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.
236 questions
2
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1
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Definition of inverse Laplace transform [duplicate]
Although there is a ready-made code for the inverse Laplace transform in Mathematica, I want to manually write the code to define the inverse Laplace transform so I can modify it.
This is my attempt:
<...
- 109
1
vote
0
answers
127
views
What are possible ways to avoid DiracDelta[ω] in Fourier transforms? [closed]
I found one related question here, but I am looking for other possible solutions for this DiracDelta function in the Fourier transform.
...
- 125
1
vote
0
answers
107
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BilateralLaplaceTransform is seemingly broken?
It does not work for me even on the simplest elementary functions.
BilateralLaplaceTransform[Sin[t], t, x]
...
- 3,862
4
votes
2
answers
504
views
Wave equation via Laplace transform
Solve, via Laplace transforms
\begin{align}
w_{tt}&=w_{xx}\\
w(0,t)&=g(t), \lim_{x \rightarrow \infty}w(x,t)=0, \: x,t\geq 0\\
w(x,0)&=0=w_t(x,0)
\end{align}
where $w$ is the ...
9
votes
1
answer
224
views
InverseMellinTransform unexpected result after Simplify
I'm having unexpected difference applying inverse Mellin transform after a simple algebraic rearrangement (1/(-1 + 2 s) vs ...
- 7,681
2
votes
1
answer
127
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Handling of inverse Laplace transform of $e^{a s}$ when $a>0$
Do not know if this was asked before.
Why in V 14.1 this works
ClearAll[a, s];
InverseLaplaceTransform[Exp[a*s], s, t, Assumptions -> a > 0];
% /. a -> 1
...
- 156k
2
votes
1
answer
188
views
Errors with the numerical inverse Laplace transform
I want to calculate numerically the inverse Laplace transform by the "Piessens" method of a complicated function given in terms of a numerical integral. However errors occur for functions ...
- 174
1
vote
2
answers
142
views
Same integral giving different results
I am trying to solve the following integral using Mathematica
$\int_{0}^1 dz \int_{0}^{1} dz' \exp(-k|z-z'|)\cos[\pi p z] \cos[\pi q z]$,
with $p,q\in \mathbb{Z}$. To do so, I am doing the following:
<...
- 595
2
votes
1
answer
193
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LaplaceTransform and DSolve handles PDE well in version 14, but not in 13.3
I'm facing an error when trying to solve a simple Laplace-transformed PDE with one basic boundary condition, followed by applying the InverseLaplaceTransform. The ...
6
votes
1
answer
282
views
Laplace transform on boundary condition of PDE results in D::ivar warning
Bug introduced in 14.0, persisting through 14.3.
I am trying to convert my boundary condition to its Laplace transform as in this picture:
I have tried to do so using the following code:
...
2
votes
1
answer
284
views
Inverse Laplace transform does not give a soluton
I am trying to derive the inverse Laplace transform of the following Laplace transform:
$$
\mathcal{L}(d, \sigma; t) = \left(\sec \left(\frac{\pi d}{2}\right) \left(\left(\sigma ^4 t^2-1\right)^{-d/2}...
- 25
4
votes
0
answers
220
views
0
votes
2
answers
176
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Integration with respect to functions
I observe some strange behaviour of Mathematica when pulling functions into the integrator. I know that Mathematica's capabilities are limited when computing Stieltje or Lebesgue Integrals. But even ...
- 187
1
vote
2
answers
193
views
Different solutions between Mathematica and Wolfram Alpha
I started to use Mathematica just recently and tried to do the following simple integral:
The result I get from
Integrate[1/(Exp[x]+1),x]
is
...
- 71
3
votes
2
answers
170
views
InverseLaplaceTransform returns the input
Initially, I was trying to invert the following expression:
$$ \frac{e^{-a\sqrt s}}{s-c} $$ and got the following result:
...
- 31
1
vote
1
answer
194
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Partial integro-differential equation
I want to solve the following partial integro-differential equation for $\delta(x,t)$:
$$
1-B \cdot \frac{\partial \delta(x, t)}{\partial t}=\frac{1}{\pi} \int_{-1}^1 \frac{\partial \delta(s, t)}{\...
