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MaplePrimes Questions

Identifying Series...

Asked by: Don_Caraota 150
sum series summation
August 23 2020
2  3

Hello,

My question is quite simple yet maybe a bit hard since I understand that messing around with series can lead to dead ends.

Given a certain general term, for instance sum(x^n/n!,n=0..infinity): Can Maple recognize which function does it come from (in case such analytic function is known)? in the example I would expect it to return exp(x).

Thank you very much.

Set Maple to degree instead of radian...

Asked by: faizfrhn 25
degree trig radians
August 23 2020
0  3

Hi,

I wonder if we can change the way Maple works so that all calculation can be done in degree instead of radian?

I know that we can *pi/180 to set it to degree but if the calculation lots of sin, cos, tan etc. it would be really hard.

Hope you can help me. Thanks for your participation and assistance.

Thank you.


Kind regards

Faiz Farhan

How to graph function cos(pi/4)?...

Asked by: paulmcquad 60
user-error plotting cos
August 23 2020
0  3

I'm trying to graph this function but it comes out a straight line.

Thanks in Advance.

plot(cos(Pi/4), -2*Pi .. 2*Pi)

plot(cos((1/4)*Pi), -2*Pi .. 2*Pi)

  

``


 

Download Graph-cos.mw

Selecting numbers from random bins...

Asked by: erik10 721
sample statistics
August 23 2020
2  10

Dear people at Mapleprimes

My situation is the following: I have a number of bins (n bins), which all contain a specific number. I want to choose at random from these bins a number of times (N times), but in such a way that it is not equal likely that I choose from bin1, bin2 ...

Example: I have 5 bins: The 5 bins contains the numbers 50, 60, 70, 80, 90 respectively. The probabilities of choosing from these 5 bins should be 0.10, 0.30, 0.20, 0.25, 0.15 respectively. I make N = 100 selections. The output is expected to be a list with 100 numbers, for example: [ 50, 80, 80, 60, 60, 90, 50, ... , 70]. 

I could make it work myself by picking a random number from the uniform distribution from 0 to 1 and deciding if the a random number is between 0 and 0.10, between 0.10 and 0.40, between 0.40 and 0.60, etc. and then choosing the number in the corresponding bin ... I just wonder if Maple has an elegant way to to it? 

Regards,

Erik

 

Why isn't "evalm" working?...

Asked by: Shiryui 5
linalg deprecated
August 23 2020
1  3

Can anyone kindly tell me why isn't "evalm" working even with all the neccessary varaibles have corresponding values? 

Thank you!

Here is my code:

XMLTools / file formatting...

Asked by: Anthrazit 750
xml export xmltools
August 22 2020
0  2

Question regarding output formatting of xml files.

The WriteFile command writes the xmldocument / xmltree out to a file completly, and closes the file. The backdraw is that it isn't formatted nicely.

The PrintToFile command writes out nicely formatted, but the file isn't closed. And the XMLProcessingInstruction is also missing.

If I use Print(xmltree) before the writefile command, I get an error (not a xmltree anymore).

So how do I get nicely formatted XML files?

File open dialogue box?...

Asked by: Anthrazit 750
open maplets
August 22 2020
1  1

Are there any examples of worksheets / workbooks that have a file open / save dialogue box?

I'm playing around trying to save data from a worksheet, and would like to have a dialogue box where the user can pick a directory and a name for storing the data.

Encountered this error---Error, (in Optimization:-...

Asked by: ogunmiloro 120
optimization ode syntax
August 21 2020
0  3


 

> 

restart; with(plots); with(Optimization); with(Statistics)
> 

Optimization[Interactive]();
> 

`Πh` := .1097250; mu := 0.863781636e-1; `Πm` := .15151; `μb` := .3333; k1 := .5905; k2 := .6905; k3 := .6302; omega := .701; Nh := 500; Nb := 500

.1097250

  

0.863781636e-1

  

.15151

  

.3333

  

.5905

  

.6905

  

.6302

  

.701

  

500

  

500

 (1)
> 

times := [5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 255, 260, 265];

[5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 255, 260, 265]

 (2)
> 

C[f] := [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 8, 12, 12, 22, 30, 40, 44, 51, 65, 70, 97, 111, 131, 135, 174, 184, 210, 214, 232, 238, 254, 276, 285, 305, 318, 323, 343, 373, 407, 442, 493, 542, 627, 665, 782, 873, 981, 1095, 1182, 1273, 1337, 1532, 1728, 1932, 2170, 2388, 2558, 2802, 2950, 3145, 3526, 3912, 4151, 4399, 4641, 4787, 4971, 5162, 5445, 5621, 5959, 6175, 6401, 6677, 7016, 7261, 7526, 7839, 8068, 8344, 8733, 8915, 9302, 9855, 10162, 10819, 11166, 11516, 11844, 12233, 12486, 12801, 13464, 13873, 14554, 15181, 15682, 16085, 16658, 17148, 17735]:
> 

DE1 := diff(St(T), T) = `Πh`-a1*`βx`*St(T)*[(k1*IT(T)+k2*Mi(T)+omega(k3*Ib(T)+k2*Mi(T)))/Nh]-(mu+beta2*Ev(T))*St(T)+sigma*Th(T):
> 

res := dsolve({DE1, DE2, DE3, DE4, DE5, DE6, DE7, ics}, parameters = [a1, `βx`, beta2, sigma, ro, `αo`, alpha1, r1, ko, eo, a2, `βy`], numeric);

