6
votes
0answers
123 views

How to solve a time-dependent Schrodinger equation in periodic Dirac delta potential

I'm trying to solve a 1D time-dependent Schrodinger equation: $$ i\frac{\partial \psi(x,t)}{\partial t}=\left[-\frac{1}{2} \frac{\partial^2}{\partial x^2}+V(x)+F(t)*x\right]\psi(x,t) $$ where $V(x)$ ...
0
votes
1answer
92 views

numerical update rule for discretized hawkes excitation process

So I think I am just misunderstanding some simple notation or something and would appreciate some help. I am trying to replicate this model in an agent based model, but I cannot seem to figure out the ...
0
votes
1answer
25 views

need to construct a function satisfying wave equation.

I need an example of a function that satisfies wave equation and that vanishes beyond certain range. I mean if $f(x,t)$ is a function of space and time, then $f(x, t) = 0, $ for $x < a(t) $ and ...
2
votes
0answers
28 views

Solution to the “cubic” Helmholtz equation

What is known about the solutions of the differential equation in three-dimensions $$ \nabla^2 \phi = -\kappa^2 (\phi + (1/3!)\phi^3) $$ Without the cubic term, this gives a linear operator ...
3
votes
1answer
78 views

Solving Wave Equation with Initial Values

I am trying to solve the wave equation: $u_{tt}$ = $u_{xx}$ With initial values: $u(x,0) =\begin{cases} x^3 - x, &\text{for }|x|\le 1,\ \\0, &\text{for }|x|\ge1\end{cases}$ $u_t(x,0) ...
0
votes
0answers
44 views

How to solve this differential equation of the second order

Do you know how to solve this equation? I'm a physicist student and I have initial equation, condition and answer. Unfortunately I need an explanation how this answer was got. I am mew to such ...
1
vote
1answer
34 views

1-D heat diffusion in a bar with fixed temperatures at the edges

I'm trying to calculate the temperature gradient in a bar of metal with a heater at either end. Initially, the bar is at room-temperature $T_0$, then at $t=0$ the heaters are turned on: $u(0,t)=T_1$ ...
0
votes
1answer
56 views

Well-posed problem

In the definition of a well-posed problem it states that a problem is well posed if: 1.A solution exists. 2.The solution is unique. 3.The solution's behaviour changes continuously with the initial ...
7
votes
2answers
179 views

Wave-Particle Duality in PDE?

I am reading Arnold's Lectures on Partial Differential Equations. It is definitely a good book, yet sometimes I am a little bit confused. One theme of the first chapter seems to be From the ...
1
vote
2answers
191 views

Derive the equation of motion from Lagrangian of a particle moving in an electromagnetic field

I really don't even know where to start with this question any help would go very very far. Thank you. A particle with charge $q$ moving in an electromagnetic field is described by the Lagrangian ...
6
votes
1answer
74 views

Solution form for Stokes flows

If $p:\mathbb{R^3} \rightarrow \mathbb{R} $ and $u: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ satisfy: $$\nabla p-\nabla^2u=0$$ $$\nabla\cdot u=0$$ How can we prove that every solution is of the form: ...
3
votes
1answer
93 views

Solving a PDE arising from physics

Is there a way to find an analytic solution to the following PDE? $i \partial _t \psi = - \gamma \partial _x ^2 \psi - c x $cos$(\omega t) \psi $, where $\psi (x,t)$ is defined (in $x$) on the ...
2
votes
1answer
62 views

How to solve ${\partial^2y\over \partial x^2}={1\over v^2}{\partial^2y\over \partial t^2}$

How to solve this Differential Equation $${\partial^2y\over \partial x^2}={1\over v^2}{\partial^2y\over \partial t^2}$$ I found this as the partial differential equation of progressive waves. The ...
2
votes
2answers
40 views

particle models with network interaction - simultaneous estimation

I am working on an application in behavioral ecology that combines network and particle interaction models. I have not been able to find any articles that simultaneously estimate these types of models ...
4
votes
1answer
67 views

Diffusive/Dispersive character of discretization schemes

I am not sure if this is the correct place to post my question, so please correct me if there is a better site. I recently started applying some discretization schemes such as Upwind, Lax-Wendroff ...
1
vote
1answer
109 views

Analytical Solution for Elastic Bar under applied end velocity

Say, a thin long rod is occupying the space $[0,L]$. It's isotropic, linear elastic, homogeneous. The partial differential equations for stress $\sigma(x,t)$ and displacement $u(x,t)$ are as follows ...
1
vote
3answers
268 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
votes
2answers
149 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
122 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 ...
7
votes
2answers
357 views

Euler-Lagrange equations of the Lagrangian related to Maxwell's equations

Clarification on Lagrangian mechanics would be much appreciated: Suppose $$L(\phi,\,\,\phi_{,i},\,\,A_i, \dot A_i)=|\dot A+\nabla\phi|^2-|\nabla \times A|^2-c\phi+d\cdot A$$ Are the corresponding ...
3
votes
1answer
329 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
2answers
77 views

Hint for integral

Could someone provide a hint as to why $$\nabla \cdot \vec a(\vec x) = -i\,\,\,b\,\,\,c(\vec x)$$ where $b$ is a constant, $i$ is $\sqrt {-1}$, implies that $$2\int d^3x \,\,x_ia_j(\vec ...
2
votes
1answer
126 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
211 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
289 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 ...
2
votes
0answers
71 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 ...
2
votes
1answer
253 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
465 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
114 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
1k 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 ...
4
votes
1answer
275 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 ...
5
votes
1answer
258 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
324 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
145 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
172 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. ...
14
votes
2answers
795 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
132 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
316 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
3k 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
294 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
422 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
960 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
142 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 ...
3
votes
2answers
597 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 ...
11
votes
2answers
3k 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 ...
7
votes
3answers
926 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
205 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
441 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: $$u(x,t)=\sum^\infty_{n=-\infty}\cos k_nx(C_n\cos k_nt+D_n\sin k_nt)$$ ...
0
votes
0answers
157 views

The quantum harmonic oscillator [closed]

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.
9
votes
1answer
642 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 ...