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

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2
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0answers
14 views

Analytic solution to Poisson equation

I need to find the analytic solution to this equation, in order to compare it with solution I get from using a numerical solution. However, I have not been able to find the solution. I think I can't ...
2
votes
1answer
10 views

About fractional Sobolev space

I'm reading a paper used Sobolev space $H^s(\Omega)$ , I only know the definition of these space when $\Omega=R^n$ which used Fourier Transform, what if $\Omega$ is a bounded open set? And I also ...
1
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0answers
15 views

Expansion wave in PDEs

The book I'm using is "Applied partial differential equations" written by Richard Haberman. On page 556, he gives the definition of expansion waves as "The distance between $$p(_{x_{1}},0)$$ and ...
0
votes
1answer
33 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 ...
2
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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
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0answers
20 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
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0answers
16 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 ...
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0answers
19 views
0
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1answer
22 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
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0answers
9 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 ...
0
votes
3answers
2k views

How to solve the non-homogeneous PDEs: $u_x + u_y=2u,\ u(x,0)=h(x)$?

I have this first-order non-homogeneous partial differential equation with initial condition: $u_x + u_y=2u,\ u(x,0)=h(x)$ The following was what I tried: By the method of characteristic curves, we ...
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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
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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
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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 ...
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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
1answer
35 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
2answers
45 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: ...
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0answers
22 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 ...
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 ...
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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 ...
0
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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. ...
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 $$\nabla^2\phi=\rho$$ ...
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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 ...
1
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2answers
63 views

Eliminate the arbitrary funcion - PDE first order

I'm heading the book Elements Of Partial Differential Equations -Sneddon 1957. At chapter two exists this exercise "Eliminate the arbitrary function $f$ fron the equation $$ z= ...
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
1answer
19 views

Probability density function for a PDE with random inputs

I am looking for a general method or alternatively few textbook examples of deriving a probability density function for a solution of partial differential equation with random inputs in the equation ...
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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'?
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43 views
+50

Energy inequalities with negative sobolev number.

Let $\phi\in H^{s}$ such that the following energy inequality is true: $$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\cdot)\|_s \, dt $$ where $P$ is a strictly hyperbolic linear operator. For ...
1
vote
3answers
39 views

Solutions of the Laplace equation

How do I find solutions $u=f(r)$ of the two-dimensional Laplace equation $u_{xx}+u_{yy}=0$ that depend only on the radial coordinate $r= \sqrt{x^2+y^2}$
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0answers
16 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 $$ ...
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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
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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
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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?
1
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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
2answers
398 views

Heat equation and semigroup theory.

Theorem: Let $X$ be a Banach space, $\{T(t)\}_{t\geq 0}$ a $C_0$-semigroup on $X$ and $U_0\in X$. If $A:D(A)\subset X\to X$ is the infinitesimal generator of $\{T(t)\}_{t\geq0}$, then the function ...
5
votes
1answer
75 views
+50

Nonlinear heat equation $u_{t} = \Delta(u^{4})$

Consider the nonlinear heat equation $u_{t} = \Delta(u^{4})$ in $\{x \in \mathbb{R}^{3}: |x| < 1\}$ with $u = 0$ on $\{x \in \mathbb{R}^{3}: |x| = 1\}$. The problem I am working on is to show that ...
6
votes
1answer
151 views

Does $\operatorname{div}\left(\nabla G +xG\right)=0\Longleftrightarrow \nabla G +xG=0$?

Let $G$ be a function of $\mathcal{C}^2(\textbf{R}^d$;$\textbf{R}^*_+)$ such that $G \in \operatorname{L}^1(\textbf{R}^d)$. I read on the Internet that one has the following equivalence ...
3
votes
2answers
307 views

Does a closed form solution to 1-D heat diffusion equation with Neumann and convective Boundary conditions exist?

this problem has been eluding me and I've begun to wonder if a solution exists, and hoping I've simply overlooked something. Using the normalized diffusion equation I have $$ {\partial{\rm ...
0
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1answer
47 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
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 ...
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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 ...
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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 $$u=xG\left(\frac{x}{y}\right)$$
1
vote
2answers
38 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 ...
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 ...
3
votes
2answers
31 views

Find a harmonic function in the cylindrical shell between $r=a$ and $r=b$

Calculate $\phi$, satisfying $\nabla^2 \phi=0$ between the two cylinders $r=a$, on which $\phi=0$, and $r=b>a$, on which $\phi=V$. I calculate it and found the solution is ...
0
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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?
2
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1answer
54 views
+150

Why do we take the odd extension?

When we have the initial and boundary value problem $$u_{tt}(x,t)-c^2u_{xx}(x,t)=0, x>0, t>0 \\ u(0,t)=0 \\ u(x,0)=f(x), x \geq 0 \\ u_t(x,0)=g(x), x \geq 0$$ can we apply Green's theorem or ...