Product Suggestions

Help Menu Bugs

by: Samuel Hollis 100 Product: Maple 2016

August 15 2016

2 1

Using a mouse's back/forward buttons or alt+ left arrow/ alt+right arrow did not work in Maple 2015 and they still don't work in Maple 2016.

Also, if you select anywhere in the search bar, it automatically selects all the words. I sometimes (most the time) want to change only one word.

Im using x64 Windows 10 and it also did it when running Windows 8/8.1 . Does anybody else have this problem?

Can it be fixed please if not?

Make Maple touch friendly!

by: Samuel Hollis 100 Products: Maple 2016 , MapleSim 2016

August 13 2016

1 2

If you try and scroll a page up or down, it just selects the content. How hard would it be to fix this?

Primitive words in StringTools

by: JamieSimpson 5 Product: Maple

July 11 2016

1 1

It would be nice if the StringTools package had a function giving the number of primitive words of length n on an alphabet of size a. The formula for this is well-known but tedious to code.

Linkage mechanisms

by: one man 1115 Product: Maple 15

July 10 2016

1 5

        General description of the method of solving underdetermined systems of equations. As a particular application of the idea proposed a universal method kinematic analysis for all kinds of spatial and planar link mechanisms with any number degrees of freedom.  The method can be used for powerful CAD linkages.
        http://www.maplesoft.com/applications/view.aspx?SID=154228

Hot key F1 does not work in 2016.1

by: Jean-Michel 230 Product: Maple 2016

April 22 2016

1 1

Hello All,

(I also sent this fact to Maplesoft Support).

Since I updayed to 2016.1 the F1 key does bring a menu witch send to..F5 only.

No way to have a "full" Help Menu.(See the attached file)

I guess a silly bug jumped in :)

Kind regards,

Jean-Michel

Error and nonsense in eulermac outputs

by: vv 11935 Product: Maple

March 16 2016

1 3

eulermac(1/(n*ln(n)^2),n=2..N,1); #Error
Error, (in SumTools:-DefiniteSum:-ClosedForm) summand is singular in the interval of summation

eulermac(1/(n*ln(n)^2+1),n=2..N,1); #nonsense

Calculating linkage mechanisms

by: one man 1115 Product: Maple 17

March 04 2016

3 12

The method of solving underdetermined systems of equations, and universal method for calculating link mechanisms. It is based on the Draghilev’s method for solving systems of nonlinear equations.
When calculating link mechanisms we can use geometrical relationships to produce their mathematical models without specifying the “input link”. The new method allows us to specify the “input link”, any link of mechanism.

Example.
Three-bar mechanism. The system of equations linkages in this mechanism is as follows:

f1 := x1^2+(x2+1)^2+(x3-.5)^2-R^2;
f2 := x1-.5*x2+.5*x3;
f3 := (x1-x4)^2+(x2-x5)^2+(x3-x6)^2-19;
f4 := sin(x4)-x5;
f5 := sin(2*x4)-x6;

Coordinates green point x'i', i = 1..3, the coordinates of red point x'i', i = 4..6.
Set of x0'i', i = 1..6 searched arbitrarily, is the solution of the system of equations and is the initial point for the solution of the ODE system. The solution of ODE system is the solution of system of equations linkages for concrete assembly linkage.
Two texts of the program for one mechanism. In one case, the “input link” is the red-green, other case the “input link” is the green-blue.
After the calculation trajectories of points, we can always find the values of other variables, for example, the angles.
Animation displays the kinematics of the mechanism.
MECAN_3_GR_P_bar.mw
MECAN_3_Red_P_bar.mw

(if to use another color instead of color = "Niagara Dark Orchid", the version of Maple <17)

Method_Mechan_PDF.pdf

Calculating linkage mechanisms

by: one man 1115 Product: Maple 17

March 04 2016

3 12

The method of solving underdetermined systems of equations, and universal method for calculating link mechanisms. It is based on the Draghilev’s method for solving systems of nonlinear equations.
When calculating link mechanisms we can use geometrical relationships to produce their mathematical models without specifying the “input link”. The new method allows us to specify the “input link”, any link of mechanism.