proc (x_rkf45) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_rkf45) else _xout := evalf(x_rkf45) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _dtbl, _dat, _vmap, _x0, _y0, _val, _dig, _n, _ne, _nd, _nv, _pars, _ini, _par, _i, _j, _k, _src; option `Copyright (c) 2002 by Waterloo Maple Inc. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _pars := [a1 = a1, `&beta;x` = `&beta;x`, beta2 = beta2, sigma = sigma, ro = ro, `&alpha;o` = `&alpha;o`, alpha1 = alpha1, r1 = r1, ko = ko, eo = eo, a2 = a2, `&beta;y` = `&beta;y`]; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 20, [( 1 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 2 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 3 ) = ([0, 0, 0, Array(1..0, 1..2, {}, datatype = float[8], order = C_order)]), ( 4 ) = (Array(1..53, {(1) = 7, (2) = 7, (3) = 0, (4) = 0, (5) = 12, (6) = 0, (7) = 0, (8) = 0, (9) = 0, (10) = 0, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 0, (19) = 30000, (20) = 0, (21) = 0, (22) = 1, (23) = 4, (24) = 0, (25) = 1, (26) = 15, (27) = 1, (28) = 0, (29) = 1, (30) = 3, (31) = 3, (32) = 0, (33) = 1, (34) = 0, (35) = 0, (36) = 0, (37) = 0, (38) = 0, (39) = 0, (40) = 0, (41) = 0, (42) = 0, (43) = 1, (44) = 0, (45) = 0, (46) = 0, (47) = 0, (48) = 0, (49) = 0, (50) = 50, (51) = 1, (52) = 0, (53) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = .0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = .0, (7) = .0, (8) = 0.10e-5, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = 1.0, (14) = .0, (15) = .49999999999999, (16) = .0, (17) = 1.0, (18) = 1.0, (19) = .0, (20) = .0, (21) = 1.0, (22) = 1.0, (23) = .0, (24) = .0, (25) = 0.10e-14, (26) = .0, (27) = .0, (28) = .0}, datatype = float[8], order = C_order)), ( 6 ) = (Array(1..19, {(1) = 1240., (2) = 1380., (3) = 1245., (4) = 1300., (5) = 1400., (6) = 1500., (7) = 1270., (8) = Float(undefined), (9) = Float(undefined), (10) = Float(undefined), (11) = Float(undefined), (12) = Float(undefined), (13) = Float(undefined), (14) = Float(undefined), (15) = Float(undefined), (16) = Float(undefined), (17) = Float(undefined), (18) = Float(undefined), (19) = Float(undefined)})), ( 7 ) = ([Array(1..4, 1..7, {(1, 1) = .0, (1, 2) = .203125, (1, 3) = .3046875, (1, 4) = .75, (1, 5) = .8125, (1, 6) = .40625, (1, 7) = .8125, (2, 1) = 0.6378173828125e-1, (2, 2) = .0, (2, 3) = .279296875, (2, 4) = .27237892150878906, (2, 5) = -0.9686851501464844e-1, (2, 6) = 0.1956939697265625e-1, (2, 7) = .5381584167480469, (3, 1) = 0.31890869140625e-1, (3, 2) = .0, (3, 3) = -.34375, (3, 4) = -.335235595703125, (3, 5) = .2296142578125, (3, 6) = .41748046875, (3, 7) = 11.480712890625, (4, 1) = 0.9710520505905151e-1, (4, 2) = .0, (4, 3) = .40350341796875, (4, 4) = 0.20297467708587646e-1, (4, 5) = -0.6054282188415527e-2, (4, 6) = -0.4770040512084961e-1, (4, 7) = .77858567237854}, datatype = float[8], order = C_order), Array(1..6, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = 1.0, (2, 1) = .25, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = 1.0, (3, 1) = .1875, (3, 2) = .5625, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = 2.0, (4, 1) = .23583984375, (4, 2) = -.87890625, (4, 3) = .890625, (4, 4) = .0, (4, 5) = .0, (4, 6) = .2681884765625, (5, 1) = .1272735595703125, (5, 2) = -.5009765625, (5, 3) = .44921875, (5, 4) = -0.128936767578125e-1, (5, 5) = .0, (5, 6) = 0.626220703125e-1, (6, 1) = -0.927734375e-1, (6, 2) = .626220703125, (6, 3) = -.4326171875, (6, 4) = .1418304443359375, (6, 5) = -0.861053466796875e-1, (6, 6) = .3131103515625}, datatype = float[8], order = C_order), Array(1..6, {(1) = .0, (2) = .386, (3) = .21, (4) = .63, (5) = 1.0, (6) = 1.0}, datatype = float[8], order = C_order), Array(1..6, {(1) = .25, (2) = -.1043, (3) = .1035, (4) = -0.362e-1, (5) = .0, (6) = .0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 1.544, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = .9466785280815533, (3, 2) = .25570116989825814, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = 3.3148251870684886, (4, 2) = 2.896124015972123, (4, 3) = .9986419139977808, (4, 4) = .0, (4, 5) = .0, (5, 1) = 1.2212245092262748, (5, 2) = 6.019134481287752, (5, 3) = 12.537083329320874, (5, 4) = -.687886036105895, (5, 5) = .0, (6, 1) = 1.2212245092262748, (6, 2) = 6.019134481287752, (6, 3) = 12.537083329320874, (6, 4) = -.687886036105895, (6, 5) = 1.0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = -5.6688, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = -2.4300933568337584, (3, 2) = -.20635991570891224, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = -.10735290581452621, (4, 2) = -9.594562251021896, (4, 3) = -20.470286148096154, (4, 4) = .0, (4, 5) = .0, (5, 1) = 7.496443313968615, (5, 2) = -10.246804314641219, (5, 3) = -33.99990352819906, (5, 4) = 11.708908932061595, (5, 5) = .0, (6, 1) = 8.083246795922411, (6, 2) = -7.981132988062785, (6, 3) = -31.52159432874373, (6, 4) = 16.319305431231363, (6, 5) = -6.0588182388340535}, datatype = float[8], order = C_order), Array(1..3, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 10.126235083446911, (2, 2) = -7.487995877607633, (2, 3) = -34.800918615557414, (2, 4) = -7.9927717075687275, (2, 5) = 1.0251377232956207, (3, 1) = -.6762803392806898, (3, 2) = 6.087714651678606, (3, 3) = 16.43084320892463, (3, 4) = 24.767225114183653, (3, 5) = -6.5943891257167815}, datatype = float[8], order = C_order)]), ( 9 ) = ([Array(1..7, {(1) = .1, (2) = .1, (3) = .1, (4) = .1, (5) = .1, (6) = .1, (7) = .1}, datatype = float[8], order = C_order), Array(1..7, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0}, datatype = float[8], order = C_order), Array(1..7, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0}, datatype = float[8], order = C_order), Array(1..7, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0}, datatype = float[8], order = C_order), Array(1..7, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0}, datatype = float[8], order = C_order), Array(1..7, 1..7, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (1, 7) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0, (2, 7) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = .0, (3, 7) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (4, 4) = .0, (4, 5) = .0, (4, 6) = .0, (4, 7) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = .0, (5, 4) = .0, (5, 5) = .0, (5, 6) = .0, (5, 7) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = .0, (6, 4) = .0, (6, 5) = .0, (6, 6) = .0, (6, 7) = .0, (7, 1) = .0, (7, 2) = .0, (7, 3) = .0, (7, 4) = .0, (7, 5) = .0, (7, 6) = .0, (7, 7) = .0}, datatype = float[8], order = C_order), Array(1..7, 1..7, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (1, 7) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0, (2, 7) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = .0, (3, 7) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (4, 4) = .0, (4, 5) = .0, (4, 6) = .0, (4, 7) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = .0, (5, 4) = .0, (5, 5) = .0, (5, 6) = .0, (5, 7) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = .0, (6, 4) = .0, (6, 5) = .0, (6, 6) = .0, (6, 7) = .0, (7, 1) = .0, (7, 2) = .0, (7, 3) = .0, (7, 4) = .0, (7, 5) = .0, (7, 6) = .0, (7, 7) = .0}, datatype = float[8], order = C_order), Array(1..7, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (4, 4) = .0, (4, 5) = .0, (4, 6) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = .0, (5, 4) = .0, (5, 5) = .0, (5, 6) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = .0, (6, 4) = .0, (6, 5) = .0, (6, 6) = .0, (7, 1) = .0, (7, 2) = .0, (7, 3) = .0, (7, 4) = .0, (7, 5) = .0, (7, 6) = .0}, datatype = float[8], order = C_order), Array(1..7, {(1) = 0, (2) = 0, (3) = 0, (4) = 0, (5) = 0, (6) = 0, (7) = 0}, datatype = integer[8]), Array(1..19, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = .0, (14) = .0, (15) = .0, (16) = .0, (17) = .0, (18) = .0, (19) = .0}, datatype = float[8], order = C_order), Array(1..19, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = .0, (14) = .0, (15) = .0, (16) = .0, (17) = .0, (18) = .0, (19) = .0}, datatype = float[8], order = C_order), Array(1..19, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = .0, (14) = .0, (15) = .0, (16) = .0, (17) = .0, (18) = .0, (19) = .0}, datatype = float[8], order = C_order), Array(1..19, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = .0, (14) = .0, (15) = .0, (16) = .0, (17) = .0, (18) = .0, (19) = .0}, datatype = float[8], order = C_order)]), ( 8 ) = ([Array(1..19, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = .0, (14) = .0, (15) = .0, (16) = .0, (17) = .0, (18) = .0, (19) = .0}, datatype = float[8], order = C_order), Array(1..19, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = .0, (14) = .0, (15) = .0, (16) = .0, (17) = .0, (18) = .0, (19) = .0}, datatype = float[8], order = C_order), Array(1..7, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0}, datatype = float[8], order = C_order)]), ( 11 ) = (Array(1..6, 0..7, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (1, 7) = .0, (2, 0) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0, (2, 7) = .0, (3, 0) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = .0, (3, 7) = .0, (4, 0) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (4, 4) = .0, (4, 5) = .0, (4, 6) = .0, (4, 7) = .0, (5, 0) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = .0, (5, 4) = .0, (5, 5) = .0, (5, 6) = .0, (5, 7) = .0, (6, 0) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = .0, (6, 4) = .0, (6, 5) = .0, (6, 6) = .0, (6, 7) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = Ev(T), Y[2] = IT(T), Y[3] = Ib(T), Y[4] = Mi(T), Y[5] = Ms(T), Y[6] = St(T), Y[7] = Th(T)]`; YP[1] := -Y[1]*Y[17]+Y[2]*Y[13]+Y[3]*Y[14]; YP[2] := Y[8]*Y[9]*Y[6]*[0.1181000000e-2*Y[2]+0.1381000000e-2*Y[4]]-(0.863781636e-1+Y[12]+Y[13])*Y[2]; YP[3] := Y[8]*Y[9]*Y[6]*[0.8835404000e-3*Y[3]+0.9680810000e-3*Y[4]]-(0.863781636e-1+Y[15]+Y[14]+Y[16])*Y[3]; YP[4] := (1/500)*Y[18]*Y[19]*Y[5]*(.5905*Y[2]+.6905*Y[3]+Y[1])-.3333*Y[4]; YP[5] := .15151-(1/500)*Y[18]*Y[19]*Y[5]*(.5905*Y[2]+.6905*Y[3]+Y[1])-.3333*Y[5]; YP[6] := .1097250-Y[8]*Y[9]*Y[6]*[0.1181000000e-2*Y[2]+0.1381000000e-2*Y[4]+0.1402000000e-2]-(0.863781636e-1+Y[10]*Y[1])*Y[6]+Y[11]*Y[7]; YP[7] := Y[12]*Y[2]+Y[15]*Y[3]-(0.863781636e-1+Y[11])*Y[7]; 0 end proc, -1, 0, 0, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = Ev(T), Y[2] = IT(T), Y[3] = Ib(T), Y[4] = Mi(T), Y[5] = Ms(T), Y[6] = St(T), Y[7] = Th(T)]`; YP[1] := -Y[1]*Y[17]+Y[2]*Y[13]+Y[3]*Y[14]; YP[2] := Y[8]*Y[9]*Y[6]*[0.1181000000e-2*Y[2]+0.1381000000e-2*Y[4]]-(0.863781636e-1+Y[12]+Y[13])*Y[2]; YP[3] := Y[8]*Y[9]*Y[6]*[0.8835404000e-3*Y[3]+0.9680810000e-3*Y[4]]-(0.863781636e-1+Y[15]+Y[14]+Y[16])*Y[3]; YP[4] := (1/500)*Y[18]*Y[19]*Y[5]*(.5905*Y[2]+.6905*Y[3]+Y[1])-.3333*Y[4]; YP[5] := .15151-(1/500)*Y[18]*Y[19]*Y[5]*(.5905*Y[2]+.6905*Y[3]+Y[1])-.3333*Y[5]; YP[6] := .1097250-Y[8]*Y[9]*Y[6]*[0.1181000000e-2*Y[2]+0.1381000000e-2*Y[4]+0.1402000000e-2]-(0.863781636e-1+Y[10]*Y[1])*Y[6]+Y[11]*Y[7]; YP[7] := Y[12]*Y[2]+Y[15]*Y[3]-(0.863781636e-1+Y[11])*Y[7]; 0 end proc, -1, 0, 0, 0, 0, 0, 0]), ( 20 ) = ([]) ] )) ] ); _y0 := Array(0..19, {(1) = 0., (2) = 1240., (3) = 1380., (4) = 1245., (5) = 1300., (6) = 1400., (7) = 1500., (8) = 1270., (9) = undefined, (10) = undefined, (11) = undefined, (12) = undefined, (13) = undefined, (14) = undefined, (15) = undefined, (16) = undefined, (17) = undefined, (18) = undefined, (19) = undefined}); _vmap := array( 1 .. 