Maple Questions and Posts

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Dear All

Is there anybody who is working on contruction of optimal Lie algebra using Maple packages like DifferentialGeometry and LieAlgebra, I tried to find commands for constructing algebra in these packages but could not find such commands. I am sure these are only package that might help me. Following is Lie algebra whose optimal system is required:

 

with(PDEtools, SymmetryCommutator, InfinitesimalGenerator):

S[1], S[2], S[3], S[4], S[5], S[6], S[7], S[8], S[9], S[10], S[11] := [_xi[x] = 0, _xi[y] = 0, _xi[t] = 0, _eta[u] = 0, _eta[v] = 1], [_xi[x] = 0, _xi[y] = t, _xi[t] = 0, _eta[u] = 0, _eta[v] = x], [_xi[x] = 0, _xi[y] = y, _xi[t] = 2*t, _eta[u] = -2*u, _eta[v] = -v], [_xi[x] = 0, _xi[y] = 0, _xi[t] = 0, _eta[u] = 0, _eta[v] = t], [_xi[x] = 0, _xi[y] = 0, _xi[t] = 0, _eta[u] = 1, _eta[v] = y], [_xi[x] = 0, _xi[y] = 0, _xi[t] = 1, _eta[u] = 0, _eta[v] = 0], [_xi[x] = 1, _xi[y] = 0, _xi[t] = 0, _eta[u] = 0, _eta[v] = 0], [_xi[x] = t, _xi[y] = 0, _xi[t] = 0, _eta[u] = 1, _eta[v] = 0], [_xi[x] = y, _xi[y] = 0, _xi[t] = 0, _eta[u] = 0, _eta[v] = 2*x], [_xi[x] = x, _xi[y] = 0, _xi[t] = -t, _eta[u] = 2*u, _eta[v] = 2*v], [_xi[x] = 0, _xi[y] = 1, _xi[t] = 0, _eta[u] = 0, _eta[v] = 0]

[_xi[x] = 0, _xi[y] = 0, _xi[t] = 0, _eta[u] = 0, _eta[v] = 1], [_xi[x] = 0, _xi[y] = t, _xi[t] = 0, _eta[u] = 0, _eta[v] = x], [_xi[x] = 0, _xi[y] = y, _xi[t] = 2*t, _eta[u] = -2*u, _eta[v] = -v], [_xi[x] = 0, _xi[y] = 0, _xi[t] = 0, _eta[u] = 0, _eta[v] = t], [_xi[x] = 0, _xi[y] = 0, _xi[t] = 0, _eta[u] = 1, _eta[v] = y], [_xi[x] = 0, _xi[y] = 0, _xi[t] = 1, _eta[u] = 0, _eta[v] = 0], [_xi[x] = 1, _xi[y] = 0, _xi[t] = 0, _eta[u] = 0, _eta[v] = 0], [_xi[x] = t, _xi[y] = 0, _xi[t] = 0, _eta[u] = 1, _eta[v] = 0], [_xi[x] = y, _xi[y] = 0, _xi[t] = 0, _eta[u] = 0, _eta[v] = 2*x], [_xi[x] = x, _xi[y] = 0, _xi[t] = -t, _eta[u] = 2*u, _eta[v] = 2*v], [_xi[x] = 0, _xi[y] = 1, _xi[t] = 0, _eta[u] = 0, _eta[v] = 0]

(1)

