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Rouben Rostamian

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These are answers submitted by Rouben Rostamian

Working file...

Rouben Rostamian 7711
November 03 2019
0 0

Is your code attempting to guess what the last few notes of the melody are?  If that is so, then you should explain how that is supposed to work since the code that you have presented does not succeed.

In the meantime, I have edited your code by removing the guessing parts and adding the missing parts of the melody.  Since the final phrase of the melody calls for a B-flat note, I have added that to your set of notes.

Working code:   Download mw.mw

 

In addition to the answers suggested......

Rouben Rostamian 7711
November 01 2019
2 0 
div := (a,b) -> a/b:                                                         
div(a,b);                                                                    
                                      a/b

div_list := x -> x[1]/x[2]:                                                  
div_list([a,b]);                                                             
                                      a/b

 

Perhaps this is what you are looking for...

Rouben Rostamian 7711
October 31 2019
0 0 
restart;
de1 := diff(x1(t),t) = 3*x1(t) + 2*x2(t):
de2 := diff(x2(t),t) = 5*x1(t) + 1*x2(t):
ic := seq([x1(0)=s, x2(0)=0], s=-1..1, 0.1),
      seq([x1(0)=0, x2(0)=s], s=-1..1, 0.1):
DEtools:-DEplot({de1, de2}, [x1(t),x2(t)], t=-1..1, [ic],
    x1(t)=-1..1, x2(t)=-1..1, linecolor=black,
    thickness=1, arrows=none);

Change of variables...

Rouben Rostamian 7711
October 29 2019
4 1

I will use subscripts to indicate partial derivatives, as in u__x for `∂`(u)/`∂`(x).
You are looking for the formula
u__x(x, y) = `u__ξ`(xi, eta)*`ξ__x`(x, y)+`u__η`(xi, eta)*`η__x`(x, y).
That is not quite wrong but it has at least a couple of difficulties.

First, you use the symbol u with two different meanings since your
u(xi, eta) really means u(xi(x, y), eta(x, y)).  A human can interpret
that by allowing for some sloppiness, but we shouldn't expect
Maple to get sloppy.

Second, your formula as a change of variables is incomplete.  In a
proper change of variables we expect to see the old variables on one
side and the new variables on the other side.  In your formula the
variables x and y occur on both sides.

OK, let's do it the right way.

Let the variables x, y and xi, eta be related through
"x = A(xi,eta),"
"y=B(xi,eta)."
Differentiating these with respect to x we get
"1=`A__xi`(xi,eta) `xi__x`+`A__eta`(xi,eta) `eta__x `,"
"0 =`B__xi`(xi,eta) `xi__x`+`B__eta`(xi,eta) `eta__x `."
We solve this as a linear system for `ξ__x` and `η__x` and obtain
`ξ__x` = `B__η`/Delta, `η__x` = -`B__ξ`/Delta,   where  Delta = `A__ξ`*`B__η`-`B__ξ`*`A__η`.

Now let "u(x,y)=u(A(xi,eta),B(xi,eta)) = U(xi,eta)."  Differentiating
this with respect to x we get

u__x(x, y) = `U__ξ`(xi, eta)*`ξ__x`+`U__η`(xi, eta)*`η__x` and `U__ξ`(xi, eta)*`ξ__x`+`U__η`(xi, eta)*`η__x` = `U__ξ`(xi, eta)*`B__η`(xi, eta)/Delta-`U__η`(xi, eta)*`B__ξ`(xi, eta)/Delta
.
That's the correct form of the change of variables formula.
Note that x and y occur on the left, while xi and eta occur
on the right.  Also observe the introduction of the symbols
U, A, B  to avoid notational ambiguity.

Now that we know how to do this by hand, let's see what
we get with Maple's dchange.

> 

restart;
> 

PDEtools:-dchange({x=A(xi,eta), y=B(xi,eta), u(x,y)=U(xi,eta)},
                      diff(u(x,y),x), {xi,eta,U});

(diff(U(xi, eta), eta))*(diff(B(xi, eta), xi))/((diff(A(xi, eta), eta))*(diff(B(xi, eta), xi))-(diff(A(xi, eta), xi))*(diff(B(xi, eta), eta)))-(diff(U(xi, eta), xi))*(diff(B(xi, eta), eta))/((diff(A(xi, eta), eta))*(diff(B(xi, eta), xi))-(diff(A(xi, eta), xi))*(diff(B(xi, eta), eta)))

We see that Maple's result agrees with our manual solution.

