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

learn more… | top users | synonyms (2)

0
votes
0answers
21 views

Is it possible to solve this set of equations?

Let's have system of equations: $$ \tag 1 [\nabla \times \mathbf E ] = -\frac{\partial \mathbf B}{\partial t} , $$ $$ \tag 2 [\nabla \times \mathbf B] = \sigma \mathbf E + A(\mu \mathbf K + C \mathbf ...
0
votes
0answers
4 views

What are difference among natural boundary, exit boundary, regular boundary and killing boundary??

In the paper i'm reading, they used the terminologies, natural boundary, exit boundary, regular boundary and killing boundary. I can't find the difference of them and definition of them. Tell me ...
0
votes
0answers
7 views

Numerical Overflow in Dirichlet Boundary Value Problem On High Dimensional State

I am using multigrid methods to solve a quasilinear parabolic pde with Dirichlet boundaries. The problem is too long to reproduce here, but my question is more practical than theoretical: The state ...
2
votes
0answers
23 views

A particular case of Gelfand triple

I am currently working on hyperbolic equations on bounded domains. For this reason, I am considering in particular functions $u$ such that: $u \in L^2(0;T;H^1_0(\Omega)), \quad u' \in ...
0
votes
0answers
15 views

A Free Boundary Problem

Is there any special way to solve such a problem. Any idea would be appreciated. At least does anybody know which method is useful to solve this problem numerically? Is it even solvable numerically? ...
1
vote
1answer
17 views

Connections between SDE and PDE

I have encountered a number of situations where the solution of a PDE and a certain expectation associated to a Markov process are equal. Two examples include: The heat equation $u_t = \Delta u$ ...
0
votes
1answer
19 views

Solving the integral which shows the second moment of subtracting two Beta-distributed Random Variables

Peace be upon you In my project I needed to find the second moment of the subtraction of two Beta-distributed random variables. I have computed it and reached to the following integral which I should ...
0
votes
0answers
10 views

Estimates for the wave equation

Spose $ u $ solves the wave equation on $ U \subset \mathbb{R}^3 $ with initial conditions $ u (x, 0) = g(x)$ and $ u_t(x,0) = h(x)$, where lower script indicates partial differentiation. Then we have ...
0
votes
1answer
18 views

Reference request: functional analysis results used in Taubes paper (1980)

I'm studying Taubes paper 'Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations'. I'm looking for a reference of three following theorems: Let $f(x)$ be a convex funtional ...
0
votes
0answers
13 views

find coefficient-functions s.t. PDE has no global solution

Consider the PDE \begin{align*} a_1(x)u_{x_1} + a_2(x)u_{x_2} & = 0 \quad \mathrm{in}\ \mathbb R^2 \\ u & = u_0 \quad\mathrm{on}\ \Gamma \end{align*} with $\Gamma=\{(x_1,0)|x_1\in\mathbb ...
0
votes
0answers
27 views

How to solve parabolic equation via implicit Euler in 2 dimensions?

I have the following parabolic equation: $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial ^2u}{\partial y^2} $$ over domain $(x,y)\in [0,10] \times [0,10]$ ...
0
votes
1answer
24 views

Laplace equation - PDE

For the PDE:$$u_{xx}+u_{yy}=0$$ $$u(0,y)=\sin\pi y, \ u(1,y)=0$$ $$u(x,0)=u(x,1)=0.$$ I have that $u(x,y)=X(x)Y(y)$, then $$-Y(y)=\mu Y(y), \ Y(0)=Y(1)=0$$ $$X''(x)=\mu X(x), \ X(1)=0.$$ Thus, ...
1
vote
1answer
27 views

How to deal with $u_{ttt}$ in derivatives estimates of $u_{tt}$ if $u_{ttt}$ is not defined?

