Tagged Questions

1
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3answers
88 views

Thermodynamics for math majors

I'm about to wrap a course in partial differential equations. We've discussed the heat/wave equations and introductory Fourier Analysis. I'd like to do some reading into the field of thermodynamics. ...
0
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2answers
86 views

Give a physical explanation for why the Neumann Problem has no solution?

Give a physical explanation for why the Neumann Problem $$ U_{xx}+U_{yy}=q(x,y) $$ $$ \nabla U(p)\cdot n(p)=g(p) \quad \forall p\in C $$ on $D$ for Poissons equation, has no solution, unless we ...
0
votes
1answer
64 views

Find all solutions of the 1-D heat equation of a specific form

I'd like to find all solutions of $u_t$ = $u_{xx}$ of the form $$u = \left(\frac{1}{\sqrt{t}}\right)v\left(\frac{x}{2\sqrt{t}}\right).$$ I know that this problem reduces to solving a second order ...
-4
votes
1answer
117 views

Mathematics solving by software [closed]

Is there any way to reconstruct the mathematics from this paper ? It will be convenient for me if there is any program or software that can do this.
1
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1answer
208 views

Energy of wave equation decreasing

I have problems checking that the energy $E(t)=\frac{1}{2}\int_I(u_t^2+c^2u_x^2)dx$ on an open interval $I\subset \mathbb R$, such that $u(0,x)=0$ and $u_t(0,x)=0$ for $x\in\mathbb R\setminus I$ is ...
2
votes
1answer
92 views

The PDE $u_t = -\Delta^2 u -\Delta u + f$

Does the PDE $u_t = -\Delta^2 u -\Delta u + f$ have a physical use or meaning? I am asking specifically about the the Laplace term after the biLaplace term.. is it unusual or "unnecessary" in some way ...
1
vote
1answer
91 views

Discontinuity of double-layer potentials

I'm currently reading about solutions to boundary-value problems for Laplace's equation, and I'm a bit confused with regards to the discontinuity properties of double-layer potentials. So the text ...
0
votes
1answer
186 views

damped wave equation

For $t>0$, $x$ in a compact Riemannian manifold $(M,g)$, and $a\in C^\infty(M)$, $a\geq0$, $(\partial_t^2+a\partial_t-\Delta_g)u=0$ is called the damped wave equation. My question is...why is the ...
1
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0answers
57 views

Differential equations with different constants for different sub-domains

I remember that when I was studying differential equations, there was an example with solutions of the form $f(x) + C_1$ for $x>0$ and $f(x)+C_2$ for $x<0$ where $C_1$ and $C_2$ may be different ...
1
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0answers
101 views

Finding the 'inhomogeneous' plane wave solutions of the wave equation via Fourier analysis

When one solves the wave equation $$ ( \partial_t^2 - v^2 \nabla^2) \mathbf{E}(\mathbf{x},t) = 0 $$ in $\mathbb{R}^3 \times \mathbb{R} $ using the Fourier transform method, the general solution is ...
0
votes
1answer
108 views

Effects of gravity on diffusion [closed]

I'm just now learning the diffusion model and it seems that we aren't taking into account the acceleration due to gravity of the particles. Is this a shortcoming of the model or irrelevant? I don't ...
1
vote
2answers
82 views

Basic conceptual diffusion problem

Suppose that some particles which are suspended in a liquid medium would be pulled down at the constant velocity V by gravity in the absence of diffusion. Taking into account the diffusion, find the ...
2
votes
2answers
539 views

Wave Equation Neumann Boundary Conditions

I am studying basic PDEs and I would like to ask a thing I can't understand. I would really appreciate a piece of advice. I must compute the solution $u(x,t)$ of a 1-D wave equation with Neumann ...
3
votes
1answer
158 views

Expressing the wave equation solution by separation of variables as a superposition of forward and backward waves.

(From an exercise in Pinchover's Introduction to Partial Differential Equations). $$u(x,t)=\frac{A_0 + B_0 t}{2}+\sum_{n=1}^{\infty} \left(A_n\cos{\frac{c\pi nt}{L}}+ B_n\sin{\frac{c\pi ...
3
votes
1answer
187 views

Physical interpretation: weighted eigenvalues of the Laplacian with a potential

I've already posted this question on Physics.SE, but I thougth it could be useful to ask also here. No problem if moderators will ask me to cancel this thread... But, please, have mercy! :-D Let ...
4
votes
1answer
143 views

Choosing the sign of the separation constant for a vibrating string

Suppose we have this PDE problem $$\frac{\partial^2 \psi}{\partial x^2}=\frac{1}{c^2}\frac{\partial^2 \psi}{\partial t^2}$$ $$\psi(0,t)=\psi(L,t)=0$$ It represents the vibrations of a string tightly ...
1
vote
0answers
123 views

The wave equation in action

The vibration of a piano string is governed by the wave equation $u_{tt} - c^2u_{xx} = 0$ where $c$ is related to the tension and the mass density. Suppose a string is hit by a hammer on the interval ...
2
votes
3answers
150 views

Non-uniqueness of the solution of the equation for a plucked string

I'm a bit confused about what is written in this PDF (in page 2). The author asserts that the differential equation $y'' +y = 0$ with boundary conditions $y(0)=0=y(\pi)$ has infinitely many solutions. ...
13
votes
2answers
512 views

Why are harmonic functions called harmonic functions?