 


 

Workaround...

Rouben Rostamian 7711
October 21 2019
1 0

What you have observed is a shortcoming (a polite way of saying a bug) of pdsolve() and needs to be corrected.  A temporary workaround is to replace the four occurrences of Pi in the boundary conditions and the two occurrences of Pi in the range specification with 1.0*Pi.

Aside: The range specification is redundant since pdsolve() picks up the range from the boundary conditions.

Sliding on a surface...

Rouben Rostamian 7711
October 18 2019
3 1

Carl has shown how to find the path of the steepest descent on a surface.  I want to point out that a particle that slides down a surface (without friction) does not necessarily follow the path of steepest descent—the particle's inertia pulls it away.

In the attached worksheet I calculate and plot on a paraboloid (a) the path of steepest descent in red (as Carl has done), and (b) the path followed by a point mass in blue.  The figure below, extracted from the worksheet, shows that the two paths are not the same.

 

Worksheet: paticle-sliding-on-a-paraboloid.mw

 

Reference...

Rouben Rostamian 7711
October 15 2019
2 3

This was asked here about a year ago and I posted this answer.

  
Reference: https://www.mapleprimes.com/questions/225760-Spinning-T-Handle-In-Space#answer253863

Note added later:

Oops, I just noticed that you are asking for a solution in MapleSim.  I don't have MapleSim so I cannot help you there.  Sorry.  Perhaps someone else can.

 

Another way...

Rouben Rostamian 7711
October 13 2019
0 0

As Carl said, there are very many ways of doing this.  Here is an alternative.

> 

restart;

Replace this with your function of current versus torque and speed

> 

F := proc(torque, speed)
  return torque*speed;
end proc:

Tabulate:

> 

torque_max := 100:
speed_max := 4000:
printf("%15s%15s%15s\n", "torque", "speed", "current");
for i from 0 to 10 do
  torque := i/10*torque_max;
  for j from 0 to 10 do
    speed := j/10*speed_max;
    printf("%15g%15g%15g\n", torque, speed, F(torque, speed));
  end do:
end do:

         torque          speed        current
              0              0              0
              0            400              0
              0            800              0
             ... (snipped)
             90           3600         324000
             90           4000         360000
            100              0              0
            100            400          40000
            100            800          80000
            100           1200         120000
            100           1600         160000
            100           2000         200000
            100           2400         240000
            100           2800         280000
            100           3200         320000
            100           3600         360000
            100           4000         400000

  

 

Download mw.mw

Animation...

Rouben Rostamian 7711
October 09 2019
2 1

It would be difficult to produce a truly periodic solution.  The cage has its own natural frequency (which depends on the amplitude) and the hamster has its own independent frequency.  The overall effect is the combination of two frequencies which is not necessarily periodic.  If you add some damping in the cage's motion, and if the hamster's motion is periodic, then the cage's motion will converge to a periodic solution in the long run.

In the attached worksheet I derive the equation of motion (without damping) assuming that the hamster movies sinusoidally on the cage's floor, solve the equation, and produce an animation.  The motion is not quite periodic but it comes close.  That's why the animation (which is played in an infinite loop) gives the appearance of a periodic motion.

Worksheet:  hamster_cage.mw

 

Introduce f(t)...

Rouben Rostamian 7711
October 07 2019
2 0

Let f(t) be int(c(h),h=0..t).  Then we have D(f)=c, and f(0)=0, D(f)(0)=c(0).  Then you may express your differential equation in terms of f, as follows.

Nota bene: I have inserted a missing multiplication sign (marked in red) in your equation.  Be sure to verify that's what is intended.