G[1] := InfinitesimalGenerator(S[1], [u(x, y, t), v(x, y, t)]); 1; G[2] := InfinitesimalGenerator(S[2], [u(x, y, t), v(x, y, t)]); 1; G[3] := InfinitesimalGenerator(S[3], [u(x, y, t), v(x, y, t)]); 1; G[4] := InfinitesimalGenerator(S[4], [u(x, y, t), v(x, y, t)]); 1; G[5] := InfinitesimalGenerator(S[5], [u(x, y, t), v(x, y, t)]); 1; G[6] := InfinitesimalGenerator(S[6], [u(x, y, t), v(x, y, t)]); 1; G[7] := InfinitesimalGenerator(S[7], [u(x, y, t), v(x, y, t)]); 1; G[8] := InfinitesimalGenerator(S[8], [u(x, y, t), v(x, y, t)]); 1; G[9] := InfinitesimalGenerator(S[9], [u(x, y, t), v(x, y, t)]); 1; G[10] := InfinitesimalGenerator(S[10], [u(x, y, t), v(x, y, t)]); 1; G[11] := InfinitesimalGenerator(S[11], [u(x, y, t), v(x, y, t)])

proc (f) options operator, arrow; diff(f, v) end proc

 

proc (f) options operator, arrow; t*(diff(f, y))+x*(diff(f, v)) end proc

 

proc (f) options operator, arrow; y*(diff(f, y))+2*t*(diff(f, t))-2*u*(diff(f, u))-v*(diff(f, v)) end proc

 

proc (f) options operator, arrow; t*(diff(f, v)) end proc

 

proc (f) options operator, arrow; diff(f, u)+y*(diff(f, v)) end proc

 

proc (f) options operator, arrow; diff(f, t) end proc

 

proc (f) options operator, arrow; diff(f, x) end proc

 

proc (f) options operator, arrow; t*(diff(f, x))+diff(f, u) end proc

 

proc (f) options operator, arrow; y*(diff(f, x))+2*x*(diff(f, v)) end proc

 

proc (f) options operator, arrow; x*(diff(f, x))-t*(diff(f, t))+2*u*(diff(f, u))+2*v*(diff(f, v)) end proc

 

proc (f) options operator, arrow; diff(f, y) end proc

(2)

``

 

Download Lie_Algebra_Classification.mwLie_Algebra_Classification.mw

   
 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(2)
 

``

Regards

Hello, guys

         I am a totally new comer in this erea. I get a new module file and want to implement it into maple,here is the link:

         http://cpc.cs.qub.ac.uk/summaries/ADWE

         After I downloaded the package, I didn't know the next step and I can't find a perfect solution to this case, maybe it is too easy.

         Can some cool guys can help me, THX.

The question in the title has been raised before over the years, but has maybe not received enough attention.
Reraising the question was motivated by a comment by Kitonum to a recent post on improved integration results in Maple 2016:
http://mapleprimes.com/maplesoftblog/202910-New-And-Improved-Integration-Results

Consider the following session.
restart;
assume(n=0);
n^2; #Returns n^2
eval(%);#Returns n^2
sin(n*Pi); # Returns 0
sin(n); # Returns sin(n)
eval(%); # Returns sin(n)
ln(n); #Returns ln(n)
ln(n*exp(1)); # Returns ln(n*exp(1))
expand(%); # Error, (in ln) numeric exception: division by zero
ln((n+1)*exp(1)); Returns ln((n+1)*exp(1))
expand(%); # Returns ln(n+1)+1
sqrt(n^2); # Returns 0
sqrt(n); # Returns n^(1/2)
eval(%,n=n^2); # Returns (n^2)^(1/2)
simplify(%); #Returns 0: simplify doesn't help in the examples above.
##################
We see that assume n=0 certainly doesn't imply that expressions always will be evaluated at n=0, but sometimes it appears that it does.
So what is the intended behavior when assuming equality?
##
Several years ago (Maple 14 or earlier) I overloaded assuming so that equality assumptions were handled by eval.
There was a discussion at the time in MaplePrimes about this. Shall try to find the link.

