Mariusz Iwaniuk

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These are answers submitted by Mariusz Iwaniuk

Do you have any reason to think there is a closed form? Most nonlinear ODE's don't have one.

With numerics,see attached file:
 

Download ODE.mw

You have a BVP type ODE ,RK methods dosen't work(in Maple) for yours ICS and BC.


 

restart

DE := diff(f(x), x, x, x)+(1/2)*(diff(f(x), x, x))*(cos(alpha)*x+sin(alpha)*f(x)) = 0

diff(diff(diff(f(x), x), x), x)+(1/2)*(diff(diff(f(x), x), x))*(cos(alpha)*x+sin(alpha)*f(x)) = 0

(1)

alpha := (1/2)*Pi

(1/2)*Pi

(2)

ics := f(0) = 0, (D(f))(0) = 1, (D(f))(20) = 0

f(0) = 0, (D(f))(0) = 1, (D(f))(20) = 0

(3)

sol := dsolve([DE, ics], numeric, method = bvp)

proc (x_bvp) local res, data, solnproc, _ndsol, outpoint, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then outpoint := evalf[_EnvDSNumericSaveDigits](x_bvp) else outpoint := evalf(x_bvp) end if; data := Array(1..4, {(1) = proc (outpoint) local X, Y, YP, yout, errproc, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; X := Vector(26, {(1) = .0, (2) = .7740276521903071, (3) = 1.5509825989797759, (4) = 2.3308983059361585, (5) = 3.113909662707222, (6) = 3.9014225800794105, (7) = 4.693952164204047, (8) = 5.491595526837149, (9) = 6.2936856079399695, (10) = 7.098221028140669, (11) = 7.905109527264662, (12) = 8.714340235562716, (13) = 9.524617524969804, (14) = 10.335254722758192, (15) = 11.14625230894588, (16) = 11.957542794627765, (17) = 12.768893990799713, (18) = 13.580286548493516, (19) = 14.391720133302103, (20) = 15.20317024911012, (21) = 16.01462180657243, (22) = 16.8260748057475, (23) = 17.635392539683593, (24) = 18.434295256637704, (25) = 19.222354441049287, (26) = 20.0}, datatype = float[8], order = C_order); Y := Matrix(26, 3, {(1, 1) = .0, (1, 2) = 1.0, (1, 3) = -.4437483631867375, (2, 1) = .6441085952500252, (2, 2) = .6716037030622702, (2, 3) = -.388537166894675, (3, 1) = 1.0593772338870806, (3, 2) = .411915568539731, (3, 3) = -.2772636888974909, (4, 1) = 1.3074339266596293, (4, 2) = .23766008890198692, (4, 3) = -.1740048342296656, (5, 1) = 1.4484429056919907, (5, 2) = .13200360213914475, (5, 3) = -.10118230657337712, (6, 1) = 1.526293002425342, (6, 2) = 0.7158430915633403e-1, (6, 3) = -0.5624459816685278e-1, (7, 1) = 1.568464356108971, (7, 2) = 0.382292301392669e-1, (7, 3) = -0.3043735342537328e-1, (8, 1) = 1.591019979362346, (8, 2) = 0.20207534170443832e-1, (8, 3) = -0.16202417731846604e-1, (9, 1) = 1.6029701258095048, (9, 2) = 0.10608057138220823e-1, (9, 3) = -0.8537241259630802e-2, (10, 1) = 1.6092514574255978, (10, 2) = 0.55479598742602144e-2, (10, 3) = -0.44737002335379205e-2, (11, 1) = 1.6125418345657971, (11, 2) = 0.2893322331968495e-2, (11, 3) = -0.23355264917579296e-2, (12, 1) = 1.6142608964062477, (12, 2) = 0.15052758771365673e-2, (12, 3) = -0.12157884123578575e-2, (13, 1) = 1.6151559409876364, (13, 2) = 0.7822331360649307e-3, (13, 3) = -0.6320368044368434e-3, (14, 1) = 1.6156211589498495, (14, 2) = 0.4062857014228102e-3, (14, 3) = -0.32838621353109055e-3, (15, 1) = 1.6158628400264388, (15, 2) = 0.2109074168308063e-3, (15, 3) = -0.170545763768325e-3, (16, 1) = 1.6159883120134977, (16, 2) = 0.10941637613550126e-3, (16, 3) = -0.8854477959120226e-4, (17, 1) = 1.6160533911316586, (17, 2) = 0.56721956490906675e-4, (17, 3) = -0.4596713887725025e-4, (18, 1) = 1.616087112321174, (18, 2) = 0.29365452008446224e-4, (18, 3) = -0.2386212194685309e-4, (19, 1) = 1.6161045534488754, (19, 2) = 0.1516390815449538e-4, (19, 3) = -0.12386593625953825e-4, (20, 1) = 1.6161135426663806, (20, 2) = 0.7791938637060515e-5, (20, 3) = -0.6429639956506369e-5, (21, 1) = 1.6161181444123491, (21, 2) = 0.3965292776274608e-5, (21, 3) = -0.33374881762639542e-5, (22, 1) = 1.6161204687065325, (22, 2) = 0.1978961788333754e-5, (22, 3) = -0.17324140387128301e-5, (23, 1) = 1.61612160879516, (23, 2) = 0.9498247656501698e-6, (23, 3) = -0.9008088934281642e-6, (24, 1) = 1.6161221331825066, (24, 2) = 0.41960008638368255e-6, (24, 3) = -0.47235494169304415e-6, (25, 1) = 1.616122343924887, (25, 2) = 0.1442650915773907e-6, (25, 3) = -0.249867370049047e-6, (26, 1) = 1.616122394182121, (26, 2) = .0, (26, 3) = -0.1332923050844365e-6}, datatype = float[8], order = C_order); YP := Matrix(26, 3, {(1, 1) = 1.0, (1, 2) = -.4437483631867375, (1, 3) = .0, (2, 1) = .6716037030622702, (2, 2) = -.388537166894675, (2, 3) = .12513006438547686, (3, 1) = .411915568539731, (3, 2) = -.2772636888974909, (3, 3) = .146863419900776, (4, 1) = .23766008890198692, (4, 2) = -.1740048342296656, (4, 3) = .11374991183732477, (5, 1) = .13200360213914475, (5, 2) = -.10118230657337712, (5, 3) = 0.7327839706888009e-1, (6, 1) = 0.7158430915633403e-1, (6, 2) = -0.5624459816685278e-1, (6, 3) = 0.4292286830314631e-1, (7, 1) = 0.382292301392669e-1, (7, 2) = -0.3043735342537328e-1, (7, 3) = 0.23869951970994643e-1, (8, 1) = 0.20207534170443832e-1, (8, 2) = -0.16202417731846604e-1, (8, 3) = 0.12889185162671348e-1, (9, 1) = 0.10608057138220823e-1, (9, 2) = -0.8537241259630802e-2, (9, 3) = 0.6842471348008241e-2, (10, 1) = 0.55479598742602144e-2, (10, 2) = -0.44737002335379205e-2, (10, 3) = 0.3599654310453068e-2, (11, 1) = 0.2893322331968495e-2, (11, 2) = -0.23355264917579296e-2, (11, 3) = 0.1883067086848176e-2, (12, 1) = 0.15052758771365673e-2, (12, 2) = -0.12157884123578575e-2, (12, 3) = 0.981299846186562e-3, (13, 1) = 0.7822331360649307e-3, (13, 2) = -0.6320368044368434e-3, (13, 3) = 0.5104189998045043e-3, (14, 1) = 0.4062857014228102e-3, (14, 2) = -0.32838621353109055e-3, (14, 3) = 0.2652738574441266e-3, (15, 1) = 0.2109074168308063e-3, (15, 2) = -0.170545763768325e-3, (15, 3) = 0.1377892810985819e-3, (16, 1) = 0.10941637613550126e-3, (16, 2) = -0.8854477959120226e-4, (16, 3) = 0.7154366445459707e-4, (17, 1) = 0.56721956490906675e-4, (17, 2) = -0.4596713887725025e-4, (17, 3) = 0.3714267533160009e-4, (18, 1) = 0.29365452008446224e-4, (18, 2) = -0.2386212194685309e-4, (18, 3) = 0.1928163387547276e-4, (19, 1) = 0.1516390815449538e-4, (19, 2) = -0.12386593625953825e-4, (19, 3) = 0.10009015180312396e-4, (20, 1) = 0.7791938637060515e-5, (20, 2) = -0.6429639956506369e-5, (20, 3) = 0.5195514104089411e-5, (21, 1) = 0.3965292776274608e-5, (21, 2) = -0.33374881762639542e-5, (21, 3) = 0.26968875992109285e-5, (22, 1) = 0.1978961788333754e-5, (22, 2) = -0.17324140387128301e-5, (22, 3) = 0.1399894894119178e-5, (23, 1) = 0.9498247656501698e-6, (23, 2) = -0.9008088934281642e-6, (23, 3) = 0.7279083590320563e-6, (24, 1) = 0.41960008638368255e-6, (24, 2) = -0.47235494169304415e-6, (24, 3) = 0.3816916379941305e-6, (25, 1) = 0.1442650915773907e-6, (25, 2) = -0.249867370049047e-6, (25, 3) = 0.2019081198770065e-6, (26, 1) = .0, (26, 