Preben Alsholm

11749 Reputation

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15 years, 300 days

MaplePrimes Activity


These are replies submitted by Preben Alsholm

@Rouben Rostamian  In 2D in versions before Maple 2016 a dialogue box comes up when you do

f(x):=x^2

In that box it says that parts of the expression are ambiguous and asks the user to choose between function definition and remember table or rtable assignment. Function definition is first and preselected, so just hitting Enter would choose that.

This box disappeared in Maple 2016 where a function definition is the result, but there is no option function_assign.
Same in Maple 2017. Putting in option function_assign happened in Maple 2018.
Removing the dialogue box may have been done (?) because it annoyed users of 2D, but could be brought back I suppose.

The current situation isn't all that bad, I think. If you start by using the arrow definition as in f:= x -> x^2 then it is now possible to make a remember table assignment to f as in f(Pi):=1. If x is just a name (and not a constant like Pi), however, assigning f(t):=t^4  to the remember table for f doesn't make much sense to me, so here a dialogue box could (should) pop up asking the user to choose.

With all that said I never use 2D math input myself, so don't have any of those problems.
 

 

By doing
 

f:=subsop(3=(operator,arrow, function_assign),eval(f));

after having done f:=x->sqrt(x), the 2D assignment f(x):=x^2*sin(x) works as desired.
This is revealed in the help page for Typesetting, Settings, where it states:

When a function is created for the first time using this syntax, the generated procedure will contain the option function_assign. This results in any subsequent assignments involving  "f(args)"  to be interpreted as function assignments. If f has already been defined as a procedure but does not have option function_assign, then any assignment with "f(args)"  on the left-hand side is interpreted as a remember-table assignment regardless of the functionassign setting.[my emphasis].

I guess that that means: If you start by using the "arrow" definition for a given function f, stick to it if you decide to change the definition of f. You can't have it both ways. If you define a new function g by using g(x):=whatever(x), then that will result in a procedure (by default).
 

@acer I like your version. It easily allows extension to include c__ (double underscore) instead of c[]:
 

replace_all_C:=proc(expr::anything,c::name,{useindex::truefalse:=false})
  if useindex then
    subsindets(expr, ':-suffixed(_C, posint)',
           u->c[parse(convert(u,string)[3..])])
  else
    subsindets(expr, ':-suffixed(_C, posint)',
           u->cat(c,"__",parse(convert(u,string)[3..])))
  end if
end proc:

Examples:
 

sol:=dsolve(diff(x(t),t$3) = x(t));
replace_all_C(sol,a)
replace_all_C(sol,c,useindex);

While this allows for different symbols for different outputs from dsolve it doesn't change the numbering. This could easily be added:
 

replace_all_C:=proc(expr::anything,c::name,{useindex::truefalse:=false,add::posint:=0})
  if useindex then
    subsindets(expr, ':-suffixed(_C, posint)',
           u->c[parse(convert(u,string)[3..])+add])
  else
    subsindets(expr, ':-suffixed(_C, posint)',
           u->cat(c,"__",parse(convert(u,string)[3..])+add))
  end if
end proc:

Examples:
 

replace_all_C(sol,a,add=3);
replace_all_C(sol,a,useindex,add=3);

 

@tomleslie I don't know anything about skin friction or Nusselt number, but certainly you can plot the solution in phase space:
 

plots:-odeplot(sol,[theta(eta),g(eta),f(eta)],0..10);

Whether that is anywhere near what he is thinking about is another question.

@ecterrab I tried getting Physics Updates 783 from the cloud from within Maple 2020.1.1.

It stopped in the middle of the process as Tom Leslie described. I didn't wait for a crash.

Then I created the folder structure as you described. I went to
https://www.maplesoft.com/products/maple/features/physicsresearch.aspx
and downloaded the file Physics Updates_1598849555273.maple.
I put that file in the lib folder and the version.txt into the Physics Updates folder.
I didn't get or see the other 2 files and was actually sure something was wrong because of that long number on the .maple file.
OK then I copied the override_maple.txt file from Maple 2019.

