ecterrab

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These are replies submitted by ecterrab

@Rouben Rostamian  

Enjoyable reading as usual, only one comment: convert(%, set) is not necessary, dsolve accepts differential equations also directly in matricial form.

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

@otaviosama 

You realize you sent a document with 23 pages. Man, it shouldn't be that way. Think about. Could you please indicate one page where I can see the formulas you would like help proving?

@mwahab 

Having m and/or k as exponents in powers, or as coefficients of non-linear terms, makes no difference: they are just parameters, things that don't depend on the independent or dependent variables of the problem. You can try replacing pde (equation (1) in the post) by pde := U[t] + U[]^m*U[x] + U[]^k = 0 and run the same computation. Of course, the problem is more complicated, and so are the infinitesimals. You can then indicate u as the dependent variable, or the list [u, k, m] as I did in answer to the recent question in Mapleprimes, or replace the functionality (x, t) by (x, t, k, m)as I did in the post above. It all works (Maple 2019, latest Physics Updates), and what you do depends on what you want to achieve.

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

@otaviosama 

My impression is that you are not typing the right equations. For instance: your algebra rules involve J[y], but your expected result (righ-hand side of the equation you added in red in the original post and after the post) has no J[y]. (Also: when changing the worksheet, do not rewrite the original, but attach a new one so that we can follow and find things right away.)

Likewise, although I understood your now CommuRules are not the ones for the Pauli sigma matrices, from your title, if you are computing with angular momentum operators, their algebra rules and identities are all shown in this Wikipedia page and they all check OK when entered and simplified on a Maple worksheet.

Would you mind revise your input and, if necessary, please create an image from the book (also what book) you are taking these formulas and paste the image here? That may help to identify a typo or what is going on. From what I've seen so far, Maple's result is just correct, and not what you expect.

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

@mwahab 

I looked again at your worksheet. This is not a Differential Geometry problem, but one of algebraic manipulation of simple form. You can do what you want, entirely, using PDEtools only, and that will avoid the confusion caused by the occurrence of diff(u(x,y,z),y,y) together with u[y,y] in the same expression representing the same thing. The PDEtools commands you want: diff_table (makes your input tremendously simpler), ToJet / FromJet (see the notation options you have to match the one you show in your question) and of course, dcoeffs (use it the way I indicated in my first answer). And then you won't need to write a program or anything of that sort in order to "get the coefficients of u(x,y,z) and/or v(x,y,z)", including of course the coefficients of their derivatives (that is why it is called 'd'coeffs).

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

@otaviosama 

In that case (remember also please for other questions posted here), it is important for you to indicated what you expect as right value, especially if you say "[Maple] Physics doesn't return the right value". Also, in your post I read PauliSigmaRules - that can produce missunderstanding if you (e.g. now) say the J[n] do not obey the anticommutation rules of Pauli sigma Matrices - i.e. the rules you posted as PauliSigmaRules are not actually the Pauli sigma rules.

Please update your worksheet with whatever missing information and we (re)start from there - I'm sure we can help you. Best.

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

@Rouben Rostamian  

Thanks for the feedback; this is already adjusted in v.450.

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

@Fereydoon_Shekofte @Christopher2222

I perceive this the same way as Fereydoon_Shekofte, indeed. And those are, for me, the main reason for a Stack-Exchange Maple forum. There is a TON of mathematics capabilities in the Maple system. Many of them don't even exist in Mathematica, while they would be of use for many people working with mathematics. Yet nobody knows about that - we only advertise it in Mapleprimes where - mostly - only exists Maple users see it.

Regarding the (for me genuine) concern about duplication, I imagine that temporary solutions (e.g. 1 or 2 starting years) on the lines of what Daniel Skoog suggested, or even having every SE-Maple automatically reposted here in Mapleprimes for those who prefer to see it here, and even perhaps the other way around too with automatic reposts in the SE-Maple would suffice.

Regarding searching (another genuine concern) the search field of Mapleprimes could easily (optionally or not) also search in SE-Maple. Maybe it is even possible to have it the other way around too. That looks to me as a matter technically easy to solve.

Another issue that I would mention is the possibility of posting the contents of a Maple worksheet in SE-Maple. I don't know how that could be resolved, but I am sure there is a solution (maybe a tweaked pdf format ... something to think, not a serious obstacle I think).

@nm and  acer

Thanks. From what you posted, I can see we can try. I do not see the roadmap, though; the main questions that spring to my head: "what are they expecting?" Is it a number of questions per day? Per week? Is it a number of visitors per week? Is it a number of posts? And whatever the qualifier is, is it along one month? One semester? One year? And when do they give the membership status? In a month? Six months? A year? 

