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[38]: https://www.wolfram.com/mathematica/core-areas/Charles Pooh2022-08-05T21:37:19ZWolfram R&D LIVE: Everything to know about Mellin-Barnes Integrals - PartII
https://community.wolfram.com/groups/-/m/t/2861119
*MODERATOR NOTE: This is the notebook used in the livestream "Everything to know about Mellin-Barnes Integrals - Part II" on Wednesday, March 22 -- a part of Wolfram R&D livestream series announced and scheduled here: https://wolfr.am/RDlive For questions about this livestream, please leave a comment below.*
&[Wolfram Notebook][1]
[1]: https://www.wolframcloud.com/obj/9c7910a4-9774-4c32-9b56-d001426ca8bfOleg Marichev2023-03-28T17:35:42ZBrachistochrone Problem: shortest time to slide on a curve
https://community.wolfram.com/groups/-/m/t/2859015
![enter image description here][1]
&[Wolfram Notebook][2]
[1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=9843fall.gif&userId=20103
[2]: https://www.wolframcloud.com/obj/fe054a06-4dd1-43ef-8712-403b1ecbf790Loras Tyrell2023-03-25T16:01:29ZHow does Mathematica compute a derivative numerically by Euler sums?
https://community.wolfram.com/groups/-/m/t/2860686
I have a question: How does Mathematica compute a derivative numerically by Euler sums in the function "ND", which is found in the numerical calculus package? Which formula is used? Thank you very much!Ralph Trenkler2023-03-28T07:29:21ZSolving a Sturm-Liouville BVP in two ways
https://community.wolfram.com/groups/-/m/t/2860094
&[Wolfram Notebook][1]
[1]: https://www.wolframcloud.com/obj/4dbe5ee0-0ce6-4e8e-b8a6-cc6fd69ddd6fAthanasios Paraskevopoulos2023-03-27T21:29:57ZIntegrate doesn't give a result for non-linear functions
https://community.wolfram.com/groups/-/m/t/2859776
I use Acegen for Finitie Element formulation that gives out residuals and tangents by taking in the field values as inputs. For this in my residual formulation I need to integrate a nonlinear term over the element Area(2D Element). For some reason, this doesn't give a result. I isolated the problem in a smaller version of the code as shown below:
SetOptions[EvaluationNotebook[]]
ClearAll["Global`*"]
<< AceGen`; NAME = "QIntegrate";
SMSInitialize[NAME, "Language" -> "Matlab", "Mode" -> "Debug"];
(*Initialization*)
SMSModule[NAME, Real[ y$$[2], ue$$[4], a$$, b$$, f$$, intf$$],
"Input" -> {y$$, ue$$, a$$},
"Output" -> {f$$, intf$$}];
(*copy Acegen variables to Mathematica variables*)
{y1,
y2} \[RightTee] SMSReal[Table[y$$[i], {i, 2}]];
ue \[DoubleRightTee] SMSReal[Table[ue$$[i], {i, 4}]];
ue1 \[DoubleRightTee] SMSReal[ue$$[1]];
ue2 \[DoubleRightTee] SMSReal[ue$$[2]];
ue3 \[DoubleRightTee] SMSReal[ue$$[3]];
ue4 \[DoubleRightTee] SMSReal[ue$$[4]];
a \[DoubleRightTee] SMSReal[a$$];
(*Shape Functions*)
sf \[DoubleRightTee] ConstantArray[1, 4];
sf[[1]] \[DoubleRightTee] 0.25*(1 - y1)*(1 - y2);
sf[[2]] \[DoubleRightTee] 0.25*(1 + y1)*(1 - y2);
sf[[3]] \[DoubleRightTee] 0.25*(1 + y1)*(1 + y2);
sf[[4]] \[DoubleRightTee] 0.25*(1 - y1)*(1 + y2);
(*Field value interpolated at gauss point*)
uGP \[DoubleRightTee] sf . ue;
(*Non-linear function definition*)
f \[DoubleRightTee] Exp[uGP];
(*Integrate the function over the area of the element after restoring y1 and y2 dependencies*)
intf = Integrate[
SMSSmartRestore[f, y1 | y2], {y1, -1, 1}, {y2, -1, 1}];
(*export the output variables/copy mathematica variables to AceGen \
variables*)
SMSExport[f, f$$];
SMSExport[intf, intf$$];
SMSWrite[NAME, "LocalAuxiliaryVariables" -> True];
FilePrint[StringJoin[NAME, ".m"]]
This should technically run since the function is integrable. But the code doesnt give any result. Just says:
Expression contains part/parts that can not be numerically evaluated.