To check the point on the belonging to the segment I use the algorithm shown in the example. This is an example of intersection of the two segments in 2d. (We not check for parallelism.) We find the point of intersection of the corresponding lines and solve the equation f1 with respect to t and f2 with respect to tt. If 0 <= t <= 1, then the point belongs to the first segment, and if 0 <= tt <= 1 then the point belongs to the second segment.
(Similarly we can check point on the belonging to the segment in 3d.)
In the example point belongs to the second segment, but not the first. These segments do not intersect.
Question: Is there a function in Maple to find the intersection of the segments or to check on the belonging segment point, to make shorter?

segments_intersection.mw 

 

I mean

J := int((x^2+2*x+1+(3*x+1)*sqrt(x+ln(x)))/(x*sqrt(x+ln(x))*(x+sqrt(x+ln(x)))), x);

Of course, with Maple.

Dear ones.hello!
  I want to export a plot to a specified path but i dont want do enter it in plotsetup command.
For example:


plotsetup(bmp,plotoutput="F:/testplot.bmp",plotoptions="height=100,width=400");
plot(sin(x),x=-Pi..Pi);


exports the bmp file to a location already provided (bold one).
Is there a way to initiate save/browse windows instead of writing the adress of the location inside the plotsetup ?
I know how to do it in mathematica and i want a similar thing in maple

Dear All

I have an numerical integral to solve over set of 10 discrete values. The problem is, this integral have two dependent variables. Please see following:

Numerical_Integration.mw

How to Solve following integral numerically over discrete data values?

I[k] = int(rho(x)*x^4*eta[k](x)^2, x)

I[k] = int(rho(x)*x^4*eta[k](x)^2, x)

(1)

where we have data values are given as under

x=1, 2, 3, 4, 5, 6, 7, 8, 9, 10

rho(x)= 0.045, 0.0459, 0.0564, 0.05689, 0.06015, 0.06235, 0.0654, 0.0687, 0.07012, 0.07251

eta[k](x)= 1.15, 1.256, 1.56, 1.85, 1.86, 2.01, 2.35, 2.56, 2.86, 2.901

 

Download Numerical_Integration.mw

Regards

I want to plot a function given different value combinations of parameters. I used the following code and it doesn't work. Could anybody please help?

The attached worksheet shows a small selection of new and improved results in integration for Maple 2016. Note that integration is a vast topic, so there will always be more improvements that can be made, but be sure that we are working on them.

Maple2016_Integration.mw

A selection of new and improved integration results for Maple 2016

New answers in Maple 2016

 

 

Indefinite integrals:

 

int(sqrt(1+sqrt(z-1)), z);

(4/5)*(1+(z-1)^(1/2))^(5/2)-(4/3)*(1+(z-1)^(1/2))^(3/2)

(1.1)

int(arctan((-1+sec(x))^(1/2))*sin(x), x);

-arctan((-(1/sec(x)-1)*sec(x))^(1/2))/sec(x)+(1/2)*(-1+sec(x))^(1/2)/sec(x)+(1/2)*arctan((-1+sec(x))^(1/2))

(1.2)

int(((1+exp(I*x))^2+(1+exp(-I*x))^2)/(1-2*c*cos(x)+c^2), x);

-x-2*x/c-x/c^2+I*exp(I*x)/c-I*exp(-I*x)/c-I*c*ln(exp(I*x)-1/c)/(c-1)-I*ln(exp(I*x)-1/c)/(c-1)-I*ln(exp(I*x)-1/c)/(c*(c-1))-I*ln(exp(I*x)-1/c)/(c^2*(c-1))+I*c*ln(-c+exp(I*x))/(c-1)+I*ln(-c+exp(I*x))/(c-1)+I*ln(-c+exp(I*x))/(c*(c-1))+I*ln(-c+exp(I*x))/(c^2*(c-1))

(1.3)

int(x^4/arccos(x)^(3/2),x);

(1/4)*(-x^2+1)^(1/2)/arccos(x)^(1/2)-(1/4)*2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)*arccos(x)^(1/2)/Pi^(1/2))+(3/8)*sin(3*arccos(x))/arccos(x)^(1/2)-(3/8)*2^(1/2)*Pi^(1/2)*3^(1/2)*FresnelC(2^(1/2)*3^(1/2)*arccos(x)^(1/2)/Pi^(1/2))+(1/8)*sin(5*arccos(x))/arccos(x)^(1/2)-(1/8)*2^(1/2)*Pi^(1/2)*5^(1/2)*FresnelC(2^(1/2)*5^(1/2)*arccos(x)^(1/2)/Pi^(1/2))

