Hi

You pose several questions.

First of all, when Maple returns ODESolStruc it means it succeeded in reducing the ODE order, but not in solving the reduced ODE. So, unlike the DESol case you compare with, the ODESolStruc contains information about the ODE solution, a reduction of order, not just the ODE passed to dsolve. The reduce ODE is of course a simpler problem, and the tools to construct a full solution to the original problem in case you find a solution to the reduced ODE are there: DEtools[buildsol]. All this is explained in ?ODESolStruc.

Second: the output you got from odetest can be seen as an indication that something is wrong, but not necessarily with the solution - it could also be with the testing approach, any of the intermediate simplifications, etc. Computer Algebra systems are complex systems with all of their parts interdependent. In this example you posted, the reduction of order computed by dsolve is actually correct, but an intermediate call to solve within the algorithm to test ODESolStruc answers returns NULL, and from there odetest is unable to confirm the solution. Note that in the help page for odetest it tells that the command either confirms the solution is correct by returning 0, or otherwise it only means it didn't succeed in verifying the solution - not necessarily that the solution is incorrect, even when, most of the time, one thing implies the other one.

To feel the issue, try this: a) compute symmetry infinitesimals, b) use them to reduce the ODE order, and you will see basically the same solution returned by dsolve, just written slightly different (one simplification less); and this solution verifies OK via odetest. To accomplish this, enter

X := DEtools[symgen](ode);  # pair of Lie symmetry infinitesimals

sol := DEtools[reduce_order](ode, X);  # reduction of order almost identical to the one returned by dsolve

odetest(sol, ode);

                                                 0

All what happened is that sol above has one simplification less, and with that all the intermediate steps succeed - included that call to solve mentioned - and hence a full verification by odetest proceeds normally. To verify that sol above is actually the same as the solution returned by dsolve enter

simplify(normal(sol, expanded) - normal(sol_by_dsolve, expanded))

                                                 0

So: the output you got from odetest as "not zero", that you posted, reveals a weakness in the testing code only. I already fixed that weakness, and uploaded the improvement to the usual place, the Maplesoft R&D webpage for Differential Equations and Mathematical Functions. If you update your library with the download available in that page, odetest returns 0 for the solution returned by dsolve as well, instead of the "not zero" expression you posted.

Third: I see you would like to have a different typesetting for ODESolStruc. You can see 'ODESolStruc' using lprint - no need to convert to a string - I also prefer the current typesetting of it, but in any case you can always remove the current one by entering

unassign(`print/ODESolStruc`)

And that will suffice. If in addition you want to program the typesetting of ODESolStruc in any other way, you can create the corresponding routine and assign it to `print/ODESolStruc`. All this is explained in ?print.

Summarizing: ODESolStruc does mean the equation got solved to some point, not all the way; ODESolStruc conveys more information than DESol; the solution returned by dsolve for your example, presented as a reduction of order, is correct; to have odetest verifying it you'd need to update your library as said above; you can remove or redo the typesetting of ODESolStruc in any desired way.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Hi nm

Good catch. It's fixed now, the fix is available as usual in the Maplesoft R&D webpage for Differential Equations and Mathematical Functions. The zip available for download that includes the fix also includes instructions to install it.

After you install the fix:

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Hi

Indeed, when dsolve returns DEsol, or Mathematica's DSolve returns DifferentialRoot, you can consider the system actually did not solve the ODE: the returned structure does not really contain more information than the given ODE. For linear equations, however, Maple's dsolve always returns DESol when no solutions were found, because, as it is the case of RootOf representations of solutions for non-differential solutions, in the case of linear ODEs Maple is able to do some manipulations with the DESol representation of the solution, as for instance computing series.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Hi

I updated the Maplesoft R&D Physics webpage with a new version of Physics, the issue is resolved. At the end it is is also a link to your worksheet reviewed, with your computation running with the fix in place.

Details.