- 11
0
votes
1
answer
160
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On the Laplace transform of Beta function
While trying to evaluate the Laplace transform below
$$I = \int_{0}^{\infty}e^{-st}B(\frac{1}{2}-it,\frac{3}{2}+it)\mathrm{d}t,$$
invoking
...
user96517
3
votes
3
answers
311
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Wolfram Alpha and Wolfram Engine produce different Laplace transforms
Consider the following Laplace transform:
$$\mathcal{L}\{\dfrac{\sin{2x}}{x}\}$$
To calculate it, I'd write LaplaceTransform[Sin[2 x] / x, x, p] into both Wolfram ...
3
votes
1
answer
134
views
Solving Laplace Transform IVP [closed]
I want to solve the IVP Laplace transform for the following:
$$y''-2y'+y=3e^t$$ with $$y(0)=1, \; y'(0)=1.$$
How would I input this in Wolfram Language? I've tried a bunch of different things and ...
- 33
1
vote
1
answer
107
views
Is there an option for InverseLaplaceTransform to make Mathematica use the convolution theorem when feasible?
By default, it appears that Mathematica won't use the convolution theorem to write an inverse Laplace transform in the form of a convolution of two functions.
For example, ...
- 455
4
votes
1
answer
190
views
Strange result simplifying higher order BesselJ [duplicate]
Consider the following integral:
$$\int_0^1 Z_n^m(r)\ J_m(\rho r) r dr$$.
The solution of this should contain a single Bessel function:
$$(-1)^{(m-n)/2}\ J_{n+1} (\rho)/\rho$$ (see https://www.osti....
1
vote
0
answers
131
views
How can I write the code for an inverse integral transform?
How can I code any type of integral transform with its inverse integral transform in Mathematica which gives the correct results and properties? Below is the example for Elzaki transform:
...
5
votes
1
answer
147
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InverseFourierSinTransform on Mathematica did not give a result
I am trying to solve the linear Schrödinger equation with the Fourier transform. I have difficulty making the corresponding graph when I solve the problem numerically. Can you explain to me what is my ...
1
vote
1
answer
169
views
How to deduce analytical solution from numerical solution
I came across one such integral in my calculations, for which there is no analytical solution. But it exists numerical solution, so how can I derive analytical solution of this integral from numerical ...
- 193
3
votes
2
answers
264
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Question about numerical integration of a double integral
I am trying to numerically integrate the following double integral in Mathematica for different values of t.
...
- 139
0
votes
1
answer
170
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Taking the inverse Laplace Transform as a vector operation
I am solving a system of first-order equations using matrix operations and the Laplace Transform. I begin with the matrix equation that represents the solution to my system, like this:
$$ \underline{\...
- 201
8
votes
4
answers
789
views
Inverse Laplace Transform of Hypergeometric Function
Any tips how to massage the following to get computed by Mathematica for $p>1$? I suspect the result should be expressible in terms of exponential integral
...
- 7,681
5
votes
1
answer
149
views
NIntegrate with highly oscillatory Bessel and hypergeometric integrands
I am trying to compute the following double integral
int2[n_] := NIntegrate[
n^2 * u * BesselJ[0, u]^n * r^2 * BesselJ[0, n*r*u],
{r, 0, 1}, {u, 0, Infinity}]
for ...
- 153
1
vote
2
answers
101
views
InverseFourierSinTransform in mathematika failed to give a result
I have to use InverseFourierSinTransForm in Mathematika for the function u[ω,t] but infortunately it does not work.It gives back the same!
I tried it without the assumptions but it does not work again!...
- 43
2
votes
1
answer
177
views
Derive Parseval's theorem in one dimension
Parseval's theorem (in one dimension) is a fundamental result in the theory of Fourier transforms. If $f(t) \Leftrightarrow F(\omega )$ are Fourier transform pairs and $t$ (time) and $\omega$ (...
- 43k
1
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1
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199
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Evaluating Fourier transform in mathematica [closed]
I am trying to evaluate the expression
FourierTransform[[m (a^2 - t^2 - I g t)]^-1, t, ω]
in Mathematica. It gives me the error message that "Syntax: "[m(...
- 167
0
votes
1
answer
104
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Handling singularities like (x-y) in the denominator while evaluating double integrals
I have to solve an Integral of the following type
...
2
votes
0
answers
88
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Efficient powers of DPR1 matrices
I'm looking to compute the following quantity for $A$ where $A$ is a convergent positive definite $d\times d$ diagonal + rank1 matrix (DPR1).
$$f(s)=\operatorname{Tr}(A^s)$$
Earlier answer answer by ...
