2answers
149 views

How to model multi-step cell differentiation

Can I better explain cell lineages using PDEs or stochastic?
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1answer
364 views

Heat transfer in a cylinder. Inhomogeneous PDE

I have a problem of heat distribution in a solid cylinder with the heater in the middle, which I take as $\exp(-r^2)$. $$\frac{\partial u(t,r)}{\partial t}=a^2\frac{\partial^2 u(t,r)}{\partial ...
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1answer
58 views

How should I understand $\frac{\partial^2 v_i}{\partial x_j\partial x_j}(x)=\frac{\partial p}{\partial x_i}(x)$?

The formula is from the first paragraph in the paper "Second Kind Integral Equation Formulation of Stokes' Flows Past a Particle of Arbitrary Shape" by Power and Miranda: ... the governing ...
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1answer
139 views

First order partial differential equation question

The population density $u$ of a species at age $y$ and time $t$ is given by the equation $u_{t} + u_{y} = \frac{-u}{(L-y)}$ $t\geq 0$ and $0 \leq y < L$ Initial conditions are $y = 0, u(t,0) = ...
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2answers
155 views

Laplace's equation

I am very confused. So I have Laplace's equation $\nabla^2\phi(x,y)=0$ and B.C.'s $\phi(x,0)=f(x); \,\,\,\,\,\, \phi(x,1)\equiv0$ where I have to solve it by Fourier transform. So I take the ...
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0answers
435 views

Fundamental solution of Heat equation with non-homogenous Dirichlet boundary condition

Suppose $\Omega = [0, 1]^3$ is a cube in $R^3$. Consider the problem $$u_t = \Delta u$$ $$u(0,x) = f(x)$$ $$u \mid_{\partial \Omega} = 0$$ Then by taking the convolution of $f$ with the fundamental ...
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1answer
1k views

Schrödinger versus heat equations

I'm trying to solve the initial value problem $(i\partial_t+\Delta_x)u(t,x)=0$, $u(0,x)=f(x)$ for the Schrödinger equation ($t\in\mathbb{R}$, $x\in\mathbb{R}^n$, $f$ Schwartz). I know that a ...
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1answer
143 views

Basic First-order quasi-linear PDEs question, involving characteristics

I would be incrediabely gratful if someone could go through step by step and explain how to do this question, as i'm rather stuck -and the lecture notes have a lot to be desired! 'Find $z(x,y)$ ...
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1answer
235 views

Separation of Variables/Boundary Value Problem

So, my problem is as follows: $$ U_{xx} +U_{yy}=0; U(x,0)=0, U_x (0,y)=0 ,U_x (L,y)=0 $$ I need to use Separation of Variables, which I did, and got: $$ U(x,y) = ...
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0answers
112 views

One question on 1st-order PDE

Given a smooth vector field $\mathbf{b}$ on $\mathbb{R}^n$, let $\mathbf{x}(s)=\mathbf{x}(s,x,t)$ solve the ODE $$\dot{\mathbf{x}}=\mathbf{b}(\mathbf{x}) (s\in\mathbb{R}), x(t)=x.$$ (a) ...
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2answers
415 views

Does Uniform Boundedness in the Sobolev Space $W^{1,2}$ and Convergence in $L^p$ $(1 \leq p < 6)$ Imply Convergence in $L^6$?

Let $B$ denote the open unit ball in $\mathbb{R}^3$. I want to either prove or disprove that a sequence of functions $u_m$ in the Sobolev space $W^{1,2}(B)$ which is uniformly bounded in the ...
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1answer
142 views

How one can find solution of PDE of the forms

I am a mathematical physics student and I had the following question in my mind from few weeks. I couldn't find any solutions. I am very thankful to this site and I hope, I expect some good reasonable ...
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1answer
138 views

What is the spherical means method which is used in solving a wave equation of second order?

Respected one, I want to know what the spherical means method is to solve a wave equation of two dimensions.
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3answers
504 views

A harmonic function which is bounded by $\ln(|x|)$ at infinity

I think I can prove that a harmonic function $u$ on $\mathbf{R}^n$ which satisfies $|u(x)|\leqslant C \ln(|x|+1)$ is constant. But what can we say about $u$ when the absolute value sign of $u$ is ...