2) = -0.1332923050844365e-6, (26, 3) = 0.1077083396095566e-6}, datatype = float[8], order = C_order); errproc := proc (x_bvp) local outpoint, X, Y, yout, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; Digits := 15; outpoint := evalf(x_bvp); X := Vector(26, {(1) = .0, (2) = .7740276521903071, (3) = 1.5509825989797759, (4) = 2.3308983059361585, (5) = 3.113909662707222, (6) = 3.9014225800794105, (7) = 4.693952164204047, (8) = 5.491595526837149, (9) = 6.2936856079399695, (10) = 7.098221028140669, (11) = 7.905109527264662, (12) = 8.714340235562716, (13) = 9.524617524969804, (14) = 10.335254722758192, (15) = 11.14625230894588, (16) = 11.957542794627765, (17) = 12.768893990799713, (18) = 13.580286548493516, (19) = 14.391720133302103, (20) = 15.20317024911012, (21) = 16.01462180657243, (22) = 16.8260748057475, (23) = 17.635392539683593, (24) = 18.434295256637704, (25) = 19.222354441049287, (26) = 20.0}, datatype = float[8], order = C_order); Y := Matrix(26, 3, {(1, 1) = .0, (1, 2) = .0, (1, 3) = 0.20280224164799158e-8, (2, 1) = 0.2803183042184232e-8, (2, 2) = 0.4991625955056737e-9, (2, 3) = -0.727581254546996e-9, (3, 1) = 0.4557363240240356e-8, (3, 2) = -0.3673387937871442e-8, (3, 3) = 0.22534652705375846e-8, (4, 1) = -0.6829185436526684e-9, (4, 2) = -0.35100840159303387e-9, (4, 3) = 0.5074433102697723e-9, (5, 1) = -0.15488026580064915e-8, (5, 2) = 0.8484090566812387e-9, (5, 3) = -0.6752145145589206e-9, (6, 1) = -0.4604601478218892e-9, (6, 2) = 0.890154623250492e-11, (6, 3) = 0.13709221767661697e-10, (7, 1) = -0.6102545293905585e-10, (7, 2) = -0.359282005111915e-9, (7, 3) = 0.3599498064607342e-9, (8, 1) = -0.197391568402355e-9, (8, 2) = -0.22894407799722967e-9, (8, 3) = 0.2362860013795752e-9, (9, 1) = -0.379205537675534e-9, (9, 2) = -0.4645276431659482e-10, (9, 3) = 0.56088543823463375e-10, (10, 1) = -0.4586441799614866e-9, (10, 2) = 0.38725458120158544e-10, (10, 3) = -0.314617130477522e-10, (11, 1) = -0.4635264865059751e-9, (11, 2) = 0.4950536586519704e-10, (11, 3) = -0.4553885618285106e-10, (12, 1) = -0.44282417345093263e-9, (12, 2) = 0.3285439737447645e-10, (12, 3) = -0.3110667146083322e-10, (13, 1) = -0.42268582319118604e-9, (13, 2) = 0.14947302529329693e-10, (13, 3) = -0.1408832217518466e-10, (14, 1) = -0.41044864593046706e-9, (14, 2) = 0.3808087869611605e-11, (14, 3) = -0.3002389201268037e-11, (15, 1) = -0.40506874691322416e-9, (15, 2) = -0.9736695377936347e-12, (15, 3) = 0.2018171226319354e-11, (16, 1) = -0.40362183595006497e-9, (16, 2) = -0.19677817488290347e-11, (16, 3) = 0.32115428642925657e-11, (17, 1) = -0.40378868758539834e-9, (17, 2) = -0.13624753324824084e-11, (17, 3) = 0.264529324558413e-11, (18, 1) = -0.4042910528678276e-9, (18, 2) = -0.4291900330509504e-12, (18, 3) = 0.15970666505751784e-11, (19, 1) = -0.40462961957185725e-9, (19, 2) = 0.28994823578818636e-12, (19, 3) = 0.6648002719431414e-12, (20, 1) = -0.40471401510303397e-9, (20, 2) = 0.6698102275694219e-12, (20, 3) = 0.3296256527574159e-13, (21, 1) = -0.404620155256271e-9, (21, 2) = 0.7665813776661401e-12, (21, 3) = -0.3108701481375216e-12, (22, 1) = -0.4044554481338332e-9, (22, 2) = 0.6842809730923993e-12, (22, 3) = -0.4457130232957266e-12, (23, 1) = -0.40430214896595403e-9, (23, 2) = 0.5155422812351513e-12, (23, 3) = -0.45403866728024595e-12, (24, 1) = -0.40420708479840935e-9, (24, 2) = 0.3246479611190476e-12, (24, 3) = -0.39934428675880875e-12, (25, 1) = -0.4041836916238162e-9, (25, 2) = 0.14769412605155626e-12, (25, 3) = -0.32296317050584663e-12, (26, 1) = -0.40423664955744696e-9, (26, 2) = .0, (26, 3) = -0.24728279984657615e-12}, datatype = float[8], order = C_order); if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "right" then return X[26] elif outpoint = "order" then return 8 elif outpoint = "error" then return HFloat(4.557363240240356e-9) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [3, 26, [f(x), diff(f(x), x), diff(diff(f(x), x), x)], X, Y] else return ('procname')(x_bvp) end if end if; if outpoint < X[1] or X[26] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[26] end if; V := array([1 = 4, 2 = 0]); if Digits <= trunc(evalhf(Digits)) then L := Vector(4, 'datatype' = 'float'[8]); yout := Vector(3, 'datatype' = 'float'[8]); evalhf(`dsolve/numeric/lagrange`(26, 3, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(3, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(26, 3, X, Y, outpoint, yout, L, V) end if; [x = outpoint, seq('[f(x), diff(f(x), x), diff(diff(f(x), x), x)]'[i] = yout[i], i = 1 .. 3)] end proc; if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "method" then return "bvp" elif outpoint = "right" then return X[26] elif outpoint = "order" then return 8 elif outpoint = "error" then return HFloat(4.557363240240356e-9) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [3, 26, "depnames", X, Y, YP] else error "non-numeric value" end if end if; if outpoint < X[1] or X[26] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[26] end if; if Digits <= trunc(evalhf(Digits)) and (_EnvInFsolve <> true or _EnvDSNumericSaveDigits <= trunc(evalhf(Digits))) then V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false)  ] ); L := Matrix(7, 2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0, (3, 1) = .0, (3, 2) = .0, (4, 1) = .0, (4, 2) = .0, (5, 1) = .0, (5, 2) = .0, (6, 1) = .0, (6, 2) = .0, (7, 1) = .0, (7, 2) = .0}, datatype = float[8], order = C_order); yout := Vector(3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8]); evalhf(`dsolve/numeric/hermite`(26, 3, X, Y, YP, outpoint, var(yout), var(L), var(V))) else if _EnvInFsolve = true then Digits := _EnvDSNumericSaveDigits end if; V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false)  ] ); L := Matrix(7, 2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0., (3, 1) = 0., (3, 2) = 0., (4, 1) = 0., (4, 2) = 0., (5, 1) = 0., (5, 2) = 0., (6, 1) = 0., (6, 2) = 0., (7, 1) = 0., (7, 2) = 0.}, order = C_order); yout := Vector(3, {(1) = 0., (2) = 0., (3) = 0.}); `dsolve/numeric/hermite`(26, 3, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 3)] end proc, (2) = Array(0..0, {}), (3) = [x, f(x), diff(f(x), x), diff(diff(f(x), x), x)], (4) = 0}); solnproc := data[1]; if not type(outpoint, 'numeric') then if outpoint = "solnprocedure" then return eval(solnproc) elif member(outpoint, ["start", "left", "right", "errorproc", "rawdata", "order", "error"]) then return solnproc(x_bvp) elif outpoint = "sysvars" then return data[3] elif procname <> unknown then return ('procname')(x_bvp) else _ndsol := pointto(data[2][0]); return ('_ndsol')(x_bvp) end if end if; try res := solnproc(outpoint); [x = res[1], seq('[f(x), diff(f(x), x), diff(diff(f(x), x), x)]'[i] = res[i+1], i = 1 .. 3)] catch: error  end try end proc