> 

ode := diff(c(t),t) = (ln(c(t)) + w - p*c(t)) *
           (c(t) * (t + 1/int(c(h), h =0..t)) + int(c(h), h=0..t)) /
           (p - 1/c(t));

diff(c(t), t) = (ln(c(t))+w-p*c(t))*(c(t)*(t+1/(int(c(h), h = 0 .. t)))+int(c(h), h = 0 .. t))/(p-1/c(t))

> 

diff(f(t),t,t) = (ln(diff(f(t),t)) + w - p*diff(f(t),t)) *
         ( diff(f(t),t) * (diff(f(t),t)*(t+1/f(t)) + f(t))) /
         (p - 1/diff(f(t),t));

diff(diff(f(t), t), t) = (ln(diff(f(t), t))+w-p*(diff(f(t), t)))*(diff(f(t), t))*((diff(f(t), t))*(t+1/f(t))+f(t))/(p-1/(diff(f(t), t)))


 

Download mw.mw

Perhaps this...

Rouben Rostamian 7711
October 04 2019
0 3 
plots:-gradplot3d(r^2+1, r=0..1, phi=0..2*Pi, theta=0..Pi/2, coords=spherical,
  view=[-1.5..1.5, -1.5..1.5, 0..1.5], scaling=constrained, axes=normal,
  grid=[3,10,5], arrows=THIN);

How about this?...

Rouben Rostamian 7711
September 26 2019
1 0

vars := [x,y,z];
seq([seq(cat(v,i), i=1..nops(vars))], v in vars);
 

A skeleton solution...

Rouben Rostamian 7711
September 23 2019
1 0

This skeleton example shows how to do what you want.  You can jazz it up as needed.

> 

restart;
> 

de := diff(x(t),t) = -3*x(t) + add(Dirac(t-i), i=1..3);

diff(x(t), t) = -3*x(t)+Dirac(t-1)+Dirac(t-2)+Dirac(t-3)

> 

ic := x(0)=1;

x(0) = 1

> 

dsol := dsolve({de,ic}, numeric);

