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

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What are good resources for learning Numerical methods for Partial Differential Equations?

I'm having an undergraduate course on Numerical Solutions to Ordinary and Partial Differential Equations. I need online resources to supplement my study preferably videos and books. I want to build a ...
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10 views

Laplace equation on semidisk

I am interested in the solution of the following boundary value problem on the semidisk $D=\{(r,\theta): 0<r<1, 0<\theta<\pi\}$: $$u_{xx}+u_{yy}=0 \mbox{ in } D, $$ $$u(1,\theta)=0 \mbox{ ...
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22 views

Finding an upper bound on the gradient of the solution to the heat equation

I have a function $u:\mathbb{R}^n\times [0,T]\to \mathbb{R}$ that solves the heat equation $u_t=\Delta u$, is bounded, and $u(x,0)=g(x)$. I need to show that $$\max|\nabla u(x,t)|\leq ...
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How can I write this in Divergence form

Consider the PDE $u_{xx}-(yu_y)_x-y(u_x)_y+yu_y+(y^2+\frac{1}{H^2(x)})u_{yy}$ I need to write this in divergence form. That is, I need to write it in the form $\sum_{i,j}\frac{\partial}{\partial ...
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6 views

finite difference scheme for nonlinear partial differential equations

I have the following second order partial differential equation (PDE) on $[0,T] \times \mathbb{R},~ T >0 $: \begin{equation} \left(1 + \frac{1}{(1 + b f)^2}\right) \frac{\partial f}{\partial t} ...
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Finite Difference Scheme for a PDE on non-rectilinear coordinates

Consider poisson's equation on the domain $0 \leq x \leq 1$ $0 \leq y \leq H(x)$. Change the coordinates to $\xi=x$, $\eta=y/H(x)$. Construct a FDS that gives a positive definite symmetric matrix. ...
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Pseudo-monotone operators research paper question

Hi I just want to know if anyone can see how the result (2.34) is obtained in the following research paper http://caa.epfl.ch/publications/9-Boccardo-Dacorogna1984.pdf. Thanks, I know that it is a ...
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16 views

Elliptic regularity of Dirichlet problem

Suppose $\Omega\subset\mathbb{C}$ is a simply connected domain with $C^\infty$ boundary. Consider the following Dirichlet problem $$\Delta u |_{\Omega}=0 $$ $$u|_{\partial\Omega}=f$$ Under what ...
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9 views

What is the motivation for characterizing second order linear PDEs as hyperbolic, elliptic, or parabolic?

I see the connection between the PDEs and the equations of conic sections, but why is that important? I am under the impression that one of the big differences between the wave equation and the heat ...
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21 views

Find a bound for the summation $\sum_{j=k}^J jc^{k - j - 1} $

The problem: I've hit what might be a dead-end. If it is true, I would like to show that for $c \in (0,1)$ and $1 \leq k \leq J$, the sum $$ \sum_{j=k}^J jc^{1-j-k} = \sum_{n=1}^{J+1-k} (k-n-1) ...
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Applications of PDEs

I teach an undergrad ODE course. As I have completed basically all the material, I thought it would be nice to give the students a brief introduction to PDEs. At the end of the lecture, I said that ...
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22 views

An application of the mountain pass lemma

I am trying to show the existence of classical solution for the following problem using the mountain pass theorem : $$ \left\{ \begin{array}{ccccccc} u^{''} + \lambda u + u³ = 0 (0<t<\pi)\\ ...
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1answer
25 views

Integrating the Gaussian kernel with absolute value

How do I integrate: $$\dfrac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty}|y|e^{\frac{-(\xi-y)^2}{4t}}dy,$$ in terms of of the error function, erf$(x)=\dfrac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt$. I have ...
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7 views

Obtaining characteristic v on Cauchy Problem

$(x-y)p+(y-x-z)q=z$ Find the integral surface which the curves it passes are $z=1$ and $x^2+y^2=1$ Here is my try. $$\frac{dx}{x-y}=\frac{dy}{y-x-z}=\frac{dz}{z}$$ So we have ...
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49 views

Second Derivative of Eigenvalue

Consider the potential $V_t=tx$ and the corresponding Schrödinger operator $H_t=- \frac{\partial ^2}{\partial x^2}+V_t$ on $L^2([0,R])$ with Neumann or Dirichlet boundary conditions. Let ...
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21 views

Final value problem PDE

I need help unterstanding and solving the following problem $\rho c_p \frac{\partial\lambda}{\partial t} + k \frac{\partial^2\lambda}{\partial x^2} + 2 [T(x,t,q(t)) - Y(t)] \delta(x-x_{s}) = 0 $ ...
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1answer
42 views

How to solve Cauchy problem?