(4)

plots:-odeplot(sol, [[x, f(x)], [x, diff(f(x), x)]], x = 0 .. 20)

 

``


 

Download BVP.mw

 

After eliminating  syntax errors:

restart

kernelopts(version)

`Maple 2019.1, X86 64 WINDOWS, May 21 2019, Build ID 1399874`

(1)

with(plots)

Eq := diff(f(eta), `$`(eta, 3))+(diff(f(eta), `$`(eta, 2)))*f(eta)-(diff(f(eta), eta))^2-M*(diff(f(eta), `$`(eta, 1)))-A*(diff(f(eta), `$`(eta, 1))+(1/2)*eta*(diff(f(eta), `$`(eta, 2)))) = 0

diff(diff(diff(f(eta), eta), eta), eta)+(diff(diff(f(eta), eta), eta))*f(eta)-(diff(f(eta), eta))^2-M*(diff(f(eta), eta))-A*(diff(f(eta), eta)+(1/2)*eta*(diff(diff(f(eta), eta), eta))) = 0

(2)

M := 1

1

(3)

A := 0

0

(4)

bcs := f(0) = 0, (D(f))(0) = 1, ((D@@2)(f))(0) = 0

f(0) = 0, (D(f))(0) = 1, ((D@@2)(f))(0) = 0

(5)

dsys := {Eq, bcs}

{diff(diff(diff(f(eta), eta), eta), eta)+(diff(diff(f(eta), eta), eta))*f(eta)-(diff(f(eta), eta))^2-(diff(f(eta), eta)) = 0, f(0) = 0, (D(f))(0) = 1, ((D@@2)(f))(0) = 0}

(6)

dsol := dsolve(dsys, numeric); odeplot(dsol, [eta, f(eta)], 0 .. 2, color = red, axes = box)

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 := []; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 26, [( 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..63, {(1) = 3, (2) = 3, (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) = 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, (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) = .0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = 0.2523829377920773e-2, (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..3, {(1) = .0, (2) = 1.0, (3) = .0}, datatype = float[8], order = C_order)), ( 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..3, {(1) = .1, (2) = .1, (3) = .1}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, 1..3, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0}, datatype = float[8], order = C_order), Array(1..3, 1..3, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, 1..3, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0}, datatype = float[8], order = C_order), Array(1..3, 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}, datatype = float[8], order = C_order), Array(1..3, {(1) = 0, (2) = 0, (3) = 0}, datatype = integer[8]), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..6, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = 0, (2) = 0, (3) = 0}, datatype = integer[8])]), ( 8 ) = ([Array(1..3, {(1) = .0, (2) = 1.0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = 1.0, (2) = .0, (3) = 2.0}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..3, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (2, 0) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (3, 0) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (4, 0) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (5, 0) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = .0, (6, 0) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = f(eta), Y[2] = diff(f(eta),eta), Y[3] = diff(diff(f(eta),eta),eta)]`; YP[3] := -Y[1]*Y[3]+Y[2]^2+Y[2]; YP[1] := Y[2]; YP[2] := Y[3]; 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] = f(eta), Y[2] = diff(f(eta),eta), Y[3] = diff(diff(f(eta),eta),eta)]`; YP[3] := -Y[1]*Y[3]+Y[2]^2+Y[2]; YP[1] := Y[2]; YP[2] := Y[3]; 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..3, {(1) = 0., (2) = 0., (3) = 1.}); _vmap := array( 1 .. 3, [( 1 ) = (1), ( 2 ) = (2), ( 3 ) = (3)  ] ); _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) = [eta, f(eta), diff(f(eta), eta), diff(diff(f(eta), eta), eta)], (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

 

 

``


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kernelopts(version)

`Maple 2019.1, X86 64 WINDOWS, May 21 2019, Build ID 1399874`

(1)

convert(BesselI(alpha, x), Sum, include = powers, dummy = j)

Sum(x^(alpha+2*j)/(2^(alpha+2*j)*GAMMA(1+alpha+j)*GAMMA(j+1)), j = 0 .. infinity)

(2)

simplify(diff(op(1, Sum(x^(alpha+2*j)/(2^(alpha+2*j)*GAMMA(1+alpha+j)*GAMMA(j+1)), j = 0 .. infinity)), `$`(x, n)))

2^(-alpha-2*j)*GAMMA(alpha+2*j+1)*x^(alpha+2*j-n)/(GAMMA(1+alpha+j)*GAMMA(j+1)*GAMMA(alpha+2*j-n+1))

(3)

simplify(sum(2^(-alpha-2*j)*GAMMA(alpha+2*j+1)*x^(alpha+2*j-n)/(GAMMA(1+alpha+j)*GAMMA(j+1)*GAMMA(alpha+2*j-n+1)), j = 0 .. infinity))

2^(-alpha)*x^(alpha-n)*hypergeom([(1/2)*alpha+1, (1/2)*alpha+1/2], [1+alpha, (1/2)*alpha-(1/2)*n+1, (1/2)*alpha-(1/2)*n+1/2], (1/4)*x^2)/GAMMA(alpha-n+1)

(4)

Diff(BesselI(alpha, x), `$`(x, n)) = 2^(-alpha)*x^(alpha-n)*hypergeom([(1/2)*alpha+1, (1/2)*alpha+1/2], [1+alpha, (1/2)*alpha-(1/2)*n+1, (1/2)*alpha-(1/2)*n+1/2], (1/4)*x^2)/GAMMA(alpha-n+1)

Diff(BesselI(alpha, x), `$`(x, n)) = 2^(-alpha)*x^(alpha-n)*hypergeom([(1/2)*alpha+1, (1/2)*alpha+1/2], [1+alpha, (1/2)*alpha-(1/2)*n+1, (1/2)*alpha-(1/2)*n+1/2], (1/4)*x^2)/GAMMA(alpha-n+1)

(5)

NULL

alpha := -1/3

evalf(seq(eval(diff(BesselI(alpha, x), `$`(x, n)), x = 1), n = 1 .. 5))

.3720709970, 1.079429974, -.584246954, 3.903867245, -14.00900558

(6)

evalf(seq(eval(2^(-alpha)*x^(alpha-n)*hypergeom([(1/2)*alpha+1, (1/2)*alpha+1/2], [1+alpha, (1/2)*alpha-(1/2)*n+1, (1/2)*alpha-(1/2)*n+1/2], (1/4)*x^2)/GAMMA(alpha-n+1), x = 1), n = 1 .. 5))

.3720709970, 1.079429975, -.5842469547, 3.903867243, -14.00900558

(7)

NULL

NULL derivative  

plot([eval(2^(-alpha)*x^(alpha-n)*hypergeom([(1/2)*alpha+1, (1/2)*alpha+1/2], [1+alpha, (1/2)*alpha-(1/2)*n+1, (1/2)*alpha-(1/2)*n+1/2], (1/4)*x^2)/GAMMA(alpha-n+1), n = 4/5), eval(2^(-alpha)*x^(alpha-n)*hypergeom([(1/2)*alpha+1, (1/2)*alpha+1/2], [1+alpha, (1/2)*alpha-(1/2)*n+1, (1/2)*alpha-(1/2)*n+1/2], (1/4)*x^2)/GAMMA(alpha-n+1), n = 9/10), eval(2^(-alpha)*x^(alpha-n)*hypergeom([(1/2)*alpha+1, (1/2)*alpha+1/2], [1+alpha, (1/2)*alpha-(1/2)*n+1, (1/2)*alpha-(1/2)*n+1/2], (1/4)*x^2)/GAMMA(alpha-n+1), n = 99/100), diff(BesselI(alpha, x), x)], x = 1/10 .. 2, legend = [typeset("Curve: ", n = 8*(1/10)), typeset("Curve: ", n = 9/10), typeset("Curve: ", n = 99/100), typeset("Curve: ", n = 1)], linestyle = [1, 2, 3, 4], view = [1/10 .. 2, -3 .. 1])

 

``


 

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kernelopts(version)

`Maple 2019.1, X86 64 WINDOWS, May 21 2019, Build ID 1399874`

(1)

Digits := 14

14

(2)

with(VectorCalculus)

pde := Laplacian(u(r, t), 'cylindrical'[r, theta, z]) = diff(u(r, t), t)

(diff(u(r, t), r)+r*(diff(diff(u(r, t), r), r)))/r = diff(u(r, t), t)

(3)

iv := [u(1, t) = 0, u(4, t) = 0, u(r, 0) = r]

[u(1, t) = 0, u(4, t) = 0, u(r, 0) = r]

(4)

dsol := pdsolve(pde, iv, numeric):-value(output = listprocedure)