As you noticed there was a weakness that I'd briefely describe as: SubstituteTensor(T[m,n] A[m] = B[n], T[i,j] A[j]) was not substituting because the position of the free index 'n' in  T[m,n] A[m] (this is left-hand side of the substitution equation), is not the same as the position of the free index 'i' in the target expression T[i,j] A[j].

The issue is that, as you said, T is symmetric (BTW there are two different ways of indicating symmetry, that work differently; more about this in the last paragraph), and so T[n, m] = T[m, n] and one expects that the substitution will be performed - i.e. the order of the indices in T should not be an issue.

The implementation however is not as straightforward in that we are talking of something like subs(A*B = C, A*D*F*B), a sort of algsubs operation (i.e. A*B is not an operand of A*D*F*B), and in addition there are the free and repeated tensor indices that work like different kinds of dummies, plus the symmetry properties under permutation of the indices in the substitution equation and in the target expression.

Anyway it is now working. And yes, the key to crack the problem is in the use of this new routine Physics:-Library:-RepositionRepeatedIndicesAsIn, that I still need to document.

A comment on how to indicate that at tensor is symmetric; there are three ways:

1. Define(eta, symmetric)
2. Define(eta = ... tensorial expression or Matrix, symmetric)
3. Define(eta = ... tensorial expression or Matrix)

The 1st form permits you to compute with eta[i, j] <> eta[j, i] and you can apply the symmetry afterwards, when you prefer, using Simplify, i.e Simplify(eta[i, j] - eta[j, i]) returns 0.

The 2nd form automatically normalizes eta[j, i] -> eta[i, j], it is more convenient most of the time, not all the times. But if the right-hand side of the equation passed to Define is (anti)symmetric under permutation of the indices, one would expect that it is not necessary to additionally specify (anti)'symmetric' (as you did in your worksheet).

The 3rd form does not include the word 'symmetric' in the definition, and also automatically assumes eta is symmetric if the code can prove that the tensorial expression on the right-hand side is actually symmetric, and if so this definition will work as 2. However, due to a weakness in the code this automatic determiination of symmetries was not working when the right-hand side was an (anti)symmetric matrix (2 indices) or (anti)symmetric Array (as many indices as you want).

With the changes introduced today: the 3rd form assumes (anti)symmetric automatically if the corresponding matrix or array in the right-hand-side is (anti)symmetric, so this lateral issue is fixed and you do not need to specify (anti)'symmetric' additionally (you still can do that, to override if something is not working as you expect). Substitute tensor works fine also regardless of the position of the free indices provided that the substitution equation can be mapped into the target equation using symmetries (by using this new Reposition routine), and so it works fine, in the same way, whether you entered your definition using the form 1., 2. or 3.

Maple_Question_7.7.14_(reviewed).mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Hi

Thanks for pointing at this one, it is fixed, the fix is available for download at Maplesoft's R&D Physics webpage, and this is your worksheet running your computation with the fix in place: Maple_Question_7.3.14_(reviewed).mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Hi

When you say that an ODE solution involves an 'arbitrary constant' - say _C1 - the only requirement on _C1 is that  d/dx _C1 = 0. So, if y(x, _C1) is a solution, y(x, _C2) where _C2 = -_C1, or for the case _C2 = F(_C1) and F is an arbitrary function (i.e. can be any function), are all different ways of expressing the same solution, because both _C1 and _C2 satisfy d/dx _C = 0. For details see ?dsolve,details, the section on Arbitrary Constants.

Having in mind what the meaning of an arbitrary constant is, going to your question, y(x)=a+_C1e^-(1/bc) and y(x)=a-_C1e^-(1/bc) are just two different ways of expressing the same solution (just replace the arbitrary constant '_C1' by the arbitrary constant _C2 = -_C1), as opposed to one being right and the other one being wrong.