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1
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Improving quality of plot with bad numeric performance
I have a plot which suffers from poor numeric accuracy, any tips on how I can improve the quality of this plot?
...
- 7,681
1
vote
0
answers
105
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 $k\approx 1000$?
<...
- 7,681
1
vote
0
answers
103
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Incorrect result of InverseFourierTransform
When executing in 13.2 on Windows 10, the command
FourierTransform[DiracDelta@Cos[x], x, s]
results in
...
1
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0
answers
48
views
0
votes
0
answers
67
views
Hilbert Transform in mathematica [duplicate]
I would like to do a Hilbert transformation in Mathematica on a function. However, it does not seem to give a right result.
The Hilbert transform is given by
So I did :
...
- 629
5
votes
1
answer
403
views
Incorrect result of FourierTransform
Let us consider in 13.2 on Windows 10
FourierTransform[1/Sinh[x]^2, x, k]
-((2 + k \[Pi] Coth[(k \[Pi])/2])/Sqrt[2 \[Pi]])
...
6
votes
1
answer
680
views
Wrong result of Laplace Transformation [closed]
I am trying to calculate the Laplace Transformation of the following function:
$$f(x) = \theta(t+1)-\theta(t-1)$$
where $\theta(t)$ is the Heaviside step function defined as:
$${\displaystyle \theta(x)...
8
votes
1
answer
394
views
Still bug in Integrate. 3
Let us consider in version 13.1 on Windows 10
r = Integrate[1/(x - a)/Sqrt[1 - x^2], {x, -1, 1},
Assumptions -> a \[Element] Reals]
...
0
votes
0
answers
124
views
LaplaceTransform doesn't work
I have code:
A = 1;
ω0 = 5*10^6;
τ0 = 3*10^-3*Sqrt[2/Log[2]]/(5.85*10^3);
s0[t_] = A*E^(-t^2/τ0^2)*Cos[ω0*t];
Phi[ω_] = LaplaceTransform[s0[t], t, ω]
The last ...
2
votes
0
answers
92
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Is this a bug in InverseLaplaceTransform or LaplaceTransform?
Let us consider in version 13.1 on Windows 10
ClearAll["Global`*"]; a = InverseLaplaceTransform[s*Log[(s - 1)/(s + 1)], s, x]
...
4
votes
1
answer
279
views
Inverse Laplace Algorithm used in Mathematica [closed]
I have a general question. I want to know what algorithm (The name of this numerical method) is already used to calculate a numerical Inverse Laplace in Mathematica.
I used the function ...
- 141
2
votes
1
answer
82
views
Validation of the Laplace inversion and storage in a table or an array
I tried to validate this function from 0 to 50 but it takes very long time, is there a faster way to validate this function for t from 0 to 50 and add them in a list?
...
- 141
1
vote
0
answers
266
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Inverse Laplace in mathematica
I can't even get an inverse Laplace for this expression numerically in mathematica,
is there a way to inverse this equation below?
I have also tried to use fixt talbot package for a numerical ...
- 141
0
votes
2
answers
219
views
Inverse Triple Laplace Transform of $\frac{-1}{s^2_{1} + s^2_{2} + s^2_{3}}$
I want to find the inverse triple Laplace transform of $L^{-1}_{x_{3}} L^{-1}_{x_{2}} L^{-1}_{x_{1}} \left[ \frac{-1}{s^2_{1} + s^2_{2} + s^2_{3}} \right]$. I did
\begin{align*}
L^{-1}_{x_{3}} L^{-1}...
0
votes
1
answer
203
views
Fourier Transform of Integral Expression
I am trying to Fourier transform an expression containing an integral like this:
FourierTransform[Integrate[f[v]*Cos[w[v]*t],{v,-v_0,v_0}],t,k]
where
...
- 1
2
votes
0
answers
64
views
Replace subexpression with variable in result from Laplace transform
I'd like to clean up the result I obtained from an inverse Laplace transform:
First of all, I'd like to replace the square root expressions in the hyperbolic function arguments (part encircled in ...
- 21
0
votes
0
answers
192
views
Implementing the 3D Radon transform
I am wondering how to implement the Radon transform, the 3D Radon transform, that is, given a 'density' function $f: \mathbb{R}^3\to \mathbb{R}$ The Radon transform of $f$ is $$Rf(s,w)= \int_{x\cdot w=...
- 109