[r = proc () option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; evalf(args[1]) end proc, t = proc () option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; evalf(args[2]) end proc, u(r, t) = proc () local tv, xv, solnproc, stype, ndsol, vals; option `Copyright (c) 2001 by Waterloo Maple Inc. All rights reserved.`; Digits := trunc(evalhf(Digits)); solnproc := proc (tv, xv) local INFO, errest, nd, dvars, dary, daryt, daryx, vals, msg, i, j; option `Copyright (c) 2001 by Waterloo Maple Inc. All rights reserved.`; table( [( "soln_procedures" ) = array( 1 .. 1, [( 1 ) = (18446744419940228278)  ] ) ] ) INFO := table( [( "eqndep" ) = [1], ( "solvec3" ) = Vector(21, {(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, (20) = .0, (21) = .0}, datatype = float[8]), ( "extrabcs" ) = [0], ( "PDEs" ) = [(diff(u(r, t), r)+r*(diff(diff(u(r, t), r), r)))/r-(diff(u(r, t), t))], ( "dependson" ) = [{1}], ( "method" ) = theta, ( "solmat_i2" ) = 0, ( "spaceadaptive" ) = false, ( "solution" ) = Array(1..3, 1..21, 1..1, {(1, 1, 1) = .0, (1, 2, 1) = .0, (1, 3, 1) = .0, (1, 4, 1) = .0, (1, 5, 1) = .0, (1, 6, 1) = .0, (1, 7, 1) = .0, (1, 8, 1) = .0, (1, 9, 1) = .0, (1, 10, 1) = .0, (1, 11, 1) = .0, (1, 12, 1) = .0, (1, 13, 1) = .0, (1, 14, 1) = .0, (1, 15, 1) = .0, (1, 16, 1) = .0, (1, 17, 1) = .0, (1, 18, 1) = .0, (1, 19, 1) = .0, (1, 20, 1) = .0, (1, 21, 1) = .0, (2, 1, 1) = .0, (2, 2, 1) = .0, (2, 3, 1) = .0, (2, 4, 1) = .0, (2, 5, 1) = .0, (2, 6, 1) = .0, (2, 7, 1) = .0, (2, 8, 1) = .0, (2, 9, 1) = .0, (2, 10, 1) = .0, (2, 11, 1) = .0, (2, 12, 1) = .0, (2, 13, 1) = .0, (2, 14, 1) = .0, (2, 15, 1) = .0, (2, 16, 1) = .0, (2, 17, 1) = .0, (2, 18, 1) = .0, (2, 19, 1) = .0, (2, 20, 1) = .0, (2, 21, 1) = .0, (3, 1, 1) = .0, (3, 2, 1) = .0, (3, 3, 1) = .0, (3, 4, 1) = .0, (3, 5, 1) = .0, (3, 6, 1) = .0, (3, 7, 1) = .0, (3, 8, 1) = .0, (3, 9, 1) = .0, (3, 10, 1) = .0, (3, 11, 1) = .0, (3, 12, 1) = .0, (3, 13, 1) = .0, (3, 14, 1) = .0, (3, 15, 1) = .0, (3, 16, 1) = .0, (3, 17, 1) = .0, (3, 18, 1) = .0, (3, 19, 1) = .0, (3, 20, 1) = .0, (3, 21, 1) = .0}, datatype = float[8], order = C_order), ( "timeadaptive" ) = false, ( "totalwidth" ) = 6, ( "spacevar" ) = r, ( "depeqn" ) = [1], ( "solmatrix" ) = Matrix(21, 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, (8, 1) = .0, (8, 2) = .0, (8, 3) = .0, (8, 4) = .0, (8, 5) = .0, (8, 6) = .0, (9, 1) = .0, (9, 2) = .0, (9, 3) = .0, (9, 4) = .0, (9, 5) = .0, (9, 6) = .0, (10, 1) = .0, (10, 2) = .0, (10, 3) = .0, (10, 4) = .0, (10, 5) = .0, (10, 6) = .0, (11, 1) = .0, (11, 2) = .0, (11, 3) = .0, (11, 4) = .0, (11, 5) = .0, (11, 6) = .0, (12, 1) = .0, (12, 2) = .0, (12, 3) = .0, (12, 4) = .0, (12, 5) = .0, (12, 6) = .0, (13, 1) = .0, (13, 2) = .0, (13, 3) = .0, (13, 4) = .0, (13, 5) = .0, (13, 6) = .0, (14, 1) = .0, (14, 2) = .0, (14, 3) = .0, (14, 4) = .0, (14, 5) = .0, (14, 6) = .0, (15, 1) = .0, (15, 2) = .0, (15, 3) = .0, (15, 4) = .0, (15, 5) = .0, (15, 6) = .0, (16, 1) = .0, (16, 2) = .0, (16, 3) = .0, (16, 4) = .0, (16, 5) = .0, (16, 6) = .0, (17, 1) = .0, (17, 2) = .0, (17, 3) = .0, (17, 4) = .0, (17, 5) = .0, (17, 6) = .0, (18, 1) = .0, (18, 2) = .0, (18, 3) = .0, (18, 4) = .0, (18, 5) = .0, (18, 6) = .0, (19, 1) = .0, (19, 2) = .0, (19, 3) = .0, (19, 4) = .0, (19, 5) = .0, (19, 6) = .0, (20, 1) = .0, (20, 2) = .0, (20, 3) = .0, (20, 4) = .0, (20, 5) = .0, (20, 6) = .0, (21, 1) = .0, (21, 2) = .0, (21, 3) = .0, (21, 4) = .0, (21, 5) = .0, (21, 6) = .0}, datatype = float[8], order = C_order), ( "depdords" ) = [[[2, 1]]], ( "indepvars" ) = [r, t], ( "solvec5" ) = 0, ( "depords" ) = [[2, 1]], ( "solmat_i1" ) = 0, ( "initialized" ) = false, ( "pts", r ) = [1, 4], ( "fdepvars" ) = [u(r, t)], ( "startup_only" ) = false, ( "solvec1" ) = Vector(21, {(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, (20) = .0, (21) = .0}, datatype = float[8]), ( "vectorproc" ) = proc (v, vp, vpp, t, x, k, h, n, vec) local _s1, _s2, _s3, _s4, _s5, _s6, xi; _s3 := -2*k; _s4 := -4*h^2; _s5 := -h*k; _s6 := 4*h^2*k; vec[1] := 0; vec[n] := 0; for xi from 2 to n-1 do _s1 := -vp[xi-1]+vp[xi+1]; _s2 := vp[xi-1]-2*vp[xi]+vp[xi+1]; vec[xi] := (_s2*_s3*x[xi]+_s4*vp[xi]*x[xi]+_s1*_s5)/(_s6*x[xi]) end do end proc, ( "adjusted" ) = false, ( "explicit" ) = false, ( "solspace" ) = Vector(21, {(1) = 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, (20) = .0, (21) = 4.0}, datatype = float[8]), ( "banded" ) = true, ( "solmat_ne" ) = 0, ( "solvec2" ) = Vector(21, {(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, (20) = .0, (21) = .0}, datatype = float[8]), ( "periodic" ) = false, ( "linear" ) = true, ( "errorest" ) = false, ( "rightwidth" ) = 0, ( "IBC" ) = b, ( "t0" ) = 0, ( "spacepts" ) = 21, ( "mixed" ) = false, ( "norigdepvars" ) = 1, ( "theta" ) = 1/2, ( "depshift" ) = [1], ( "stages" ) = 1, ( "soltimes" ) = Vector(3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8]), ( "solmat_is" ) = 0, ( "intspace" ) = Matrix(21, 1, {(1, 1) = .0, (2, 1) = .0, (3, 1) = .0, (4, 1) = .0, (5, 1) = .0, (6, 1) = .0, (7, 1) = .0, (8, 1) = .0, (9, 1) = .0, (10, 1) = .0, (11, 1) = .0, (12, 1) = .0, (13, 1) = .0, (14, 1) = .0, (15, 1) = .0, (16, 1) = .0, (17, 1) = .0, (18, 1) = .0, (19, 1) = .0, (20, 1) = .0, (21, 1) = .0}, datatype = float[8], order = C_order), ( "timeidx" ) = 2, ( "depvars" ) = [u], ( "leftwidth" ) = 