7, [( 1 ) = (1), ( 2 ) = (2), ( 3 ) = (3), ( 4 ) = (4), ( 5 ) = (5), ( 6 ) = (6), ( 7 ) = (7) ] ); _x0 := _dtbl[1][5][5]; _n := _dtbl[1][4][1]; _ne := _dtbl[1][4][3]; _nd := _dtbl[1][4][4]; _nv := _dtbl[1][4][16]; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then if _Env_smart_dsolve_numeric = true or _dtbl[1][4][10] = 1 then if _xout = "left" then if type(_dtbl[2], 'table') then return _dtbl[2][5][1] end if elif _xout = "right" then if type(_dtbl[3], 'table') then return _dtbl[3][5][1] end if end if end if; return _dtbl[1][5][5] elif _xout = "method" then return _dtbl[1][15] elif _xout = "storage" then return evalb(_dtbl[1][4][10] = 1) elif _xout = "leftdata" then if not type(_dtbl[2], 'array') then return NULL else return eval(_dtbl[2]) end if elif _xout = "rightdata" then if not type(_dtbl[3], 'array') then return NULL else return eval(_dtbl[3]) end if elif _xout = "enginedata" then return eval(_dtbl[1]) elif _xout = "enginereset" then _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); return NULL elif _xout = "initial" then return procname(_y0[0]) elif _xout = "laxtol" then return _dtbl[`if`(member(_dtbl[4], {2, 3}), _dtbl[4], 1)][5][18] elif _xout = "numfun" then return `if`(member(_dtbl[4], {2, 3}), _dtbl[_dtbl[4]][4][18], 0) elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return procname(_y0[0]), [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _dtbl[4] <> 2 and _dtbl[4] <> 3 or _x0-_dtbl[_dtbl[4]][5][1] = 0. then error "no information is available on last computed point" else _xout := _dtbl[_dtbl[4]][5][1] end if elif _xout = "function" then if _dtbl[1][4][33]-2. = 0 then return eval(_dtbl[1][10], 1) else return eval(_dtbl[1][10][1], 1) end if elif _xout = "map" then return copy(_vmap) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); if _par <> [] then `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) end if; `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars); if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') and _dtbl[1][4][10] <> 1 then procname("right") := _y0[0]; procname("left") := _y0[0] end if; if _xout = "initial" then return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)] elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] else return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)], [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] end if elif _xin = "eventstop" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then return 0 end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 10 and 10 <= _dtbl[5-_i][4][9] then _i := 5-_i; _dtbl[4] := _i; _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) elif 10 <= _dtbl[_i][4][9] then _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) else return 0 end if elif _xin = "eventstatus" then if _nv = 0 then error "this solution has no events" end if; _i := [selectremove(proc (a) options operator, arrow; _dtbl[1][3][1][a, 7] = 1 end proc, {seq(_j, _j = 1 .. round(_dtbl[1][3][1][_nv+1, 1]))})]; return ':-enabled' = _i[1], ':-disabled' = _i[2] elif _xin = "eventclear" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then error "no events to clear" end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 10 and 10 < _dtbl[5-_i][4][9] then _dtbl[4] := 5-_i; _i := 5-_i end if; if _dtbl[_i][4][9] < 10 then error "no events to clear" elif _nv < _dtbl[_i][4][9]-10 then error "event error condition cannot be cleared" else _j := _dtbl[_i][4][9]-10; if irem(round(_dtbl[_i][3][1][_j, 4]), 2) = 1 then error "retriggerable events cannot be cleared" end if; _j := round(_dtbl[_i][3][1][_j, 1]); for _k to _nv do if _dtbl[_i][3][1][_k, 1] = _j then if _dtbl[_i][3][1][_k, 2] = 3 then error "range events cannot be cleared" end if; _dtbl[_i][3][1][_k, 8] := _dtbl[_i][3][1][_nv+1, 8] end if end do; _dtbl[_i][4][17] := 0; _dtbl[_i][4][9] := 0; if _dtbl[1][4][10] = 1 then if _i = 2 then try procname(procname("left")) catch: end try else try procname(procname("right")) catch: end try end if end if end if; return elif type(_xin, `=`) and member(lhs(_xin), {"eventdisable", "eventenable"}) then if _nv = 0 then error "this solution has no events" end if; if type(rhs(_xin), {('list')('posint'), ('set')('posint')}) then _i := {op(rhs(_xin))} elif type(rhs(_xin), 'posint') then _i := {rhs(_xin)} else error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; if select(proc (a) options operator, arrow; _nv < a end proc, _i) <> {} then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _k := {}; for _j to _nv do if member(round(_dtbl[1][3][1][_j, 1]), _i) then _k := `union`(_k, {_j}) end if end do; _i := _k; if lhs(_xin) = "eventdisable" then _dtbl[4] := 0; _j := [evalb(assigned(_dtbl[2]) and member(_dtbl[2][4][17], _i)), evalb(assigned(_dtbl[3]) and member(_dtbl[3][4][17], _i))]; for _k in _i do _dtbl[1][3][1][_k, 7] := 0; if assigned(_dtbl[2]) then _dtbl[2][3][1][_k, 7] := 0 end if; if assigned(_dtbl[3]) then _dtbl[3][3][1][_k, 7] := 0 end if end do; if _j[1] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[2][3][4][_k, 1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to defined init `, _dtbl[2][3][4][_k, 1]); _dtbl[2][3][1][_k, 8] := _dtbl[2][3][4][_k, 1] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to rate hysteresis init `, _dtbl[2][5][24]); _dtbl[2][3][1][_k, 8] := _dtbl[2][5][24] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to initial init `, _x0); _dtbl[2][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to fireinitial init `, _x0-1); _dtbl[2][3][1][_k, 8] := _x0-1 end if end do; _dtbl[2][4][17] := 0; _dtbl[2][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("left")) end if end if; if _j[2] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[3][3][4][_k, 2], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to defined init `, _dtbl[3][3][4][_k, 2]); _dtbl[3][3][1][_k, 8] := _dtbl[3][3][4][_k, 2] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to rate hysteresis init `, _dtbl[3][5][24]); _dtbl[3][3][1][_k, 8] := _dtbl[3][5][24] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to initial init `, _x0); _dtbl[3][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to fireinitial init `, _x0+1); _dtbl[3][3][1][_k, 8] := _x0+1 end if end do; _dtbl[3][4][17] := 0; _dtbl[3][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("right")) end if end if else for _k in _i do _dtbl[1][3][1][_k, 7] := 1 end do; _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); _dtbl[4] := 0; if _dtbl[1][4][10] = 1 then if _x0 <= procname("right") then try procname(procname("right")) catch: end try end if; if procname("left") <= _x0 then try procname(procname("left")) catch: end try end if end if end if; return elif type(_xin, `=`) and lhs(_xin) = "eventfired" then if not type(rhs(_xin), 'list') then error "'eventfired' must be specified as a list" end if; if _nv = 0 then error "this solution has no events" end if; if _dtbl[4] <> 2 and _dtbl[4] <> 3 then error "'direction' must be set prior to calling/setting 'eventfired'" end if; _i := _dtbl[4]; _val := NULL; if not assigned(_EnvEventRetriggerWarned) then _EnvEventRetriggerWarned := false end if; for _k in rhs(_xin) do if type(_k, 'integer') then _src := _k elif type(_k, 'integer' = 'anything') and type(evalf(rhs(_k)), 'numeric') then _k := lhs(_k) = evalf[max(Digits, 18)](rhs(_k)); _src := lhs(_k) else error "'eventfired' entry is not valid: %1", _k end if; if _src < 1 or round(_dtbl[1][3][1][_nv+1, 1]) < _src then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _src := {seq(`if`(_dtbl[1][3][1][_j, 1]-_src = 0., _j, NULL), _j = 1 .. _nv)}; if nops(_src) <> 1 then error "'eventfired' can only be set/queried for root-finding events and time/interval events" end if; _src := _src[1]; if _dtbl[1][3][1][_src, 2] <> 0. and _dtbl[1][3][1][_src, 2]-2. <> 0. then error "'eventfired' can only be set/queried for root-finding events and time/interval events" elif irem(round(_dtbl[1][3][1][_src, 4]), 2) = 1 then if _EnvEventRetriggerWarned = false then WARNING(`'eventfired' has no effect on events that retrigger`) end if; _EnvEventRetriggerWarned := true end if; if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then _val := _val, undefined elif type(_dtbl[_i][3][4][_src, _i-1], 'undefined') or _i = 2 and _dtbl[2][3][1][_src, 8] < _dtbl[2][3][4][_src, 1] or _i = 3 and _dtbl[3][3][4][_src, 2] < _dtbl[3][3][1][_src, 8] then _val := _val, _dtbl[_i][3][1][_src, 8] else _val := _val, _dtbl[_i][3][4][_src, _i-1] end if; if type(_k, `=`) then if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then error "cannot set event code for a rate hysteresis event" end if; userinfo(3, {'events', 'eventreset'}, `manual set event code `, _src, ` to value `, rhs(_k)); _dtbl[_i][3][1][_src, 8] := rhs(_k); _dtbl[_i][3][4][_src, _i-1] := rhs(_k) end if end do; return [_val] elif type(_xin, `=`) and lhs(_xin) = "direction" then if not member(rhs(_xin), {-1, 1, ':-left', ':-right'}) then error "'direction' must be specified as either '1' or 'right' (positive) or '-1' or 'left' (negative)" end if; _src := `if`(_dtbl[4] = 2, -1, `if`(_dtbl[4] = 3, 1, undefined)); _i := `if`(member(rhs(_xin), {1, ':-right'}), 3, 2); _dtbl[4] := _i; _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if; return _src elif _xin = "eventcount" then if _dtbl[1][3][1] = 0 or _dtbl[4] <> 2 and _dtbl[4] <> 3 then return 0 else return round(_dtbl[_dtbl[4]][3][1][_nv+1, 12]) end if else return "procname" end if end if; if _xout = _x0 then return [_x0, seq(evalf(_dtbl[1][6][_vmap[_i]]), _i = 1 .. _n-_ne)] end if; _i := `if`(_x0 <= _xout, 3, 2); if _xin = "last" and 0 < _dtbl[_i][4][9] and _dtbl[_i][4][9] < 10 then _dat := eval(_dtbl[_i], 2); _j := _dat[4][20]; return [_dat[11][_j, 0], seq(_dat[11][_j, _vmap[_i]], _i = 1 .. _n-_ne-_nd), seq(_dat[8][1][_vmap[_i]], _i = _n-_ne-_nd+1 .. _n-_ne)] end if; if not type(_dtbl[_i], 'array') then _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if end if; if _xin <> "last" then if 0 < 0 then if `dsolve/numeric/checkglobals`(op(_dtbl[1][14]), _pars, _n, _y0) then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars, _i) end if end if; if _dtbl[1][4][7] = 0 then error "parameters must be initialized before solution can be computed" end if end if; _dat := eval(_dtbl[_i], 2); _dtbl[4] := _i; try _src := `dsolve/numeric/SC/IVPrun`(_dat, _xout) catch: userinfo(2, `dsolve/debug`, print(`Exception in solnproc:`, [lastexception][2 .. -1])); error end try; if _src = 0 and 10 < _dat[4][9] then _val := _dat[3][1][_nv+1, 8] else _val := _dat[11][_dat[4][20], 0] end if; if _src <> 0 or _dat[4][9] <= 0 then _dtbl[1][5][1] := _xout else _dtbl[1][5][1] := _val end if; if _i = 3 and _val < _xout then Rounding := -infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further right of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further right of %1, maxfun limit exceeded (see <a href='http://www.maplesoft.com/support/help/search.aspx?term=dsolve,maxfun' target='_new'>?dsolve,maxfun</a> for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further right of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further right of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif 10 < _dat[4][9] then if _dat[4][9]-10 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-10 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-10 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further right of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-10, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further right of %1", evalf[8](_val) end if elif _i = 2 and _xout < _val then Rounding := infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further left of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further left of %1, maxfun limit exceeded (see <a href='http://www.maplesoft.com/support/help/search.aspx?term=dsolve,maxfun' target='_new'>?dsolve,maxfun</a> for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further left of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further left of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif 10 < _dat[4][9] then if _dat[4][9]-10 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-10 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-10 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further left of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-10, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further left of %1", evalf[8](_val) end if end if; if _EnvInFsolve = true then _dig := _dat[4][26]; _dat[4][26] := _EnvDSNumericSaveDigits; _Env_dsolve_SC_native := true; if _dat[4][25] = 1 then _i := 1; _dat[4][25] := 2 else _i := _dat[4][25] end if; _val := `dsolve/numeric/SC/IVPval`(_dat, _xout, _src); _dat[4][25] := _i; _dat[4][26] := _dig; [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] else Digits := _dat[4][26]; _val := `dsolve/numeric/SC/IVPval`(eval(_dat, 2), _xout, _src); [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] end if end proc, (2) = Array(0..0, {}), (3) = [T, Ev(T), IT(T), Ib(T), Mi(T), Ms(T), St(T), Th(T)], (4) = [a1 = a1, `&beta;x` = `&beta;x`, beta2 = beta2, sigma = sigma, ro = ro, `&alpha;o` = `&alpha;o`, alpha1 = alpha1, r1 = r1, ko = ko, eo = eo, a2 = a2, `&beta;y` = `&beta;y`]}); _vars := _dat[3]; _pars := map(rhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_rkf45, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_rkf45, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_rkf45, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_rkf45, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_rkf45), 'string') = rhs(x_rkf45); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_rkf45), 'string') = rhs(x_rkf45)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_rkf45) else _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_rkf45) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error end try end proc

