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

learn more… | top users | synonyms (2)

0
votes
0answers
19 views

Solving the Wave Equation using Fourier Transforms

The problem is: \begin{equation} u_{tt} -c^2u_{xx} - a^2 u = 0 \end{equation} with $\hspace{2mm}-\infty < x < \infty $, $ \hspace{2mm} u(x,t) \hspace{2mm}$ bounded as $ x \rightarrow \pm ...
0
votes
1answer
28 views

If the integrals of a harmonic function over horizontal lines are uniformly bounded, it is identically zero

Let $u\colon\mathbb{R}^2\rightarrow\mathbb{R}$ be a harmonic function, such that $$\int\limits_{-\infty}^{+\infty} \lvert u(x,y)\rvert dx < C,$$ where $C>0$ is a constant not depending on ...
0
votes
0answers
15 views

Heat Equation in spherical coordinates

Consider the problem of a sphere of material that starts at a non-uniform temperature, $T = r^{2}$ and is covered with insulation on the outer surface so that no heat gets out. We take the coordinate ...
0
votes
1answer
19 views

Is there any nonnegative $u\in C^2(\mathbb{R}^n)$ with $-\Delta u=1$ in $\mathbb{R}^n$?

Is there any nonnegative $u\in C^2(\mathbb{R}^n)$ with $\Delta u=-1$ in $\mathbb{R}^n$? I think not, but how can we prove it? Let's assume that such a solution exists. Let $R>0$ and $B_R:=B_R(0)$ ...
0
votes
0answers
6 views

Problem on Majda's Vorticity and Incompressible Flow

I'm reading Majda's book, on page 110, I cannot understand how to get $v\in C_W (0,T;H^{m})$ from the above. And he wrote "$[\phi,v^{\epsilon}]\to[\phi,v]$ uniformly on $[0,T]$ " two times, do the ...
1
vote
0answers
6 views

How to define boundary conditions for a sphere to run reaction-diffusion equations on its surface?

I'm in a Biology lab, and we managed to simulate reaction-diffusion equations on a torus using periodic boundary conditions for a 2D matrix. We want to try doing the same on a sphere, but I'm a ...
0
votes
0answers
9 views

Solutions of $u(x)=\int_{\mathbb R^n} |x-y|^p u(y)^{-q} dy$ are bounded away from zero

In one of research papers I am interested in, see this link or this if you cannot access, there is a lemma, Lemma 5.1, saying that if $n\geq 1$, $p,q>0$ and $u$ a non-negative Lebesgue measureable ...
0
votes
1answer
14 views

PDE Boundary conditions for characteristic solution for when x=0 and t is some constant

Solve $$\frac{dw}{dt}+4\frac{dw}{dx}=0$$ With initial condition $$w(0,t)=Sin(3t)$$ The characteristic equations are: $$ \frac{dt}{dt}=1, \frac{dx}{dt}=4, \frac{dw}{dt}=0$$ The characteristic ...
0
votes
0answers
19 views

Question on: gradient & laplace operator

given the function $f(x,y,) = x^2-y^2$. the gradient should be given by $grad f = (2x, -2y)$. If I'm drawing single of these vectors, I only get the ones on the positive x-axis. Is this correct? ...
0
votes
0answers
21 views

a question about prove an exponential matrix function can be infinitely differentiable

If I have an exponential matrix exp(t(U+sH)), can someone tell me what is the dirivative with respect to s? I am really confused. (where U and H are matrices,and s,t are real numbers). Thus,if I let ...
0
votes
1answer
34 views

Difficult integral $\frac{du}{u}=\left(\frac{x+y}{x}\right)dx$ in PDE

The linear problem is given as $$x\frac{\text{$\delta $u}\backslash }{\text{$\delta $x}}\text{+y}\frac{\text{$\delta $u}\backslash }{\text{$\delta $y}}\text{=(x+y)u}$$ with $u = 1$ on $x=1$ with ...
1
vote
1answer
16 views

Determine the Fourier transform of $f(x) &

f(x)=1 if |x| < a or f(x) = 0 if |x| > a We use the formula $$ {1\over 2\pi} \int_{\infty}^\infty f(\bar x)e^{i\omega \bar x} $$ So is $f(\bar x)$ the same as $f(x)$ ?? In an answer they ...
0
votes
0answers
9 views

What it means for a Jacobian determinant to be zero in the context of PDEs and their solution?