1, ( "solmat_v" ) = Vector(126, {(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, (20) = .0, (21) = .0, (22) = .0, (23) = .0, (24) = .0, (25) = .0, (26) = .0, (27) = .0, (28) = .0, (29) = .0, (30) = .0, (31) = .0, (32) = .0, (33) = .0, (34) = .0, (35) = .0, (36) = .0, (37) = .0, (38) = .0, (39) = .0, (40) = .0, (41) = .0, (42) = .0, (43) = .0, (44) = .0, (45) = .0, (46) = .0, (47) = .0, (48) = .0, (49) = .0, (50) = .0, (51) = .0, (52) = .0, (53) = .0, (54) = .0, (55) = .0, (56) = .0, (57) = .0, (58) = .0, (59) = .0, (60) = .0, (61) = .0, (62) = .0, (63) = .0, (64) = .0, (65) = .0, (66) = .0, (67) = .0, (68) = .0, (69) = .0, (70) = .0, (71) = .0, (72) = .0, (73) = .0, (74) = .0, (75) = .0, (76) = .0, (77) = .0, (78) = .0, (79) = .0, (80) = .0, (81) = .0, (82) = .0, (83) = .0, (84) = .0, (85) = .0, (86) = .0, (87) = .0, (88) = .0, (89) = .0, (90) = .0, (91) = .0, (92) = .0, (93) = .0, (94) = .0, (95) = .0, (96) = .0, (97) = .0, (98) = .0, (99) = .0, (100) = .0, (101) = .0, (102) = .0, (103) = .0, (104) = .0, (105) = .0, (106) = .0, (107) = .0, (108) = .0, (109) = .0, (110) = .0, (111) = .0, (112) = .0, (113) = .0, (114) = .0, (115) = .0, (116) = .0, (117) = .0, (118) = .0, (119) = .0, (120) = .0, (121) = .0, (122) = .0, (123) = .0, (124) = .0, (125) = .0, (126) = .0}, datatype = float[8], order = C_order, attributes = [source_rtable = (Matrix(21, 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, (8, 1) = .0, (8, 2) = .0, (8, 3) = .0, (8, 4) = .0, (8, 5) = .0, (8, 6) = .0, (9, 1) = .0, (9, 2) = .0, (9, 3) = .0, (9, 4) = .0, (9, 5) = .0, (9, 6) = .0, (10, 1) = .0, (10, 2) = .0, (10, 3) = .0, (10, 4) = .0, (10, 5) = .0, (10, 6) = .0, (11, 1) = .0, (11, 2) = .0, (11, 3) = .0, (11, 4) = .0, (11, 5) = .0, (11, 6) = .0, (12, 1) = .0, (12, 2) = .0, (12, 3) = .0, (12, 4) = .0, (12, 5) = .0, (12, 6) = .0, (13, 1) = .0, (13, 2) = .0, (13, 3) = .0, (13, 4) = .0, (13, 5) = .0, (13, 6) = .0, (14, 1) = .0, (14, 2) = .0, (14, 3) = .0, (14, 4) = .0, (14, 5) = .0, (14, 6) = .0, (15, 1) = .0, (15, 2) = .0, (15, 3) = .0, (15, 4) = .0, (15, 5) = .0, (15, 6) = .0, (16, 1) = .0, (16, 2) = .0, (16, 3) = .0, (16, 4) = .0, (16, 5) = .0, (16, 6) = .0, (17, 1) = .0, (17, 2) = .0, (17, 3) = .0, (17, 4) = .0, (17, 5) = .0, (17, 6) = .0, (18, 1) = .0, (18, 2) = .0, (18, 3) = .0, (18, 4) = .0, (18, 5) = .0, (18, 6) = .0, (19, 1) = .0, (19, 2) = .0, (19, 3) = .0, (19, 4) = .0, (19, 5) = .0, (19, 6) = .0, (20, 1) = .0, (20, 2) = .0, (20, 3) = .0, (20, 4) = .0, (20, 5) = .0, (20, 6) = .0, (21, 1) = .0, (21, 2) = .0, (21, 3) = .0, (21, 4) = .0, (21, 5) = .0, (21, 6) = .0}, datatype = float[8], order = C_order))]), ( "solvec4" ) = 0, ( "allocspace" ) = 21, ( "matrixproc" ) = proc (v, vp, vpp, t, x, k, h, n, mat) local _s1, _s2, xi; _s1 := 4*h^2; _s2 := -(h^2+k)/(h^2*k); mat[3] := 1; mat[6*n-3] := 1; for xi from 2 to n-1 do mat[6*xi-3] := _s2; mat[6*xi-4] := -(h-2*x[xi])/(_s1*x[xi]); mat[6*xi-2] := (h+2*x[xi])/(_s1*x[xi]) end do end proc, ( "erroraccum" ) = true, ( "ICS" ) = [r], ( "minspcpoints" ) = 4, ( "maxords" ) = [2, 1], ( "multidep" ) = [false, false], ( "vectorhf" ) = true, ( "timevar" ) = t, ( "spaceidx" ) = 1, ( "inputargs" ) = [(diff(u(r, t), r)+r*(diff(diff(u(r, t), r), r)))/r = diff(u(r, t), t), [u(1, t) = 0, u(4, t) = 0, u(r, 0) = r]], ( "bandwidth" ) = [1, 2], ( "BCS", 1 ) = {[[1, 0, 1], b[1, 0, 1]], [[1, 0, 4], b[1, 0, 4]]}, ( "timestep" ) = .150000000000000, ( "eqnords" ) = [[2, 1]], ( "autonomous" ) = true, ( "matrixhf" ) = true, ( "spacestep" ) = .150000000000000 ] ); if xv = "left" then return INFO["solspace"][1] elif xv = "right" then return INFO["solspace"][INFO["spacepts"]] elif tv = "start" then return INFO["t0"] elif not (type(tv, 'numeric') and type(xv, 'numeric')) then error "non-numeric input" end if; if xv < INFO["solspace"][1] or INFO["solspace"][INFO["spacepts"]] < xv then error "requested %1 value must be in the range %2..%3", INFO["spacevar"], INFO["solspace"][1], INFO["solspace"][INFO["spacepts"]] end if; dary := Vector(3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8]); daryt := 0; daryx := 0; dvars := []; errest := false; nd := nops(INFO["depvars"]); if dary[nd+1] <> tv then try `pdsolve/numeric/evolve_solution`(INFO, tv) catch: msg := StringTools:-FormatMessage(lastexception[2 .. -1]); if tv < INFO["t0"] then error cat("unable to compute solution for %1<%2:
", msg), INFO["timevar"], INFO["failtime"] else error cat("unable to compute solution for %1>%2:
", msg), INFO["timevar"], INFO["failtime"] end if end try end if; if dary[nd+1] <> tv or dary[nd+2] <> xv then `pdsolve/interp2dto0d`(3, INFO["soltimes"], INFO["spacepts"], INFO["solspace"], nops(INFO["depvars"]), INFO["solution"], true, tv, xv, dary); if errest then `pdsolve/interp2dto0d`(3, INFO["soltimes"], INFO["spacepts"], INFO["err_t"], nops(INFO["depvars"]), INFO["solution"], true, tv, xv, daryt); `pdsolve/interp2dto0d`(3, INFO["soltimes"], INFO["spacepts"], INFO["err_x"], nops(INFO["depvars"]), INFO["solution"], true, tv, xv, daryx) end if end if; dary[nd+1] := tv; dary[nd+2] := xv; if dvars = [] then [seq(dary[i], i = 1 .. INFO["norigdepvars"])] else vals := NULL; for i to nops(dvars) do j := eval(dvars[i]); try if errest then vals := vals, evalhf(j(tv, xv, dary, daryt, daryx)) else vals := vals, evalhf(j(tv, xv, dary)) end if catch: userinfo(5, `pdsolve/numeric`, `evalhf failure`); try if errest then vals := vals, j(tv, xv, dary, daryt, daryx) else vals := vals, j(tv, xv, dary) end if catch: vals := vals, undefined end try end try end do; [vals] end if end proc; stype := "2nd"; if nargs = 1 then if args[1] = "left" then return solnproc(0, "left") elif args[1] = "right" then return solnproc(0, "right") elif args[1] = "start" then return solnproc("start", 0) else error "too few arguments to solution procedure" end if elif nargs = 2 then if stype = "1st" then tv := evalf(args[1]); xv := evalf(args[2]) else tv := evalf(args[2]); xv := evalf(args[1]) end if; if not (type(tv, 'numeric') and type(xv, 'numeric')) then if procname <> unknown then return ('procname')(args[1 .. nargs]) else ndsol := pointto(solnproc("soln_procedures")[1]); return ('ndsol')(args[1 .. nargs]) end if end if else error "incorrect arguments to solution procedure" end if; vals := solnproc(tv, xv); vals[1] end proc]