proc (x_rkf45) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_rkf45) else _xout := evalf(x_rkf45) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _dtbl, _dat, _vmap, _x0, _y0, _val, _dig, _n, _ne, _nd, _nv, _pars, _ini, _par, _i, _j, _k, _src; option `Copyright (c) 2002 by Waterloo Maple Inc. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _pars := []; _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 ) = ([Array(1..13, 1..21, {(1, 1) = 1.0, (1, 2) = 2.0, (1, 3) = .0, (1, 4) = .0, (1, 5) = 2.999999600004, (1, 6) = .0, (1, 7) = 1.0, (1, 8) = undefined, (1, 9) = undefined, (1, 10) = 1.0, (1, 11) = undefined, (1, 12) = undefined, (1, 13) = undefined, (1, 14) = undefined, (1, 15) = undefined, (1, 16) = undefined, (1, 17) = undefined, (1, 18) = undefined, (1, 19) = undefined, (1, 20) = undefined, (1, 21) = undefined, (2, 1) = 2.0, (2, 2) = 2.0, (2, 3) = .0, (2, 4) = .0, (2, 5) = 2.999999999996, (2, 6) = .0, (2, 7) = 1.0, (2, 8) = undefined, (2, 9) = undefined, (2, 10) = 1.0, (2, 11) = undefined, (2, 12) = undefined, (2, 13) = undefined, (2, 14) = undefined, (2, 15) = undefined, (2, 16) = undefined, (2, 17) = undefined, (2, 18) = undefined, (2, 19) = undefined, (2, 20) = undefined, (2, 21) = undefined, (3, 1) = 3.0, (3, 2) = 2.0, (3, 3) = .0, (3, 4) = .0, (3, 5) = 3.000000000004, (3, 6) = .0, (3, 7) = 1.0, (3, 8) = undefined, (3, 9) = undefined, (3, 10) = 1.0, (3, 11) = undefined, (3, 12) = undefined, (3, 13) = undefined, (3, 14) = undefined, (3, 15) = undefined, (3, 16) = undefined, (3, 17) = undefined, (3, 18) = undefined, (3, 19) = undefined, (3, 20) = undefined, (3, 21) = undefined, (4, 1) = 4.0, (4, 2) = 2.0, (4, 3) = .0, (4, 4) = .0, (4, 5) = 3.000000399996, (4, 6) = .0, (4, 7) = 1.0, (4, 8) = undefined, (4, 9) = undefined, (4, 10) = 1.0, (4, 11) = undefined, (4, 12) = undefined, (4, 13) = undefined, (4, 14) = undefined, (4, 15) = undefined, (4, 16) = undefined, (4, 17) = undefined, (4, 18) = undefined, (4, 19) = undefined, (4, 20) = undefined, (4, 21) = undefined, (5, 1) = 5.0, (5, 2) = 2.0, (5, 3) = .0, (5, 4) = .0, (5, 5) = 1.999999700003, (5, 6) = .0, (5, 7) = 1.0, (5, 8) = undefined, (5, 9) = undefined, (5, 10) = 1.0, (5, 11) = undefined, (5, 12) = undefined, (5, 13) = undefined, (5, 14) = undefined, (5, 15) = undefined, (5, 16) = undefined, (5, 17) = undefined, (5, 18) = undefined, (5, 19) = undefined, (5, 20) = undefined, (5, 21) = undefined, (6, 1) = 6.0, (6, 2) = 2.0, (6, 3) = .0, (6, 4) = .0, (6, 5) = 1.999999999997, (6, 6) = .0, (6, 7) = 1.0, (6, 8) = undefined, (6, 9) = undefined, (6, 10) = 1.0, (6, 11) = undefined, (6, 12) = undefined, (6, 13) = undefined, (6, 14) = undefined, (6, 15) = undefined, (6, 16) = undefined, (6, 17) = undefined, (6, 18) = undefined, (6, 19) = undefined, (6, 20) = undefined, (6, 21) = undefined, (7, 1) = 7.0, (7, 2) = 2.0, (7, 3) = .0, (7, 4) = .0, (7, 5) = 2.000000000003, (7, 6) = .0, (7, 7) = 1.0, (7, 8) = undefined, (7, 9) = undefined, (7, 10) = 1.0, (7, 11) = undefined, (7, 12) = undefined, (7, 13) = undefined, (7, 14) = undefined, (7, 15) = undefined, (7, 16) = undefined, (7, 17) = undefined, (7, 18) = undefined, (7, 19) = undefined, (7, 20) = undefined, (7, 21) = undefined, (8, 1) = 8.0, (8, 2) = 2.0, (8, 3) = .0, (8, 4) = .0, (8, 5) = 2.000000299997, (8, 6) = .0, (8, 7) = 1.0, (8, 8) = undefined, (8, 9) = undefined, (8, 10) = 1.0, (8, 11) = undefined, (8, 12) = undefined, (8, 13) = undefined, (8, 14) = undefined, (8, 15) = undefined, (8, 16) = undefined, (8, 17) = undefined, (8, 18) = undefined, (8, 19) = undefined, (8, 20) = undefined, (8, 21) = undefined, (9, 1) = 9.0, (9, 2) = 2.0, (9, 3) = .0, (9, 4) = .0, (9, 5) = .999999800002, (9, 6) = .0, (9, 7) = 1.0, (9, 8) = undefined, (9, 9) = undefined, (9, 10) = 1.0, (9, 11) = undefined, (9, 12) = undefined, (9, 13) = undefined, (9, 14) = undefined, (9, 15) = undefined, (9, 16) = undefined, (9, 17) = undefined, (9, 18) = undefined, (9, 19) = undefined, (9, 20) = undefined, (9, 21) = undefined, (10, 1) = 10.0, (10, 2) = 2.0, (10, 3) = .0, (10, 4) = .0, (10, 5) = .999999999998, (10, 6) = .0, (10, 7) = 1.0, (10, 8) = undefined, (10, 9) = undefined, (10, 10) = 1.0, (10, 11) = undefined, (10, 12) = undefined, (10, 13) = undefined, (10, 14) = undefined, (10, 15) = undefined, (10, 16) = undefined, (10, 17) = undefined, (10, 18) = undefined, (10, 19) = undefined, (10, 20) = undefined, (10, 21) = undefined, (11, 1) = 11.0, (11, 2) = 2.0, (11, 3) = .0, (11, 4) = .0, (11, 5) = 1.000000000002, (11, 6) = .0, (11, 7) = 1.0, (11, 8) = undefined, (11, 9) = undefined, (11, 10) = 1.0, (11, 11) = undefined, (11, 12) = undefined, (11, 13) = undefined, (11, 14) = undefined, (11, 15) = undefined, (11, 16) = undefined, (11, 