I'm new to this problem. Here is the question. $$(y+xz)z_x+(x+yz)z_y=z^2-1$$ Find the integral surface which the curves it passes are $y=1$ and $z=x^2$ By Lagrange system i found $u$ and $v$. We ...
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29 views

Partial Derivative With Respect to $t$

What is $\frac{\partial v}{\partial t}$ if $v$ can be defined as $v(x,t,\zeta)=w(x(3t)^{-1/3},\zeta (3t)^{1/3})$?
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11 views

Finite Element Method Weak Formulation

I have a question about the weak formulation of a PDE in finite element analysis. Suppose we have the following two-dimensional PDE: $$ \Delta \cdot u(x,y) = q(x,y) $$ where $q$ is given, $u$ is ...
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27 views

Regularity of Dirichlet Eigenvalues on Lipschitz Domain

What kind of regularity do we generally have for weak solutions to the Dirichlet problem? $$(\Delta+\lambda)u=0 \textrm{ in }U$$ $$u=0 \textrm{ on }\partial U $$ where $U$ is a planar domain with ...
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1answer
11 views

Solving the reaction-diffusion equation for a single species

$$ \frac{\partial u}{\partial t} =k\Delta u+ru. $$ Where all of the bounds are $0$. Please help! Very new to PDE's and don't understand how to solve this. I know that I need to use separation of ...
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6 views

Modified Symplectic Euler

Simple harmonic motion: $y'= -z $, $z'= f(y)$ and the modified Symplectic Euler equation are $$y'=-z+\frac {1}{2} hf(y)$$ $$y'=f(y)+\frac {1}{2} hf_y z$$ deduce that the coresponding approximate ...
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1answer
39 views

How to determine where a non-linear PDE is elliptic, hyperbolic, or parabolic?

I'm trying to understand the classification of PDEs into the categories elliptic, hyperbolic, and parabolic. Frustratingly, most of the discussions I've found are "definition by examples.'' I think ...
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1answer
44 views

Neumann's problem necessary and sufficient condition (Evans PDE)

Let $U$ be connected. A function $u \in H^1(U)$ is a weak solution of Neumann's problem \begin{equation} (*)\qquad\left\{ \begin{array}{rl} -\Delta = f & \text{in } U \\ \frac{\partial ...
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43 views

Show Green's function solves Poisson's equation

I am reading Walter Strauss's book Introduction to PDE. The definition of Green's function is as follows: The Green's function $G(x)$ for the operator $-\Delta$ and the domain $D \subset ...
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1answer
28 views

Implicit Euler Scheme and stability

Find the fixed points of the implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1}) \end{equation} when applied to the differential equation $y'=y(1-y)$ and investigate their ...
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20 views

A predictor-corrector method

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} ...
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13 views

Exist a solution of partial functional equation?

Let $\alpha\in \mathbb{R}, f$ is a function in $L^2(0,1)\longrightarrow \mathbb{R}$ and consider the equation (find $u$) $$\begin{cases} -u''+ \alpha u = f \quad 0<x<1\\ u(0) = 0, u'(1) = 0\\ ...
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1answer
19 views

Chain or product rule for heat diffusion equation

A portion of the heat diffusion equation for a 1-D solid is given as: $$\frac{1}{r} \frac{\partial}{\partial r} \left(r \; k \frac{\partial T}{\partial r} \right)$$ Apparently this can be expanded ...
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1answer
23 views

$1^{st}$ order PDE in population system

Here is the age-structured continuous population partial differential equation: \begin{equation} \left\{ \begin{array}{lcl} \frac{\partial p(a,t)}{\partial a}+\frac{\partial p(a,t)}{\partial t} = ...
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30 views

Alternative proof of Heisenberg uncertainty principle - step 1

I'm student of physics having problem with pdes. I have rapidly decreasing function in $\mathbb{R}^d$ st $\int{|u|^2}dx=1$ and function $v=e^{i\langle\psi\rangle x}u(x+\langle x\rangle)$, where ...
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Semilinear Poisson Equation Using Direct Method of Calculus of Variations

The following problem comes from: http://people.physics.anu.edu.au/~gvn105/analyticMethPDE.pdf 12.9 Exercises 12.3: Let $\Omega$ be a bounded domain in the plane with smooth boundary. Let $f$ be ...
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1answer
23 views

PDE $ u_{x}+u_{t}+f(x)*u=0$

How would I solve this pde using characteristic line? $u_{x}+u_{t}+f(x)u=0$---arbitrary function f $u(x,0)=u_{0}(x)$---$u_{0}$ can be any value $u(0,t)=\varphi(t)$---non-homogeneous where $u(x,t)\ge ...
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1answer
26 views

Checking a solution of a PDE

I have the following PDE: \begin{equation} -yu_x + xu_y = 0 \quad\text{where } u(0, y) = f(y) \end{equation} I derived a solution as follows: \begin{align} -yu_x + xu_y =& 0 \\ \iff& ...
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Determining the expressions of the coefficients of the full Fourier series from the Complex Series