The point is that without such a roadmap - with explicit qualifiers known - it is not possible to estimate the chances of success (this resembles me giving a problem to a student and not providing him all the information to solve it. I heated that when I was a student, and not doing that was my main flag when giving courses at university.)

Without further information, just in this dark, I can tell you my estimation: many Maple users would be interested in a stack-exchange Maple forum. I foresee the volume of questions/answers and posts getting bigger rapidly if compared with what we see in this moment in Mapleprimes.

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

Adam Ledger's proposal is quite appropriate, I think; stack exchange seems to me the way to go nowadays, and am willing to put some weight/time on this. I am not familiar with the details, maybe you nm could summarize here, kinda brief/succint road-map, what would be necessary for success? Thanks.

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

@Area51 

Yes, you can define (add) tensors using tensorial expressions as definitions: the left-hand side (lhs) is the tensor being defined, the right-hand side (rhs) is the tensorial expression. And yes, the rhs can also have contracted indices - the only requirement is that the free indices of the lhs and rhs are the same, naturally. For all that, see the help page for ?Physics:-Define. I also see you used Box; there is Physics:-dAlembertian, the galilean one, but you can define one in curved spaces as explained in the previous sentence.

If you don't know how to formulate your problem on a Maple worksheet using Physics, try to approximate as much as possible the input lines representing your problem, and please post the resulting worksheet and questions here again.

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

You need to post the problem, explicitly.

@a_simsim 

Andras

I mentioned I was going to write again here regarding your suggestion concerning Den Iseger's algorithm. We gave a look at the paper, with Katherina, and the three algorithms implemented work pretty well with all the examples, Table 1: Analytical test functions, and Table 4: Continuous Nonanalytical test functions, in both cases getting better accuracy; but do not handle those of Table 2: Two Discontinuous test functions. In brief, for these two discontinuous examples it is not possible to determine whether the accuracy is or not increasing, and then the numerical evaluation routine gives up. We are short of time but will see if we can still give Iseger's algorithm a try. If so, I will post again here. Thanks again for the reference.

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

@Mariusz Iwaniuk 

Thanks for your feedback. I looked at your seven examples. Your example 5 is taken from Table 2 of the paper by Iseger mentioned by Andras, a discontinuous example. Then examples 1, 2, 4 and 7 (with regards to tackling them using the methods implemented) are basically of the same type as example 5. You see that by looking at the form of the Laplace transform being inverted, or at the exact form of the inverse Laplace transform. These examples - the one from Iseger's paper is the best representative, I think -  result uncomputable (using the implemented algorithms) because for discontinuous inverse Laplace transforms it is sometimes not possible to determine whether the accuracy is or not increasing. We were yesterday discussing with Katherina the possibility of implementing one more algorithm for this case (basically, what Andras suggested).

Then your example 6 is a piecewise branching at t < Pi, and you ask the numerical evaluation at t = Pi. Such a case is always difficult, for any numerical algorithm. Your example 3 is however, different: the expression being (Laplace) numerically inverted involves the Zeta function, whose numerical evaluation is itself a difficult problem, and the expected inverse Laplace transform is equal to harmonic(floor(exp(t)), 2). I am not a numerical analyst (more like a theoretical physicist), but I tend to think that numerical methods for problems like your 3 and 6, where you have discrete numerical functions, or where you ask for the numerical evaluation at where the discrete function branches, are not the standard problem, and in any case, not the problems I am targetting in a first implementation round.

Summarizing, if we have time, we will try to implement Iseger's algorithm as suggested by Andras and with that handle the two discontinuous examples of Table 2 of Iseger's paper (that would handle your examples 1, 2, 4, 5 and 7), Your examples 6 and 3 may or not be automatically solved at that point; otherwise they will require computing the exact transform first.

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

@a_simsim 

Hi Andras,
Thanks for the reference. Below is a worksheet with the testing example - comparing numerical inversion algorithms - of the paper you just mentioned, But the values of the numerical parameters they used are not shown. Do you have them?

Example from: Bruno Josso - Leif Larse, "Laplace transform numerical inversion"

 

In what follows, each Eq.(number) indicated right before an input line corresponds to the equation number shown in the paper.


