Dear maple user  any one suggest me how to solve  second order coupled differential equation using galerkin finite element method for 8 elements and 10 elements using maple codes

 

I thought I remembed how to do this once in Maple, or asking something like this here, but may be it was something similar. But I am not able to figure it now or remember.

Given an expression, I want to find all occuranes of a pattern in it.  Not just one.  So this is like using patmatch but over and over again untill all patterns found. I'll give an example to make it easy to explain.

Given

expr := y^2*sin(1/y) + y^(3/2) + y + x*y^7;

I want to find all patterns of form y^n  so the result should be 

{y^(3/2), y^7, y^2, 1/y, y}

This below is how it is done in Mathematica, but having hard time translating this code to Maple.

The last line below does the actual repeated pattern matching. That line was not written by me. It is something from Mathematica forum at stackexchange and I use it all the time and it works well.

ClearAll[x,y,n]
expr = y^2*Sin[1/y] + y^(3/2) + y + x*y^7;
pat = y^n_.;
Last@Reap[expr /. a : pat :> Sow[a], _, Sequence @@ #2 &]

I looked at hastype also. But hastype will only tell me if the pattern is there or not. May be I did not use it right.

restart;
expr := y^2*sin(1/y) + y^(3/2) + y + x*y^7;
hastype(expr,symbol^(anything));

Gives true

I tried patmatch, but again, it only find one:

restart;
expr := y^2*sin(1/y) + y^(3/2) + y + x*y^7;
if patmatch(expr,a::anything+y^(n::anything)+b::anything,'la') then
   assign(la);
   n;
fi;

And the above is not even robust for finding one. Since I have to change the pattern again to account for multiplication/addition cases between terms. 

Is it possible to do in Maple the same thing I show in Mathematica? I am sure it is possible, but can't now find the right function in Maple.

Maple 2019.1

Should I adjust numcpus while running thread command

I want to extract lists that are term by term smaller than a fix one.

To be clearer, I want to create something like

if a[i] <= b[i] for all i=1..min(nops(a),nops(b)) then ...

I tried with the forall call, but it doesn't work:

if forall( i=1..min(nops(a),nops(b)),a[i] <= b[i]) then ...
 

I also tried the assuming call with the is call. It works, but doesnt't give me the result I want:

 

if is(a[i] <= b[i]) assuming(1 <= i, i <= min(nops(a),nops(b))) then...

Is there a way to have a chain of conditions in a if statement like I am trying to do?

I experienced a significant obstacle while trying to generate independent random samples with Statistics:-Sample on different nodes of a Grid multi-processing environment. After many hours of trial-and-error, I discovered an astonishing workaround, and I achieved excellent time and memory performance. Since this seems like a generally useful computation, I thought that it was worthy of a Post.

This Post is also worth reading to learn how to use Grid when you need to initialize a substantial environment on each node before using Grid:-Map or Grid:-Seq.

All remaining details are in the following worksheet.
 