 (3)
> 

sse := proc (a1, `&beta;x`, beta2, sigma, ro, `&alpha;o`, alpha1, r1, ko, eo, a2, `&beta;y`) local testRes; res(parameters = [a1, `&beta;x`, beta2, sigma, ro, `&alpha;o`, alpha1, r1, ko, eo, a2, `&beta;y`]); try testRes := add((C__f[i]-rhs(res(times[i])[3]))^2, i = 1 .. numelems(times)) catch "cannot evaluate the solution": testRes := 1000000000000 end try; return testRes end proc
> 

sse := proc (a1, `&beta;x`, beta2, sigma, ro, `&alpha;o`, alpha1, r1, ko, eo, a2, `&beta;y`) local testRes; res(parameters = [a1, `&beta;x`, beta2, sigma, ro, `&alpha;o`, alpha1, r1, ko, eo, a2, `&beta;y`]); try testRes := add((C__f[i]-rhs(res(times[i])[3]))^2, i = 1 .. numelems(times)) catch "cannot evaluate the solution": testRes := 1000000000000 end try; return testRes end proc:
> 

optPars := Minimize(('sse')(a1, `&beta;x`, beta2, sigma, ro, `&alpha;o`, alpha1, r1, ko, eo, a2, `&beta;y`), initialpoint = {`&beta;x` = 10.021, `&beta;y` = 10.0212, a1 = 10.0114, a2 = 10.0119, alpha1 = 10.0432, beta2 = 10.0111, eo = 10.0114, ko = 10.0211, r1 = 10.0651, ro = 10.0601, sigma = 10.01030, `&alpha;o` = 10.0431}, assume = nonnegative, optimalitytolerance = 0.1e-4)

Error, (in Optimization:-NLPSolve) could not store [HFloat(515427.81882622803)]-HFloat(27861.617865768) in a floating-point rtable

  
> 

Statistics:-ExponentialFit([`$`(1 .. nops(C__f))], C__f, T)

Error, (in Statistics:-ExponentialFit) invalid input: for this calling sequence, use a two-column Matrix

  
> 

``


 

Download wrktrc.mw

How does one reference spacetime-metrics in Physic...