Let $l \gt 0$ be a positive real number, and $\phi:[-l,l]\rightarrow\mathbf{R}$ be the function defined by: $$\forall x\epsilon[-l,l], \phi(x)=e^x $$ (1) Calculate the coefficients of the Full ...
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1answer
26 views

Burger's Equation Shock Solutions

So what I'm confused about is how you go about finding shock waves. So suppose we are given the Cauchy problem for Burger's equation $u_t + uu_x = 0$ with $u(x, 0) = 1$ for $x \le 0$ and $u(x, 0) = ...
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1answer
15 views

Characteristics PDE with discontinuity

So there is an example problem in the textbook and I really just don't understand what's going on, both mathematically and conceptually. The problem is solve $z_x + 2zz_y = 1$ with boundary ...
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14 views

Symmetry results of solutions for elliptic equations

In the celebrated paper of Gidas, Ni, and Nirenberg, see http://web.math.unifi.it/users/magnanin/Dott/BibliografiaCorso/GidasNiNirenberg79.pdf ,certain symmetry results of positive solutions of ...
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12 views

$C^\infty$ approximations of $f(r) = |r|^{m-1}r$

Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$. Is it possible to find $C^\infty$ functions $f_n$, such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$, $f_n' \to f'$ uniformly on ...
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Fourier transform for pde

I solved the following PDE: $u_{t}-(u_{t})_{x}+(b+1)uu_{x}=bu_{x}u_{xx}+uu_{xxx}$ numerically, using Fourier Transform method. For this i wrote it in the following way: ...
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Establishing a certain bound

If I know the bound of the certain derivatives: $$|\dfrac{\partial^k}{\partial x^{k_1}_1\cdots \partial x^{k_n}_n}V(x)|\leq M\epsilon^{-k} \exp(-m \epsilon^{-1}g(x)),$$ where $k=k_1+\cdots+k_n$ and ...
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1answer
39 views

Deriving the PDE for basket option

The payoff for basket option is max($w_1S_1+w_2S_2 -k,0)$. Using Ito's formula, I need to derive the PDE, where $dS_1 = rS_1dt + \sigma_1 S_1dW_1$ $dS_2 = rS_2dt + \sigma_2 S_2dW_2$ I need some ...
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1answer
31 views

Let $a,b,c \,$ be continuous functions defined on $\Bbb R^2$.

I am stuck on the following problem: Let $a,b,c \,$ be continuous functions defined on $\Bbb R^2$.Let $V_1,V_2,V_3$ be non-empty subsets of $\Bbb R^2$ such that $V_1 \cup V_2 \cup V_3 =\Bbb R^2 $ ...
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1answer
38 views

Combining two partial differential equations into one

I have the equation $$ \frac {\partial v}{\partial t}= \gamma \left(1-\frac{v}{v_0}\right)+\alpha \left(1-\frac{v}{v_0}\right)\rho-\beta (\rho-\rho_0) $$ and the mass conservation equation $$ ...
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29 views

Laplacian eigenvalue problem

I'm working through a PDE problem and we are given the eigenvalue problem $-\Delta u = \lambda u$ with $\frac{\partial u}{\partial n} = 0$ along the boundary given by the rectangle $\Omega = (0, \pi)$ ...
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1answer
38 views

Non-linear Partial Differential Equations

So I know how to solve linear partial differential equations but I am stuck on this new type of problem that is a nonlinear pde. The question is: Determine the solution of $\frac{\partial ...
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Positivity of principal eigenvalue for $L\phi=-\triangle \phi + \nabla \cdot( u \phi )$

EDIT: This question is still unresolved as of April 17. The two answers provide useful work in the right direction, but neither resolves the question. A counterexample should have $u \in ...
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1answer
15 views

Separation of variables for a non homogeneous PDE $u_t-ku_{xx} = f(x,t),\quad u(0,t)=u(L,t)=0,\quad u(x,0)=\phi(x)$

Separation of variables for a non homogeneous PDE. I found this problem on this page http://www.math.psu.edu/wysocki/M412/Notes412_10.pdf Consider the problem on $(x,t) \in (0,L)\times (0,\infty)$ ...
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1answer
24 views

Proof of the Unsöld's Theorem (the sum of spherical harmonics)

There is an identity concerning spherical harmonics that plays a pretty important role in atomic physics. Thanks to wikipedia (http://en.wikipedia.org/wiki/Spherical_harmonic) I know that its name is ...
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1answer
33 views
+50

Solving the LaPlace PDE by complete induction

I have a problem with an induction step that appears to be bothersome for me to comprehend it. Problem: Show that $f(x_1, \dots ,x_n)=(x_1^2+ \dots + x_n^2)^{1- n/2}$ for $(x_1^2 + \dots + ...