The book mentioned that if the Jacobian determinant is zero then no solution exists in the neighbourhood of the boundary curves. What does this means in simplified terms? What are boundary curves? I ...
1
vote
2answers
41 views

Calculating $a_n$ in $\sum_{n=1}^\infty a_n \sin(\frac{n \pi}{2})=T_0$

I'm looking to solve the following when $T_0$ is a constant: $$\sum_{n=1}^\infty a_n \sin\left(\frac{n \pi}{2}\right)=T_0$$ If it matters this was reached from the following: ...
-1
votes
0answers
19 views

Eliminate the arbitrary Function of PDE

I need to solve this problem; Eliminate the arbitrary function $f$ from the equation: $f(x^2+y^2+z^2,z^2-2xy)=0$ I try this solution $u= x^2+y^2+z^2, \quad $ $\quad v=z^2-2xy, \quad$ so ...
0
votes
1answer
22 views

Proof of uniqueness for the Poisson equation

Show that the following problem has at most one solution: Given a continuous function $\rho(x,y,z)$ which is zero for $x^2+y^2+z^2>a^2>0$, find $\phi$ such that $$\nabla^2\phi=\rho$$ ...
0
votes
1answer
21 views

What is vectors straddle a plane mean?

There is a condition in a paper, saying that two vectors straddle a plane. How can we transfer this condition to a equation? Because I have another 5 equations and need this one to solve 6 unknowns. ...
-2
votes
0answers
27 views

Initial and Boundary value problem [on hold]

Solve the following initial-boundary value problem: $$\frac{\partial u}{\partial t} = a^{2}\frac{\partial ^{2}u}{\partial x^{2}}-b(u-u_{0}), t>0, 0 <x<L, \\ u(0,t) = u(L,t) = u_{0}, t > ...
2
votes
1answer
26 views

Definition of 'blow up' in the context of PDEs

What exactly is 'blow up'? Is there a proper well-defined definition for this term? What does it means mathematically? Does it implies 'infinity'?
2
votes
3answers
40 views

solving a second order nonlinear pde

I would like to solve the following PDE, $$f_{y}^{2} = 2 f f_{yy}$$ where $f= f(x,y)$ is a real function of two variables $x,y$. My solution : derivative of $f_{y}^{2}$ with respect to $y$ is itself, ...
0
votes
0answers
15 views

Can someone check my work on the PDE five-point scheme problem?

I'm working on a practice exam for an upcoming final exam next week, but unfortunately the professor did not release solutions to the practice exam. I was hoping someone here could verify that I did ...
0
votes
1answer
13 views

Superlinearity in the definition of the Legendre transform

Suppose the Lagrangian $L:\Bbb{R}^n\to\Bbb{R}$ satisfies the following conditions: $L$ is convex $$ \lim_{|v|\to\infty}\frac{L(v)}{|v|}=+\infty $$ Define the Legendre transform of $L$ as $$ ...
0
votes
0answers
11 views

Laplace equation in polar coordinate

\begin{array}{*{20}{c}} {\Delta u = 0}\\ {u = V;r = b}\\ {u + \frac{{Va\sin n\theta }}{{r(\log b - \log a)}} = 0;r = a} \end{array} I want to solve the above Laplace's equation in polar coordinates at ...
3
votes
0answers
29 views

