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

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If solution of NLS is smooth and decaying at infinity; how to justify it satisfy the conservation law?

We consider the cubic nonlinear Shr\"odinger equation(NLS) $iu_{t}+\frac{1}{2} \Delta u = |u|^{2}u, \ u(x, 0)= u_0(x), (x\in \mathbb R^{d}, t\in \mathbb R)$ I have been trying to understand the ...
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9 views

weak solution to one dimension conservation law

Suppose $u:\Bbb{R}\times[0,\infty)\to\Bbb{R}$ is a continuous function such that for all $v\in C_c^\infty(\Bbb{R}\times[0,\infty))$ $$ \int^{\infty}_0 \int^{+\infty}_{-\infty} \Big(u(x,t) ...
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16 views

For which $(x_1,x_2)$ is this a solution to the minimal surface equation?

Let $u(x_1,x_2):=arcosh(\sqrt{x_1^2+x^2})$ then I want to find out for which $(x_1,x_2)$ this is a solution to the minimal surface equation in two dimensions that you can find for example here. ...
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21 views

Compute the solutions of the following equation in Fourier space:

$$\frac{d^3u}{dx^3} − αxu = 0, x ∈ R, $$ where $ α > 0$ is some constant and $u(x)$ is assumed to satisfy $\int_R u(x) dx = π.$ I know this is a ODE so this is what I came up with so far: ...
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1answer
20 views

Existence of Solutions to PDEs - How do I know I've got them all?

I'm taking a very computational course in partial differential equations. Because of this emphasis, I'm feeling very underwhelmed by the course, and have a lot of questions that really aren't ...
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2answers
47 views

Need a textbook for math course

The undergrad course is called intro the applied math, and it covers: "The unit introduces some of the principal mathematical techniques such as difference equations, differential equations and ...
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20 views

Explicit numerical method for solving second order PDE

I'm interested in solving PDEs of the following form $\frac{\partial^2}{\partial t^2} G(t,t^{\prime}) = f\left(t,t^{\prime},\frac{\partial}{\partial t} G(t,t^{\prime}), G(t,t^{\prime})\right), \qquad ...
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12 views

Heat Equation: Remains finite?

I am doing a PDE question. It's about heat equation, spherical coordinates (the usual stuff). The boundary condition is $\frac {\partial T}{\partial r} (1,t) = 0 $ and it also said for $T$ to remain ...
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16 views

Homogeneous Dilation of the Domain in the Free Membrane Problem

Consider the Neumann boundary value problem of the Laplace operator: $$ \begin{cases} \Delta u+\mu u=0,&\text{in }D,\\ \frac{\partial u}{\partial n}=0,&\text{on }\partial D. \end{cases} $$ Let ...
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1answer
22 views

Heat kernel properties

I'm having problem with the heat equation in $\mathbb{R}^n$; specifically in proving the following: let $f\in L^1(\mathbb{R}^n)$ and: $$ u(x,t)=H_{\sqrt{t}}\star f(x)=(4\pi ...
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19 views

Solve pde using laplace?

I have to solve the following pde using Laplace transforms: $xw_x + w_t= xt$ i.c: w(x,0)= 0 Firstly, transforming the above wrt t, i get: $\bar{w_x} + s\bar{w}/x = 1/s^2$ But, in the textbook, the ...
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33 views

PDE question: heat equation (third order??)

I am familiar with the usual heat equation, however, my lecturer gave me this problem and it does not look like anything I have ever seen (in my whole entire life and I am not just being dramatic). ...
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16 views

Can we use method of reflection to find Green's function in infinite strip?

I have learned how to use method of reflection to find Green's function of Laplacian equation for Dirichlet problem in half-space or quadrant in my undergraduate pde course. Now I am wondering how to ...
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56 views

PDE Heat Equation with Variable Coefficient {Second ODE Variable Coefficient}

Another PDE question: If I have a non constant coefficients in my heat equation (PDE), how do I solve it? For example we have: $\frac {\partial T}{\partial t} =\frac {\partial ^2 T}{\partial r^2} + ...
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1answer
14 views

$\Gamma$-convergence (Gamma-convergence) and PDEs?