How to use Statistics:-Sample in the `Grid` environment Author: Carl Love <carl.j.love@gmail.com> 1 August 2019   I experienced a significant obstacle while trying to generate indenpendent random samples with Statistics:-Sample on the nodes of a multi-processor Grid (on a single computer). After several hours of trial-and-error, I discovered that two things are necessary to do this: 1.  The random number generator needs to be seeded differently in each node. (The reason for this is easy to understand.) 2.  The random variables generated by Statistics:-RandomVariable need to have different names in each node. This one is mind-boggling to me. Afterall, each node has its own kernel and so its own memory It's as if the names of random variables are stored in a disk file which all kernels access. And also the generator has been seeded differently in each node.   Once these things were done, the time and memory performance of the computation were excellent. >  restart : >  Digits:= 15 : >  #Specify the size of the computation: (n1,n2,n3):= (100, 100, 1000): # n1 = size of each random sample; # n2 = number of samples in a batch; # n3 = number of batches. >  # #Procedure to initialize needed globals on each node: Init:= proc(n::posint) local node:= Grid:-MyNode();    #This is wrapped in parse so that it'll act globally. Otherwise, an environment    #variable would be reset when this procedure ends.    parse("Digits:= 15;", 'statement');    randomize(randomize()+node); #Initialize independent RNG for this node.    #If repeatability of results is desired, remove the inner randomize().    (:-X,:-Y):= Array(1..n, 'datatype'= 'hfloat') $ 2;    #Perhaps due to some oversight in the design of Statistics, it seems necessary that    #r.v.s in different nodes **need different names** in order to be independent:    N||node:= Statistics:-RandomVariable('Normal'(0,1));    :-TRS:= (X::rtable)-> Statistics:-Sample(N||node, X);    #To verify that different names are needed, change N||node to N in both lines.    #Doing so, each node will generate identical samples!    #Perform some computation. For the pedagogical purpose of this worksheet, all that    #matters is that it's some numeric computation on some Arrays of random Samples.    :-GG:= (X::Array, Y::Array)->       evalhf(          proc(X::Array, Y::Array, n::posint)          local s, k, S:= 0, p:= 2*Pi;             for k to n do                s:= sin(p*X[k]);                  S:= S + X[k]^2*cos(p*Y[k])/sqrt(2-sin(s)) + Y[k]^2*s             od          end proc          (X, Y, n)       )          ;    #Perform a batch of the above computations, and somehow numerically consolidate the    #results. Once again, pedagogically it doesn't matter how they're consolidated.      :-TRX1:= (n::posint)-> add(GG(TRS(X), TRS(Y)), 1..n);        #It doesn't matter much what's returned. Returning `node` lets us verify that we're    #actually running this on a grid.    return node end proc : The procedure Init above uses the :- syntax to set variables globally for each node. The variables set are X, Y, N||node, TRS, GG, and TRX1. Names constructed by concatenation, such as N||node, are always global, so :- isn't needed for those. >  # #Time the initialization: st:= time[real]():    #Send Init to each node, but don't run it yet:    Grid:-Set(Init)    ;    #Run Init on each node:    Nodes:= Grid:-Run(Init, [n1], 'wait'); time__init_Grid:= time[real]() - st; The only purpose of array Nodes is that it lets us count the nodes, and it lets us verify that Grid:-MyNode() returned a different value on each node. >  num_nodes:= numelems(Nodes); >  #Time the actual execution: st:= time[real]():    R1:= [Grid:-Seq['tasksize'= iquo(n3, num_nodes)](TRX1(k), k= [n2 $ n3])]: time__run_Grid:= time[real]() - st >  #Just for comparison, run it sequentially: st:= time[real]():    Init(n1): time__init_noGrid:= time[real]() - st; st:= time[real]():    R2:= [seq(TRX1(k), k= [n2 $ n3])]: time__run_noGrid:= time[real]() - st; R1 and R2 will be different because different random numbers were used, but they should have similar histograms. >  plots:-display(    Statistics:-Histogram~(       <R1 | R2>, #side-by-side plots       'title'=~ <<"With Grid\n"> | <"Without Grid\n">>,       'gridlines'= false    ) ); (Plot output deleted because MaplePrimes cannot handle side-by-side plots!) They look similar enough to me!   Let's try to quantify the benefit of using Grid: >  speedup_factor:= time__run_noGrid / time__run_Grid; Express that as a fraction of the theoretical maximum speedup: >  efficiency:= speedup_factor / num_nodes; I think that that's really good!   The memory usage of this code is insignificant, which can be verified from an external memory monitor such as Winodws Task Manager. It's just a little bit more than that needed to start a kernel on each node. It's also possible to measure the memory usage programmatically. Doing so for a Grid:-Seq computation is a little bit beyond the scope of this worksheet. >

 

Download GridRandSample.mw

Here are the histograms:

 

Hello,

How to show the y-axis numbers in the second plot ?   y is in [0,3]. 

I try the following but it shows a negative axis. 

display(P, L, axis = [tickmarks = [5, subticks = 3], thickness = 2]);  this is not correct

The file is attached.

Thanks

 

Download Curves.mw

 

I do not understand why the following is not working. I have a package which has a module inside it. The module is exported by the package.

Inside that module, there is a proc, which is also exported by the module.

So why can't one call the proc from outside the package? What Am I doing wrong? and how to correct it? I'd like to be able to call the proc directly. 

A:=module()
  option package;
  export B;
  B := module()
      export foo;

      foo:=proc()
           print("in foo");
      end proc;

  end module;
end module;   

Now when typing (A::B):-foo() Maple is not happy and says Error, module does not export `foo`

I tried different syntax from the above, but can't get it to work. For example A::B:-foo() gives Error, `B` does not evaluate to a module

 

Maple 2019.1

P; Q; lma; [3 -42] [-, ---] [2 25 ] [22649 -2304] [-----, -----] [5523 1841 ] 1.627112258 with(geometry); point(Pp, P[1], P[2]); point(Qp, Q[1], Q[2]); ellipse(p, ['foci' = [Pp, Qp], 'MajorAxis' = lma]); Error, (in geometry:-ellipse) the given polynomial/equation is not an algebraic representation of a ellipse; I can't understand this error

Hi I’m a student, running Maple on a student license, and I have an issue with the way the Maple 2019 setup is.