17) = undefined, (11, 18) = undefined, (11, 19) = undefined, (11, 20) = undefined, (11, 21) = undefined, (12, 1) = 12.0, (12, 2) = 2.0, (12, 3) = .0, (12, 4) = .0, (12, 5) = 1.000000199998, (12, 6) = .0, (12, 7) = 1.0, (12, 8) = undefined, (12, 9) = undefined, (12, 10) = 1.0, (12, 11) = undefined, (12, 12) = undefined, (12, 13) = undefined, (12, 14) = undefined, (12, 15) = undefined, (12, 16) = undefined, (12, 17) = undefined, (12, 18) = undefined, (12, 19) = undefined, (12, 20) = undefined, (12, 21) = undefined, (13, 1) = 12.0, (13, 2) = .0, (13, 3) = 100.0, (13, 4) = .0, (13, 5) = .0, (13, 6) = .0, (13, 7) = .0, (13, 8) = undefined, (13, 9) = undefined, (13, 10) = 0.10e-6, (13, 11) = undefined, (13, 12) = .0, (13, 13) = undefined, (13, 14) = .0, (13, 15) = .0, (13, 16) = undefined, (13, 17) = undefined, (13, 18) = undefined, (13, 19) = undefined, (13, 20) = undefined, (13, 21) = undefined}, datatype = float[8], order = C_order), proc (t, Y, Ypre, n, EA) EA[1, 8+2*n] := 1; EA[2, 8+2*n] := 1; EA[3, 8+2*n] := 1; EA[4, 8+2*n] := 1; EA[5, 8+2*n] := 1; EA[6, 8+2*n] := 1; EA[7, 8+2*n] := 1; EA[8, 8+2*n] := 1; EA[9, 8+2*n] := 1; EA[10, 8+2*n] := 1; EA[11, 8+2*n] := 1; EA[12, 8+2*n] := 1; 0 end proc, proc (e, t, Y, Ypre) return 0 end proc, Array(1..12, 1..2, {(1, 1) = undefined, (1, 2) = undefined, (2, 1) = undefined, (2, 2) = undefined, (3, 1) = undefined, (3, 2) = undefined, (4, 1) = undefined, (4, 2) = undefined, (5, 1) = undefined, (5, 2) = undefined, (6, 1) = undefined, (6, 2) = undefined, (7, 1) = undefined, (7, 2) = undefined, (8, 1) = undefined, (8, 2) = undefined, (9, 1) = undefined, (9, 2) = undefined, (10, 1) = undefined, (10, 2) = undefined, (11, 1) = undefined, (11, 2) = undefined, (12, 1) = undefined, (12, 2) = undefined}, datatype = float[8], order = C_order)]), ( 4 ) = (Array(1..63, {(1) = 1, (2) = 1, (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) = 12, (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) = 0.18508082104752334e-1, (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..1, {(1) = 1.0}, 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..1, {(1) = .1}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..1, {(1, 1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..1, {(1, 1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..1, {(1, 1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = 0}, datatype = integer[8]), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = 0}, datatype = integer[8])]), ( 8 ) = ([Array(1..1, {(1) = 1.0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = -3.0}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..1, {(1, 1) = .0, (2, 0) = .0, (2, 1) = .0, (3, 0) = .0, (3, 1) = .0, (4, 0) = .0, (4, 1) = .0, (5, 0) = .0, (5, 1) = .0, (6, 0) = .0, (6, 1) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = x(t)]`; YP[1] := -3*Y[1]+piecewise(X < 4999999/5000000, 0, X < 1, -24999995000000+25000000000000*X, X < 5000001/5000000, 25000005000000-25000000000000*X, 0)+piecewise(X < 19999997/10000000, 0, X < 2, -199999970000000/9+(100000000000000/9)*X, X < 20000003/10000000, 200000030000000/9-(100000000000000/9)*X, 0)+piecewise(X < 7499999/2500000, 0, X < 3, -18749997500000+6250000000000*X, X < 7500001/2500000, 18750002500000-6250000000000*X, 0); 0 end proc, -1, 0, 0, 0, 0, proc (t, Y, Ypre, n, EA) EA[1, 8+2*n] := 1; EA[2, 8+2*n] := 1; EA[3, 8+2*n] := 1; EA[4, 8+2*n] := 1; EA[5, 8+2*n] := 1; EA[6, 8+2*n] := 1; EA[7, 8+2*n] := 1; EA[8, 8+2*n] := 1; EA[9, 8+2*n] := 1; EA[10, 8+2*n] := 1; EA[11, 8+2*n] := 1; EA[12, 8+2*n] := 1; 0 end proc, proc (e, t, Y, Ypre) return 0 end proc, 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)]`; YP[1] := -3*Y[1]+piecewise(X < 4999999/5000000, 0, X < 1, -24999995000000+25000000000000*X, X < 5000001/5000000, 25000005000000-25000000000000*X, 0)+piecewise(X < 19999997/10000000, 0, X < 2, -199999970000000/9+(100000000000000/9)*X, X < 20000003/10000000, 200000030000000/9-(100000000000000/9)*X, 0)+piecewise(X < 7499999/2500000, 0, X < 3, -18749997500000+6250000000000*X, X < 7500001/2500000, 18750002500000-6250000000000*X, 0); 0 end proc, -1, 0, 0, 0, 0, proc (t, Y, Ypre, n, EA) EA[1, 8+2*n] := 1; EA[2, 8+2*n] := 1; EA[3, 8+2*n] := 1; EA[4, 8+2*n] := 1; EA[5, 8+2*n] := 1; EA[6, 8+2*n] := 1; EA[7, 8+2*n] := 1; EA[8, 8+2*n] := 1; EA[9, 8+2*n] := 1; EA[10, 8+2*n] := 1; EA[11, 8+2*n] := 1; EA[12, 8+2*n] := 1; 0 end proc, proc (e, t, Y, Ypre) return 0 end proc, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0) ] )) ] ); _y0 := Array(0..1, {(1) = 0.}); _vmap := array( 1 .. 1, [( 1 ) = (1) ] ); _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)], (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