User subroutine: QIntegrate
Error in user input parameters for function: SMSExport
Input parameter: {LARGE EXPRESSION}
Parts that can not be evaluated: {Undefined}
Events: 0
Version: 7.505 Linux (16 Aug 22) (MMA 13.) Module: SMSExport
See also: Symbolic-Numeric Interface AceGen Troubleshooting Continue
I tried many things such as: defining the exponent as a polynomial function instead of a vector multiplication
uGP \[DoubleRightTee]
0.25*(1 - y1)*(1 - y2)*ue1 + 0.25*(1 + y1)*(1 - y2)*ue2 +
0.25*(1 + y1)*(1 + y2)*ue3 + 0.25*(1 - y1)*(1 + y2)*ue4;
using SMSPower instead
f \[DoubleRightTee] SMSPower[E, uGP];
It only gives a result for:
intf = Integrate[f, {y1, -1, 1}, {y2, -1, 1}];
but here it considers the exponent as independent of y1 and y2 and gives out wrong result.
or
f \[DoubleRightTee] Exp[a];
where a is an independent variable.
and surprisingly f=sf works with array output
f \[DoubleRightTee] Exp[sf];
I'm not able to understand the issue. Please suggest a resolution to this. I'm not sure where the problem is occuring and my entire work depends on this. Any help is extremely appreciated. I've also attached the notebook file for referance. Thank You!!Sai Sudhir Chalavadi2023-03-27T14:03:18ZDifferent results when solving a DE using symbolic and numerical methods
https://community.wolfram.com/groups/-/m/t/2858283
I'm getting different results when solving a differential equation using symbolic and numerical methods. Attached is a notebook for the community to evaluate.
Sincerely,
SinvalSinval Santos2023-03-24T16:30:46Z1D HeatTransfer problem reformulation
https://community.wolfram.com/groups/-/m/t/2850551
Hi, I would like to solve the 1D Heat Transfer problem for $ T(x,t) $ on a rod of length $ L $
that has a sinus temperature at one end and is isolated at the other end:
(IBVP 1): $ T_t - a^2 T_xx = 0 $ with
BC: $ T(x=0,t) = T_1 sin(wt), T_x(x=L,t) = 0, $ and IC: $T(x,t=0) = T_0 $
This is easily solved using the NDSolveValue command (see below).
As I am finally interested in an analytic solution of (IBVP 1),
in a frist step I reformulated the initial boundary value problem using
$T(x,t) = U(x,t) + f(t),$ with $ f(t) = T_1 sin(wt) $
arriving at the inhomogeneous (IBVP 2) with homogeneous BCs for $ U(x,t):$
(IBVP 2): $ U_t - a^2 U_xx = Q $ with $ Q=- df(t)/dt,$
BC: $U(x=0,t) = 0, U_x(x=L,t) = 0, $ IC: $ U(x,t=0) = T_0$
To confirm the identity of (IBVP 1) with (IBVP 2), I implemented both with the NDSolve command (see below).
Surprisingly, the solutions are clearly different.
Is there an error in my arguments? Is there a problem with my implementation of the numerical solution in NDSolve?
I noticed that an additional factor of 1000 to the souce term $ Q $ makes both solutions more similar.
This brings me to the question of why the source term (as the time derivative of $f(t)$) is so very small .......?
Analytically, (IBVP 2) could be further analysed by Fourier sin transformation.
If there is a known analytical solution of IBVP 1, I would be very happy for a hint to a reference.
I would be very grateful for any suggestions. Thank you!