Now it appears to work.
The uninstall_manifest.mtxt file from Maple 2019 I didn't copy, so I don't have that in 2020 at this time.


 

@Kitonum Your procedure P cannot handle constants of type name.
Here is a revised version.
 

restart;
interface(typesetting=extended);
###
P:=proc(ode)
  local idiff,t;
  idiff:=convert(indets(ode,specfunc(diff)),D);
  t:=op~(idiff);
  if numelems(t)<>1 then error "Only one independent variable allowed. Got %1", t end if; 
  t:=op(t);
  Typesetting:-Settings(usedot=false,prime=t,typesetprime=true);
  ode
end proc:
###
ode1:=diff(y(t),t$2)+a*diff(y(t),t)+y(t)= b*diff(z(t),t);
P(ode1);
ode2:=diff(y(t),t$2)+a*diff(y(s),s)+y(t)= b*diff(z(t),t)
P(ode2); # error by design

 

@acer 

With floatPi = false (my setting) I don't get any problems, but JacobiTheta3 is not evaluated.
With floatPi = true I get a loss of kernel connection for your worksheet and for edgar's as well.

@janhardo As dharr said I changed all the cases of the double quotes (") into % and corrected the obvious error in defining sys (twice), where x and y should be x(t) and y(t) (and always should). The worksheet worked in my Maple 2020.
If you are in fact using a release beginning with Maple V you should not replace " by %, but the t's on x and y have to be put into sys.

Thank you for this fascinating mechanism. Theo's last name is Jansen, not Jensen, according to Wikipedia. He is Dutch, not Danish as I had hoped.

@Arif Ullah khan In similar problems N has been a substitute for infinity. I suppose that is not the case here?

@Arif Ullah khan You give some partial results for gamma = 1 and rho = nu:
Yet your original values were:
para:={nu=3, gamma=10} and rho = 2.

@vv Your example doesn't really suggest that does it? The values at t = 0 for U and X are certainly different:

 

restart;

Digits:=30:
  

ode := X(t)*diff(X(t), t, t) - diff(X(t), t)/(1 + diff(X(t), t)^2) - (100000000000000*t^14 + 7000000000000*t^12 + 110000000000*t^10 - 500000000*t^8 - 43000000*t^6 - 170000*t^4 - 1700*t^2 - 1)/(2*(100*t^2 + 1)^4*(50000000*t^8 + 2000000*t^6 + 35000*t^4 + 100*t^2 + 1));
          

X(t)*(diff(diff(X(t), t), t))-(diff(X(t), t))/(1+(diff(X(t), t))^2)-(1/2)*(100000000000000*t^14+7000000000000*t^12+110000000000*t^10-500000000*t^8-43000000*t^6-170000*t^4-1700*t^2-1)/((100*t^2+1)^4*(50000000*t^8+2000000*t^6+35000*t^4+100*t^2+1))

U := t -> t/((t*10)^2+1):
          

simplify(eval(ode, X=U)), [U(0), D(U)(0)];
 

0, [0, 1]

SOL:=dsolve([ode, X(1/2)=0.0211219739941351352807211288991, D(X)(1/2)=0.0540921883052830726848992435887], numeric,abserr=1e-15,relerr=1e-13);
          