Asked by: prldb 50
metric physics
August 21 2020
1  2

The Help page Physics/tensors-a complete guide states that spacetime metrics from Kramer et al. are referenced by chapter, section, and equation number, e.g., g_[[12, 16, 1]]. But there is no section 16 in Ch 12 and equations within each chapter are numbered sequentially without reference to section. By playing around it seems that in fact the first number is chapter, the second number is equation number, and the third number refers to subcases of the metric, when they are specified in the text. Is that correct?

Also, the output I get from say g_[[27, 27, 1]], or any other attempt  made, is just the metric, without any specification of the coordinates etc, which the Help pages susggest should be part of the output.

From Fortran to Maple...

Asked by: Gonzalo Garcia 150
programming
August 21 2020
1  28

Hi!

The algorithm in this PDF is written in Fortran, but unfortunately I do not known this programming languaje (actually, I am not an "expert" in progamation).

Algoritmo_Fortram.pdf  [removed by moderator. © Institute of Mathematics AS CR, 1980]

 

https://dml.cz/handle/10338.dmlcz/103855

Could someone please write this algorithm in Maple? Or, at least, indicate me how to do.

Many thanks in advance for your comments.

Unstability of the eigenvectors of some general ma...

Asked by: petit loup 40
eigenvectors
August 21 2020
0  9
 

 Good morning all.
Consider $A=[a_{i,j}]$, a $18\times 18$ matrix (for example); the integer entries are randomly chosen in $[[-5,5]]$ (for example). In general, $A$ has distinct eigenvalues.

I seek the eigen-elements of $B=Transpose(A^{-1})A$ with $10$ significand digits (for example); in general, $B$ is diagonalizable ($P^{-1}BP=D$, a diagonal complex matrix). I use the command $evalf(Eigenvectors(?))$.

Many randomized tests require working with hundreds of Digits. The worst one requires $629$ digits!! Moreover (in this test), when $Digits:=400$, the condition number of $P$ is $10^{118}$ and , with $Digits:=619$, the condition number drops to $376$.

I am surprised by this instability. In particular, this method seems to be unusable when $n=100$.
Does there exist a method (using maple) which allows to solve the problem without dragging behind me a multitude of digits ? (perhaps with iterations...)
Thanks in advance.

 

solving system of partial differential equation...

Asked by: bikramphy 70
pde pdsolve differential-equations
August 21 2020
0  0

I want to to solve the system of partial differential equation using maple. I tried it but I am not able to solve it ... please help.

the equations are as follows

 


 

``

Finding transformation eqn between zero and harmonic with conformal1

``

 

> 

restart

``

> 

with(PDEtools)
> 

sys := {(diff(Phi(r1, r2, r4), r1))^2-(diff(R(r1, r2, r4), r1))^2 = cos(T(r1, r2, r4))^2, (diff(Phi(r1, r2, r4), r2))^2-(diff(R(r1, r2, r4), r2))^2 = -cos(T(r1, r2, r4))^2, -(diff(R(r1, r2, r4), r2))*(diff(R(r1, r2, r4), r1))+(diff(Phi(r1, r2, r4), r2))*(diff(Phi(r1, r2, r4), r1)) = 0, -(diff(R(r1, r2, r4), r4))*(diff(R(r1, r2, r4), r1))+(diff(Theta(r1, r2, r4), r1))*(diff(T(r1, r2, r4), r4))+(diff(Phi(r1, r2, r4), r4))*(diff(Phi(r1, r2, r4), r1)) = 0, -(diff(R(r1, r2, r4), r4))*(diff(R(r1, r2, r4), r2))+(diff(Theta(r1, r2, r4), r2))*(diff(T(r1, r2, r4), r4))+(diff(Phi(r1, r2, r4), r4))*(diff(Phi(r1, r2, r4), r2)) = 0, (R(r1, r2, r4)^2-Phi(r1, r2, r4)^2)*(diff(T(r1, r2, r4), r4))^2+2*(diff(Theta(r1, r2, r4), r4))*(diff(T(r1, r2, r4), r4))+(diff(Phi(r1, r2, r4), r4))^2-(diff(R(r1, r2, r4), r4))^2 = 0, diff(T(r1, r2, r4), r1) = 0, diff(T(r1, r2, r4), r2) = 0, diff(T(r1, r2, r4), r4) = cos(T(r1, r2, r4))^2}

{(diff(Phi(r1, r2, r4), r1))^2-(diff(R(r1, r2, r4), r1))^2 = cos(T(r1, r2, r4))^2, (diff(Phi(r1, r2, r4), r2))^2-(diff(R(r1, r2, r4), r2))^2 = -cos(T(r1, r2, r4))^2, -(diff(R(r1, r2, r4), r2))*(diff(R(r1, r2, r4), r1))+(diff(Phi(r1, r2, r4), r2))*(diff(Phi(r1, r2, r4), r1)) = 0, -(diff(R(r1, r2, r4), r4))*(diff(R(r1, r2, r4), r1))+(diff(Theta(r1, r2, r4), r1))*(diff(T(r1, r2, r4), r4))+(diff(Phi(r1, r2, r4), r4))*(diff(Phi(r1, r2, r4), r1)) = 0, -(diff(R(r1, r2, r4), r4))*(diff(R(r1, r2, r4), r2))+(diff(Theta(r1, r2, r4), r2))*(diff(T(r1, r2, r4), r4))+(diff(Phi(r1, r2, r4), r4))*(diff(Phi(r1, r2, r4), r2)) = 0, (R(r1, r2, r4)^2-Phi(r1, r2, r4)^2)*(diff(T(r1, r2, r4), r4))^2+2*(diff(Theta(r1, r2, r4), r4))*(diff(T(r1, r2, r4), r4))+(diff(Phi(r1, r2, r4), r4))^2-(diff(R(r1, r2, r4), r4))^2 = 0, diff(T(r1, r2, r4), r1) = 0, diff(T(r1, r2, r4), r2) = 0, diff(T(r1, r2, r4), r4) = cos(T(r1, r2, r4))^2}

 (1)

``

> 

declare(Phi(r1, r2, r4), R(r1, r2, r4), T(r1, r2, r4), Theta(r1, r2, r4))

` Phi`(r1, r2, r4)*`will now be displayed as`*Phi

  

` R`(r1, r2, r4)*`will now be displayed as`*R

  

` T`(r1, r2, r4)*`will now be displayed as`*T

  

` Theta`(r1, r2, r4)*`will now be displayed as`*Theta

 (2)

``

> 

cases := [PDEtools:-casesplit({(diff(Phi(r1, r2, r4), r1))^2-(diff(R(r1, r2, r4), r1))^2 = cos(T(r1, r2, r4))^2, (diff(Phi(r1, r2, r4), r2))^2-(diff(R(r1, r2, r4), r2))^2 = -cos(T(r1, r2, r4))^2, -(diff(R(r1, r2, r4), r2))*(diff(R(r1, r2, r4), r1))+(diff(Phi(r1, r2, r4), r2))*(diff(Phi(r1, r2, r4), r1)) = 0, -(diff(R(r1, r2, r4), r4))*(diff(R(r1, r2, r4), r1))+(diff(Theta(r1, r2, r4), r1))*(diff(T(r1, r2, r4), r4))+(diff(Phi(r1, r2, r4), r4))*(diff(Phi(r1, r2, r4), r1)) = 0, -(diff(R(r1, r2, r4), r4))*(diff(R(r1, r2, r4), r2))+(diff(Theta(r1, r2, r4), r2))*(diff(T(r1, r2, r4), r4))+(diff(Phi(r1, r2, r4), r4))*(diff(Phi(r1, r2, r4), r2)) = 0, (R(r1, r2, r4)^2-Phi(r1, r2, r4)^2)*(diff(T(r1, r2, r4), r4))^2+2*(diff(Theta(r1, r2, r4), r4))*(diff(T(r1, r2, r4), r4))+(diff(Phi(r1, r2, r4), r4))^2-(diff(R(r1, r2, r4), r4))^2 = 0, diff(T(r1, r2, r4), r1) = 0, diff(T(r1, r2, r4), r2) = 0, diff(T(r1, r2, r4), r4) = cos(T(r1, r2, r4))^2}, caseplot)]

`========= Pivots Legend =========`

  

p1 = diff(R(r1, r2, r4), r2)

  

p2 = diff(Phi(r1, r2, r4), r1)

  

p3 = (diff(Phi(r1, r2, r4), r2))^2+cos(T(r1, r2, r4))^2

  

p4 = diff(Phi(r1, r2, r4), r2)

  

p5 = diff(R(r1, r2, r4), r1)

  

 