(1.4)

 

Definite integrals:

int(arcsin(sin(z)), z=0..1);

1/2

(1.5)

int(sqrt(1 - sqrt(1+z)), z=0..1);

((4/5)*I)*(2^(1/2)-1)^(3/2)*2^(1/2)+((8/15)*I)*(2^(1/2)-1)^(3/2)

(1.6)

int(z/(exp(2*z)+4*exp(z)+10),z = 0 .. infinity);

(1/20)*dilog((I*6^(1/2)-3)/(-2+I*6^(1/2)))-((1/60)*I)*6^(1/2)*dilog((I*6^(1/2)-3)/(-2+I*6^(1/2)))+(1/20)*dilog((I*6^(1/2)+3)/(2+I*6^(1/2)))+((1/60)*I)*6^(1/2)*dilog((I*6^(1/2)+3)/(2+I*6^(1/2)))+((1/120)*I)*6^(1/2)*ln(2+I*6^(1/2))^2-((1/120)*I)*6^(1/2)*ln(2-I*6^(1/2))^2+(1/40)*ln(2+I*6^(1/2))^2+(1/40)*ln(2-I*6^(1/2))^2+(1/60)*Pi^2

(1.7)

simplify(int(sinh(a*abs(x-y)), y=0..c, 'method'='FTOC'));

(1/2)*(piecewise(x < 0, 0, 0 <= x, 2*exp(-a*x))+piecewise(x < 0, 0, 0 <= x, -4)+2*piecewise(c <= x, -cosh(a*(-x+c))/a, x < c, (cosh(a*(-x+c))-2)/a)*a-exp(-a*x)+piecewise(x < 0, 0, 0 <= x, 2*exp(a*x))+4-exp(a*x))/a

(1.8)

int(ln(x+y)/(x^2+y), [x=0..infinity, y=0..infinity]);

infinity

(1.9)


Definite integrals with assumptions on the parameters:

int(x^(-ln(x)),x=0..b) assuming b > 0;

(1/2)*erf(ln(b)-1/2)*Pi^(1/2)*exp(1/4)+(1/2)*Pi^(1/2)*exp(1/4)

(1.10)

int(exp(-z)*exp(-I*n*z)*cos(n*z),z = -infinity .. infinity) assuming n::integer;

undefined

(1.11)


Integral of symbolic integer powers of sin(x) or cos(x):

int(sin(x)^n,x) assuming n::integer;

` piecewise`(0 < n, -(Sum((Product(1+1/(n-2*j), j = 1 .. i))*sin(x)^(n-2*i-1), i = 0 .. ceil((1/2)*n)-1))*cos(x)/n+(Product(1-1/(n-2*j), j = 0 .. ceil((1/2)*n)-1))*x, n < 0, (Sum((Product(1-1/(n+2*j+1), j = 0 .. i))*sin(x)^(n+2*i+1), i = 0 .. -ceil((1/2)*n)-1))*cos(x)/n+(Product(1+1/(n+2*j-1), j = 1 .. -ceil((1/2)*n)))*ln(csc(x)-cot(x)), x)

(1.12)

int(cos(x)^n,x) assuming n::negint;

-(Sum((Product(1-1/(n+2*j+1), j = 0 .. i))*cos(x)^(n+2*i+1), i = 0 .. -ceil((1/2)*n)-1))*sin(x)/n+(Product(1+1/(n+2*j-1), j = 1 .. -ceil((1/2)*n)))*ln(sec(x)+tan(x))

(1.13)

int(cos(x)^n,x) assuming n::posint;