> 

plots:-odeplot(dsol, t=0..4, view=0..1);


 

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Maybe this will help...

Rouben Rostamian 7711
September 21 2019
1 2

I have made many corrections and changes to your worksheet.  This may or may not be what you really need, therefore look at everything closely!

> 

restart;
> 

varepsilon := 300;

300

> 

alpha := 15;

15

> 

T := 5e6;

0.5e7

> 

L := 2e4;

0.2e5

> 

g := x -> piecewise(0 < x and x <= 5000, 0, 5000 < x and x <= 9000, 0.005*x - 25, 9000 < x and x <= 16000, (-1)*0.00286*x + 45.76, 16000 < x and x <= 20000, 0);

g := proc (x) options operator, arrow; piecewise(0 < x and x <= 5000, 0, 5000 < x and x <= 9000, 0.5e-2*x-25, 9000 < x and x <= 16000, (-1)*0.286e-2*x+45.76, 16000 < x and x <= 20000, 0) end proc

> 

Em := 2*int(g(x)*sin(m*Pi*x/L), x = 0 .. L)/L;

-0.8105694688e-8*(1570796327.*m*cos(1.413716694*m)-2500000000.*sin(1.413716694*m)+2500000000.*sin(.7853981635*m))/m^2-0.1159114341e-8*(-0.1099557429e11*m*cos(1.413716694*m)+0.1000000000e11*sin(2.513274123*m)-0.1000000000e11*sin(1.413716694*m))/m^2

Note: I have inserted a negative sign in the exponent because I think that's what you want.

> 

add(Em*exp(-m^2*Pi^2*t*(alpha - varepsilon*m^2*Pi^2/L^2)/L^2)*sin(m*Pi*x/L), m = 1 .. 60):
f := unapply(%, [x,t]):
> 

plot(f(x,10), x=0..L, color="red");

> 

plot3d(f(x,t), x=0..L, t=0..T);


 

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Your f is a function of two variables.&n...

Rouben Rostamian 7711
September 19 2019
0 0

Your f is a function of two variables.  What do you mean by "the derivative of f"?

The partial derivative of f with respect to its first variable (which is t) is written D[1](f).  To evaluate that derivative at the point (t[n-1],w[n-1]), we write
D[1](f)(t[n-1],w[n-1]).

If you need the derivative of f with respect to its second variable (which is y), then you will need
D[2](f)(t[n-1],w[n-1]).

I don't know which of two you need in your code because it's not clear to me what it is attempting to do.

 

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