Edgardo S. Cheb-Terrab
Physics, Maplesoft

Hi

The problem is fixed, the fix uploaded and available at the Maplesoft's R&D Differential Equations and Mathematical Functions webpage. Your equation however is currently beyond Maple's pdsolve capabilities for exactly solving problems with boundary conditions, for reason mentioned by Preben. Just to be clear: that doesn't mean the problem cannot be solved using a human-guided approach. For example: you may want to try solving the PDE exactly - the solution involves and arbitrary function _F1 as shown by Preben - then reverse the RootOf using DEtools[remove_RootOf] then try giving 'mapping values' to _F1 such that you can match your boundary conditions.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Hi
The question, however, is not about what system is better in general, but about what would be “the Maple strengths compared to Matlab and Mathematica”, and as such it looks to me properly presented.

I cannot present a complete list answering your question but here are some Maple strengths compared to Mathematica or Matlab that in my opinion are indisputable, relevant functionality that, somehow surprisingly, basically only exists in Maple.

• The Maple Differential Equation programs. Give a look at the commands listed in DEtools and PDEtools, then search for similar functionality in Mathematica: mostly none of it exists. These Maple commands are at the root of the much better performance of Maple for the exact solving of ODEs and PDEs. The results of the comparison Maple versus Mathematica in solving Kamkes ODEs are hard to believe, and yet all reproducible. An equivalent comparison for PDEs does not exist yet, but if produced it would show even more disparity between what Maple and Mathematica can do: Mathematica can handle only a very tiny fraction of the PDE problems that Maple can handle. Not less important: in Maple you have commands for performing mostly all the solving steps of rather varied different DE approaches. That allows one to tackle a problem that is beyond the computer abilities by using a human-guided approach. These Maple commands, that do not exist in Mathematica, are also key when teaching DE courses.
  
  
• The Maple Physics package. Give a look at Physics. Now open Mathematica and search for anything that would look similar. There is basically nothing. From Vector Analysis using coordinate free algebraic vectors, to noncommutative quantum operators, from 3D Euclidean tensors to tensors in curved 3+1 spacetimes, or Dirac’s notation for quantum mechanics, functional differentiation, the list is immense. Mathematica only started with tensors, and recently, and in a way that, frankly speaking, resembles more FORTRAN syntax than standard textbook or paper and pencil notation, not to mention that the functionality is fragmentary, for instance nothing for general relativity. Physics is an area where Maple’s strengths compared to Mathematica contrast even more than in DEs. You may also want to give a look at the Virtual User Summit webinar on Physics, a 24 minutes presentation showing the use of the package in mechanics, quantum mechanics and relativity.
  
  
• The Maple conversion network for mathematical functions. Check convert/to_special_function, as well as the whole FunctionAdvisor project. Now open Mathematica and search for this functionality. Again, you won't find it. In Maple things are also not restricted to this network of conversions or the FunctionAdvisor. Maple also has the ability to represent as PDE systems with rational coefficients almost any linear and nonlinear algebraic expression involving almost any mathematical function. The same regarding Maple’s ability to perform symbolic differentiation, e.g. d^n/dz^n cos(z) (n is ‘a symbol’, not a number), or to compute branch cuts of algebraic expressions, all of this only exists in Maple.
  
  
• The Maple DifferentialGeometry package. This is by all means the most resourceful mathematical software available for the area, with over 250 commands covering a wide range of topics, from basic jet calculus to the realm of the mathematics behind general relativity, thorough documentation for all of its commands, 19 differential geometry Lessons from beginner to advanced level and 5 Tutorials illustrating the use of the package in applications. I am repeating, but again: search for something resembling this in Mathematica. This functionality is inexistent in that system.
  
  
• The Maple ability to perform differential algebra. I am talking both about the standard differential algebra, and the completely non-standard ability to perform differential algebra in the presence of non-rational objects that include the whole set of mathematical functions. Search for something resembling that in Mathematica and again you will find that there is none. This by itself already puts Maple’s potential for algebraic computations beyond what Mathematica could do. Differential elimination can be used as a key tool in most non-trivial algebraic problems one could imagine. By the way this differential algebra capabilites are at the root of the advantages Maple has in differential equations and differential geometry.
  