My question is about the applying calculus of variations to solving Partial Differential Equations. In particular, what is the idea behind using $\Gamma$-convergence to find weak solutions of PDEs? ...
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8 views

Given a piecewise initial condition, how can the characteristic curve x be sketched when the solution x does not contain u terms?

The charac equation for x: $$\frac{\text{dx}}{\text{d$\tau $}}\text{=2t}$$ The solution x is $$x=t^2+x_0$$ Note that $$\tau=t$$ There is a problem. In order to sketch x, I require some ...
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1answer
23 views

Solving ODES in PDE

The PDE given as: $$t^2u_t-\text{yu}_x+\text{xu}_y\text{=0}$$ The characteristic equations are: $$\frac{\text{dt}}{\text{dt}}=t^2$$ $$\frac{\text{dx}}{\text{dt}}\text{=-y}$$ ...
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21 views

What method do we use to find the solution?

Find the solution of the initial and boundary value problem $$u_t(x,t)-u_{xx}(x,t)=0, x>0, t>0, \\ u(x,0)=f(x), x>0,\\ u(0,t)=0, t>0 $$ (The solution should be expressed as an integral ...
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28 views

heat equation pde Newton law of cooling boundary condition discrepancy

If $T(t)$ denotes the temperature of an object in an environment with temperature $T_0$, then Newton's law of cooling says $$ \frac{dT}{dt} = -k(T(t) - T_0).\quad (*) $$ This is an ODE. Consider now ...
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1answer
19 views

Harmonic function satisfying given condition

In trying to solve a homework problem I end up having the equation $\Delta f=0$ knowing that $f(x)=f(\||x\||^2)$ where $x=(x_1,\dots,x_n)$. In 2-d it leads me to $f_{x_1} + f_{x_2} + x_1 f_{x_1x_1} + ...
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61 views

What type of equation is this?

Is this equation an ODE or PDE $$ \frac{d^3u}{dx^3}−αxu=0, x∈R $$ The only thing given is $\int_R u(x) =\pi $ and $α>0$ is some constant. I have to find the solution using fourier ...
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2answers
59 views

Wave Equation Partial Differential EEquation

Basically I got a simple wave equation with an extra twist. The PDE is $\frac {\partial^2 y}{\partial t^2} = c^2\frac {\partial^2 y}{\partial x^2} + L $ with homogeneous boundary condition As usual, ...
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Proper writing of IBVP PDE & Finite Difference Implementation

I've seen a few examples (see slide 4 ) of the 2D heat equation described as $$ \begin{cases} u_t(t,x,y) = \nabla^2 u(t,x,y), \quad t > 0, \quad (x,y) \in (0,L) \times (0,H), \\ u(t,0,y) = ...
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1answer
10 views

Dirac Delta function to solve PDE in the sense of distribution

For each Borel set $A$ and $x\in\mathbb{R}^n$, denote Dirac measure centered at $x$ as \begin{equation} \delta_x(A)=\begin{cases} 1 & \mbox{ if } x\in A \\ 0 & \mbox{ otherwise}. \end{cases} ...
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2answers
42 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 ...
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1answer
14 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 ...
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19 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 ...
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31 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 ...
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1answer
40 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 ...
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2answers
90 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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1answer
26 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 $B_R:=B_R(0)$ ...
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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 ...
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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 ...
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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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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 ...
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23 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? ...
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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 ...
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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 ...
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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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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 ...
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2answers
65 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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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 ...
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1answer
26 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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1answer
23 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. ...
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29 views

Initial and Boundary value problem [closed]

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 > ...
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1answer
27 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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3answers
44 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, ...
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1answer
17 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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14 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
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31 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 $x_0$. ...