I'm sorry if my answer is a bit hard to understand, I don't really know how to explain it.
 

My Maple 2018 expired and I was forced to upgrade to Maple 2019, however for me there is a huge difference in how the setup is in therms of “math” and “text” mode. 

 

Before, I was able to easily switch between “text” and “math” mode in a paragraph (the red “>”) in my worksheet, by using “cmd+R” and “cmd+T”. So I would have paragraphs for text, where i could switch between “math” and “text” for writing, and then I would have paragraphs only for math for when I would calculate things. (I’ve tried to screenshot how it would typically look)

 

Mt problem with Maple 2019, is that when I switch a paragraph into text and start writing and want to switch to “math” by using “cmd+R”, Maple executes the formula whenever I press “enter”, and give me an output I don’t want.

 

How do I change the settings to get back to the way they were on Maple 2018 and the years before this?

Thanks in advance.

(I know I can put a ":" after the "math" part in my text, but i don't wan't to do it every time)

 

 

 

When making a proc that accepts positional and keyword based arguments, Maple allows the caller to put the keyword arguments before or after the positional arguments, or in between.

Is there a way to force the keyword arguments (if they are used) to only be placed after the (required) positional arguments during the call? 

An example will make it more clear. Given this proc

foo := proc(a::integer, b::integer, {c::integer:= 1, d::integer:= 1},$)
    print("a=",a," b=",b," c=",c," d=",d);
end proc:

It can be called as any one of these cases

foo(1,2);                                #case 1                                             
foo(1, 2,'c' = 1, 'd' = 2);              #case 2
foo('c' = 1, 1, 'd' = 2, 2);             #case 3
foo('c' = 1, 1, 2);                      #case 4
foo(1, 'c'=1, 2);                        #case 5
#more combinations possible

Is it possible to change how proc is defined, so that Maple will accept case1, and case 2 only in the above when the call is made and give an error otherwise?

i.e. only allow keyword arguments after all positional arguments.  I read https://www.maplesoft.com/support/help/Maple/view.aspx?path=argument_processing  and it says

After all arguments matching keyword parameters have been processed, matching of required positional and optional or expected ordered parameters is carried out

So from above, it looks like what I am asking for is not really possible. But I thought to ask, in case there is some way.

I find ability to put keyword arguments in the call anywhere and in-between required positional arguments confusing that is all.

Maple 2019.1

I am trying to expand a multivariable (more specifically 4 variables) function in powers of one of its variables when it goes to infinity.

However, the result I get is always zero, even if I input (or not) values for some of the other variables.

Can anybody help?

series_expansion.mw

P.s.: I want to do the same for the other two functions I defined in the worksheet as well.

Is there a command or method in Maple to list all initially protected names?

This page https://www.maplesoft.com/support/help/Maple/view.aspx?path=UndocumentedNames list undocumented protected names, and this page https://www.maplesoft.com/support/help/Maple/view.aspx?path=initialconstants  lists initially known names, which I assume are all protected.

But what about a list of all protected names? sin,cos, eval, uneval, etc.. any name that can't be assigned to. In my search so far, I could not find how to find these names. There are 100's of such names. Can one get them all in a list to look at?

Using Maple 2019.1

I want to solve the following system using PDE Solve command but finding an error. Please help me in this regard. Thanks!
 

