Questions tagged [laplace-transform]

Use for the Laplace integral transformation.

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Laplace transform of triple convolution involving Heaviside function — stems from multi-state random walk

I'm a bit stuck on this problem stemming from a multi-state random walk. I have a function of the form $$C(t) = \int_0^t dt' \theta(t'-T)A(t')B(t-t')$$ and I'd like to calculate its Laplace transform....
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Diffusion in a Semi-Infinite Medium: Breakaup of a Duhamel Convolution Integral

For a diffusion problem in a semi-infinite domain with a transient boundary condition, the temperature profile can be obtained from Duhamel's Principle as $$ \theta(r,t)= \int_0^t \frac{\partial \phi}...
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85 views

Diffusion equation with time-dependent boundary condition

I was trying to solve this 1D diffusion problem \begin{equation} \dfrac{\partial^2 T}{\partial \xi^2} = \dfrac{1}{\kappa_S}\dfrac{\partial T}{\partial t}\, , \label{eq_diff_xi} \end{equation} with ...
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205 views

Density of states for 3D simple harmonic oscillator

I have the thermal partition function and the density of states for the 3D simple harmonic oscillator, which are given below $$ Z(\beta) = \frac { 1 } { \left( 2 \sinh \left( \frac { \beta \omega } { ...
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144 views

Laplace transform on thermodynamics equation

I'm trying to create a simple model of a single-flash geothermal plant which consist of 3 main parts (flash-separator, turbine, and condenser). Is there any way to create a transfer function using ...
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84 views

Why is the Laplace Transform essentially never used when dealing with problems involving resonance?

Both the Laplace Transform and the Fourier Transform can be applied to a PDE, for example the wave equation, and used to derive a solution to the equation. But I never see the Laplace Transform used ...
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Laplace Transform Density of States & Partition function

I am currently going through Pathria's Statistical Mechanics text , and under the Canonical Ensemble description, the author stresses that the partition function of a continuous system is the Laplace ...
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105 views

Fourier Transforms of Harmonic Functions

Suppose you are presented with the equation ($D=3$) \begin{equation} \nabla^2 A(x) = \nabla^2 B(x). \end{equation} Decompose $A$ and $B$ into their Fourier components, \begin{equation} A(x) = \int d^...
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Why is there a sudden change in current between $t=0^{-}$ and $t=0^{+}$ when an active inductor is connected in series with a relaxed inductor?

Let us take the following question as an example: For the above question I drew the corresponding Laplace transform diagram, as follows (didn't draw the switch since it basically open circuit after $...
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87 views

Difference between Fourier and Laplace transforms in analyzing data

I have a set of displacement-time graphs from an experiment to convert to the frequency domain. Both the Fourier and Laplace transform seem to do this, so what's the difference between them (...
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240 views

What is this equation, written on a wall? [closed]

I've also asked this in MathOverflow, but since the equation in question is related to acceleration and physical phenomenon, I figure this is also a good place. I was asked to ID the following, but ...
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Question about the Radiative Transfer Equation (RTE) Phase function term

I am new to this subject. I am trying to understand what is the purpose of the phase function $$ ∂u/∂t=-v∘∇u-v( μ_{a}+μ_{s})u+∫vμ_s p_{s} (Ω^{'}→Ω)u(r,Ω^{'},t)ⅆΩ^{'}+q(r,Ω,t) $$ Were u= angular ...
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Why is the impedance of a resistor the same in time and frequency domains if the Laplace transform of a constant is not the same constant?

I'm sure this is a silly question, but if the Laplace transform of a constant is not a constant, e.g. $$\mathfrak{L}[1] = \frac{1}{s}$$ then how come the impedance of a resistor is the same in the ...
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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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640 views

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 ...