[`casesplit/ans`([diff(Theta(r1, r2, r4), r4) = (1/2)*(Phi(r1, r2, r4)^2*cos(T(r1, r2, r4))^4-R(r1, r2, r4)^2*cos(T(r1, r2, r4))^4-(diff(Phi(r1, r2, r4), r4))^2+(diff(R(r1, r2, r4), r4))^2)/cos(T(r1, r2, r4))^2, diff(Theta(r1, r2, r4), r1) = (-(diff(Phi(r1, r2, r4), r4))*(diff(Phi(r1, r2, r4), r1))*(diff(Phi(r1, r2, r4), r2))^2-(diff(Phi(r1, r2, r4), r4))*(diff(Phi(r1, r2, r4), r1))*cos(T(r1, r2, r4))^2+(diff(Phi(r1, r2, r4), r1))*(diff(Phi(r1, r2, r4), r2))*(diff(R(r1, r2, r4), r4))*(diff(R(r1, r2, r4), r2)))/(cos(T(r1, r2, r4))^2*(diff(Phi(r1, r2, r4), r2))^2+cos(T(r1, r2, r4))^4), diff(Theta(r1, r2, r4), r2) = (-(diff(Phi(r1, r2, r4), r4))*(diff(Phi(r1, r2, r4), r2))+(diff(R(r1, r2, r4), r4))*(diff(R(r1, r2, r4), r2)))/cos(T(r1, r2, r4))^2, diff(diff(R(r1, r2, r4), r4), r4) = -R(r1, r2, r4)*cos(T(r1, r2, r4))^4-2*cos(T(r1, r2, r4))*sin(T(r1, r2, r4))*(diff(R(r1, r2, r4), r4)), diff(R(r1, r2, r4), r1) = (diff(Phi(r1, r2, r4), r2))*(diff(Phi(r1, r2, r4), r1))/(diff(R(r1, r2, r4), r2)), (diff(R(r1, r2, r4), r2))^2 = (diff(Phi(r1, r2, r4), r2))^2+cos(T(r1, r2, r4))^2, diff(diff(Phi(r1, r2, r4), r4), r4) = -Phi(r1, r2, r4)*cos(T(r1, r2, r4))^4-2*(diff(Phi(r1, r2, r4), r4))*cos(T(r1, r2, r4))*sin(T(r1, r2, r4)), diff(diff(Phi(r1, r2, r4), r2), r4) = -(diff(Phi(r1, r2, r4), r2))*cos(T(r1, r2, r4))*sin(T(r1, r2, r4)), diff(diff(Phi(r1, r2, r4), r2), r2) = 0, (diff(Phi(r1, r2, r4), r1))^2 = (diff(Phi(r1, r2, r4), r2))^2+cos(T(r1, r2, r4))^2, sin(T(r1, r2, r4))^2 = -cos(T(r1, r2, r4))^2+1, diff(T(r1, r2, r4), r4) = cos(T(r1, r2, r4))^2, (diff(T(r1, r2, r4), r1))*sin(T(r1, r2, r4)) = 0, (diff(T(r1, r2, r4), r2))*sin(T(r1, r2, r4)) = 0, diff(T(r1, r2, r4), r1) = 0, diff(T(r1, r2, r4), r2) = 0], [diff(R(r1, r2, r4), r2) <> 0, diff(Phi(r1, r2, r4), r1) <> 0]), `casesplit/ans`([diff(Theta(r1, r2, r4), r4) = (1/2)*(Phi(r1, r2, r4)^2*cos(T(r1, r2, r4))^4-R(r1, r2, r4)^2*cos(T(r1, r2, r4))^4-(diff(Phi(r1, r2, r4), r4))^2+(diff(R(r1, r2, r4), r4))^2)/cos(T(r1, r2, r4))^2, diff(Theta(r1, r2, r4), r1) = (diff(R(r1, r2, r4), r4))*(diff(R(r1, r2, r4), r1))/cos(T(r1, r2, r4))^2, diff(Theta(r1, r2, r4), r2) = -(diff(Phi(r1, r2, r4), r4))*(diff(Phi(r1, r2, r4), r2))/cos(T(r1, r2, r4))^2, diff(diff(R(r1, r2, r4), r4), r4) = -R(r1, r2, r4)*cos(T(r1, r2, r4))^4-2*cos(T(r1, r2, r4))*sin(T(r1, r2, r4))*(diff(R(r1, r2, r4), r4)), diff(R(r1, r2, r4), r2) = 0, (diff(R(r1, r2, r4), r1))^2 = -cos(T(r1, r2, r4))^2, diff(diff(Phi(r1, r2, r4), r4), r4) = -Phi(r1, r2, r4)*cos(T(r1, r2, r4))^4-2*(diff(Phi(r1, r2, r4), r4))*cos(T(r1, r2, r4))*sin(T(r1, r2, r4)), diff(Phi(r1, r2, r4), r1) = 0, (diff(Phi(r1, r2, r4), r2))^2 = -cos(T(r1, r2, r4))^2, sin(T(r1, r2, r4))^2 = -cos(T(r1, r2, r4))^2+1, diff(T(r1, r2, r4), r4) = cos(T(r1, r2, r4))^2, (diff(T(r1, r2, r4), r1))*sin(T(r1, r2, r4)) = 0, (diff(T(r1, r2, r4), r2))*sin(T(r1, r2, r4)) = 0, diff(T(r1, r2, r4), r1) = 0, diff(T(r1, r2, r4), r2) = 0], [diff(R(r1, r2, r4), r1) <> 0, diff(Phi(r1, r2, r4), r2) <> 0])]

 (3)

``

> 

map(length, cases)

[2101, 1405]

 (4)
> 

sys1 := op(1, cases[2])

[diff(Theta(r1, r2, r4), r4) = (1/2)*(Phi(r1, r2, r4)^2*cos(T(r1, r2, r4))^4-R(r1, r2, r4)^2*cos(T(r1, r2, r4))^4-(diff(Phi(r1, r2, r4), r4))^2+(diff(R(r1, r2, r4), r4))^2)/cos(T(r1, r2, r4))^2, diff(Theta(r1, r2, r4), r1) = (diff(R(r1, r2, r4), r4))*(diff(R(r1, r2, r4), r1))/cos(T(r1, r2, r4))^2, diff(Theta(r1, r2, r4), r2) = -(diff(Phi(r1, r2, r4), r4))*(diff(Phi(r1, r2, r4), r2))/cos(T(r1, r2, r4))^2, diff(diff(R(r1, r2, r4), r4), r4) = -R(r1, r2, r4)*cos(T(r1, r2, r4))^4-2*cos(T(r1, r2, r4))*sin(T(r1, r2, r4))*(diff(R(r1, r2, r4), r4)), diff(R(r1, r2, r4), r2) = 0, (diff(R(r1, r2, r4), r1))^2 = -cos(T(r1, r2, r4))^2, diff(diff(Phi(r1, r2, r4), r4), r4) = -Phi(r1, r2, r4)*cos(T(r1, r2, r4))^4-2*(diff(Phi(r1, r2, r4), r4))*cos(T(r1, r2, r4))*sin(T(r1, r2, r4)), diff(Phi(r1, r2, r4), r1) = 0, (diff(Phi(r1, r2, r4), r2))^2 = -cos(T(r1, r2, r4))^2, sin(T(r1, r2, r4))^2 = -cos(T(r1, r2, r4))^2+1, diff(T(r1, r2, r4), r4) = cos(T(r1, r2, r4))^2, (diff(T(r1, r2, r4), r1))*sin(T(r1, r2, r4)) = 0, (diff(T(r1, r2, r4), r2))*sin(T(r1, r2, r4)) = 0, diff(T(r1, r2, r4), r1) = 0, diff(T(r1, r2, r4), r2) = 0]

 (5)

``

> 

sys2 := op(1, cases[1])