Eq.(10)

q__D := proc (t) options operator, arrow; Gamma*exp(-mu*t) end proc


Eq.(11)

Gamma := 1/((1/2)*ln(4*A__D/(exp(gamma)*C__A))+S)

Eq.(12)

A__D := A/`r__&omega;`^2


Eq.(13)

mu := 2*Pi/(A__D*Gamma)


Formula within text, between Eq.(13) and Eq.(14)

`#mover(mi("F"),mo("&circ;"))` := proc (p) options operator, arrow; Gamma/(p+mu) end proc

proc (p) options operator, arrow; Gamma/(p+mu) end proc

(1)

Check it out:

`#mover(mi("F"),mo("&circ;"))`(p)

1/(((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S)*(p+2*Pi*`r__&omega;`^2*((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S)/A))

(2)

Load the inttrans package

with(inttrans)

 

Eq.(14), that is:  `#mover(mi("F"),mo("&circ;"))`(p) = laplace(q__D(t), t, p), and q__D(t) = invlaplace(`#mover(mi("F"),mo("&circ;"))`(p), p, t)

`#mover(mi("F"),mo("&circ;"))`(p) = laplace(q__D(t), t, p)

1/(((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S)*(p+2*Pi*`r__&omega;`^2*((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S)/A)) = 1/((-(1/2)*gamma+S+(1/2)*ln(4*A/(`r__&omega;`^2*C__A)))*(p+Pi*`r__&omega;`^2*(-gamma+2*S+ln(4*A/(`r__&omega;`^2*C__A)))/A))

(3)

simplify((lhs-rhs)(1/(((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S)*(p+2*Pi*`r__&omega;`^2*((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S)/A)) = 1/((-(1/2)*gamma+S+(1/2)*ln(4*A/(`r__&omega;`^2*C__A)))*(p+Pi*`r__&omega;`^2*(-gamma+2*S+ln(4*A/(`r__&omega;`^2*C__A)))/A))))

0

(4)

q__D(t) = invlaplace(`#mover(mi("F"),mo("&circ;"))`(p), p, t)NULL

exp(-2*Pi*`r__&omega;`^2*((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S)*t/A)/((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S) = 2*exp((gamma-2*S-ln(4*A/(`r__&omega;`^2*C__A)))*`r__&omega;`^2*t*Pi/A)/(-gamma+2*S+ln(4*A/(`r__&omega;`^2*C__A)))

(5)

simplify((lhs-rhs)(exp(-2*Pi*`r__&omega;`^2*((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S)*t/A)/((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S) = 2*exp((gamma-2*S-ln(4*A/(`r__&omega;`^2*C__A)))*`r__&omega;`^2*t*Pi/A)/(-gamma+2*S+ln(4*A/(`r__&omega;`^2*C__A)))))

0

(6)

Eq.(15)

(%limit = limit)(p*`#mover(mi("F"),mo("&circ;"))`(p), p = 0)

%limit(p/(((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S)*(p+2*Pi*`r__&omega;`^2*((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S)/A)), p = 0) = 0

(7)

Eq.(16): this other limit is more difficult to compute without further information on the values of the parameters involved; according to the paper, the right-hand side is expected to be equal to 0

`assuming`([(%limit = limit)(q__D(t), t = infinity)], [real, Gamma > 0])

%limit(exp(-2*Pi*`r__&omega;`^2*((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S)*t/A)/((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S), t = infinity) = limit(exp(-2*Pi*`r__&omega;`^2*((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S)*t/A)/((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S), t = infinity)

(8)

The problem now is that, in order to reproduce Figure 3 of the paper, that is, to compare the analytical solution q__D(t) = GAMMA*exp(-mu*t) with a numerical Laplace inversion of `#mover(mi("F"),mo("&circ;"))`(p) (i.e., numerically compute the right-hand side of q__D(t) = invlaplace(`#mover(mi("F"),mo("&circ;"))`(p), p, t)), we need numbers for all these letters (parameters of the model) you see in (8)

remove(type, `minus`(indets(%limit(exp(-2*Pi*`r__&omega;`^2*((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S)*t/A)/((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S), t = infinity) = limit(exp(-2*Pi*`r__&omega;`^2*((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S)*t/A)/((1/2)*ln(4*A/(`r__&omega;`^2*exp(gamma)*C__A))+S), t = infinity), symbol), {t}), constant)

{A, C__A, S, `r__&omega;`}

(9)

I do not see the values of these parameters in the paper. Without them it is not possible to run this numerical inversion of q__D(t). Do you have the values of these parameters?

NULL


 

Download Example_from_Laplace_transform_numerical_inversion.mw

 

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

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