tomleslie

Use IntegrationTools:-Change()...

Answered: tomleslie 13876

December 05 2020

1 0

as in the attached (although it isn't reall necessary in order to obtain the "answer"

>	# # Direct solution # restart; int( (1+ln(x))/x, x=1..exp(1) );

(1)

>	# # With change of variables # restart; expr:=Int( (1+ln(x))/x, x=1..exp(1) ); IntegrationTools:-Change( expr, isolate( u=1+ln(x), x) ); value(%);

Int((1+ln(x))/x, x = 1 .. exp(1))

(2)

>

Download simpint.mw

Numerous...

Answered: tomleslie 13876

December 04 2020

2 0

typos and logical problems - but maybe the attached covers what you want?

>

restart:

>

t := 6*Pi; lambda := .1; N := 5; M := sqrt(N(N+1))*exp(I*phi); omegap := 1;

eq1:=-2*(n(phi,delta)-N)+(u(phi,delta)-M)*exp(-2*I*omegap*t/lambda)+((z(phi,delta)-conjugate(M))*exp(2*I*omegap*t/lambda)):
eq2:=-2*(1-I*delta)*u(phi,delta)+2*(n(phi,delta)-N)*exp(2*I*omegap*t/lambda)+2*M:
eq3:=-2*(1+I*delta)*z(phi,delta)+2*(n(phi,delta)-N)*exp(-2*I*omegap*t/lambda)+2*conjugate(M):
convert(eq1, rational):
convert(eq2, rational):
convert(eq3, rational):

f1:=evalc(eq1):f2:=evalc(eq2): f3:=evalc(eq3):
f1:= solve( [f1, f2, f3],
[n(phi, delta), u(phi, delta),z(phi, delta) ]
)[];

(1)

>

#
# The solution for n(phi,delta) is real.
#
  plot3d( eval(n(phi,delta), f1),delta = -10 .. 10, phi = 0 .. 2*Pi);
#
# Solutions for z(phi,delta) and u(phi,delta)
# are complex -not surehow you want to deal with these
#
  plot3d( abs(eval(z(phi,delta), f1)),delta = -10 .. 10, phi = 0 .. 2*Pi);
  plot3d( abs(eval(u(phi,delta), f1)),delta = -10 .. 10, phi = 0 .. 2*Pi);

>

Download somePlots.mw

global versus local...

Answered: tomleslie 13876

December 04 2020

2 0

The symbols returned by the command

sol(parameters=data)

are local to the procedure 'sol' whereas the symbols within the list 'data' are global.

See the execution group I added in the attached

>

restart

>	interface(version);

(1)

>	sys := { M__p * diff(x(t), t) = K*x(t), x(0) = 0};

${M__p*(diff(x(t), t)) = K*x(t), x(0) = 0}$

(2)

>	dsolve(sys);

(3)

>	params := [M__p, K]; sol := dsolve(sys, numeric, parameters=params);

(4)

>	data := [params[1] = 10., params[2] = 10000.]; instance := sol(parameters=data); whattype(%); sol(0.5);

(5)

>	type( lhs(instance[1]), `local`); type( lhs(data[1]), `local`);

(6)

>	lprint(data); lprint(lhs~(data));

[M__p = 10., K = 10000.]
[M__p, K]

>	lprint(instance); lprint(lhs~(instance));

[M__p = 10., K = 10000.]
[M__p, K]

>	lhs~(data); lhs~(instance); zip((u,v)->evalb(u=v), lhs~(data), lhs~(instance))

(7)

>	zip((u,v) -> evalb(u=v), params, lhs~(instance))

(8)

>	zip((u,v) -> evalb(u=v), params, lhs~(data))

(9)

>	evalb(lhs(data[1])=lhs(instance[1]));

(10)

>

Download globLoc.mw

Ugly - but works?...

Answered: tomleslie 13876

December 03 2020

3 2

I spent a while trying to do this calculation in an "elegant" manner - and failed - so I decided to apply "brute force and ignorance" and came up with the attached.

At least this comes up with the required answers - alhtough there must be a better way to achieve this!!!

[Preliminary analysis of the Question by Carl Love:

In the worksheet below, the OP numerically solves a BVP on an interval a..b, for several values of a parameter. Using the numeric solution procedures returned by dsolve, he wants to numerically integrate an algebraic expression (named P1 below) that uses several of the BVP's dependent variables as well as its independent variable. This integral is to be done on an interval c..d, where a < c < d < b (all inequalities strict). He's having trouble coming up with the correct syntax for this.

I (Carl) am confident that the dsolve calls are all syntactically correct and that they return solution procedures in the usual manner without raising any issues, numerical or otherwise. Whether the integrations will ultimately require some adjustments to dsolve's error-control options, I don't know at this point. This Question is only about syntax required to perform the integrations. Any error-control adjustments, if they're needed at all, will be handled later.

I (Carl) have only edited the the OP's worksheet to remove a large amount of content that is irrelevant to Answering this Question. I haven't changed any of the content; I've only deleted. Of course, I already have several ideas about what needs to be done, but up until this point I haven't acted as an Answerer but rather as a Moderator and Editor making this Question suitable for posting.

--Carl Love]

>

restart;

>	eq1:=diff(f(y), y$4)+Uhs*diff(E(y),y$3)-(diff(f(y), y$2))+(diff(theta(y), y$1))= 0:

>	eq2:=diff(theta(y), y$2)+(diff(f(y), y$2)+1)^2+1+diff(theta(y),y$2) = 0:

>	E:=y->zeta(cosh(k/2(h1+h2-2y)))/(cosh(k/2(h1-h2))):

>	bcs:= f(h1) = -(1/2)(Q-1-d), f(h2) = (1/2)(Q-1-d), (D(f))(h1) = -1, (D(f))(h2) = -1, theta(h1) = 0, theta(h2) = 1:

>	epsilon1:=0.1: d:=1: omega:=Pi/6: h1:=-(1+epsilon1sin(2Pix)): h2:=d+epsilon2sin(2Pix+omega): F:= Q-1-d: epsilon2:=0.5: x:=1: alpha:=Pi/6:

>	d1 := subs( Uhs =-2, zeta=3, k=1, [eq1, eq2, bcs ] );

[diff(diff(diff(diff(f(y), y), y), y), y)+6*sinh(.1250000000-y)/cosh(1.125000000)-(diff(diff(f(y), y), y))+diff(theta(y), y) = 0, 2*(diff(diff(theta(y), y), y))+(diff(diff(f(y), y), y)+1)^2+1 = 0, f(-1.) = -(1/2)*Q+1, f(1.250000000) = (1/2)*Q-1, (D(f))(-1.) = -1, (D(f))(1.250000000) = -1, theta(-1.) = 0, theta(1.250000000) = 1]

(1)

>	P1:= eval( diff(f(y), y$3)+Uhs*diff(E(y),y$2)-(diff(f(y), y$1)+1)+(theta(y))+sin(alpha), { Uhs =-2, zeta=3, k=1 } );

diff(diff(diff(f(y), y), y), y)-3.524364340*cosh(-.1250000000+y)-(diff(f(y), y))-1/2+theta(y)

(2)

>

  interface(rtablesize=20):
  ans:= Matrix(13,2):
  ec:=0.5:
  j:= 1:
  for Q from -3 to 3 by ec do
      F2[Q]:= dsolve(d1, numeric, output=listprocedure):
      PP:= unapply( rhs(F2[Q][5])(z)-3.524364340*cosh(-0.1250000000 + rhs(F2[Q][1])(z))-rhs(F2[Q][3])(z)-1/2+rhs(F2[Q][6])(z),z);
      ans[j, 1..2]:= <Q, evalf(Int(PP(y), y=0..1))>;
      j:=j+1;
  od:
  ans;
  plot(ans[..,1], ans[..,2]);
  interface(rtablesize=10):

>

Download ugly.mw

Look at CodeTools:-Usage...

Answered: tomleslie 13876

December 01 2020

1 1

with the option 'iterations'. This will execute the subpplied command the number of times specified by the iterations option, and then (roughly speaking) divide the total times taken by the number of iterations - so you get a pretty good average.

See the attached example

Download timeIt.mw

How about...

Answered: tomleslie 13876

December 01 2020

0 3

the attached

Download intU.mw

My $0.02...

Answered: tomleslie 13876

December 01 2020

1 0

(most of which is equivalent to vv's answer (I think).

FIrst of all your original code doesn't work in Maple 2019. You don't see an error message, but you don't get an "export" either, so I would say that having the error message appearing in Maple 2020 is an impovement

With the worksheet

Download testInc.mw

and the test files

A.mpl

  module A()

$include "C:/Users/TomLeslie/Desktop/B.mpl";

         export boo:=proc(x)
	                  x^3;
	             end proc:
					 
  end module;

and B.mpl

  export foo:= proc(x)
                    return x^2;
               end proc;

then everything works as I would expect.

Obviously you will have to changethe filepaths in testInc.mw and A.mpl to something appropriate for your installation

It depends...

Answered: tomleslie 13876

November 29 2020

4 5

on whether

you just want to create an appropriate data structure for use in subsequent calculations, and are happy with Maple's "normal", pretty-printed output, or
for some "display-only" purpose you want to make your data structure look "pretty". To do this you can use the DocumentTools package. This allows you to make a table look pretty much any way you want. The version in the attached is just one of gazillion possibilities

Both options are given in the attached, although the rendering on this site will not show the output of the second group as part of the worksheet.

If you download/re-execute the code, you should get something similar to the table shown in the attached gif

>

restart:

>

  f:=(x,t)->x*t;
  g:=(x,t)->x^2*t;interface(rtablesize=20):
  A:=Array( [ [`x`,`t`,`f(x,t)`, `g(x,t)`],
              seq
              ( seq
                ( [i, j, f(i,j), g(i,j)],
                  j=0.125..0.875, 0.25
                ),
                i=0.125..0.875, 0.25
              )
            ]
          );
  interface(rtablesize=10):

Array(%id = 36893488148141253324)

(1)

>

  with(DocumentTools):
  with(DocumentTools:-Layout):
  T := Table( interior=all,
              exterior=all,
              widthmode=pixels,
              width=400,
              alignment=center,
              Row( seq
                   ( Cell
                     ( Font
                       ( sprintf( cat( "%", length(A[1,j]), "s"), A[1,j]),
                         family="Times New Roman",
                         bold=true,
                         italic=true,
                         size=16,
                         color="red"
                       ),
                       fillcolor="Silver"
                     ),
                     j=1..4
                   )
                 ),
              seq( `if`( irem(i,4)=2,
                         Row( Cell
                              ( Font
                                ( sprintf( "%5.3f", A[i,1]),
                                  italic=true,
                                  color=MediumBlue
                                )
                              ),
                              Cell
                              ( Font
                                ( sprintf( "%5.3f", A[i,2]),
                                  italic=true,
                                  color=MediumBlue
                                )
                              ),
                              Cell
                              ( Font
                                ( sprintf( "%8.6f", A[i,3]),
                                  bold=true,
                                  color=Indigo
                                ),
                                fillcolor=turquoise
                              ),
                              Cell
                              ( Font
                                ( sprintf( "%12.9f", A[i,4]),
                                  bold=true,
                                  color=Indigo
                                ),
                                fillcolor=turquoise
                              )
                            ),
                         Row( sprintf(" "),
                              Cell
                              ( Font
                                ( sprintf( "%5.3f", A[i,2]),
                                  italic=true,
                                  color=MediumBlue
                                )
                              ),
                              Cell
                              ( Font
                                ( sprintf( "%8.6f", A[i,3]),
                                  bold=true,
                                  color=Indigo
                                ),
                                fillcolor=turquoise
                              ),
                              Cell
                              ( Font
                                ( sprintf( "%12.9f", A[i,4]),
                                  bold=true,
                                  color=Indigo
                                ),
                                fillcolor=turquoise
                              )
                            )
                       ),
                   i=2..17
                 )
               ):
  InsertContent(Worksheet( T )):

>

Download pretty.mw

You can use it now!...

Answered: tomleslie 13876

November 28 2020

0 0

Just try, for example, 2**8.

If you type ** in the MAple help search box, then you ought to be directed to the 'type/arithop' page where it states

Note: type(expr, `^`) is equivalent to type(expr, `**`).

Multiple solutions...

Answered: tomleslie 13876

November 26 2020

1 1

most of them complex! See tha attached. You have found the first solution given in sol2 below

Download sols.mw

Two problems...