Suppose we have proved the equation $$u_{tt}-u_{xx}=0\quad\text{in}\quad(0,T)\times(0,\ell)$$ with some boundary and initial conditions has an unique solution $u\in C^2(0,T,H^2(0,\ell))$ and we need a ...
0
votes
0answers
10 views

Unbounded derivative conservation law

Following the result in Implicit function theorem for conservation law, how does the derivative become unbounded if $u_0'f''(u) >= 0$ ?
0
votes
0answers
14 views

Partial differential equation question (using chain rule?)

How do I do this? Find how the equation $\dfrac{\partial^2 w}{\partial x \partial y} +w=0$ will be transformed if we look for the particular solution w=f(u) where $u=(x-x_0)(y-y_0)$. I got $uf_{uu} ...
1
vote
0answers
11 views

How to interpret dual space convergence in norm $L^2(0,T;V^*)$ (sobolev--bochner spaces)

Let $V \subset H \subset V^*$ be a Gelfand triple. Define $W = \{ u \in L^2(0,T;V) \mid u' \in L^2(0,T;V^*)$. A result is that $C^\infty([0,T];V)$ is dense in $W$. So given $w \in W$, there exists ...
1
vote
1answer
32 views

Analytic solution for a type of PDE systems

Peace be upon you, I have the following system of partial differential equations \begin{align*} \begin{cases} \frac{\partial}{\partial a}S(a,b,c,d)=f_1(a)\\ \frac{\partial}{\partial ...
2
votes
0answers
22 views

How can two PDE's have the same classical solution, but different weak solutions?

For example: The implicit solution for the inviscid Burgers' Equation, in two forms: $u_y+(\frac{1}{2} u^2)_x = 0$ and $(u^2)_y+(\frac{2}{3} u^3)_x = 0$ share the same smooth solutions, but they ...
1
vote
0answers
52 views

Derivatives of Differential operators

Let $V=H^1(\Omega)$ and $U = L^{\infty}(\Omega)^2$. Take for example, a differential operator, $L:U\times V \rightarrow V^* $ such that: \begin{equation} L(u,y) = \Delta y + \vec{u}(x)\nabla y ...
2
votes
1answer
22 views

How to identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$? (measurability of function)

How can we identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$? For my understanding: $Q_T:=(0, T)\times \Omega$; DEF1: $L^2(Q_T)=\{u: (0, T)\times \Omega \to \mathbb{R}, \mbox{measurable and} ...
1
vote
0answers
20 views

Inviscid Burger's equation solution

I managed to understand the method of characteristics to get the solution of the transport equation. However, I am getting stucked finding the final solution for the Burger's equation given the ...
1
vote
1answer
36 views

My attempt to prove an inequality get stuck——————where do I go wrong?

Hi, there. Bellow is my attempt. I don't know if I have gone in the wrong way and I am stuck. My attempt: Using Green's representation formula, $u(y)=\int_{\partial \Omega}u \frac{\partial ...
1
vote
0answers
23 views

Fundamental solution of Laplacian on manifold

I'm looking for a reference for the result that there exists a fundamental solution for the Laplacian on a flat torus $$\Delta \Gamma(x-y) = \delta(x-y), \quad x,y \in \mathbb T^2.$$ and that, ...
1
vote
1answer
31 views

Fundamental solution of heat equation on a compact Riemannian manifold

Let $(M,g)$ be a compact Riemannian manifold of $m$ dimensional. Then there exists a sequence $(\phi_i, \lambda_i)_{i\in\mathbb{N}}\subset C^\infty(M)\times\mathbb{R}_{\geq0}$ such that ...
1
vote
0answers
24 views

How to solve fourth order pde similar to biharmonic equation.

I'm trying to solve a fourth order pde similar to the biharmonic equation $ 0=\frac{\partial ^4}{\partial x^4}u(x,y)+Q\frac{\partial ^4}{\partial x^2 \partial y^2}u(x,y)+\frac{\partial ^4}{\partial ...
2
votes
1answer
26 views

Weak form of $-\frac{d}{dx} \left[(1+x) \frac{du}{dx}\right]$

I am doing some self studying on J.N. Reddy's book, An Introduction to the finite element method, 1st edition. (This was the only one available at the local library). In exercise 2.12, one reads ...
0
votes
0answers
19 views

Crank-Nicolson method for solving nonlinear parabolic PDEs

Is the Crank-Nicolson method appropriate for solving a system of nonlinear parabolic PDEs like $\partial u/\partial t - a\Delta u + u^4 = 0$ ? I tried to apply this method for solving such system but ...
1
vote
0answers
36 views

Question regarding the dual space of $H_0^1(\Omega)$

Given $\Omega\in R^N$ open bounded with smooth boundary. We define $H^{-1}$ to be the dual space of $H_0^1(\Omega)$ and from Evan's PDE book, chapter $5$, we know that for any $f\in H^{-1}$, there ...
0
votes
0answers
36 views

Prove $U$ is subharmonic?

My attempt: Integration by parts says $\int u \triangle \varphi=\int\triangle u \varphi$. We know the left hand side is always $\ge 0$, and hence $\int\triangle u \varphi \ge 0$, since $\varphi \ge ...
1
vote
0answers
12 views

Existence of fundamental solutions of Heston's PDE

The Heston's PDE is a variable coefficient linear PDE. I know the existence of fundamental solution of constant coefficient linear PDE has been proofed by Bernard Malgrange and Leon Ehrenpreis in 1954 ...
1
vote
1answer
26 views

Define a “rotation of u” by $R_{A}u\doteq u\circ A $ with A an orthogonal $n\times n$ matrix and “$\circ$” means composition.

Show that $\Delta (u\circ A )=(\Delta u)\circ A$. (Note: $u(x)\in C^{m\geqslant 2} (\mathbb{R}^{n}) $) I tried to treat u as merely a number in $\mathbb{R}^{n}$ and then take the Laplacian to each ...
1
vote
0answers
5 views

Where to learn perturbation theory for pde (in introductory level)? [Reference Request]

Recently I've been reading the text by Falow 'PDEs for Scientist and Enginieers'. In the latter sections is contained 'Perturbation method'. This one gives only kind of computational techniques; no ...
2
votes
1answer
31 views

Help solving $\frac{\partial^3u}{\partial x^3}=\frac{\partial u}{\partial t}$ using Fourier transforms

I am trying to solve $$\frac{\partial^3u}{\partial x^3}=\frac{\partial u}{\partial t}$$ $x\in\mathbb{R},\:t >0$. Subject to the conditions $u(x,0)=f(x),\:u,\:\frac{\partial u}{\partial ...
1
vote
1answer
22 views

Regularity of semilinear PDE

I'm reading Evans' PDE book (second edition) and I tried to solve this problem but I'm a little confused Problem 7, chapter 6: Let $u\in H^1(\mathbb{R}^n)$ have compact support and be a weak ...
0
votes
0answers
15 views

Speed of Fisher Kolmogorov Wave equation in Matlab [on hold]

I have a working code for the Fisher Kolmogorov equation. Now I want to find the speed at which the wave is propagating. And that calculation would be : (x coordinate of U at time t+ delta t) - x ...
0
votes
0answers
23 views

A finite vibrating string satisfies the following initial value problem, find a solution using separation of variables

We have $u_{xx} = u_{tt}$ on $0<x<L, t\geq 0$ with boundary conditions $u(0, t) = 0$ and $u(L, t)=0$ The string is released from rest with initial displacement $u(x,0) = \left\{\begin{matrix} ...
0
votes
0answers
5 views

Exercise needed for weak solution of elliptic equations

I'm trying to find more exercise for weak theorem for 2nd order elliptic PDEs, like the exercise in chapter 6 on Evans's PDE book. Any suggestions other than Evans or Gilbarg-Trudinger? Thx!
0
votes
0answers
28 views

space of solutions of a PDE

so I have just completed part (c) and I'm now on part (d). To fill you in, I have found that $K = -\pi^2 (n^2+m^2)$ for some $n,m \in \mathbb{Z}$ and $p = n\pi, q = m\pi$ now I don't really ...
0
votes
1answer
21 views

Asymptotic bounds for the solutions of 3d wave equation

Let $u$ solve the 3-d wave equation: $u_{tt}-\Delta u =0$ such that $u=g$ and $u_t=h$ for $t=0$ and where $g$ and $h$ are both assumed to be compactly supported and smooth. I have shown that there ...
0
votes
0answers
7 views

Accessible resources to learn about bicharacteristic strips

I'm taking an introductory course in PDEs and, once seen the method of characteristics, the professor briefly talked about bicharacteristic strips and micro-local analysis. I'd hate to pass by such a ...
2
votes
0answers
17 views

Newtonian potential for ellipsoid

Is there an explicit expression of the Newtonian potential for ellipsoid? As the expression for ball is clear by its symmetry. Definition of Newtonian potential of ellipsoid $\Omega$ at x is defined ...
0
votes
0answers
17 views

A Direct Proof of Representation Theorem for Positive Harmonic Functions in the Half Plane?

Does anyone know a direct proof of this representation theorem for non-negative harmonic functions in the half-plane that doesn't appeal to a similar result in the unit disk? Also, does anyone who ...
0
votes
1answer
26 views

Strange thing about Weak Maximum\Minimum Principle?

I feel confused about this problem. I think it is obvious using Weak Maximum\Minimum principles. Since for harmonic functions. If $\Omega$ is bounded and $u\in C^2(\Omega) \cap C^0(\overline ...
0
votes
0answers
14 views

Coupled second order partial

I have a set of coupled differential equations $\frac{\partial f}{\partial t} + a_1 \frac{\partial^2 f}{\partial x^2} = -b f$ $\frac{\partial g}{\partial t} + a_2 \frac{\partial^2 g}{\partial x^2} = ...
0
votes
0answers
31 views

Proof using convolution?

there. I am a novice in graduate school. This is the first time I learn PDE in graduate level. I found it so hard. I am going to have a test next week and I am so worried about it. Since I always ...
0
votes
1answer
20 views

Solving the PDE $\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial ^2 \phi}{\partial y^2} = K \phi$ using the separation of variables

I'm trying to solve $\dfrac{\partial^2 \phi}{\partial x^2} + \dfrac{\partial ^2 \phi}{\partial y^2} = K \phi$ with $K$ constant I let $\phi = XY$ then got $X''/X + Y''/Y = K$ but I'm not sure where ...
0
votes
0answers
12 views

finding ODEs satisfied by X and Y for a PDE

for the following PDE separate the variables using $\phi(x,y) = X(x)Y(y)$ and find the ODEs satisfied by $X $ and $Y$ PDE: $$ \dfrac{\partial ^2 \phi}{\partial x^2} = \dfrac{\partial^2 \phi}{\partial ...
0
votes
1answer
8 views

Maximal principle: proof for subsolution

I have recently gone through the statement that a subsolution $v$ satisfies the maximal principle: $\sup_{\Omega T} =\sup_{\partial\Omega T}$. So if $u(x,t)>0$ is a supersolution, then how can we ...
0
votes
0answers
18 views

Inverse Fourier Transform of $\cos(c\omega t)$ and $\sin(c\omega t)$

I'm just needing a bit of help to understand the derivation of the inverse fourier transform of $\cos(c\omega t)$ and $\sin(c\omega t)$, in deriving D'Alembert's solution to the wave equation. I get ...
3
votes
1answer
37 views

Solving Laplace's equation

Given Laplace's equation $u_{xx} + u_{yy} = 0$ and 2 boundary conditions $u(x,y)=x^2y$ $u(cos\theta,sin\theta) = 1 + cos\theta$ in the unit disc. I want to solve laplace equation. After using ...