  
• The Maple readability and traceability of Maple programs. You can give a look - literally read - basically any Maple program. Only a tiny bit of kernel commands, typically of little interest, are not readable. Everything else is readable. Wow! In fact I *never* studied a single page of Maple. Instead I learned tons just giving a look at the existing Maple programs and their help pages - and tracing them while running: try trace(command), then execute command(whatever) and you actually see how things are computed, all the steps, you can even use stopat to perform one step at a time, think, do your experiments, execute the next step. Maple is an incredibly open language. Now open Mathematica and search for any of this. In brief: none.

In summary, for physics (all kinds of problems), differential equations (all kinds of problems), mathematical function relations (their inter-connections or their representation with differential equations), differential geometry (all kinds of problems), differential algebra (all kinds of problems) or reading and tracing the actual programs that perform your computations, Maple has dramatic advantages regarding Mathematica or Matlab.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Hi

Note first that r[0]_ is not correct input: in 2D input mode it results in the product of r[0] with `_` while in 1D input mode it interrupts with an error. You can correct that input using either r_[0] or better: r__0_.

Besides that there is an issue regarding a Taylor expansion of the Norm of a vector: you know, the Taylor expansion of f(x-a) around x=a involves the derivatives of f at 0, but d/dr_ Norm(r_) = r_/Norm(r_), so this derivative at 0 would result in division by 0, so you cannot expand the Norm in Taylor series around a 0 of its argument.

Regarding this other question on a generic expansion of a vector function around a vectorial argument, I agree with you, this would be a nice and relevant feature that is not implemented in this moment. I'm adding this to the list of things being developed.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Hi Fred

Thanks for pointing to the problem. There was a typo introduced in a previous change such that the computation was performed but the result not returned - resulting in what you noticed: the computation returned unperformed. It is fixed now, the fix is available to everybody in the usual place, the Maplesoft R&D Physics webpage. The PhysicsUpdates18.mw worksheet that comes within the zip with the update illustrates the fix.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Hi USPAS2014

(what a name! :) PlotExpression is a procedure I use frequently, here it is:

> PlotExpression := f -> plots[plotcompare](f, 0, _rest, 'expression_plot', 5);

Now: other people mentioned this approx 1 month and a half ago - at that time I updated the mini-course in Mapleprimes including the definition of PlotExpression above directly in the worksheet, so perhaps you have the previous version of the mini-course?

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

PS: I don't always monitor Mapleprimes; in cases like this one please feel free to write to physics@maplesoft.com and I will receive your email directly in my mailbox.

Hi

What Maple are you using? Although GRTensor was the sate-of-the are for many years, nowadays in Maple there are two packages, Physics and DifferentialGeometry, with which you can now perform General Relativity computations more comprehensively than with GRTEnsor. Feel free to present your problem, maybe we can help you showing how you tackle it with these Maple packages.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

@trace 

mm_(reviewed).mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Hi

In your worksheet critical.mw, in the input that results in (3), an incomplete solution, instead of having only {x1,x2,x3,x4,x5} as solving variables, include in that set m, as in

> solve({…same system…}, {m, x1, x2, x3, x4, x5});

and you will receive, exactly, the solutions (critical lines) you were expecting. So there is no weakness in solve as some people thought. It is in fact a rather sophisticated/powerful command.

Alternatives to solve, using a different solving engine: PDEtools:-Solve, and to split into cases there is also PDEtools:-casesplit, both using the rifsimp routines of DEtools, or optionally the DifferentialAlgebra package. By the way there is nothing in Mathematica like these two packages, rifsimp and DifferentiAlgebra.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

By default, dsolve returns explicit results. If you prefer an implicit result, you can use the implicit option as in

> dsolve(DE, implicit);

For your DE you will receive the output you mentioned as more elegant.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft