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Questions tagged [integral-transforms]

Questions about transforms of functions by means of integration, such as Fourier transforms and Laplace transforms. Many integral transforms can be inverted by means of another integral transform.

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15 votes
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Workarounds for a possible bug in the linearity of FourierTransform

This is somewhat similar to this question, except the problem I am encountering is to do with Fourier transforms of scalar multiples of functions and their derivatives. I wish to input ...
anon1802's user avatar
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12 votes
1 answer
1k views

Implement finite Fourier transforms

Recently I came across finite Fourier transforms, which can be used for solving certain type of boundary value problem (BVP) of linear partial differential equation (PDE) with constant coefficient. ...
xzczd's user avatar
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31 votes
4 answers
13k views

Implementing discrete and continuous Hilbert transforms

What is an efficient and accurate Mathematica implementation of the Hilbert transform, for both continuous and especially discretely sampled functions? This transform relates phase and amplitude in ...
Mr.Wizard's user avatar
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23 votes
1 answer
1k views

Numerical inverse Laplace-Hankel transform

When trying to reproduce the result of this paper about numerical solution of Lamb's problem, I encountered the following double integral (to be more precise, the 0-order inverse Hankel-Laplace ...
xzczd's user avatar
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15 votes
1 answer
406 views

Possible bug with FourierTransform linearity

Each component is easily transformed but the sum is not: FourierTransform[f''[x], x, k] FourierTransform[f'[x], x, k] ...
Kuba's user avatar
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13 votes
1 answer
4k views

How can I invert a Laplace transform numerically?

I have a very complicated expression, which I want to transform using the inverse Laplace transform. The built-in function InverseLaplaceTransform doesn't work. So,...
Mary's user avatar
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6 votes
1 answer
419 views

What integrals can `FourierTransform` evaluate that `Integrate` cannot?

FourierTransform can make sense of integrals that diverge according to Integrate. ...
John Doty's user avatar
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3 votes
1 answer
692 views

Inverse Laplace transform of this complicated function

I have been solving a coupled PDE system analytically and I need to find the inverse Laplace transform of $(1)$ and get $T(x,y)$. $s$ is the Laplace domain variable and $\alpha, \beta, \gamma, T_{fi}, ...
Avrana's user avatar
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6 votes
2 answers
2k views

Bilateral Laplace Transform

I am trying to solve a linear ODE for a Kolmogorov forward equation to get a stationary distribution of a random variable. The easiest approach may be to transform the ODE with a two-sided Laplace ...
jlperla's user avatar
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18 votes
1 answer
1k views

Package for fast spherical harmonic transform in Mathematica?

Mathematica's implementation of the Fast Fourier Transform is, naturally, much faster than computing the discrete transform yourself using Sum. The analog of the ...
Jess Riedel's user avatar
  • 1,525
8 votes
2 answers
2k views

Inverse Laplace transform

Let $r=\mu = 0.15; \sigma = 0.05; T = 1; S_0 = 100; K = 95;$ Let $\nu:=\frac{2\mu}{\sigma^2}-1$ and $\eta \equiv\eta(\alpha):=-\frac{\nu}{2}+\frac{1}{2}\sqrt{\nu^2+\frac{8\alpha}{\sigma^2}}$. ...
KNN's user avatar
  • 205
5 votes
1 answer
617 views

Solving partial differential equation involving Hilbert transform

While solving one research paper published in Physical Review Letters, I came across the following equation and I am unable to solve it. $$\frac{\partial f}{\partial t}−(\mathcal{H}(f)\left(\frac{\...
Mohan Aditya Sabbineni's user avatar
4 votes
1 answer
113 views

Transform expression involving Erfc back and forth with Laplace transform and its inversion

It's a problem I encountered when answering this question and I think it's worth starting a new question for it. Consider the following expressions: ...
xzczd's user avatar
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4 votes
3 answers
317 views

Numerical Laplace transform error

I'm running Mathematica 11.3.0.0. The inverse Laplace transform of $1/(s^2-1)$, evaluated at $t=1$ gives ...
Carlo Beenakker's user avatar
4 votes
1 answer
445 views

How to speed up the integral in NDSolve?

On the MMA.SE, there have been several question on how to solve equations including integral using NDSolve, see example 1, example 2, and example 3. Many people, ...
user55777's user avatar
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1 vote
0 answers
166 views

Can this boundary value problem leading to a partio-integral DE be solved using finite Fourier or any integral transform?

I asked the three-dimensional version of this problem here which lead to a trivial solution. I have now tried it in 2-D $$\frac{\partial \theta_h}{\partial x} + b_h (\theta_h - \theta_w) = 0, \tag 1\\\...
Avrana's user avatar
  • 297
17 votes
1 answer
7k views

Computation of Hankel Transform using FFT (Fourier)

To address circularly symmetric cases of 2-D Fourier Transformations, the so-called Hankel Transform can be applied (for a detailed derivation of the relation between the 2-D Fourier transform and the ...
Rainer's user avatar
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11 votes
2 answers
2k views

Compute inverse Laplace transform with Integrate

Since inverse Laplace transform is just an integral, while we already have InverseLaplaceTransform built in, can we compute it with ...
xzczd's user avatar
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8 votes
3 answers
532 views

Inverse Laplace Transform of Hypergeometric function

Any tips how to massage the following to get computed by Mathematica for $p>1$? I suspect the result should be expressible in terms of exponential integral ...
Yaroslav Bulatov's user avatar
8 votes
2 answers
374 views

Why does InverseLaplaceTranform give an incorrect solution?

Bug introduced in 9.0.1, fixed in 11.1. I was trying to answer this question, but I ended up asking this. Method 1 A workaround is to differentiate the ODE once, ...
zhk's user avatar
  • 12k
5 votes
1 answer
887 views

How to set up a spherically symmetric Fourier transform?

I am having trouble figuring out how to set up the Fourier transform to: ...
aluuzz's user avatar
  • 67
5 votes
2 answers
152 views

How to guide Eliminate or GroebnerBasis to reduce a set of simple odes to a single ode (which can be done by Laplace Transform)

Recently, I am trying to use Eliminate or GroebnerBasis to simplify a system of ODEs. I don't want the solution of ODEs. What I ...
xinxin guo's user avatar
  • 1,401
4 votes
1 answer
315 views

Fourier transform question

Consider the fourier transform FourierTransform[1/((Cosh[x] + 1) (Cosh[x]^2 - 1)^(1/2)), x, w] If I execute the above line, Mathematica thinks for several ...
Kagaratsch's user avatar
3 votes
1 answer
260 views

Numerical Laplace Transform of InterpolatingFunction

What are some ways to find the numerical Laplace transform of an InterpolatingFunction? (I know that numerical Laplace transforms are rarely used but my application requires a numerical evaluation). ...
Rpj's user avatar
  • 249
3 votes
1 answer
821 views

Mittag Leffler function Laplace transforms with Mathematica

Mathematica seems not to to know the basic Laplace and inverse Laplace relation $$\mathcal L(E_\alpha[−λt^α],t)(s)=\frac{s^{α-1}}{λ+s^α}$$ surrounding the Mittag Leffler function (...
Mr_3_7's user avatar
  • 53
3 votes
1 answer
309 views

Bilateral ZTransform

Is there support for bilateral Z-transform in Mathematica, or a third-party package?
user avatar
2 votes
2 answers
148 views

Issues with Fourier transform in M.12.1.0

I am using M. Version: 12.1.0. In version 10, I got the result right away.So, Why mathematica (12.1.0) is not able to solve this problem? ...
Johe's user avatar
  • 83
2 votes
1 answer
1k views

Solving the heat equation on the semi-infinite rod

Cross posted in scicomp.SE. I want to test the solution which is given below is right by Mathematica. Please look the post in mathstackexhange or Please look below. Question: Solve the ...
HD239's user avatar
  • 543
1 vote
1 answer
84 views

I ask for help with commands TransferFunctionModel + StateSpaceModel

Given system of ODE: $\begin{cases} \dot{x}=G+u_1 \\ \dot{z}=-z+\frac{df}{dt} \\ \dot{G}=-G+z \cdot u_2 \end{cases}$ where $f=-x^2$, $u_1=\frac{d}{dt}(\alpha \sin(\omega \cdot t))$ and $u_2=\alpha \...
dtn's user avatar
  • 2,404
1 vote
1 answer
421 views

How to let Mathematica return impulse or Dirac delta functions when computing integrals?

For example, let's say I want to compute the (continuous-time) Fourier transform of the signal/function $\cos{(3t)}$, which is given by the following improper integral: $\displaystyle\int_{-\infty}^{\...
alejnavab's user avatar
  • 453
1 vote
0 answers
61 views

Inverting Laplace transform numerically for a set of points

Computing inverse of Laplace transform numerically in Mathematica seems very slow and requires $k$ calls for $k$ points. Is there a faster way to do it for a set of $k$ points, ie $k\approx 1000$? <...
Yaroslav Bulatov's user avatar
1 vote
2 answers
561 views

Why does InverseFourierTransform gives different result to doing it by hand? [duplicate]

When I can calculate an inverse fourier transform by the built-in function InverserFourierTransform, the result is different from what I calculate by definition ...
waynezw0618's user avatar
0 votes
0 answers
105 views

Inverse Laplace transform of powers with an arbitrary index

Tried for Inverse Laplace transform (ILT) for the following: L[s] = (L /(L + s) w + Q /(Q + s) (1 - w))^n $L[s]$ can also be written as $$L[s]=\sum_{k=0}^n\...
step-by-step's user avatar
0 votes
0 answers
222 views

Multidimensional DFT of function

Assume that we have a well-defined real-valued functionf[x,y,z] whose (inverse) Fourier transform is impossible to solve for symbolically and numerically using ...
tks's user avatar
  • 133
0 votes
1 answer
1k views

Inverse Laplace transform

For this expression: $ (1 - Exp[-Sqrt[1+s] x])/(1+s)$ How to make the Inverse Laplace Transform analytically? ...
user14613's user avatar