Here is my implementation in Mathematica:
&[Wolfram Notebook][1]
[1]: https://www.wolframcloud.com/obj/c732dc56-3d36-4d25-8ab9-3b36a95ad643Uwe Schlink2023-03-13T15:17:40ZDifference between Sum and NSum?
https://community.wolfram.com/groups/-/m/t/2856667
I have to compute this,![image][1]
I have written the Mathematica code, keeping Ta, and Tb as variables. Later we will compute prob[1,0,1,0,Ta,Tb] &[Wolfram Notebook][2]
In[75]:= prob[i_,j_,k_,l_,Ta_,Tb_]:= Sum[Binomial[ip,i]*Ta^i*(1-Ta)^(ip-i)*Binomial[jp,j]*Ta^j*(1-Ta)^(jp-j)*Binomial[kp,k]*Tb^k*(1-Tb)^(kp-k)*Binomial[lp,l]*Tb^l*(1-Tb)^(lp-l),{ip,1,Infinity},{jp,1,Infinity},{kp,1,Infinity},{lp,1,Infinity}];
In[80]:= prob[1,0,1,0,Ta,Tb]
Out[80]= ((-1+Ta) (-1+Tb))/(Ta^2 Tb^2)
My question is,
1. I have seen that if I use NSum in place of sum it takes too much to compute prob[1,0,1,0,Ta,Tb]
what is the difference between sum and NSum here?
2. what will be a good idea to use here sum or NSum as I think Sum is not giving correct answer.(or maybe I am wrong that it is giving the correct answer)
[1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=WhatsAppImage2023-03-22at16.24.07.jpg&userId=2724880
[2]: https://www.wolframcloud.com/obj/9af5c537-ff00-4ece-8a42-74c04684548bIndranil Maiti2023-03-22T15:44:40ZEmpty plot in Manipulate applied at ParametricPlot?
https://community.wolfram.com/groups/-/m/t/2855316
Hello,
I'm new to Mathematica and trying to plot a parametric equation as a solution of a system of differential equations with a parameter "a". I used ParametricNDSolve and then tried using Manipulate to change the parameter "a" with a slider. Nothing shows up on the graph.
I have successfully plotted the graph using a value for "a" and getting rid of Manipulate.
Any advice would be great!
&[Wolfram Notebook][1]
[1]: https://www.wolframcloud.com/obj/19448b1a-1b43-4bde-9594-7920e1d2822fJonathan Baker2023-03-20T01:44:40ZNDSolve error: The function doesn't have the same number of arguments
https://community.wolfram.com/groups/-/m/t/2856683
I have two functions (f[z] and gg[r]), each of one independent variable (z and r), and two ODEs (see file enclosed)
&[Wolfram Notebook][1]
in both these functions and their derivatives. I could specify more boundary conditions if I had to (currently that's not the issue), but Mathematica complains that these two functions are not functions of BOTH r and z. Is this a valid complaint, and if not, how to get around it? I have an old version of Mathematica, version 8, so if the newest version can do this and the old cannot, I will get the newer version. I am trying to solve it numerically with NDSolve, but if there was an algebraic solution that would be a bonus.
[1]: https://www.wolframcloud.com/obj/c656de65-0300-4b9e-ac14-2c6d3bf3d77cIuval Clejan2023-03-22T15:58:26ZHow to plot the result of NDEigensystem
https://community.wolfram.com/groups/-/m/t/2857124
Given the following problem
$-\frac{1}{2}y''(x)+-\frac{1}{2}[x^2+\sin^2{(\frac{2 k \pi x}{L}})]y(x)=E y(x), \quad x \in [-L,L]$
$y(-L)=y(L)=0$ where L=30 and k=2.
We want to calculate the first ten eigenvalues and the 5 first eigenfunctions. Using the command NDEigensystem.
As the following notebook shows, I have defined my equation and my Dirichlet conditions.
Then I used the NDEigensystem command and calculate the first ten eigenvalues. But I cannot figure out how I could plot only the first five eigenfunctions. Any suggestions
&[Wolfram Notebook][1]
[1]: https://www.wolframcloud.com/obj/34cbdfa5-c811-4c46-9448-0bebc3707580Athanasios Paraskevopoulos2023-03-22T09:42:42ZAnalytical and numerical solution of the normalized heat equation problem
https://community.wolfram.com/groups/-/m/t/2856871
My question is the following.
Is the procedure I am following correct?
Because I want to compare the numerical and analytical solutions for a project I am running.
&[Wolfram Notebook][1]
[1]: https://www.wolframcloud.com/obj/4f2d33ae-5ce9-44ec-ad06-e55775754625Athanasios Paraskevopoulos2023-03-21T23:37:13Z