(5)

uval := rhs(op(3, dsol))

proc () local tv, xv, solnproc, stype, ndsol, vals; option `Copyright (c) 2001 by Waterloo Maple Inc. All rights reserved.`; Digits := trunc(evalhf(Digits)); solnproc := proc (tv, xv) local INFO, errest, nd, dvars, dary, daryt, daryx, vals, msg, i, j; option `Copyright (c) 2001 by Waterloo Maple Inc. All rights reserved.`; table( [( "soln_procedures" ) = array( 1 .. 1, [( 1 ) = (18446744419940228278)  ] ) ] ) INFO := table( [( "eqndep" ) = [1], ( "solvec3" ) = Vector(21, {(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, (20) = .0, (21) = .0}, datatype = float[8]), ( "extrabcs" ) = [0], ( "PDEs" ) = [(diff(u(r, t), r)+r*(diff(diff(u(r, t), r), r)))/r-(diff(u(r, t), t))], ( "dependson" ) = [{1}], ( "method" ) = theta, ( "solmat_i2" ) = 0, ( "spaceadaptive" ) = false, ( "solution" ) = Array(1..3, 1..21, 1..1, {(1, 1, 1) = .0, (1, 2, 1) = .0, (1, 3, 1) = .0, (1, 4, 1) = .0, (1, 5, 1) = .0, (1, 6, 1) = .0, (1, 7, 1) = .0, (1, 8, 1) = .0, (1, 9, 1) = .0, (1, 10, 1) = .0, (1, 11, 1) = .0, (1, 12, 1) = .0, (1, 13, 1) = .0, (1, 14, 1) = .0, (1, 15, 1) = .0, (1, 16, 1) = .0, (1, 17, 1) = .0, (1, 18, 1) = .0, (1, 19, 1) = .0, (1, 20, 1) = .0, (1, 21, 1) = .0, (2, 1, 1) = .0, (2, 2, 1) = .0, (2, 3, 1) = .0, (2, 4, 1) = .0, (2, 5, 1) = .0, (2, 6, 1) = .0, (2, 7, 1) = .0, (2, 8, 1) = .0, (2, 9, 1) = .0, (2, 10, 1) = .0, (2, 11, 1) = .0, (2, 12, 1) = .0, (2, 13, 1) = .0, (2, 14, 1) = .0, (2, 15, 1) = .0, (2, 16, 1) = .0, (2, 17, 1) = .0, (2, 18, 1) = .0, (2, 19, 1) = .0, (2, 20, 1) = .0, (2, 21, 1) = .0, (3, 1, 1) = .0, (3, 2, 1) = .0, (3, 3, 1) = .0, (3, 4, 1) = .0, (3, 5, 1) = .0, (3, 6, 1) = .0, (3, 7, 1) = .0, (3, 8, 1) = .0, (3, 9, 1) = .0, (3, 10, 1) = .0, (3, 11, 1) = .0, (3, 12, 1) = .0, (3, 13, 1) = .0, (3, 14, 1) = .0, (3, 15, 1) = .0, (3, 16, 1) = .0, (3, 17, 1) = .0, (3, 18, 1) = .0, (3, 19, 1) = .0, (3, 20, 1) = .0, (3, 21, 1) = .0}, datatype = float[8], order = C_order), ( "timeadaptive" ) = false, ( "totalwidth" ) = 6, ( "spacevar" ) = r, ( "depeqn" ) = [1], ( "solmatrix" ) = Matrix(21, 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, (8, 1) = .0, (8, 2) = .0, (8, 3) = .0, (8, 4) = .0, (8, 5) = .0, (8, 6) = .0, (9, 1) = .0, (9, 2) = .0, (9, 3) = .0, (9, 4) = .0, (9, 5) = .0, (9, 6) = .0, (10, 1) = .0, (10, 2) = .0, (10, 3) = .0, (10, 4) = .0, (10, 5) = .0, (10, 6) = .0, (11, 1) = .0, (11, 2) = .0, (11, 3) = .0, (11, 4) = .0, (11, 5) = .0, (11, 6) = .0, (12, 1) = .0, (12, 2) = .0, (12, 3) = .0, (12, 4) = .0, (12, 5) = .0, (12, 6) = .0, (13, 1) = .0, (13, 2) = .0, (13, 3) = .0, (13, 4) = .0, (13, 5) = .0, (13, 6) = .0, (14, 1) = .0, (14, 2) = .0, (14, 3) = .0, (14, 4) = .0, (14, 5) = .0, (14, 6) = .0, (15, 1) = .0, (15, 2) = .0, (15, 3) = .0, (15, 4) = .0, (15, 5) = .0, (15, 6) = .0, (16, 1) = .0, (16, 2) = .0, (16, 3) = .0, (16, 4) = .0, (16, 5) = .0, (16, 6) = .0, (17, 1) = .0, (17, 2) = .0, (17, 3) = .0, (17, 4) = .0, (17, 5) = .0, (17, 6) = .0, (18, 1) = .0, (18, 2) = .0, (18, 3) = .0, (18, 4) = .0, (18, 5) = .0, (18, 6) = .0, (19, 1) = .0, (19, 2) = .0, (19, 3) = .0, (19, 4) = .0, (19, 5) = .0, (19, 6) = .0, (20, 1) = .0, (20, 2) = .0, (20, 3) = .0, (20, 4) = .0, (20, 5) = .0, (20, 6) = .0, (21, 1) = .0, (21, 2) = .0, (21, 3) = .0, (21, 4) = .0, (21, 5) = .0, (21, 6) = .0}, datatype = float[8], order = C_order), ( "depdords" ) = [[[2, 1]]], ( "indepvars" ) = [r, t], ( "solvec5" ) = 0, ( "depords" ) = [[2, 1]], ( "solmat_i1" ) = 0, ( "initialized" ) = false, ( "pts", r ) = [1, 4], ( "fdepvars" ) = [u(r, t)], ( "startup_only" ) = false, ( "solvec1" ) = Vector(21, {(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, (20) = .0, (21) = .0}, datatype = float[8]), ( "vectorproc" ) = proc (v, vp, vpp, t, x, k, h, n, vec) local _s1, _s2, _s3, _s4, _s5, _s6, xi; _s3 := -2*k; _s4 := -4*h^2; _s5 := -h*k; _s6 := 4*h^2*k; vec[1] := 0; vec[n] := 0; for xi from 2 to n-1 do _s1 := -vp[xi-1]+vp[xi+1]; _s2 := vp[xi-1]-2*vp[xi]+vp[xi+1]; vec[xi] := (_s2*_s3*x[xi]+_s4*vp[xi]*x[xi]+_s1*_s5)/(_s6*x[xi]) end do end proc, ( "adjusted" ) = false, ( "explicit" ) = false, ( "solspace" ) = Vector(21, {(1) = 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, (20) = .0, (21) = 4.0}, datatype = float[8]), ( "banded" ) = true, ( "solmat_ne" ) = 0, ( "solvec2" ) = Vector(21, {(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, (20) = .0, (21) = .0}, datatype = float[8]), ( "periodic" ) = false, ( "linear" ) = true, ( "errorest" ) = false, ( "rightwidth" ) = 0, ( "IBC" ) = b, ( "t0" ) = 0, ( "spacepts" ) = 21, ( "mixed" ) = false, ( "norigdepvars" ) = 1, ( "theta" ) = 1/2, ( "depshift" ) = [1], ( "stages" ) = 1, ( "soltimes" ) = Vector(3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8]), ( "solmat_is" ) = 0, ( "intspace" ) = Matrix(21, 1, {(1, 1) = .0, (2, 1) = .0, (3, 1) = .0, (4, 1) = .0, (5, 1) = .0, (6, 1) = .0, (7, 1) = .0, (8, 1) = .0, (9, 1) = .0, (10, 1) = .0, (11, 1) = .0, (12, 1) = .0, (13, 1) = .0, (14, 1) = .0, (15, 1) = .0, (16, 1) = .0, (17, 1) = .0, (18, 1) = .0, (19, 1) = .0, (20, 1) = .0, (21, 1) = .0}, datatype = float[8], order = C_order), ( "timeidx" ) = 2, ( "depvars" ) = [u], ( "leftwidth" ) = 1, ( "solmat_v" ) = Vector(126, {(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, (20) = .0, (21) = .0, (22) = .0, (23) = .0, (24) = .0, (25) = .0, (26) = .0, (27) = .0, (28) = .0, (29) = .0, (30) = .0, (31) = .0, (32) = .0, (33) = .0, (34) = .0, (35) = .0, (36) = .0, (37) = .0, (38) = .0, (39) = .0, (40) = .0, (41) = .0, (42) = .0, (43) = .0, (44) = .0, (45) = .0, (46) = .0, (47) = .0, (48) = .0, (49) = .0, (50) = .0, (51) = .0, (52) = .0, (53) = .0, (54) = .0, (55) = .0, (56) = .0, (57) = .0, (58) = .0, (59) = .0, (60) = .0, (61) = .0, (62) = .0, (63) = .0, (64) = .0, (65) = .0, (66) = .0, (67) = .0, (68) = .0, (69) = .0, (70) = .0, (71) = .0, (72) = .0, (73) = .0, (74) = .0, (75) = .0, (76) = .0, (77) = .0, (78) = .0, (79) = .0, (80) = .0, (81) = .0, (82) = .0, (83) = .0, (84) = .0, (85) = .0, (86) = .0, (87) = .0, (88) = .0, (89) = .0, (90) = .0, (91) = .0, (92) = .0, (93) = .0, (94) = .0, (95) = .0, (96) = .0, (97) = .0, (98) = .0, (99) = .0, (100) = .0, (101) = .0, (102) = .0, (103) = .0, (104) = .0, (105) = .0, (106) = .0, (107) = .0, (108) = .0, (109) = .0, (110) = .0, (111) = .0, (112) = .0, (113) = .0, (114) = .0, (115) = .0, (116) = .0, (117) = .0, (118) = .0, (119) = .0, (120) = .0, (121) = .0, (122) = .0, (123) = .0, (124) = .0, (125) = .0, (126) = .0}, datatype = float[8], order = C_order, attributes = [source_rtable = (Matrix(21, 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, (8, 1) = .0, (8, 2) = .0, (8, 3) = .0, (8, 4) = .0, (8, 5) = .0, (8, 6) = .0, (9, 1) = .0, (9, 2) = .0, (9, 3) = .0, (9, 4) = .0, (9, 5) = .0, (9, 6) = .0, (10, 1) = .0, (10, 2) = .0, (10, 3) = .0, (10, 4) = .0, (10, 5) = .0, (10, 6) = .0, (11, 1) = .0, (11, 2) = .0, (11, 3) = .0, (11, 4) = .0, (11, 5) = .0, (11, 6) = .0, (12, 1) = .0, (12, 2) = .0, (12, 3) = .0, (12, 4) = .0, (12, 5) = .0, (12, 6) = .0, (13, 1) = .0, (13, 2) = .0, (13, 3) = .0, (13, 4) = .0, (13, 5) = .0, (13, 6) = .0, (14, 1) = .0, (14, 2) = .0, (14, 3) = .0, (14, 4) = .0, (14, 5) = .0, (14, 6) = .0, (15, 1) = .0, (15, 2) = .0, (15, 3) = .0, (15, 4) = .0, (15, 5) = .0, (15, 6) = .0, (16, 1) = .0, (16, 2) = .0, (16, 3) = .0, (16, 4) = .0, (16, 5) = .0, (16, 6) = .0, (17, 1) = .0, (17, 2) = .0, (17, 3) = .0, (17, 4) = .0, (17, 5) = .0, (17, 6) = .0, (18, 1) = .0, (18, 2) = .0, (18, 3) = .0, (18, 4) = .0, (18, 5) = .0, (18, 6) = .0, (19, 1) = .0, (19, 2) = .0, (19, 3) = .0, (19, 4) = .0, (19, 5) = .0, (19, 6) = .0, (20, 1) = .0, (20, 2) = .0, (20, 3) = .0, (20, 4) = .0, (20, 5) = .0, (20, 6) = .0, (21, 1) = .0, (21, 2) = .0, (21, 3) = .0, (21, 4) = .0, (21, 5) = .0, (21, 6) = .0}, datatype = float[8], order = C_order))]), ( "solvec4" ) = 0, ( "allocspace" ) = 21, ( "matrixproc" ) = proc (v, vp, vpp, t, x, k, h, n, mat) local _s1, _s2, xi; _s1 := 4*h^2; _s2 := -(h^2+k)/(h^2*k); mat[3] := 1; mat[6*n-3] := 1; for xi from 2 to n-1 do mat[6*xi-3] := _s2; mat[6*xi-4] := -(h-2*x[xi])/(_s1*x[xi]); mat[6*xi-2] := (h+2*x[xi])/(_s1*x[xi]) end do end proc, ( "erroraccum" ) = true, ( "ICS" ) = [r], ( "minspcpoints" ) = 4, ( "maxords" ) = [2, 1], ( "multidep" ) = [false, false], ( "vectorhf" ) = true, ( "timevar" ) = t, ( "spaceidx" ) = 1, ( "inputargs" ) = [(diff(u(r, t), r)+r*(diff(diff(u(r, t), r), r)))/r = diff(u(r, t), t), [u(1, t) = 0, u(4, t) = 0, u(r, 0) = r]], ( "bandwidth" ) = [1, 2], ( "BCS", 1 ) = {[[1, 0, 1], b[1, 0, 1]], [[1, 0, 4], b[1, 0, 4]]}, ( "timestep" ) = .150000000000000, ( "eqnords" ) = [[2, 1]], ( "autonomous" ) = true, ( "matrixhf" ) = true, ( "spacestep" ) = .150000000000000 ] ); if xv = "left" then return INFO["solspace"][1] elif xv = "right" then return INFO["solspace"][INFO["spacepts"]] elif tv = "start" then return INFO["t0"] elif not (type(tv, 'numeric') and type(xv, 'numeric')) then error "non-numeric input" end if; if xv < INFO["solspace"][1] or INFO["solspace"][INFO["spacepts"]] < xv then error "requested %1 value must be in the range %2..%3", INFO["spacevar"], INFO["solspace"][1], INFO["solspace"][INFO["spacepts"]] end if; dary := Vector(3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8]); daryt := 0; daryx := 0; dvars := []; errest := false; nd := nops(INFO["depvars"]); if dary[nd+1] <> tv then try `pdsolve/numeric/evolve_solution`(INFO, tv) catch: msg := StringTools:-FormatMessage(lastexception[2 .. -1]); if tv < INFO["t0"] then error cat("unable to compute solution for %1<%2:
", msg), INFO["timevar"], INFO["failtime"] else error cat("unable to compute solution for %1>%2:
", msg), INFO["timevar"], INFO["failtime"] end if end try end if; if dary[nd+1] <> tv or dary[nd+2] <> xv then `pdsolve/interp2dto0d`(3, INFO["soltimes"], INFO["spacepts"], INFO["solspace"], nops(INFO["depvars"]), INFO["solution"], true, tv, xv, dary); if errest then `pdsolve/interp2dto0d`(3, INFO["soltimes"], INFO["spacepts"], INFO["err_t"], nops(INFO["depvars"]), INFO["solution"], true, tv, xv, daryt); `pdsolve/interp2dto0d`(3, INFO["soltimes"], INFO["spacepts"], INFO["err_x"], nops(INFO["depvars"]), INFO["solution"], true, tv, xv, daryx) end if end if; dary[nd+1] := tv; dary[nd+2] := xv; if dvars = [] then [seq(dary[i], i = 1 .. INFO["norigdepvars"])] else vals := NULL; for i to nops(dvars) do j := eval(dvars[i]); try if errest then vals := vals, evalhf(j(tv, xv, dary, daryt, daryx)) else vals := vals, evalhf(j(tv, xv, dary)) end if catch: userinfo(5, `pdsolve/numeric`, `evalhf failure`); try if errest then vals := vals, j(tv, xv, dary, daryt, daryx) else vals := vals, j(tv, xv, dary) end if catch: vals := vals, undefined end try end try end do; [vals] end if end proc; stype := "2nd"; if nargs = 1 then if args[1] = "left" then return solnproc(0, "left") elif args[1] = "right" then return solnproc(0, "right") elif args[1] = "start" then return solnproc("start", 0) else error "too few arguments to solution procedure" end if elif nargs = 2 then if stype = "1st" then tv := evalf(args[1]); xv := evalf(args[2]) else tv := evalf(args[2]); xv := evalf(args[1]) end if; if not (type(tv, 'numeric') and type(xv, 'numeric')) then if procname <> unknown then return ('procname')(args[1 .. nargs]) else ndsol := pointto(solnproc("soln_procedures")[1]); return ('ndsol')(args[1 .. nargs]) end if end if else error "incorrect arguments to solution procedure" end if; vals := solnproc(tv, xv); vals[1] end proc