Are they related to harmonic series in any way? Or something else? Wikipedia didn't help.
0
votes
1answer
103 views

Solution of the complex Ginzburg - Landau equation

Can someone show that it's possible to find a solution of the kind: $$\Phi(x,t)=R(x,t)\exp[i\Psi(x,t)]$$ of the complex Ginzburg - Landau equation: ...
0
votes
1answer
189 views

Wave equation in a medium PDE

Suppose there is a string in a medium that applies a resistant force per unit length proportional to the velocity of the string. How do you write the equation of string vibrations?
2
votes
1answer
2k views

Can't understand a simple wave equation matlab code

I'm trying to figure out how to draw a wave equation progress in a 2D graph with Matlab. I found this piece of code which effectively draw a 2D wave placing a droplet in the middle of the graph (I ...
1
vote
1answer
206 views

Heat Equation Derivation and Mean Value Theorem

Farlow book PDEs for Scientists and Engineers pg. 27 shows derivation for Heat Equation. It starts by stating Net change of heat inside $[x,x+\Delta x]$ = Net flux of heat across boundaries + Total ...
0
votes
1answer
283 views

Change of variables of PDE

I have a particle of mass $m$ that moves in 2-d in the potential $V(x,y)=\frac{1}{2}m\omega^2(6x^2-2xy+6y^2)$. I have to use the coordinates $u=\frac{x+y}{\sqrt 2}$ and $w=\frac{x-y}{\sqrt2}$ to show ...
2
votes
1answer
487 views

Schrödinger versus heat equations

I'm trying to solve the initial value problem $(i\partial_t+\Delta_x)u(t,x)=0$, $u(0,x)=f(x)$ for the Schrödinger equation ($t\in\mathbb{R}$, $x\in\mathbb{R}^n$, $f$ Schwartz). I know that a ...
0
votes
1answer
136 views

How one can find solution of PDE of the forms

I am a mathematical physics student and I had the following question in my mind from few weeks. I couldn't find any solutions. I am very thankful to this site and I hope, I expect some good reasonable ...
2
votes
2answers
378 views

Hankel function in terms of planewaves

It is well know that planewaves are a complete basis for solutions to the wave equation. Let us assume a 2D space, and at fixed temporal frequency, the equation reduces to the Helmholtz equation. In ...
9
votes
2answers
2k views

Energy norm. Why is it called that way?

Let $\Omega$ be an open subset of $\mathbb{R}^n$. The following $$\lVert u \rVert_{1, 2}^2=\int_{\Omega} \lvert u(x)\rvert^2\, dx + \int_{\Omega} \lvert \nabla u(x)\rvert^2\, dx$$ defines a norm on ...
6
votes
3answers
596 views

Energy functional in Poisson's equation: what physical interpretation?

Let's consider this boundary-value problem: $$\begin{cases} -\Delta V = \rho & \rm{in}\ \Omega \\ V=0 & \rm{on}\ \partial \Omega \end{cases}.$$ We know that this problem has a ...
1
vote
0answers
176 views

Complete vs General Integral of first order PDE

The following is an excerpt from Landau's Course on Theoretical Physics Vol.1 Mechanics: ... we should recall the fact that every first-order partial differential equation has a solution depending ...
2
votes
2answers
361 views

wave equation and superposition

If I have this equation: $$\frac{\partial^2u}{\partial x^2}=\frac{\partial^2u}{\partial t^2}$$ And this general solution: $$u(x,t)=\sum^\infty_{n=-\infty}\cos k_nx(C_n\cos k_nt+D_n\sin k_nt)$$ ...
1
vote
0answers
140 views

The quantum harmonic oscillator

I want to ask how to solve the equation $$-\frac{{\hbar}^{2}}{2m}\frac{\partial}{\partial_{r}^{2}}u(r)=(E-\frac{1}{2}Kr^{2})u(r)$$ with $K$ being a constant.
6
votes
0answers
430 views

Eigenfunctions of the Helmholtz equation in Toroidal geometry

the Helmholtz equation $$\Delta \psi + k^2 \psi = 0$$ has a lot of fundamental applications in physics since it is a form of the wave equation $\Delta\phi - c^{-2}\partial_{tt}\phi = 0$ with an ...
1
vote
3answers
341 views

Fourier analysis for waves

I'm studying physics, so I'm sorry if I'll write some inexact things in this post. I wish you can understand me. If we have 1D wave equation: $$\frac{\partial^2 \psi}{\partial ...
2
votes
1answer
730 views

Dirac's identity

Do somebody knows anything about the Dirac's identity? \begin{equation} \label{Dirac} \frac{\partial^2}{\partial x_{\mu}\partial x^{\mu}} \delta(xb_{\mu}xb^{\mu}) = -4\pi ...