Example.
Three-bar mechanism. The system of equations linkages in this mechanism is as follows:

f1 := x1^2+(x2+1)^2+(x3-.5)^2-R^2;
f2 := x1-.5*x2+.5*x3;
f3 := (x1-x4)^2+(x2-x5)^2+(x3-x6)^2-19;
f4 := sin(x4)-x5;
f5 := sin(2*x4)-x6;

Coordinates green point x'i', i = 1..3, the coordinates of red point x'i', i = 4..6.
Set of x0'i', i = 1..6 searched arbitrarily, is the solution of the system of equations and is the initial point for the solution of the ODE system. The solution of ODE system is the solution of system of equations linkages for concrete assembly linkage.
Two texts of the program for one mechanism. In one case, the “input link” is the red-green, other case the “input link” is the green-blue.
After the calculation trajectories of points, we can always find the values of other variables, for example, the angles.
Animation displays the kinematics of the mechanism.
MECAN_3_GR_P_bar.mw
MECAN_3_Red_P_bar.mw

(if to use another color instead of color = "Niagara Dark Orchid", the version of Maple <17)

Method_Mechan_PDF.pdf

Apply to be a Maple Beta Tester!

by: Daniel Skoog 1751 Product: Maple

December 22 2015

4 2

As the year draws to a close, we start looking forward to a new year and a new release of Maple. With every new release comes many new features and updates to explore.

We are looking for several new beta testers with a good working knowledge of Maple; We need your input, your ideas, and your experience with our products to help us improve the software and get it ready for general release.

There are many benefits to becoming a beta tester:

You’ll get to use the new software before anyone else does.
You’ll help us make our software better in ways that work for you.
Your suggestions could determine the future direction of the software.
You’ll get feedback right from the development team.

If you are interested in becoming a beta tester for the next version of Maple, please email: beta (at) maplesoft.com for more information.

Apply to be a Maple Beta Tester!

by: Daniel Skoog 1751 Product: Maple

December 22 2015

4 2

As the year draws to a close, we start looking forward to a new year and a new release of Maple. With every new release comes many new features and updates to explore.

We are looking for several new beta testers with a good working knowledge of Maple; We need your input, your ideas, and your experience with our products to help us improve the software and get it ready for general release.

There are many benefits to becoming a beta tester:

You’ll get to use the new software before anyone else does.
You’ll help us make our software better in ways that work for you.
Your suggestions could determine the future direction of the software.
You’ll get feedback right from the development team.

If you are interested in becoming a beta tester for the next version of Maple, please email: beta (at) maplesoft.com for more information.

New features in Mathematica 10.3.0

by: Markiyan Hirnyk 7689 Product: Maple

October 16 2015

2 16

Mathematica 10.3.0 was announced yesterday. This is the 6th release of Mathematica 10 during 16 months. I wonder its MathematicaFunctionData and FindFormula . At first sight, the former is an analog of FunctionAdvisor of Maple, but the latter isn't any analog. Also compare the outputs of

Residue[Binomial[n,k],{n,-j}]

(-1)^j/(j!*k!*(-j-k)!)

and

>`assuming`([residue(binomial(n, k), n = -j)], [integer, j > 0]);

residue(binomial(n, k), n = -j)
Let us wait for Maple 2016.

Tricking Maple to solve Differential Algebraic...

by: Dr. Venkat Subramanian 341

July 08 2015

8 37

Solving DAEs in Maple

As I had mentioned in many posts, Maple cannot solve nonlinear DAEs. As of today (Maple 2015), given a system of index 1 DAE dy/dt = f(y,z); 0 = g(y,z), Maple “extends” g(y,z) to get dz/dt = g1(y,z). So, any given index 1 DAE is converted to a system of ODEs dy/dt = f, dz/dt = g1 with the constraint g = 0, before it solves. This is true for all the solvers in Maple despite the wrong claims in the help files. It is also true for MEBDFI, the FORTRAN implementation of which actually solves index 2 DAEs directly. In addition, the initial condition for the algebraic variable has to be consistent. The problem with using fsolve is that one cannot specify tolerance. Often times one has to solve DAEs at lower tolerances (1e-3 or 1e-4), not 1e-15. In addition, one cannot use compile =true for high efficiency.