(6)

uval(4, 10^(-10))

HFloat(3.9999999959999997)

(7)

uval(1, 10^(-10))

HFloat(0.9999999989999999)

(8)

evalf((D[1](uval))(2, 4))

0.17678671166000e-1

(9)

evalf(eval(diff(uval(r, t), r), [r = 2, t = 4]))

0.17678671166000e-1

(10)

evalf(eval(diff(uval(r, t), t), [r = 2, t = 4]))

-0.54629668566000e-1

(11)

``


 

Download Derivative-numerical2.mw

Command useint in dsolve helps:


 

restart

kernelopts(version)

`Maple 2019.1, X86 64 WINDOWS, May 21 2019, Build ID 1399874`

(1)

de := x^4*(diff(y(x), `$`(x, 2)))+omega^2*y(x) = 0; bc := y(a) = 0, y(b) = 0, (D(y))(a) = 1

x^4*(diff(diff(y(x), x), x))+omega^2*y(x) = 0

 

y(a) = 0, y(b) = 0, (D(y))(a) = 1

(2)

dsol := `assuming`([dsolve([bc, de], [y(x), omega], useint)], [a > 0, b > a])

{omega = -I*b*a*ln(I*RootOf(_Z^2+1))/(-b+a), y(x) = (1/2)*b^(-b/(-b+a))*(I*RootOf(_Z^2+1)*b)^(b/(-b+a))*(-b+a)*exp(ln(x)-b*a*ln(I*RootOf(_Z^2+1))/(x*(-b+a)))/(b*ln(I*RootOf(_Z^2+1)))-(1/2)*(-b+a)*b^(-b/(-b+a))*(I*RootOf(_Z^2+1)/b)^(-b/(-b+a))*exp(ln(x)+b*a*ln(I*RootOf(_Z^2+1))/(x*(-b+a)))/(b*ln(I*RootOf(_Z^2+1)))}

(3)

simplify(convert(dsol, radical))

{omega = b*a*Pi/(-b+a), y(x) = -((1/2)*I)*b^(-a/(-b+a))*(-b+a)*((-b)^(b/(-b+a))*exp((x*(-b+a)*ln(x)-I*b*a*Pi)/(x*(-b+a)))-(-1/b)^(-b/(-b+a))*exp((x*(-b+a)*ln(x)+I*b*a*Pi)/(x*(-b+a))))/Pi}

(4)

``


 

Download solution.mw

 

simplify(int(sin(y)^(1/3)*cos(y)^3, y = 0 .. x, AllSolutions));

#piecewise(sin(x) < 0, FAIL, 0 <= sin(x), -3*(2*sin(x)^2 - 5)*sin(x)^(4/3)/20)

 

Yes, Maple can't  handle elliptic PDEs,because last Updates(Enhanced) to solving PDE's equation was in Maple 9 in year 2003.

During this time  no improvements have been made.

What a pity.

pdsolve can solve,on Maple 2019.1 with Physics:-Version(399):

PDE := diff(u(x, t), t) + 2*u(x, t)^2*diff(u(x, t), x) - diff(u(x, t), x)^2 - 1/2*diff(u(x, t), x $ 2)*u(x, t) = 0;

pdsolve([PDE, u(x, 0) = -tanh(x)]);

#u(x, t) = tanh(t - x)

sol := solve({sin(9*x - 1/3*Pi) = sin(7*x - 1/3*Pi)}, [x], explicit, allsolutions);

seq(simplify(sol[j]), j = 1 .. nops(sol));

#[x = 2*Pi*_Z13], [x = Pi*(2*_Z14 + 1)], [x = -1/48*Pi + 2*Pi*_Z15], [x = 47/48*Pi + 2*Pi*_Z15], [x = #23/48*Pi + 2*Pi*_Z15], [x = #-25/48*Pi + 2*Pi*_Z15], [x = 5/48*Pi + 2*Pi*_Z15], [x = -43/48*Pi + #2*Pi*_Z15], [x = -19/48*Pi + 2*Pi*_Z15], [x = 29/48*Pi + 2*Pi*_Z15], #[x = -7/48*Pi + 2*Pi*_Z15], [x = #41/48*Pi + 2*Pi*_Z15], [x = 17/48*Pi + 2*Pi*_Z15], [x = -31/48*Pi + 2*Pi*_Z15], [x = 11/48*Pi + #2*Pi*_Z15], [x = -37/48*Pi + 2*Pi*_Z15], [x = -13/48*Pi + 2*Pi*_Z15], [x = 35/48*Pi + 2*Pi*_Z15]