 

restart; d1 := 1; d2 := 1; AA := 0.2e-2; BB := 0.79e-1; L := 1;
with(PDEtools, casesplit, declare);
PDE1 := diff(u(x, t), t) = d1*(diff(u(x, t), x, x))-u(x, t)*v(x, t)^2+AA*(1-u(x, t)); PDE2 := diff(v(x, t), t) = d1*(diff(v(x, t), x, x))+u(x, t)*v(x, t)^2-BB*v(x, t);
                      /  2         \                           
         d            | d          |                  2        
PDE1 := --- u(x, t) = |---- u(x, t)| - u(x, t) v(x, t)  + 0.002
         dt           |   2        |                           
                      \ dx         /                           

   - 0.002 u(x, t)
                          /  2         \                   
             d            | d          |                  2
    PDE2 := --- v(x, t) = |---- v(x, t)| + u(x, t) v(x, t) 
             dt           |   2        |                   
                          \ dx         /                   

       - 0.079 v(x, t)
IBC1 := {u(0, t) = 1, u(1, t) = 1, u(x, 0) = 1-(1/2)*sin(Pi*(x-L)/(2*L))^100}; IBC2 := {v(0, t) = 0, v(1, t) = 0, v(x, 0) = (1/4)*sin(Pi*(x-L)/(2*L))^100}; dys1 := {IBC1, IBC2, PDE1, PDE2};
         /              /  2         \                           
         | d            | d          |                  2        
dys1 := < --- u(x, t) = |---- u(x, t)| - u(x, t) v(x, t)  + 0.002
         | dt           |   2        |                           
         \              \ dx         /                           

   - 0.002 u(x, t), 

                /  2         \                                     
   d            | d          |                  2                  
  --- v(x, t) = |---- v(x, t)| + u(x, t) v(x, t)  - 0.079 v(x, t), 
   dt           |   2        |                                     
                \ dx         /                                     

   /                                                           100
   |                                        1    /1           \   
  < u(0, t) = 1, u(1, t) = 1, u(x, 0) = 1 - - sin|- Pi (x - 1)|   
   |                                        2    \2           /   
   \                                                              

  \   
  |   
   >, 
  |   
  /   

   /                                                       100\ 
   |                                    1    /1           \   | 
  < v(0, t) = 0, v(1, t) = 0, v(x, 0) = - sin|- Pi (x - 1)|    >
   |                                    4    \2           /   | 
   \                                                          / 

  \ 
  | 
   >
  | 
  / 
pds := pdsolve(dys1, numeric, time = t, range = 0 .. 1);
Error, (in pdsolve) invalid input: `pdsolve/numeric` expects its 2nd argument, IBCs, to be of type {list, set}, but received time = t
p1 := pds:-plot(t = 0, numpoints = 50);

p2 := pds:-plot(t = 1/8, numpoints = 50, color = blue);

p3 := pds:-plot(t = 1/4, numpoints = 50, color = green);

plots[display]({p1, p2, p3});
Error, `pds` does not evaluate to a module
Error, `pds` does not evaluate to a module
Error, `pds` does not evaluate to a module
Error, (in plots:-display) expecting plot structures but received: {p1, p2, p3}

Hi, Is there a way in which i can solve the following optimal control problem numerically with Maple ??

where P(t)=N(t)+S(t)+A(t) and N(0)=0.4897, S(0)=0.4018, A(0)=0.1085.

μ=0.000833, d=0.000666, ε1=0.0020, ε2=0.000634, β1=0.002453, β2=0.25*0.02, γ1=0.0048, γ2=0.25*0.02+0.00013, k1=1, k2=0.001, k3=0.99.

 where 

p1,p2,p3 are transversality conditions 

p1(60)=0 
p2(60)=0 
p3(60)=0

Answers and advice are very appreciated. 

Thank you all for reading.

Benz.

Hi MaplePrimes team,

 

If I assume a plot-based function or procedure C(s,t) and P1(t) are hidden in a package somewhere I don’t care about the type of geometry (spacecurve or plot3d). Now, I just want to animate P1(t) following the variable t. How do I do?

Example 1: Curve Animation in variable t

1. Equation

C := (s,t)-> Vector([cos(s), sin(s), t*s/2]):     #Helix shape as example

 

2. Plot

P1 := t-> display(spacecurve(C(s,t), s = 0..3*2*Pi, linestyle = solid)):

 

 

Example 2: Surface Animation in variable t

1. Equation

S := (x,y,t)-> Vector([x, y, t/(1 + x^2 + y^2)]):

2. Plot

P2 := t-> display(plot3d(S(x,y,t), x = -2..2, y = -2..2)):

 

If I assume the function or procedure S(x,y,t) and P2(t) are hidden in a package somewhere I don’t care. Now, I just want to animate P2(t) following the variable t. How do I do?

 

 

Example 3: Mixing Curve and Surface Animation in variable t

The ideal/”dream” basic syntax would be:

animate(display, [P₁(t), P₂(t)], t = t₁..t₂)

But it does not work.

 

Best.

Guy.