The approach in Maple fails for many DAEs of practical interest. In addition, this can give wrong answers. For example, dy/dt = z, y^2+z^2-1=0, with y(0)=0 and z(0)=1 has a meaningful solution only from 0 to Pi/2, Maple’s dsolve/numeric will convert this to dy/dt = z and dz/dt = -y and integrate in time for ever. This is wrong as at t = Pi/2, the system becomes index 2 DAE and there is more than 1 acceptable solution.

We just recently got our paper accepted that helps Maple's dsolve and many ODE solvers in other languages handle DAEs directly. The approach is rather simple, the index 1 DAE is perturbed as dy/dt = f.S ; -mu.diff(g,t) = g. The switch function is a tanh function which is zero till t = tinit (initialization time). Mu is chosen to be a small number. In addition, lhs of the perturbed equation is simplified using approximate initial guesses as Maple cannot handle non-constant Mass matrix. The paper linked below gives below more details.

http://depts.washington.edu/maple/pubs/U_Apprach_FULL_DRAFT.pdf

Next, I discuss different examples (code attached).

Example 1: Simple DAE (one ODE and one AE), the proposed approach helps solve this system efficiently without knowing the exact initial condition.

Hint: The code is given with a semicolon at the end of the most of the statements for educational purposes for new users. Using semicolon after dsolve numeric in classic worksheet crashes the code (Maplesoft folks couldn’t fix this despite my request).

Example 2:

This is a nickel battery model (one ODE and one AE). This fails with many solvers unless exact IC is given. The proposed approach works well. In particular, stiff=true, implicit=true option is found to be very efficient. The code given in example 1 can be used to solve example 2 or any other DAEs by just entering ODEs, AEs, ICs in proper places.

Example 3:

This is a nonlinear implicit ODE (posted in Mapleprimes earlier by joha, (http://www.mapleprimes.com/questions/203096-Solving-Nonlinear-ODE#answer211682 ). This can be converted to index 1 DAE and solved using the proposed approach.

Example 4:

This example was posted earlier by me in Mapleprimes (http://www.mapleprimes.com/posts/149877-ODEs-PDEs-And-Special-Functions) . Don’t try to solve this in Maple using its dsolve numeric solver for N greater than 5 directly. The proposed approach can handle this well. Tuning the perturbation parameters and using compile =true will help in solving this system more efficiently.

Finally example 1 is solved for different perturbation parameters to show how one can arrive at convergence. Perhaps more sophisticated users and Maplesoft folks can enable this as automatically tuned parameters in dsolve/numeric.

Note:

This post should not be viewed as just being critical on Maple. I have wasted a lot of time assuming that Maple does what it claims in its help file. This post is to bring awareness about the difficulty in dealing with DAEs. It is perfectly fine to not have a DAE solver, but when literature or standard methods are cited/claimed, they should be implemented in proper form. I will forever remain a loyal Maple user as it has enabled me to learn different topics efficiently. Note that Maplesim can solve DAEs, but it is a pain to create a Maplesim model/project just for solving a DAE. Also, events is a pain with Maplesim. The reference to Mapleprimes links are missing in the article, but will be added before the final printing stage. The ability of Maple to find analytical Jacobian helps in developing many robust ODE and DAE solvers and I hope to post my own approaches that will solve more complicated systems.

I strongly encourage testing of the proposed approach and implementation for various educational/research purposes by various users. The proposed approach enables solving lithium-ion and other battery/electrochemical storage models accurately in a robust manner. A disclosure has been filed with the University of Washington to apply for a provisional patent for battery models and Battery Management System for transportation, storage and other applications because of the current commercial interest in this topic (for batteries). In particular, use of this single step avoids intialization issues/(no need to initialize separately) for parameter estimation, state estimation or optimal control of battery models.

Appendix A

Maple code for Examples 1-4 from "Extending Explicit and Linealry Implicit ODE Solvers for Index-1 DAEs", M. T. Lawder,

V. Ramadesigan, B. Suthar and V. R. Subramanian, Computers and Chemical Engineering, in press (2015).

Use y1, y2, etc. for all differential variables and z1, z2, etc. for all algebraic variables

Example 1

>

restart;

>	with(plots):

Enter all ODEs in eqode

>	eqode:=[diff(y1(t),t)=-y1(t)^2+z1(t)];

(1)

Enter all AEs in eqae

>	eqae:=[cos(y1(t))-z1(t)^0.5=0];

(2)

Enter all initial conditions for differential variables in icodes

>	icodes:=[y1(0)=0.25];

(3)

Enter all intial conditions for algebraic variables in icaes

>	icaes:=[z1(0)=0.8];

(4)

Enter parameters for perturbation value (mu), switch function (q and tint), and runtime (tf)

>	pars:=[mu=0.1,q=1000,tint=1,tf=5];

(5)

Choose solving method (1 for explicit, 2 for implicit)

>	Xexplicit:=2;

(6)

Standard solver requires IC z(0)=0.938791 or else it will fail

>	solx:=dsolve({eqode[1],eqae[1],icodes[1],icaes[1]},numeric):

Error, (in dsolve/numeric/DAE/checkconstraints) the initial conditions do not satisfy the algebraic constraints
error = .745e-1, tolerance = .559e-6, constraint = cos(y1(t))-z1(t)^.5000000000000000000000

>	ff:=subs(pars,1/2+1/2tanh(q(t-tint)));

(7)

>	NODE:=nops(eqode);NAE:=nops(eqae);

(8)

>	for XX from 1 to NODE do EQODE\|\|XX:=lhs(eqode[XX])=rhs(eqode[XX])*ff; end do;

(9)

>	for XX from 1 to NAE do EQAE\|\|XX:=subs(pars,-mu*(diff(rhs(eqae[XX])-lhs(eqae[XX]),t))=rhs(eqae[XX])-lhs(eqae[XX])); end do;

EQAE1 := -.1*sin(y1(t))*(diff(y1(t), t))-0.5e-1*(diff(z1(t), t))/z1(t)^.5 = -cos(y1(t))+z1(t)^.5

(10)

>

>	Dvars1:={seq(diff(z\|\|x(t),t)=D\|\|x,x=1..NAE)};

(11)

>	Dvars2:={seq(rhs(Dvars1[x])=lhs(Dvars1[x]),x=1..NAE)};

(12)

>	icsn:=seq(subs(y\|\|x(0)=y\|\|x(t),icodes[x]),x=1..NODE),seq(subs(z\|\|x(0)=z\|\|x(t),icaes[x]),x=1..NAE);

(13)

>	for j from 1 to NAE do

>	EQAEX\|\|j:=subs(Dvars1,eqode,icsn,Dvars2,lhs(EQAE\|\|j))=rhs(EQAE\|\|j);

>

end do:

>	Sys:={seq(EQODE\|\|x,x=1..NODE),seq(EQAEX\|\|x,x=1..NAE),seq(icodes[x],x=1..NODE),seq(icaes[x],x=1..NAE)};

$Sys := {-0.1824604200e-1-0.5590169945e-1*(diff(z1(t), t)) = -cos(y1(t))+z1(t)^.5, y1(0) = .25, z1(0) = .8, diff(y1(t), t) = (-y1(t)^2+z1(t))*(1/2+(1/2)*tanh(1000*t-1000))}$

(14)

>	if Xexplicit=1 then

>	sol:=dsolve(Sys,numeric):

>

else

>	sol:=dsolve(Sys,numeric,stiff=true,implicit=true): end if:

>

>	for XX from 1 to NODE do a\|\|XX:=odeplot(sol,[t,y\|\|XX(t)],0..subs(pars,tf),color=red); end do:

>	for XX from NODE+1 to NODE+NAE do a\|\|XX:=odeplot(sol,[t,z\|\|(XX-NODE)(t)],0..subs(pars,tf),color=blue); end do:

>	display(seq(a\|\|x,x=1..NODE+NAE),axes=boxed);

End Example 1

>

Example 2

>

restart;

>	with(plots):

>	eq1:=diff(y1(t),t)=j1*W/F/rho/V;

(15)

>	eq2:=j1+j2=iapp;

(16)

>	j1:=io1(2(1-y1(t))exp((0.5F/R/T)(z1(t)-phi1))-2y1(t)exp((-0.5F/R/T)*(z1(t)-phi1)));

(17)