 

restart

Int(d*exp(-b*x)/(a*exp(-2*b*x)+c*exp(-4*x)), x); IntegrationTools:-Change(%, x = ln(t), t); `assuming`([simplify(%, symbolic)], [0 < t]); `assuming`([value(%)], [-2*b+4 > 0, -b+3 > 0, c > 0, a > 0]); convert(%, hypergeom); INTEGRAL = simplify(eval(%, t = exp(x)))

d*(Int(t^(-b+3)/(a*t^(-2*b+4)+c), t))

 

2*d*(b-2)*t^(2*b^2/(-2*b+4)-12*b/(-2*b+4)+16/(-2*b+4))*(-b+4)*LerchPhi(-t^(-2*b+4)*a/c, 1, (b-4)/(2*b-4))/(c*(-2*b+4)^2*(b-4))

 

2*d*(b-2)*t^(2*b^2/(-2*b+4)-12*b/(-2*b+4)+16/(-2*b+4))*(-b+4)*(2*b-4)*hypergeom([1, (b-4)/(2*b-4)], [(3*b-8)/(2*b-4)], -a/(t^(2*b-4)*c))/(c*(-2*b+4)^2*(b-4)^2)

 

INTEGRAL = -d*(exp(x))^(-b)*exp(4*x)*hypergeom([1, (b-4)/(2*b-4)], [(3*b-8)/(2*b-4)], -(exp(x))^(-2*b)*exp(4*x)*a/c)/((b-4)*c)

(1)

``


 

Download integral.mw

With MMA I was able to solve  double integral.


 

restart

Digits := 14

14

(1)

P := exp(-h*Pi*v*(x+y)/b)/cosh(Pi*v*(y-x)/b)^(4+h)

exp(-h*Pi*v*(x+y)/b)/cosh(Pi*v*(y-x)/b)^(4+h)

(2)

h := -1; v := 1; b := 1; w := 1; z := 1

-1

 

1

 

1

 

1

 

1

(3)

J := proc (h, v, b, w, z) options operator, arrow; evalf(int(P, [y = -infinity .. z, x = -infinity .. w], numeric)) end proc

proc (h, v, b, w, z) options operator, arrow; evalf(int(P, [y = -infinity .. z, x = -infinity .. w], numeric)) end proc

(4)

J(h, v, b, w, z)

27.128324184182

(5)

with(MmaTranslator)

integral := FromMma("(4*b^2*(-((2^h*(E^((-2*h*Pi*v*w)/b) + E^((-2*h*Pi*v*z)/b)))/(6 + 5*h + h^2)) + (E^((4*Pi*v*w)/b) + E^((4*Pi*v*z)/b) + E^((2*Pi*v*(w + z))/b)*(2 + h))/
    (E^((h*Pi*v*(w + z))/b)*(E^((2*Pi*v*w)/b) + E^((2*Pi*v*z)/b))^2*(2 + h)*(3 + h)*Cosh[(Pi*v*(w - z))/b]^h)))/(h*Pi^2*v^2)")

4*b^2*(-2^h*(exp(-2*h*Pi*v*w/b)+exp(-2*h*Pi*v*z/b))/(h^2+5*h+6)+(exp(4*Pi*v*w/b)+exp(4*Pi*v*z/b)+exp(2*Pi*v*(w+z)/b)*(2+h))/(exp(h*Pi*v*(w+z)/b)*(exp(2*Pi*v*w/b)+exp(2*Pi*v*z/b))^2*(2+h)*(3+h)*cosh(Pi*v*(w-z)/b)^h))/(h*Pi^2*v^2)

(6)

NULL

J2 := proc (h, v, b, w, z) options operator, arrow; integral end proc

J2(h, v, b, w, z)

-4*(-(1/2)*exp(2*Pi)+(3/8)*exp(4*Pi)/(exp(-2*Pi)*(exp(2*Pi))^2))/Pi^2

(7)

evalf(-4*(-(1/2)*exp(2*Pi)+(3/8)*exp(4*Pi)/(exp(-2*Pi)*(exp(2*Pi))^2))/Pi^2)

27.128324184196

(8)

NULL

``


 

Download integral2.mw

It's seems Maple dosen't know the answer.With workaround:


 

Int(exp(-b*x)/(exp(-2*b*x)*a+exp(-4*x)), x)

Int(exp(-b*x)/(exp(-2*b*x)*a+exp(-4*x)), x)

(1)

exp(-b*x)/(exp(-2*b*x)*a+exp(-4*x)) = exp(-b*x)/(exp(-4*x)*(1+exp(-2*b*x)*a/exp(-4*x))) and exp(-b*x)/(exp(-4*x)*(1+exp(-2*b*x)*a/exp(-4*x))) = exp(-b*x)/(exp(-4*x)*(1+a*exp(-2*x*(b-2)))) and exp(-b*x)/(exp(-4*x)*(1+a*exp(-2*x*(b-2)))) = exp(-x*(b-4))/(1+a*exp(-2*x*(b-2)))

Int(exp(-x*(b-4))/(1+a*exp(-2*x*(b-2))), x) = Int(Sum(exp(-x*(b-4))*(-a*exp(-2*x*(b-2)))^n, n = 0 .. infinity), x)

Int(exp(-x*(b-4))/(1+a*exp(-2*x*(b-2))), x) = Int(Sum(exp(-x*(b-4))*(-a*exp(-2*x*(b-2)))^n, n = 0 .. infinity), x)

(2)

Sum(Int(exp(-x*(b-4))*(-a*exp(-2*x*(b-2)))^n, x), n = 0 .. infinity) = Sum(-exp(-x*(b-4))*(-a*exp(-2*x*(b-2)))^n/(2*b*n+b-4*n-4), n = 0 .. infinity)

Sum(Int(exp(-x*(b-4))*(-a*exp(-2*x*(b-2)))^n, x), n = 0 .. infinity) = Sum(-exp(-x*(b-4))*(-a*exp(-2*x*(b-2)))^n/(2*b*n+b-4*n-4), n = 0 .. infinity)

(3)

`assuming`([sum(expand(-exp(-x*(b-4))*(-a*exp(-2*x*(b-2)))^n/(2*b*n+b-4*n-4)), n = 0 .. infinity, formal)], [x > 0, a > 0, b > 0])

-exp(-b*x+4*x)*LerchPhi(-a*exp(-2*x*(b-2)), 1, (b-4)/(2*b-4))/(2*b-4)

(4)

simplify(convert(-exp(-b*x+4*x)*LerchPhi(-a*exp(-2*x*(b-2)), 1, (b-4)/(2*b-4))/(2*b-4), hypergeom))

-exp(-x*(b-4))*hypergeom([1, (b-4)/(2*b-4)], [(3*b-8)/(2*b-4)], -a*exp(-2*x*(b-2)))/(b-4)

(5)

NULL

Int(exp(-b*x)/(exp(-2*b*x)*a+exp(-4*x)), x) = -exp(-b*x+4*x)*LerchPhi(-a*exp(-2*x*(b-2)), 1, (b-4)/(2*b-4))/(2*b-4)

Int(exp(-b*x)/(exp(-2*b*x)*a+exp(-4*x)), x) = -exp(-b*x+4*x)*LerchPhi(-a*exp(-2*x*(b-2)), 1, (b-4)/(2*b-4))/(2*b-4)

(6)

Int(exp(-b*x)/(exp(-2*b*x)*a+exp(-4*x)), x) = -exp(-x*(b-4))*hypergeom([1, (b-4)/(2*b-4)], [(3*b-8)/(2*b-4)], -a*exp(-2*x*(b-2)))/(b-4)

Int(exp(-b*x)/(exp(-2*b*x)*a+exp(-4*x)), x) = -exp(-x*(b-4))*hypergeom([1, (b-4)/(2*b-4)], [(3*b-8)/(2*b-4)], -a*exp(-2*x*(b-2)))/(b-4)

(7)

NULL

evalf(eval([exp(-b*x)/(exp(-2*b*x)*a+exp(-4*x)), diff(-exp(-x*(b-4))*hypergeom([1, (b-4)/(2*b-4)], [(3*b-8)/(2*b-4)], -a*exp(-2*x*(b-2)))/(b-4), x)], [x = 2, a = 2, b = 3]))

[7.127950177, 7.127950177]

(8)

``


 

Download integral.mw

 

Try:

Explore(plots:-implicitplot(y^2 = a*x^3 + b*x^2 + c*x + d, x = -5 .. 5, y = -5 .. 5, gridrefine = 2, view = [-5 .. 5, -5 .. 5]), a = -5 .. 5, b = -5 .. 5, c = -5 .. 5, d = -5 .. 5);

func := 2*omega^2*B[1]*Zeta*omega[0] - omega^3*B[1]*I + omega*B[1]*omega[0]^2*I - 2*I*B[0]*Zeta*omega*omega[0] - B[0]*omega^2 + B[0]*omega[0]^2;

(evalc(func) assuming (B[0] in real, omega[0] in real, omega in real, Zeta in real, B[1] in real));

#2*omega^2*B[1]*Zeta*omega[0] - B[0]*omega^2 + B[0]*omega[0]^2 + (-2*Zeta*omega*B[0]*omega[0] - omega^3*B[1] + omega*B[1]*omega[0]^2)*I

 

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