Answered: tomleslie 13876

November 26 2020

1 1

exponential function incorrectly entered - fixed in the attached
one too many boundary conditions. For the order of the odes, you can only use seven, but have supplied eight. ODE system is second order in 'chi' but there are three conditions on 'chi' In the attached I removed one of these

>

2*(1+1/beta)*(eta*(diff(diff(diff(f(eta), eta), eta), eta))+diff(diff(f(eta), eta), eta))+f(eta)*(diff(diff(f(eta), eta), eta)) = 0

(1)

>

eq2 := 2*(eta*(diff(theta(eta), `$`(eta, 2)))+diff(theta(eta), eta))/Pr+f(eta)*(diff(theta(eta), eta))+2*eta*(Nb*(diff(theta(eta), eta))*(diff(chi(eta), eta))+Nt*(diff(theta(eta), eta))^2) = 0

2*(eta*(diff(diff(theta(eta), eta), eta))+diff(theta(eta), eta))/Pr+f(eta)*(diff(theta(eta), eta))+2*eta*(Nb*(diff(theta(eta), eta))*(diff(chi(eta), eta))+Nt*(diff(theta(eta), eta))^2) = 0

(2)

>

eq3 := (2*(eta*(diff(chi(eta), eta, eta))+diff(chi(eta), eta)))/Sc+f(eta)*(diff(chi(eta), eta))+2*Nt*(eta*(diff(theta(eta), eta, eta))+diff(theta(eta), eta))/(Sc*Nb)-(1/2)*sigma*(1+delta*theta(eta))^n*exp(-E/(1+delta*theta(eta)))*chi(eta) = 0
;

2*(eta*(diff(diff(chi(eta), eta), eta))+diff(chi(eta), eta))/Sc+f(eta)*(diff(chi(eta), eta))+2*Nt*(eta*(diff(diff(theta(eta), eta), eta))+diff(theta(eta), eta))/(Sc*Nb)-(1/2)*sigma*(1+delta*theta(eta))^n*exp(-E/(1+delta*theta(eta)))*chi(eta) = 0

(3)

>

bcs := f(c) = 2*s*c/Sc, (D(f))(c) = (1/2)*lambda, (D(f))(10) = (1-lambda)*(1/2), theta(c) = 1, theta(10) = 0, chi(c) = 1, (D(chi))(c) = (1/2)*lambda*c

(4)

>

(5)

>

(6)

>

Download bvpProb2.mw

You should also consider...

Answered: tomleslie 13876

November 26 2020

1 3

the StringTools:-Join() command. You will still need to 'escape' (ie backslash) the double-quote (ie ") character.

Several options, all of which produce the same result, are shown in the attached

>

  StringTools:-Join( [ "NODESteelProfiles_I:-Property",
                       "(\"allmembers\")"
                     ],
                     ""
                   );

  StringTools:-Join( [ "NODESteelProfiles_I:-Property",
                       "(\"",
                       "allmembers"
                       "\")"
                     ],
                     ""
                   );

  f:="NODESteelProfiles_I:-Property":
  g:="allmembers":
  StringTools:-Join( [ f,
                       "(\"",
                       g,
                       "\")"
                     ],
                     ""
                   );

(1)

>

Download joinStr.mw

See the attached worksheet...

Answered: tomleslie 13876

November 25 2020

2 1

Download odeSol.mw

Equation not assignment...

Answered: tomleslie 13876

November 25 2020

0 2

I think you need to assign the list to 'L', ie use ':=' rather than just '='.

Try the following

restart;
with(DifferentialGeometry):
L:= [e1+a*e3,2*e2+b*e4,e1+3*e5];
evalDG(add(eval(L[i]),i=1..numelems(L)));

More of an observation...

Answered: tomleslie 13876

November 24 2020

3 0

Sometimes (with Maple) I use dsolve() without ics/bcs, then odetest() with ics/bcs to determine the arbitrary constants. Interestingly (in this case), this approach returns the "missing" solution.

See the attached

Download trickODE.mw

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These are answers submitted by tomleslie

Use IntegrationTools:-Change()...

Numerous...

global versus local...

Ugly - but works?...

Look at CodeTools:-Usage...

How about...

My $0.02...

It depends...

You can use it now!...

Multiple solutions...

Two problems...

You should also consider...

See the attached worksheet...

Equation not assignment...

More of an observation...

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