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 := 30; 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 := []; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 26, [( 1 ) = (datatype = sfloat, order = C_order, storage = rectangular), ( 2 ) = (datatype = sfloat, order = C_order, storage = rectangular), ( 3 ) = ([0, 0, 0, Array(1..0, 1..2, {}, order = C_order)]), ( 4 ) = (Array(1..63, {(1) = 2, (2) = 2, (3) = 0, (4) = 0, (5) = 0, (6) = 0, (7) = 1, (8) = 0, (9) = 0, (10) = 0, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 1, (19) = 30000, (20) = 0, (21) = 1, (22) = 1, (23) = 4, (24) = 0, (25) = 2, (26) = 30, (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, (54) = 0, (55) = 0, (56) = 0, (57) = 0, (58) = 0, (59) = 10000, (60) = 0, (61) = 1000, (62) = 0, (63) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = .500000000000000000000000000000, (2) = 0.1e-12, (3) = 0., (4) = 0.500001e-29, (5) = .500000000000000000000000000000, (6) = 0.296338439495637901017502167099e-4, (7) = 0., (8) = 0.1e-12, (9) = 0., (10) = 0., (11) = 0., (12) = 0., (13) = 1., (14) = 0., (15) = .49999999999999, (16) = 0., (17) = 1., (18) = 1., (19) = 0., (20) = 0., (21) = 1., (22) = 1., (23) = 0., (24) = 0., (25) = 0.1e-14, (26) = 0., (27) = 0., (28) = 0.}, order = C_order)), ( 6 ) = (Array(1..2, {(1) = 0.211219739941351352807211288991e-1, (2) = 0.540921883052830726848992435887e-1}, order = C_order)), ( 7 ) = ([Array(1..4, 1..7, {(1, 1) = 0., (1, 2) = 26., (1, 3) = 39., (1, 4) = 96., (1, 5) = 104., (1, 6) = 52., (1, 7) = 104., (2, 1) = 33440., (2, 2) = 0., (2, 3) = 146432., (2, 4) = 142805., (2, 5) = -50787., (2, 6) = 10260., (2, 7) = 282150., (3, 1) = 1045., (3, 2) = 0., (3, 3) = -11264., (3, 4) = -10985., (3, 5) = 7524., (3, 6) = 13680., (3, 7) = 376200., (4, 1) = 1629155., (4, 2) = 0., (4, 3) = 6769664., (4, 4) = 340535., (4, 5) = -101574., (4, 6) = -800280., (4, 7) = 13062500.}, 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., (2, 1) = 1., (2, 2) = 0., (2, 3) = 0., (2, 4) = 0., (2, 5) = 0., (2, 6) = 4., (3, 1) = 3., (3, 2) = 9., (3, 3) = 0., (3, 4) = 0., (3, 5) = 0., (3, 6) = 32., (4, 1) = 1932., (4, 2) = -7200., (4, 3) = 7296., (4, 4) = 0., (4, 5) = 0., (4, 6) = 2197., (5, 1) = 8341., (5, 2) = -32832., (5, 3) = 29440., (5, 4) = -845., (5, 5) = 0., (5, 6) = 4104., (6, 1) = -6080., (6, 2) = 41040., (6, 3) = -28352., (6, 4) = 9295., (6, 5) = -5643., (6, 6) = 20520.}, order = C_order), Array(1..6, {(1) = 0., (2) = .386, (3) = .210, (4) = .630, (5) = 1., (6) = 1.}, order = C_order), Array(1..6, {(1) = .25, (2) = -.1043, (3) = .1035, (4) = -0.362e-1, (5) = 0., (6) = 0.}, 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) = 1.54400000000000000000000000000, (2, 2) = 0., (2, 3) = 0., (2, 4) = 0., (2, 5) = 0., (3, 1) = .946678528081553286149092671142, (3, 2) = .255701169898258116368870256812, (3, 3) = 0., (3, 4) = 0., (3, 5) = 0., (4, 1) = 3.31482518706848855878639615336, (4, 2) = 2.89612401597212315251490972823, (4, 3) = .998641913997780725707304795298, (4, 4) = 0., (4, 5) = 0., (5, 1) = 1.22122450922627482364750744120, (5, 2) = 6.01913448128775290549318365614, (5, 3) = 12.5370833293208745721773449832, (5, 4) = -.687886036105895135654889206414, (5, 5) = 0., (6, 1) = 1.22122450922627482364750744120, (6, 2) = 6.01913448128775290549318365614, (6, 3) = 12.5370833293208745721773449832, (6, 4) = -.687886036105895135654889206414, (6, 5) = 1.}, 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) = -5.66880000000000000000000000000, (2, 2) = 0., (2, 3) = 0., (2, 4) = 0., (2, 5) = 0., (3, 1) = -2.43009335683375818512768794859, (3, 2) = -.206359915708912236643547336023, (3, 3) = 0., (3, 4) = 0., (3, 5) = 0., (4, 1) = -.107352905814526218453871436335, (4, 2) = -9.59456225102189431954327672200, (4, 3) = -20.4702861480961548107719410920, (4, 4) = 0., (4, 5) = 0., (5, 1) = 7.49644331396861502581028867134, (5, 2) = -10.2468043146412183182800627326, (5, 3) = -33.9999035281990634994054417818, (5, 4) = 11.7089089320615954381414222772, (5, 5) = 0., (6, 1) = 8.08324679592241092908495062968, (6, 2) = -7.98113298806278547829882004219, (6, 3) = -31.5215943287437290544342145785, (6, 4) = 16.3193054312313618929244027333, (6, 5) = -6.05881823883405376327046553946}, 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) = 10.1262350834469120570913462177, (2, 2) = -7.48799587760763313332363461989, (2, 3) = -34.8009186155574113979166343882, (2, 4) = -7.99277170756872733717900590859, (2, 5) = 1.02513772329562064439580564715, (3, 1) = -.676280339280689777028659735967, (3, 2) = 6.08771465167860644619433469196, (3, 3) = 16.4308432089246306468843834297, (3, 4) = 24.7672251141836512529640785251, (3, 5) = -6.59438912571678161124446044889}, order = C_order)]), ( 9 ) = ([Array(1..2, {(1) = 0.100000000000000000000000000000e-1, (2) = 0.100000000000000000000000000000e-1}, order = C_order), Array(1..2, {(1) = 0., (2) = 0.}, order = C_order), Array(1..2, {(1) = 0., (2) = 0.}, order = C_order), Array(1..2, {(1) = 0., (2) = 0.}, order = C_order), Array(1..2, {(1) = 0., (2) = 0.}, order = C_order), Array(1..2, 1..2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0.}, order = C_order), Array(1..2, 1..2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0.}, order = C_order), Array(1..2, {(1) = 0., (2) = 0.}, order = C_order), Array(1..2, 1..2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0.}, order = C_order), Array(1..2, 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.}, order = C_order), Array(1..2, {(1) = 0, (2) = 0}, datatype = integer[8]), Array(1..2, {(1) = 0., (2) = 0.}, order = C_order), Array(1..2, {(1) = 0., (2) = 0.}, order = C_order), Array(1..2, {(1) = 0., (2) = 0.}, order = C_order), Array(1..2, {(1) = 0., (2) = 0.}, order = C_order), Array(1..2, {(1) = 0., (2) = 0.}, order = C_order), Array(1..4, {(1) = 0., (2) = 0., (3) = 0., (4) = 0.}, order = C_order), Array(1..2, {(1) = 0, (2) = 0}, datatype = integer[8])]), ( 8 ) = ([Array(1..2, {(1) = 0.211219739941351352807211288991e-1, (2) = 0.540921883052830726848992435887e-1}, order = C_order), Array(1..2, {(1) = 0., (2) = 0.}, order = C_order), Array(1..2, {(1) = 0.540921883052830726848992435887e-1, (2) = 4.34617387994087335834553944785}, order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..2, {(1, 1) = 0., (1, 2) = 0., (2, 0) = 0., (2, 1) = 0., (2, 2) = 0., (3, 0) = 0., (3, 1) = 0., (3, 2) = 0., (4, 0) = 0., (4, 1) = 0., (4, 2) = 0., (5, 0) = 0., (5, 1) = 0., (5, 2) = 0., (6, 0) = 0., (6, 1) = 0., (6, 2) = 0.}, order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = X(t), Y[2] = diff(X(t),t)]`; YP[2] := -(-Y[2]/(Y[2]^2+1)-(1/2)*(100000000000000*X^14+7000000000000*X^12+110000000000*X^10-500000000*X^8-43000000*X^6-170000*X^4-1700*X^2-1)/((100*X^2+1)^4*(50000000*X^8+2000000*X^6+35000*X^4+100*X^2+1)))/Y[1]; YP[1] := Y[2]; 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = X(t), Y[2] = diff(X(t),t)]`; YP[2] := -(-Y[2]/(Y[2]^2+1)-(1/2)*(100000000000000*X^14+7000000000000*X^12+110000000000*X^10-500000000*X^8-43000000*X^6-170000*X^4-1700*X^2-1)/((100*X^2+1)^4*(50000000*X^8+2000000*X^6+35000*X^4+100*X^2+1)))/Y[1]; YP[1] := Y[2]; 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0)  ] ))  ] ); _y0 := Array(0..2, {(1) = .500000000000000000000000000000, (2) = 0.211219739941351352807211288991e-1}); _vmap := array( 1 .. 2, [( 1 ) = (1), ( 2 ) = (2)  ] ); _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); _i := false; if _par <> [] then _i := `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then _i := `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) or _i end if; if _i then `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 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] < 100 and 100 <= _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 100 <= _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] < 100 and 100 < _dtbl[5-_i][4][9] then _dtbl[4] := 5-_i; _i := 5-_i end if; if _dtbl[_i][4][9] < 100 then error "no events to clear" elif _nv < _dtbl[_i][4][9]-100 then error "event error condition cannot be cleared" else _j := _dtbl[_i][4][9]-100; 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] < 100 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 _dat[17] <> _dtbl[1][17] then _dtbl[1][17] := _dat[17]; _dtbl[1][10] := _dat[10] end if; if _src = 0 and 100 < _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 ?dsolve,maxfun 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 _dat[4][9] = 6 then error "cannot evaluate the solution further right of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _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]-100, 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 ?dsolve,maxfun 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 _dat[4][9] = 6 then error "cannot evaluate the solution further left of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _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]-100, 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]; if type(_EnvDSNumericSaveDigits, 'posint') then _dat[4][26] := _EnvDSNumericSaveDigits else _dat[4][26] := Digits end if; _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, X(t), diff(X(t), t)], (4) = []}); _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 := 1; _ndsol := _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