Estimating/approximating a very high dimensional unbounded poisson's equation

Consider the poisson equation on an unbounded domain. Suppose that the solution is known to exist. $$ \Delta u=f $$ I would like to estimate the solution of the this equation at a given point $x_0$. ...
1
vote
0answers
21 views

Solving $\nabla^2U(x,y)=0$ on a donut with two inhomogeneous boundary conditions

I am given $\nabla^2U(x,y)=0$ on a donut-shaped region, with the inner circle being of radius $r_1$, and the outer circle $r_2$. In polar coordinates, the relation is ...
1
vote
2answers
27 views

book for numerical methods for solving pde

I need to find some masters-level exercises about numerical methods for solving pde. Are there any good references?
0
votes
0answers
14 views

Free harmonic vibrations of the Euler-Bernoulli equation

The Euler-Bernoulli equation describes the relation between external forces and deflections of a beam. The general formula is given by: $$ \frac {\partial ^2}{\partial x^2} \left(EI\frac{\partial ...
1
vote
1answer
26 views

Long time behavior heat equation on infinite line

We know that a solution to the Cauchy problem on $\mathbb{R}$ : $u_{xx}=u_t$ with condition $u|_{t=0}=\varphi(x)$ is of the form $$u(x,t)=\dfrac{1}{2\sqrt{\pi ...
-3
votes
0answers
22 views

General solution change of variables [on hold]

How do I show that $$\text{F} \left( \frac{x}{y},\frac{x}{u}\right)=0$$ is equal to $$u=xG\left(\frac{x}{y}\right)$$
0
votes
1answer
7 views

Curve that lies on a solution surface

Suppose the solution surface is given as $$f(x,y,u)=0$$. A curve $$C$$ lies on the solution surface. What does this means?
0
votes
1answer
18 views

Why is $\Delta u$ bounded, if $u\in C^2(\overline{\Omega})$ and $\Omega\subseteq\mathbb{R}^n$ is a bounded domain?

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain and $u\in C^2(\overline{\Omega})$. Why must $\Delta u$ be bounded?
2
votes
1answer
34 views

Large and small time PDE solution

I have the following solution for a PDE $$ u(x,t)=(2x+4t-10)+2e^{\frac{-1}{2}t}+\sum_{n=1}^{\infty} \frac{(-1)^n cos(\frac{n\pi}{2}x)e^{\frac{-1}{4}n^2\pi^2t}}{n^3\pi^3(n^2\pi^2-2)} $$ I want to ...
0
votes
1answer
21 views

How is an ODE a consistency condition?

I was reading a text on Optimal Control Theory by E. Todorov, when I came accross this passage (on page 10): An ODE is a consistency condition which singles out specific trajectories without ...
1
vote
2answers
37 views

Solving simultaneous PDEs

Given the equations (1):$$\frac{\partial u}{\partial t}+g\frac{\partial \eta}{\partial x}=0$$ and (2):$$\frac{\partial\eta}{\partial t}+H\frac{\partial u}{\partial x}=0$$ can we combine the two ...
0
votes
0answers
19 views

Using Feynamn-Kac to solve expectation

I want to solve an expectation $u(t,x)=\mathbb{E}[exp(\beta X_T)|\mathcal{F}_t]$ related to a random variable $X$ which statisfied the CIR process $$dX_s=k(\alpha-X_s)ds+\sigma\sqrt{X_s}dW_s$$, which ...
0
votes
1answer
46 views

Convergence of $\partial_{x_j} u(x,t)$ when $u$ converges in $L^2$ norm.

I hope you can help me with this question. We take $u(x,t)\in L^\infty_{loc}(\mathbb{R},H^1(M))\cap Lip_{loc}(\mathbb{R},L^2(M))$, the derivatives $\partial_{x_j} u $ exist and are continuous, i.e ...
0
votes
0answers
39 views

Diffrental equation solution [on hold]

How can I solve this equation? $$\frac{\partial f}{\partial x} =\frac{a-x}{y} \frac{\partial f}{\partial y}$$ where $a$ is a constant. So what is $f$?
2
votes
1answer
39 views

Find a solution that satisfies Laplace's equation in polar coordinates

How may I find a solution that solves Laplace's equation in polar coordinates, subject to the boundary conditions? In particular, I need to find one solution that satisfies $$\Delta u = 0,$$ subject ...
0
votes
0answers
16 views

General solution of boundary value problem

I have to find the general solution of the following boundary value problem with the use of Fourier method. $$u_t(x,t)-u_{xxt}(x,t)-u_{xx}(x,t)=0, 0<x< \pi, t>0\\u(0,t=0),t>0$$ ...
0
votes
0answers
12 views

Prove that there is at most one solution with Green's identity

Prove with the use of Green's identity that the boundary value problem $$\frac{\partial}{\partial{x}} \left( (1+x^2) ...
0
votes
1answer
29 views

Why do these equalities stand ?

In my notes there is the following theorem: Let $X_k : [a,b] \rightarrow \mathbb{R}$, $k=1, \dots , n$ an orthogonal system of functions and $X: [a,b] \rightarrow \mathbb{R}$, then $\forall c_1, ...
0
votes
0answers
19 views

Numeric Evaluation of Double Surface Integral over Greens Function with Singular Points

I'm currently using python to numerically evaluate the follow expression ...
1
vote
0answers
29 views

Why is the problem in polar coordinates in that form ?

We have the initial and boundary value problem $$u_{xx}(x,y)+u_{yy}(x,y)=0 , x^2+y^2<1 \\ u(x,y)=0 \\ u(1, \theta)=\sin{\theta}, 0< \theta< \pi$$ $$U_{\rho \rho}(\rho, \theta)+ ...
1
vote
1answer
23 views

How have we found the conditions of the problem from the graph?

In my notes there is the following : $$u_{xx}(x,y)+u_{yy}(x,y)=0$$ $$u(x,0)=f(x), 0 \leq x \leq l \\ u(0,y)=0, u(x,\pi)=0 \\ u(l,y)=0$$ How have we found these conditions from the graph?? ...
4
votes
1answer
60 views

Why are the eigenfunctions linear independent?

At a Sturm-Liouville problem how do we know that the two eigenfunctions that we have found are linear independent?? For example we have the following problem : $$X''+\lambda X=0 \\ X(0)=X(2\pi) \\ ...
0
votes
2answers
20 views

PDE Solving: Difference between Similarity Solution and Characteristics?

As far as I understand, both the method of characteristics and similarity solutions allow us to reduce certain partial differential equations to ordinary differential equations which can then be ...
0
votes
0answers
44 views

A question in PDE/Sobolev spaces [on hold]

I'll be grateful if you help me or give me hints to answer the following: Suppose $\Omega$ is a bounded domain in $\mathbb R^N$ with $C^1$ boundary $f\in L^2(\Omega)$. Prove that for every ...
0
votes
1answer
27 views

Solution of the Laplace equation in polar coordinate.

Solve the following PDE: $$\phi(r,\theta) = \begin{cases} \Delta \phi=0 & \quad \text{for $a \le r\le b$ }\\[8pt] \phi=V & \quad \text{for $r=b$} \\[8pt] \phi+ C \sin(n\theta)=0 & ...
1
vote
1answer
24 views

Boundedness of a sequence in $L^\infty(I,H^1(M))\cap\mbox{Lip}(I,L^2(M))$ implies that its temporal derivative is bounded as well

I asked my question in mathoverflow, but it seems to be inappropriate there, so I try my luck here. ...
0
votes
0answers
27 views

Can't replicate solution to a non-linear PDE ($- \square \varphi + \lambda \varphi^3 = 0$)

I am trying to replicate the solution of this paper : http://arxiv.org/pdf/0807.2179.pdf Which is, roughly, for the quartic scalar field theory, $- \square \varphi + \lambda \varphi^3 = 0$ a set ...