# Questions tagged [laplace-transform]

Use for the Laplace integral transformation.

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### Laplace Transform of time evolution of Amplitudes involving correlation functions

The correlation function is given by $$G_{ij}(t-t')= g_{i} g_{j} e^{(i\omega_{i0}-\frac{\Gamma_{0}}{2})t-(i\omega_{j0}-\frac{\Gamma_{0}}{2})t'}$$ With the time evolution of amplitudes being given by ...
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### What is the difference between scaling property when scaling factor is $-1$ and the time reversal property of Laplace transform?

I came across this question when one of my students asked me about it. Suppose that $L$ denotes the Laplace transform and that $L\{F(t)\}=f(s)$. By scaling property, we have $L\{F(-t)\}=-f(-s)$. The ...
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### Laplace transform: How to evaluate partial derivative in the denominator of a fraction?

I am solving a differential equation using the Laplace transform. However, to evaluate it I need to evaluate some strange terms. Specifically, I have a partial derivative in the denominator of the ...
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### First passage time for state transitions expressed in Laplace space?

I'm struggling to understand the reasoning between moving between two steps in a biophysical reaction scheme for a paper I am reading. For this (from the description), the probabilities over different ...
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### Unique distribution of charge on a conductor

If i place some charge on a conductor then it will distribute itself in such a way that electric field everywhere inside is zero. My text book says that only one kind of such charge distribution is ...
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### PI controller - output calculation [closed]

The propeller of an airship is connected to a DC motor. The motor is being controlled using a PI controller. The PI controller has a proportional gain of Kp = 2, and an integral gain of Ki = 0.5, and ...
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### Fourier vs. Laplace transforms

Electronics books often use Laplace to analyze circuits, while in physics we use Fourier, most of the times... if not always: from complex impedances to electromagnetism, quantum mechanics, Green ...
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### What is the physical meaning of sigma in the Laplace transform?

Let’s use a simple harmonic oscillator as an example. When we calculate the Fourier Transform (a special case of the Laplace transform) of that system we get a function that shows which frequencies of ...
1 vote
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### What is the intuitive interpretation of the transfer function of this system?

If I have the system that could be observed in the next Image: I want to know the transfer function, where the external force $f$ is the entry and $x_1$ is the output. The direction and positive ...
226 views

### Harmonic oscillator boundary condition issue for an impulse force

I am trying to solve an equation of an underdamped harmonic oscillator with a damping, and I get a weird boundary condition that perplexes me. Let me precise the issue, the equation is : \begin{...
127 views

### Finding the Frequency and Damping Ratio of a Mass Spring Damper excluding Air Resistance

In summary, I need the damping ratio and damped frequency of a system excluding air resistance, as the system will be operating in a vacuum. The system can be modeled as a mass-spring-damper, and the ...
134 views

### Using laplacian transform to derive a general solution to the quantum harmonic oscillator

I'm unhappy with the often used approach of guessing the form of the solution and working from there. In the case of the quantum harmonic oscillator, once the form \frac{d^2 \psi(x)}{...
341 views

### Transfer function of system of coupled 2nd order ODE

I am wondering how to calculate transfer function $H(s)$ of system described by 3 coupled differential equations. The pourpose of work is to calculate "Bodedx" diagram ($|H(i\omega)|(\omega)$...
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### Temperature distribution using Laplace transform

Consider room that occupies quarter space That is formed by two walls. One wall is fully insulated and has a constant temperature $0$. Another wall has a window of length L that with one edge at the ...
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### Fourier and Laplace transforms (Forster's Hydrodynamics book)

In Forster's hydrodynamics textbook, he defines the following Fourier transform and (one-sided) Laplace transform \begin{align} S(k,\omega) =& \int_{-\infty}^{\infty} dt e^{i\omega t} S(r,t) \\ \...
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### Inverse Laplace transform for Debye function

Given $$D_n(x)=\frac{n}{x^n}\int _{0}^{x}\frac{t^n}{e^t-1}dt,$$ the Debye function. Wikipedia says that it computes the Heat capacity of the Debye model. Mathematically, we are interested in the ...
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### What are the real world applications involving Laplace transforms? [closed]

Laplace transform is really interesting. Speaking about Fourier transform, there are many real world applications like we use in removal of noise and Laplace transform is again the extension of ...
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### Laplace transform for damped waves

In Zeidler's book on QFT, page 94 there is a definition for a Laplace transform that reads $$(\mathcal{L} f)(\mathcal{E}) = \int_0^\infty e^{i\mathcal{E}t/\hbar}f(t) dt,$$ ...
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### Is time constant the same for linear system for all time?

I have heard that if you take a measurement $T_1$ and wait, then take another measurement $T_2$ and then find $\Delta T = T_2 - T_1$. Then 63% of $\Delta T$ + $T_1$ will represent the measurement when ...
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### RLC - CLR - RCL Transfer function

I have problem because I don’t understand the difference between CLR RLC and LCR, they are the same no ? They have the same composant just placed in different ways, I did already found the Transfer ...
502 views

### Diffusion equation with time-dependent boundary condition

I was trying to solve this 1D diffusion problem $$\dfrac{\partial^2 T}{\partial \xi^2} = \dfrac{1}{\kappa_S}\dfrac{\partial T}{\partial t}\, , \label{eq_diff_xi}$$ with ...
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### Solving LR Circuit with Laplace Transform

I have a RLC circuit where the capacitor is connected in parallel with a resistance and inductance in series. The battery is connected "in parallel" with the capacitor and the RL branches. At t=0 the ...
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### Express Laplace transform of voltage across a capacitor in terms of charge

In Charge Tunneling Rates in Ultrasmall Junctions section 2.1, the authors consider the problem of charge relaxation in a simple circuit shown in Figure A. The implicitly use an assumption made about ...
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I'm trying to get a transfer function of $F=ma$ in the Laplace domain. This should be simple, but yet I'm confused. The transfer function is displacement over force. So, I have two approaches. First ...