[diff(Theta(r1, r2, r4), r4) = (1/2)*(Phi(r1, r2, r4)^2*cos(T(r1, r2, r4))^4-R(r1, r2, r4)^2*cos(T(r1, r2, r4))^4-(diff(Phi(r1, r2, r4), r4))^2+(diff(R(r1, r2, r4), r4))^2)/cos(T(r1, r2, r4))^2, diff(Theta(r1, r2, r4), r1) = (-(diff(Phi(r1, r2, r4), r4))*(diff(Phi(r1, r2, r4), r1))*(diff(Phi(r1, r2, r4), r2))^2-(diff(Phi(r1, r2, r4), r4))*(diff(Phi(r1, r2, r4), r1))*cos(T(r1, r2, r4))^2+(diff(Phi(r1, r2, r4), r1))*(diff(Phi(r1, r2, r4), r2))*(diff(R(r1, r2, r4), r4))*(diff(R(r1, r2, r4), r2)))/(cos(T(r1, r2, r4))^2*(diff(Phi(r1, r2, r4), r2))^2+cos(T(r1, r2, r4))^4), diff(Theta(r1, r2, r4), r2) = (-(diff(Phi(r1, r2, r4), r4))*(diff(Phi(r1, r2, r4), r2))+(diff(R(r1, r2, r4), r4))*(diff(R(r1, r2, r4), r2)))/cos(T(r1, r2, r4))^2, diff(diff(R(r1, r2, r4), r4), r4) = -R(r1, r2, r4)*cos(T(r1, r2, r4))^4-2*cos(T(r1, r2, r4))*sin(T(r1, r2, r4))*(diff(R(r1, r2, r4), r4)), diff(R(r1, r2, r4), r1) = (diff(Phi(r1, r2, r4), r2))*(diff(Phi(r1, r2, r4), r1))/(diff(R(r1, r2, r4), r2)), (diff(R(r1, r2, r4), r2))^2 = (diff(Phi(r1, r2, r4), r2))^2+cos(T(r1, r2, r4))^2, diff(diff(Phi(r1, r2, r4), r4), r4) = -Phi(r1, r2, r4)*cos(T(r1, r2, r4))^4-2*(diff(Phi(r1, r2, r4), r4))*cos(T(r1, r2, r4))*sin(T(r1, r2, r4)), diff(diff(Phi(r1, r2, r4), r2), r4) = -(diff(Phi(r1, r2, r4), r2))*cos(T(r1, r2, r4))*sin(T(r1, r2, r4)), diff(diff(Phi(r1, r2, r4), r2), r2) = 0, (diff(Phi(r1, r2, r4), r1))^2 = (diff(Phi(r1, r2, r4), r2))^2+cos(T(r1, r2, r4))^2, sin(T(r1, r2, r4))^2 = -cos(T(r1, r2, r4))^2+1, diff(T(r1, r2, r4), r4) = cos(T(r1, r2, r4))^2, (diff(T(r1, r2, r4), r1))*sin(T(r1, r2, r4)) = 0, (diff(T(r1, r2, r4), r2))*sin(T(r1, r2, r4)) = 0, diff(T(r1, r2, r4), r1) = 0, diff(T(r1, r2, r4), r2) = 0]

 (6)

``

> 

sys3 := op(2, cases[1])

[diff(R(r1, r2, r4), r2) <> 0, diff(Phi(r1, r2, r4), r1) <> 0]

 (7)

``

> 

sys4 := op(2, cases[2])

[diff(R(r1, r2, r4), r1) <> 0, diff(Phi(r1, r2, r4), r2) <> 0]

 (8)

``

> 

sol1 := dsolve(sys1, explicit)

(9)

``

> 

constraint, subsystem := selectremove(has, sys1, T)

[diff(Theta(r1, r2, r4), r4) = (1/2)*(Phi(r1, r2, r4)^2*cos(T(r1, r2, r4))^4-R(r1, r2, r4)^2*cos(T(r1, r2, r4))^4-(diff(Phi(r1, r2, r4), r4))^2+(diff(R(r1, r2, r4), r4))^2)/cos(T(r1, r2, r4))^2, diff(Theta(r1, r2, r4), r1) = (diff(R(r1, r2, r4), r4))*(diff(R(r1, r2, r4), r1))/cos(T(r1, r2, r4))^2, diff(Theta(r1, r2, r4), r2) = -(diff(Phi(r1, r2, r4), r4))*(diff(Phi(r1, r2, r4), r2))/cos(T(r1, r2, r4))^2, diff(diff(R(r1, r2, r4), r4), r4) = -R(r1, r2, r4)*cos(T(r1, r2, r4))^4-2*cos(T(r1, r2, r4))*sin(T(r1, r2, r4))*(diff(R(r1, r2, r4), r4)), (diff(R(r1, r2, r4), r1))^2 = -cos(T(r1, r2, r4))^2, diff(diff(Phi(r1, r2, r4), r4), r4) = -Phi(r1, r2, r4)*cos(T(r1, r2, r4))^4-2*(diff(Phi(r1, r2, r4), r4))*cos(T(r1, r2, r4))*sin(T(r1, r2, r4)), (diff(Phi(r1, r2, r4), r2))^2 = -cos(T(r1, r2, r4))^2, sin(T(r1, r2, r4))^2 = -cos(T(r1, r2, r4))^2+1, diff(T(r1, r2, r4), r4) = cos(T(r1, r2, r4))^2, (diff(T(r1, r2, r4), r1))*sin(T(r1, r2, r4)) = 0, (diff(T(r1, r2, r4), r2))*sin(T(r1, r2, r4)) = 0, diff(T(r1, r2, r4), r1) = 0, diff(T(r1, r2, r4), r2) = 0], [diff(R(r1, r2, r4), r2) = 0, diff(Phi(r1, r2, r4), r1) = 0]

 (10)

``

> 

sol__subsystem := dsolve(subsystem)

{Phi(r1, r2, r4) = _F1(r2, r4), R(r1, r2, r4) = _F2(r1, r4)}

 (11)

``

> 

eval(constraint, sol__subsystem)

[diff(Theta(r1, r2, r4), r4) = (1/2)*(_F1(r2, r4)^2*cos(T(r1, r2, r4))^4-_F2(r1, r4)^2*cos(T(r1, r2, r4))^4-(diff(_F1(r2, r4), r4))^2+(diff(_F2(r1, r4), r4))^2)/cos(T(r1, r2, r4))^2, diff(Theta(r1, r2, r4), r1) = (diff(_F2(r1, r4), r4))*(diff(_F2(r1, r4), r1))/cos(T(r1, r2, r4))^2, diff(Theta(r1, r2, r4), r2) = -(diff(_F1(r2, r4), r4))*(diff(_F1(r2, r4), r2))/cos(T(r1, r2, r4))^2, diff(diff(_F2(r1, r4), r4), r4) = -_F2(r1, r4)*cos(T(r1, r2, r4))^4-2*cos(T(r1, r2, r4))*sin(T(r1, r2, r4))*(diff(_F2(r1, r4), r4)), (diff(_F2(r1, r4), r1))^2 = -cos(T(r1, r2, r4))^2, diff(diff(_F1(r2, r4), r4), r4) = -_F1(r2, r4)*cos(T(r1, r2, r4))^4-2*(diff(_F1(r2, r4), r4))*cos(T(r1, r2, r4))*sin(T(r1, r2, r4)), (diff(_F1(r2, r4), r2))^2 = -cos(T(r1, r2, r4))^2, sin(T(r1, r2, r4))^2 = -cos(T(r1, r2, r4))^2+1, diff(T(r1, r2, r4), r4) = cos(T(r1, r2, r4))^2, (diff(T(r1, r2, r4), r1))*sin(T(r1, r2, r4)) = 0, (diff(T(r1, r2, r4), r2))*sin(T(r1, r2, r4)) = 0, diff(T(r1, r2, r4), r1) = 0, diff(T(r1, r2, r4), r2) = 0]

 (12)
> 

map(simplify, [diff(Theta(r1, r2, r4), r4) = (1/2)*(_F1(r2, r4)^2*cos(T(r1, r2, r4))^4-_F2(r1, r4)^2*cos(T(r1, r2, r4))^4-(diff(_F1(r2, r4), r4))^2+(diff(_F2(r1, r4), r4))^2)/cos(T(r1, r2, r4))^2, diff(Theta(r1, r2, r4), r1) = (diff(_F2(r1, r4), r4))*(diff(_F2(r1, r4), r1))/cos(T(r1, r2, r4))^2, diff(Theta(r1, r2, r4), r2) = -(diff(_F1(r2, r4), r4))*(diff(_F1(r2, r4), r2))/cos(T(r1, r2, r4))^2, diff(diff(_F2(r1, r4), r4), r4) = -_F2(r1, r4)*cos(T(r1, r2, r4))^4-2*cos(T(r1, r2, r4))*sin(T(r1, r2, r4))*(diff(_F2(r1, r4), r4)), (diff(_F2(r1, r4), r1))^2 = -cos(T(r1, r2, r4))^2, diff(diff(_F1(r2, r4), r4), r4) = -_F1(r2, r4)*cos(T(r1, r2, r4))^4-2*(diff(_F1(r2, r4), r4))*cos(T(r1, r2, r4))*sin(T(r1, r2, r4)), (diff(_F1(r2, r4), r2))^2 = -cos(T(r1, r2, r4))^2, sin(T(r1, r2, r4))^2 = -cos(T(r1, r2, r4))^2+1, diff(T(r1, r2, r4), r4) = cos(T(r1, r2, r4))^2, (diff(T(r1, r2, r4), r1))*sin(T(r1, r2, r4)) = 0, (diff(T(r1, r2, r4), r2))*sin(T(r1, r2, r4)) = 0, diff(T(r1, r2, r4), r1) = 0, diff(T(r1, r2, r4), r2) = 0])