SOL(0);


 

[t = 0., X(t) = -0.1135354542745714989743413776e-7, diff(X(t), t) = 1.00000024659796551712607712249]

plots:-odeplot(SOL, [t,X(t)-U(t)],0..0.2);

plots:-odeplot(SOL, [t,log10(abs(X(t)-U(t)))],0..0.5);

 


 

Download MaplePrimes20-08-15_ode_problem.mwMaplePrimes20-08-15_ode_problem.mw

@mmcdara I believe that your ode is unstable in the forward direction, i.e. as t -> infinity, but stable in the opposite direction.
If I'm right and you are interested in a solution on the finite interval t = 0..t0, then start the solution at t0.
Since you only know X and D(X) at 0 you could try use a shooting method.
But even if you succeed in find starting values (faking not knowing U) for a solution starting at t = t0 (t0 > 0), then that solution will only be good on t = 0..t0. Just try using X(t0) = U(t0) and D(X)(t0) = D(U)(t0) as your initial contions.
You get perfect results for t = 0..t0, but for t > t0 not so.
An example:
Here I chose t0 = 1 and plottet X(t) on 0..1.35:

@pik1432 It is easy to write down several ode systems all with equilibria coming from solving your algebraic equations.
Just multiply say 3 of these equations by minus 1 and the equilbria are the same, but the jacobian and its eigenvalues are not.
Thus stability here must refer to the solving method and obviously must refer to a numerical method as used by fsolve, not an exact as used by solve.
##################################################################################
A trivial one dimensional example is
diff(x(t),t) = x(t) which has jacobian 1 at the equilibrium x = 0 and eigenvalue 1: 0 is unstable.
diff(x(t),t) = -x(t) which has jacobian -1 at the equilibrium x = 0 and eigenvalue -1: 0 is stable.
Yet in both cases determining the equilibrium consists in solving the same basic equation x=0.
 

@pik1432 I forgot that I used expand on your f9n10 as I see that also acer does.

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