>	j2:=io2(exp((F/R/T)(z1(t)-phi2))-exp((-F/R/T)*(z1(t)-phi2)));

(18)

>	params:={F=96487,R=8.314,T=298.15,phi1=0.420,phi2=0.303,W=92.7,V=1e-5,io1=1e-4,io2=1e-10,iapp=1e-5,rho=3.4};

params := {F = 96487, R = 8.314, T = 298.15, V = 0.1e-4, W = 92.7, io1 = 0.1e-3, io2 = 0.1e-9, rho = 3.4, iapp = 0.1e-4, phi1 = .420, phi2 = .303}

(19)

>	eqode:=[subs(params,eq1)];

eqode := [diff(y1(t), t) = 0.5651477584e-2*(1-y1(t))*exp(19.46229155*z1(t)-8.174162450)-0.5651477584e-2*y1(t)*exp(-19.46229155*z1(t)+8.174162450)]

(20)

>	eqae:=[subs(params,eq2)];

(21)

>	icodes:=[y1(0)=0.05];

(22)

>	icaes:=[z1(0)=0.7];

(23)

>	solx:=dsolve({eqode[1],eqae[1],icodes[1],icaes[1]},type=numeric):

Error, (in dsolve/numeric/DAE/checkconstraints) the initial conditions do not satisfy the algebraic constraints
error = .447e9, tolerance = .880e4, constraint = -2000000*(-1+y1(t))*exp(19.46229155000000000000*z1(t)-8.174162450000000000000)-2000000*y1(t)*exp(-19.46229155000000000000*z1(t)+8.174162450000000000000)+exp(38.92458310000000000000*z1(t)-11.79414868000000000000)-exp(-38.92458310000000000000*z1(t)+11.79414868000000000000)-100000

>	pars:=[mu=0.00001,q=1000,tint=1,tf=5001];

(24)

>	Xexplicit:=2;

(25)

>	ff:=subs(pars,1/2+1/2tanh(q(t-tint)));

(26)

>	NODE:=nops(eqode):NAE:=nops(eqae);

(27)

>	for XX from 1 to NODE do EQODE\|\|XX:=lhs(eqode[XX])=rhs(eqode[XX])*ff: end do;

EQODE1 := diff(y1(t), t) = (0.5651477584e-2*(1-y1(t))*exp(19.46229155*z1(t)-8.174162450)-0.5651477584e-2*y1(t)*exp(-19.46229155*z1(t)+8.174162450))*(1/2+(1/2)*tanh(1000*t-1000))

(28)

>	for XX from 1 to NAE do EQAE\|\|XX:=subs(pars,-mu*(diff(rhs(eqae[XX])-lhs(eqae[XX]),t))=rhs(eqae[XX])-lhs(eqae[XX])); end do;

(29)

>	Dvars1:={seq(diff(z\|\|x(t),t)=D\|\|x,x=1..NAE)};

(30)

>	Dvars2:={seq(rhs(Dvars1[x])=lhs(Dvars1[x]),x=1..NAE)};

(31)

>	icsn:=seq(subs(y\|\|x(0)=y\|\|x(t),icodes[x]),x=1..NODE),seq(subs(z\|\|x(0)=z\|\|x(t),icaes[x]),x=1..NAE);

(32)

>	for j from 1 to NAE do

>	EQAEX\|\|j:=subs(Dvars1,eqode,icsn,Dvars2,lhs(EQAE\|\|j))=rhs(EQAE\|\|j);

>

end do;

(33)

>	Sys:={seq(EQODE\|\|x,x=1..NODE),seq(EQAEX\|\|x,x=1..NAE),seq(icodes[x],x=1..NODE),seq(icaes[x],x=1..NAE)};

(34)

>	if Xexplicit=1 then

>	sol:=dsolve(Sys,numeric,maxfun=0):

>

else

>	sol:=dsolve(Sys,numeric,stiff=true,implicit=true,maxfun=0):

>

end if:

>

>	for XX from 1 to NODE do a\|\|XX:=odeplot(sol,[t,y\|\|XX(t)],0..subs(pars,tf),color=red); end do:

>	for XX from NODE+1 to NODE+NAE do a\|\|XX:=odeplot(sol,[t,z\|\|(XX-NODE)(t)],0..subs(pars,tf),color=blue); end do:

>	b1:=odeplot(sol,[t,y1(t)],0..1,color=red): b2:=odeplot(sol,[t,z1(t)],0..1,color=blue):

>	display(b1,b2,axes=boxed);

>	display(seq(a\|\|x,x=1..NODE+NAE),axes=boxed);

End Example 2

>

Example 3

>

restart;

>	with(plots):

>	eq1:=diff(y1(t),t)^2+diff(y1(t),t)*(y1(t)+1)+y1(t)=cos(diff(y1(t),t));

(35)

>	solx:=dsolve({eq1,y1(0)=0},numeric):

Error, (in dsolve/numeric/make_proc) Could not convert to an explicit first order system due to 'RootOf'

>	eqode:=[diff(y1(t),t)=z1(t)];

(36)

>	eqae:=[subs(eqode,eq1)];

(37)

>	icodes:=[y1(0)=0.0];

(38)

>	icaes:=[z1(0)=0.0];

(39)

>	pars:=[mu=0.1,q=1000,tint=1,tf=4];

(40)

>	Xexplicit:=2;

(41)

>	ff:=subs(pars,1/2+1/2tanh(q(t-tint)));

(42)

>	NODE:=nops(eqode);NAE:=nops(eqae);

(43)

>	for XX from 1 to NODE do EQODE\|\|XX:=lhs(eqode[XX])=rhs(eqode[XX])*ff: end do;

(44)

>	for XX from 1 to NAE do EQAE\|\|XX:=subs(pars,-mu*(diff(rhs(eqae[XX])-lhs(eqae[XX]),t))=rhs(eqae[XX])-lhs(eqae[XX])); end do;

EQAE1 := .1*sin(z1(t))*(diff(z1(t), t))+.2*z1(t)*(diff(z1(t), t))+.1*(diff(z1(t), t))*(y1(t)+1)+.1*z1(t)*(diff(y1(t), t))+.1*(diff(y1(t), t)) = cos(z1(t))-z1(t)^2-z1(t)*(y1(t)+1)-y1(t)

(45)

>

>	Dvars1:={seq(diff(z\|\|x(t),t)=D\|\|x,x=1..NAE)};

(46)

>	Dvars2:={seq(rhs(Dvars1[x])=lhs(Dvars1[x]),x=1..NAE)};

(47)

>	icsn:=seq(subs(y\|\|x(0)=y\|\|x(t),icodes[x]),x=1..NODE),seq(subs(z\|\|x(0)=z\|\|x(t),icaes[x]),x=1..NAE);

(48)

>	for j from 1 to NAE do

>	EQAEX\|\|j:=subs(Dvars1,eqode,icsn,Dvars2,lhs(EQAE\|\|j))=rhs(EQAE\|\|j);

>

end do;

(49)

>	Sys:={seq(EQODE\|\|x,x=1..NODE),seq(EQAEX\|\|x,x=1..NAE),seq(icodes[x],x=1..NODE),seq(icaes[x],x=1..NAE)};

$Sys := {.1*(diff(z1(t), t)) = cos(z1(t))-z1(t)^2-z1(t)*(y1(t)+1)-y1(t), y1(0) = 0., z1(0) = 0., diff(y1(t), t) = z1(t)*(1/2+(1/2)*tanh(1000*t-1000))}$

(50)

>	if Xexplicit=1 then

>	sol:=dsolve(Sys,numeric):

>

else

>	sol:=dsolve(Sys,numeric,stiff=true,implicit=true):

>

end if:

>

>	for XX from 1 to NODE do a\|\|XX:=odeplot(sol,[t,y\|\|XX(t)],0..subs(pars,tf),color=red); end do:

>	for XX from NODE+1 to NODE+NAE do a\|\|XX:=odeplot(sol,[t,z\|\|(XX-NODE)(t)],0..subs(pars,tf),color=blue); end do:

>	display(seq(a\|\|x,x=1..NODE+NAE),axes=boxed);

End Example 3

>

Example 4

>

restart;

>	with(plots):

>	N:=11:h:=1/(N+1):

>	for i from 2 to N+1 do eq1[i]:=diff(y\|\|i(t),t)=(y\|\|(i+1)(t)-2y\|\|i(t)+y\|\|(i-1)(t))/h^2-y\|\|i(t)(1+z\|\|i(t));od:

>	for i from 2 to N+1 do eq2[i]:=0=(z\|\|(i+1)(t)-2z\|\|i(t)+z\|\|(i-1)(t))/h^2-(1-y\|\|i(t)^2)(exp(-z\|\|i(t)));od:

>	eq1[1]:=(3y1(t)-4y2(t)+y3(t))/(2*h)=0: eq1[N+2]:=y\|\|(N+2)(t)-1=0:

>	eq2[1]:=(3z1(t)-4z2(t)+1z3(t))/(2h)=0: eq2[N+2]:=z\|\|(N+2)(t)=0:

>	eq1[1]:=subs(y1(t)=z\|\|(N+3)(t),eq1[1]):

>	eq1[N+2]:=subs(y\|\|(N+2)(t)=z\|\|(N+4)(t),eq1[N+2]):

>	eqode:=[seq(subs(y1(t)=z\|\|(N+3)(t),y\|\|(N+2)(t)=z\|\|(N+4)(t),eq1[i]),i=2..N+1)]:

>	eqae:=[eq1[1],eq1[N+2],seq(eq2[i],i=1..N+2)]:

>	icodes:=[seq(y\|\|j(0)=1,j=2..N+1)]:

>	icaes:=[seq(z\|\|j(0)=0,j=1..N+2),z\|\|(N+3)(0)=1,z\|\|(N+4)(0)=1]:

>	pars:=[mu=0.00001,q=1000,tint=1,tf=2]:

>	Xexplicit:=2:

>	ff:=subs(pars,1/2+1/2tanh(q(t-tint))):

>	NODE:=nops(eqode):NAE:=nops(eqae):

>	for XX from 1 to NODE do

>	EQODE\|\|XX:=lhs(eqode[XX])=rhs(eqode[XX])*ff: end do:

>	for XX from 1 to NAE do

>	EQAE\|\|XX:=subs(pars,-mu*(diff(rhs(eqae[XX])-lhs(eqae[XX]),t))=rhs(eqae[XX])-lhs(eqae[XX])); end do:

>	Dvars1:={seq(diff(z\|\|x(t),t)=D\|\|x,x=1..NAE)}:

>	Dvars2:={seq(rhs(Dvars1[x])=lhs(Dvars1[x]),x=1..NAE)}:

>	icsn:=seq(subs(y\|\|x(0)=y\|\|x(t),icodes[x]),x=1..NODE),seq(subs(z\|\|x(0)=z\|\|x(t),icaes[x]),x=1..NAE):

>	for j from 1 to NAE do

>	EQAEX\|\|j:=subs(Dvars1,eqode,icsn,Dvars2,lhs(EQAE\|\|j))=rhs(EQAE\|\|j):

>

end do:

>	Sys:={seq(EQODE\|\|x,x=1..NODE),seq(EQAEX\|\|x,x=1..NAE),seq(icodes[x],x=1..NODE),seq(icaes[x],x=1..NAE)}:

>	if Xexplicit=1 then

>	sol:=dsolve(Sys,numeric,maxfun=0):

>

else

>	sol:=dsolve(Sys,numeric,stiff=true,implicit=true,maxfun=0):

>

end if:

>

>	for XX from 1 to NODE do

>	a\|\|XX:=odeplot(sol,[t,y\|\|(XX+1)(t)],1..subs(pars,tf),color=red): end do:

>	for XX from NODE+1 to NODE+NAE do

>	a\|\|XX:=odeplot(sol,[t,z\|\|(XX-NODE)(t)],1..subs(pars,tf),color=blue): end do:

>	display(seq(a\|\|x,x=1..NODE),a\|\|(NODE+NAE-1),a\|\|(NODE+NAE),axes=boxed);

End of Example 4

>

Sometimes the parameters of the switch function and perturbation need to be tuned to obtain propoer convergence. Below is Example 1 shown for several cases using the 'parameters' option in Maple's dsolve to compare how tuning parameters affects the solution

>

restart:

>	with(plots):

>	eqode:=[diff(y1(t),t)=-y1(t)^2+z1(t)]: eqae:=[cos(y1(t))-z1(t)^0.5=0]:

>	icodes:=[y1(0)=0.25]: icaes:=[z1(0)=0.8]:

>	pars:=[tf=5]:

>	Xexplicit:=2;

(51)

>	ff:=subs(pars,1/2+1/2tanh(q(t-tint))):

>	NODE:=nops(eqode):NAE:=nops(eqae):

>	for XX from 1 to NODE do EQODE\|\|XX:=lhs(eqode[XX])=rhs(eqode[XX])*ff: end do:

>	for XX from 1 to NAE do EQAE\|\|XX:=subs(pars,-mu*(diff(rhs(eqae[XX])-lhs(eqae[XX]),t))=rhs(eqae[XX])-lhs(eqae[XX])); end do:

>

>	Dvars1:={seq(diff(z\|\|x(t),t)=D\|\|x,x=1..NAE)}:

>	Dvars2:={seq(rhs(Dvars1[x])=lhs(Dvars1[x]),x=1..NAE)}:

>	icsn:=seq(subs(y\|\|x(0)=y\|\|x(t),icodes[x]),x=1..NODE),seq(subs(z\|\|x(0)=z\|\|x(t),icaes[x]),x=1..NAE):

>	for j from 1 to NAE do

>	EQAEX\|\|j:=subs(Dvars1,eqode,icsn,Dvars2,lhs(EQAE\|\|j))=rhs(EQAE\|\|j):

>

end do:

>	Sys:={seq(EQODE\|\|x,x=1..NODE),seq(EQAEX\|\|x,x=1..NAE),seq(icodes[x],x=1..NODE),seq(icaes[x],x=1..NAE)}:

>	if Xexplicit=1 then

>	sol:=dsolve(Sys,numeric,'parameters'=[mu,q,tint],maxfun=0):

>

else

>	sol:=dsolve(Sys,numeric,'parameters'=[mu,q,tint],stiff=true,implicit=true):

>

end if:

>

>	sol('parameters'=[0.1,1000,1]):

>	plot1:=odeplot(sol,[t-1,y1(t)],0..4,color=red): plot2:=odeplot(sol,[t-1,z1(t)],0..4,color=blue):

>	sol('parameters'=[0.001,10,1]):

>	plot3:=odeplot(sol,[t-1,y1(t)],0..4,color=red,linestyle=dot): plot4:=odeplot(sol,[t-1,z1(t)],0..4,color=blue,linestyle=dot):

>	display(plot1,plot2,plot3,plot4,axes=boxed);

In general, one has to decrease mu, and increase q and tint until convergence (example at t=3)

>	sol('parameters'=[0.001,10,1]):sol(3+1);

(52)

>	sol('parameters'=[0.0001,100,10]):sol(3+10);

(53)

>	sol('parameters'=[0.00001,1000,20]):sol(3+20);

(54)

>

The results have converged to 4 digits after the decimal. Of course, absolute and relative tolerances of the solvers can be modified if needed

Download SolvingDAEsinMaple.mws

Resize a code edit region

by: sbeau 29 Product: Maple 18

January 28 2015

0 0

This is a very simple suggestion that weights heavily on my enjoyment of Maple.

I'm not sure if it is only me and my students but it is really tricky to resize a code edit region. Trying to get a hold of the contour in an exercise in patience! Can anyone fix this?

Thanks!

Stéphane

correct the API for dsolve with series option

by: nm 7740

January 17 2015

1 6

I do not think the current API for dsolve when asking for series solution is done right. If one wants to obtain a series solution for an ODE, but wants different order than the default 6, now one must set this value using a global setting before making the call, like this:

eq:=diff(y(x),x$2)+y(x)=0;
Order:=10;
dsolve({eq,y(0)=1,D(y)(0)=0},y(x),type='series');

It would be better if options to calls are passed along with the other parameters in the call itself. Something like

dsolve({eq,y(0)=1,D(y)(0)=0},y(x),type='series',order=10);

This should also apply to any command that takes Order, such as series(cos(x),x=0,order=10);

Passing options and values to functions using global and environment variables is not safe and not a good way to go about it as it can cause programming errors.