[diff(Theta(r1, r2, r4), r4) = (1/2)*((_F1(r2, r4)^2-_F2(r1, r4)^2)*cos(T(r1, r2, r4))^4-(diff(_F1(r2, r4), r4))^2+(diff(_F2(r1, r4), r4))^2)/cos(T(r1, r2, r4))^2, diff(Theta(r1, r2, r4), r1) = (diff(_F2(r1, r4), r4))*(diff(_F2(r1, r4), r1))/cos(T(r1, r2, r4))^2, diff(Theta(r1, r2, r4), r2) = -(diff(_F1(r2, r4), r4))*(diff(_F1(r2, r4), r2))/cos(T(r1, r2, r4))^2, diff(diff(_F2(r1, r4), r4), r4) = -cos(T(r1, r2, r4))*(_F2(r1, r4)*cos(T(r1, r2, r4))^3+2*(diff(_F2(r1, r4), r4))*sin(T(r1, r2, r4))), (diff(_F2(r1, r4), r1))^2 = -cos(T(r1, r2, r4))^2, diff(diff(_F1(r2, r4), r4), r4) = -cos(T(r1, r2, r4))*(_F1(r2, r4)*cos(T(r1, r2, r4))^3+2*(diff(_F1(r2, r4), r4))*sin(T(r1, r2, r4))), (diff(_F1(r2, r4), r2))^2 = -cos(T(r1, r2, r4))^2, sin(T(r1, r2, r4))^2 = sin(T(r1, r2, r4))^2, diff(T(r1, r2, r4), r4) = cos(T(r1, r2, r4))^2, (diff(T(r1, r2, r4), r1))*sin(T(r1, r2, r4)) = 0, (diff(T(r1, r2, r4), r2))*sin(T(r1, r2, r4)) = 0, diff(T(r1, r2, r4), r1) = 0, diff(T(r1, r2, r4), r2) = 0]

 (13)

``

> 

eval(constraint, sol__subsystem)

[diff(Theta(r1, r2, r4), r4) = (1/2)*(_F1(r2, r4)^2*cos(T(r1, r2, r4))^4-_F2(r1, r4)^2*cos(T(r1, r2, r4))^4-(diff(_F1(r2, r4), r4))^2+(diff(_F2(r1, r4), r4))^2)/cos(T(r1, r2, r4))^2, diff(Theta(r1, r2, r4), r1) = (diff(_F2(r1, r4), r4))*(diff(_F2(r1, r4), r1))/cos(T(r1, r2, r4))^2, diff(Theta(r1, r2, r4), r2) = -(diff(_F1(r2, r4), r4))*(diff(_F1(r2, r4), r2))/cos(T(r1, r2, r4))^2, diff(diff(_F2(r1, r4), r4), r4) = -_F2(r1, r4)*cos(T(r1, r2, r4))^4-2*cos(T(r1, r2, r4))*sin(T(r1, r2, r4))*(diff(_F2(r1, r4), r4)), (diff(_F2(r1, r4), r1))^2 = -cos(T(r1, r2, r4))^2, diff(diff(_F1(r2, r4), r4), r4) = -_F1(r2, r4)*cos(T(r1, r2, r4))^4-2*(diff(_F1(r2, r4), r4))*cos(T(r1, r2, r4))*sin(T(r1, r2, r4)), (diff(_F1(r2, r4), r2))^2 = -cos(T(r1, r2, r4))^2, sin(T(r1, r2, r4))^2 = -cos(T(r1, r2, r4))^2+1, diff(T(r1, r2, r4), r4) = cos(T(r1, r2, r4))^2, (diff(T(r1, r2, r4), r1))*sin(T(r1, r2, r4)) = 0, (diff(T(r1, r2, r4), r2))*sin(T(r1, r2, r4)) = 0, diff(T(r1, r2, r4), r1) = 0, diff(T(r1, r2, r4), r2) = 0]

 (14)

``

> 

``


 

Download Finding_transformation_eqn_between_zero_and_harmonic_with_conformal1.mw

limit of the integral is function...

Asked by: Mo_Jalal 15
integration
August 20 2020
1  12

Hi 

The integration in (F2) is not work properly.

My tyr is below:

restart:with(plots):
assume(d,real):

b:=-gamma1*tau0+I*tau0*Delta-2*a*(k-1)*xr:
a:=1+I*c;
c1:=sqrt(conjugate(a))/tau0:
c2:=0.5*((conjugate(b)/sqrt(conjugate(a)))):
lambda1:=2*(1-I*d)*(t-k1)/j+(1-I*d)^2:
lambda2:=t*(1-j)*k1/j+1/sqrt(2)*(1-I*d):
lambda3:=c1*(t-k1)/j+c1*(1-I*d)+c2:
J1:=sqrt(Pi)/sqrt(2)*(1-erf(lambda2)):
J1mod:=(Re(J1))^2+(Im(J1))^2:
g1:=0.5*sqrt(Pi)*tau0*exp(c2^2)*exp(-conjugate(a)*((k-1)*xr)^2)/(sqrt(conjugate(a))):

                       
F2:=-sqrt(2)*int(exp(-x^2)*erf(sqrt(2)*c1*x+lambda3),x=-lambda2..infty);
Warning,  computation interrupted
J2:=sum(g1*J1+g1*F2,k=1..1);
J2mod:=(((Re(J2))^2+(Im(J2))^2)):
W:= unapply(no*J1mod+Omega^2*J2mod,d):
Omega:=0.01:no:=1:Delta:=0:tau0:=1:c:=0:gamma1:=0:j:=1:k1:=0:t:=2*Pi:xr:=1:
P1:=plot(W,-50..50,axes=boxed,title=tit,color=black,font=[2,3,18],thickness=2,tickmarks=[3,2],titlefont=[SYMBOL,14],font=[1,1,18],linestyle=1);

difficult integral...

Asked by: J F Ogilvie 263
integration
August 19 2020
0  4

Maple seems to have difficulty calculating this integral algebraically, although it seems not excessively complicated.

int( ((-A*omega*sin(omega*x+phi)*exp(-x/tau) - A*cos(omega*x+phi)*exp(-x/tau)/tau)^2 + 1)^(1/2), x=0..t ) assuming t>0, omega>0, tau>0, A>0

any suggestions to solve this integral?

thanks in advance

How to get convergent solution of the shear flow p...

Asked by: Arif Ullah khan 25
ode dsolve numeric
August 19 2020
1  22

Hi,

Need your help to find the convergent solution of the attached problem.

Here is some information about parameters

(0 <= gamma <= 10,      0 <= rho & nu <= 200)

 

Arif_Ullah.mw
 

 

> 

restart:
> 

with(plots):
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N:=6:
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> 

eq1 :=3*diff(f(eta),eta,eta,eta)+2*f(eta)*diff(f(eta),eta,eta)-(diff(f(eta),eta))^2;

3*(diff(diff(diff(f(eta), eta), eta), eta))+2*f(eta)*(diff(diff(f(eta), eta), eta))-(diff(f(eta), eta))^2

 (1)
> 

NULL
> 

eq2 := 3*nu*diff(g(eta),eta,eta,eta)+2*g(eta)*diff(g(eta),eta,eta)-(diff(g(eta),eta))^2;

3*nu*(diff(diff(diff(g(eta), eta), eta), eta))+2*g(eta)*(diff(diff(g(eta), eta), eta))-(diff(g(eta), eta))^2

 (2)
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> 

bc:=f(0) = 0, g(0) = 0, D(f)(0) = D(g)(0), D(D(f))(0) = -rho*nu*D(D(g))(0), D(D(f))(6) = 1, D(D(g))(N) = gamma;

f(0) = 0, g(0) = 0, (D(f))(0) = (D(g))(0), ((D@@2)(f))(0) = -rho*nu*((D@@2)(g))(0), ((D@@2)(f))(6) = 1, ((D@@2)(g))(6) = gamma

 (3)
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para:={nu=3, gamma=10};

{gamma = 10, nu = 3}

 (4)
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A1 := dsolve(subs(para,rho=2, {bc, eq1, eq2}), numeric,method = bvp[midrich],maxmesh=12500, output=array([seq( 0.01*i, i=0..100*N)])):
> 

 

Error, (in dsolve/numeric/bvp) Newton iteration is not converging

  
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pf1 := odeplot(A1, [[eta,diff(f(eta),eta), linestyle = 1, color = blue]], 0 .. N):

Error, (in plots/odeplot) input is not a valid dsolve/numeric solution

  
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pf2 := odeplot(A1, [[eta,diff(g(eta),eta), linestyle = 3, color = red]], 0 .. N):

Error, (in plots/odeplot) input is not a valid dsolve/numeric solution

  
> 

display({pf1,pf2}, axes = boxed,thickness=3,labels = [eta, "f' '"],labelfont = ["ROMAN", 22,Bold,Italic],axesfont = ["ROMAN", "ROMAN", 14,Bold],axis=[thickness=3]);

Error, (in plots:-display) expecting plot structures but received: {pf1, pf2}

  

NULL


 

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