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

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the geometry of level set of solution of elliptic PDE

Let me make my question more clear. Suppose I have a nonlinear elliptic PDE, say $$-\triangle u = u^2$$ and I am solving this problem on a nice domain, say the unit ball $B(0,1)$ and I have the zero ...
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1answer
66 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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31 views

Navier-Stokes proof - question (C) [on hold]

I just shared detailed formal proof to answer question (C) of Navier-Stokes Millennium problem. The following is brief summary of theorems and proofs made: (i) Theorem 3.1 states that curl of ...
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8 views

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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2answers
48 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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11 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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116 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
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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22 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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22 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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58 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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13 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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546 views

Prove an identity for the continuous integral solution of the conservation law

This is an exercise in Evans, Partial Differential Equations, page 164, problem 13 Assume $F(0) = 0, u$ is a continuous integral solution of the conservation law: $$ \left\{ \begin{array}{rl} u_t + ...
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19 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
481 views

How to use Fourier Transform to solve the Airy's equation?

Definition: If $f\in L^1(\mathbb{R}^n)$, the Fourier Transform of $f$ is the function $\hat{f}$ given by $$\hat{f}(y)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^n}e^{-ix\cdot ...
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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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3answers
107 views

$u_{tt} = u_{xx}$ with unusual boundary conditions

Solve using the reflection method $u_{tt} = u_{xx}$ With $ x< ct , t > 0$ for some $c \in \mathbb{R}$ And $u(ct,t) = 0, u(x,0) = f(x), u_t(x,0) = 0$ I thought of rotating the $t$ axis in ...
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1answer
624 views

Solving Wave Equations with different Boundary Conditions

Right now I'm studying the wave equation and how to solve it with different boundary conditions (i.e. $u(x,0);u(0,t);u_t(x,0);u_x(x,0);u(x,x);u_t(x,x)...$) I know how to solve it when the boundary ...
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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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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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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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402 views

Use the method of characteristics to solve nonlinear first order pde.

I find this problem challenging: Use the method of characteristics to solve $u_t+u_x^2=t$ with $u(x,0)=0$. I know I'm supposed to let $p=u_x$ and $q=u_t$. Then I get $F(x,t,u,p,q)=p^2+q-t=0$. But ...
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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 ...
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1answer
37 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 ...
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2answers
43 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
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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2answers
52 views

The mysterious $\dot{H}^{-1}$ notation.

I have encountered the $\dot{H}^{-1}$ notation in one of the SIAM Journal on Mathematical Analysis articles. It appears to be standard (or at least not uncommon) to use this one in the field, since ...
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1answer
69 views

analytical solution for linear 1st order PDE using laplace and seperation of variables

I am looking for the solution of the following pde: $\frac{\partial y(x,t)}{\partial t} = a* \frac{\partial y(z,t)}{\partial x} + b* y(x,t) + c$ and need help with the boundary and initial ...
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1answer
29 views

Solving laplace equation and decomposing

I want to solve $$ 9U_{xx}+U_{yy}=\sin (2\pi x) + \sin(2\pi y) \label{eq:1}\tag{1} $$ with $U=0$ on the boundary of the unit square. I know you would have to decompose the problem to satisfy each of ...
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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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74 views

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 ...
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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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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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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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1answer
15 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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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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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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1answer
40 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 ...
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32 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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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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17 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 ...
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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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7 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 ...
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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 ...