(Sum((Product(1+1/(n-2*j), j = 1 .. i))*cos(x)^(n-2*i-1), i = 0 .. ceil((1/2)*n)-1))*sin(x)/n+(Product(1-1/(n-2*j), j = 0 .. ceil((1/2)*n)-1))*x

(1.14)

Improved answers in Maple 2016

 

int(sqrt(1+sqrt(x)), x);

(4/5)*(1+x^(1/2))^(5/2)-(4/3)*(1+x^(1/2))^(3/2)

(2.1)

int(sqrt(1+sqrt(1+z)), z= 0..1);

-(8/15)*2^(1/2)-(8/15)*(1+2^(1/2))^(3/2)+(4/5)*(1+2^(1/2))^(3/2)*2^(1/2)

(2.2)

int(signum(z^k)*exp(-z^2), z=-infinity..infinity) assuming k::real;

(1/2)*(-1)^k*Pi^(1/2)+(1/2)*Pi^(1/2)

(2.3)

int(2*abs(sin(x*p)*sin(x)), x = 0 .. Pi) assuming p> 1;

-2*(sin(Pi*p)*signum(sin(Pi*p))*cos(Pi/p)-p*sin(Pi/p)*cos(Pi*(floor(p)+1)/p)+sin(Pi*(floor(p)+1)/p)*cos(Pi/p)*p-sin(Pi*p)*signum(sin(Pi*p))-sin(Pi*(floor(p)+1)/p)*p+sin(Pi/p)*p)/((cos(Pi/p)-1)*(p^2-1))

(2.4)

int(1/(x^4-x+1), x = 0 .. infinity);

-(sum(ln(-_R)/(4*_R^3-1), _R = RootOf(_Z^4-_Z+1)))

(2.5)


In Maple 2016, this multiple integral is computed over 3 times faster than it was in Maple 2015.

int(exp(abs(x1-x2))*exp(abs(x1-x3))*exp(abs(x3-x4))*exp(abs(x4-x2)), [x1=0..R, x2=0..R, x3=0..R, x4=0..R], AllSolutions) assuming R>0;

(1/8)*exp(4*R)-29/8+(7/2)*exp(2*R)-5*R*exp(2*R)+2*exp(2*R)*R^2-(5/2)*R

(2.6)

Austin Roche
Mathematical Software, Maplesoft

I have lots of subsections for organization and have all the outputs at the end...  is there any way to force all/any sections/subsections to remain closed while the whole page is being evaluated? I still want the section to evaluate, but I want it to stay closed.

Every time I try to type in a procedure I get the error:  

Error, unterminated procedure

immediately after typing in the first line.  How can I type in the remaining lines of my procedure?

Note:  in maple 7 (years ago) I never had this problem.

How can i find the critical points of any cubic polynomial in maple17? I want a general procedure

Hi !

I have trouble to do this stuff :

i am solving an ODE and i would like to use the result as a function.

 

example :

>>ode := diff(f(x), x) = 2*x+6;
                        d                
                       --- f(x) = 2 x + 6
                        dx               
>>init := f(0) = 12;
                           f(0) = 12
>>dsolve({init, ode});    
                      f(x) = x  + 6 x + 12

Here everything works fine...

but now i want to define g(x) = f(x)*exp(x) ...

but i can't use g(x) after :

like :

>> g := x -> f(x)*exp(x) ;
                          x -> f(x) exp(x)
g(2);
                          f(2) exp(2)
f(2);
                              f(2)

How can i do that please ??

Thanks,

 

Corentin

 

Hi Maple People

 

# Some Maple code
restart
x:= Vector(10):
y:= Vector(10):

for z from -5 to 4 do
   x[z+6]:=z^2 + 40:
   y[z+6]:=z^2 + z + 41:
end do:

plot(x,y,style=point,symbol=asterisk)

 

Regards

Matt

hi.i am a problem with calculate numeric integral.

please help me

thanks

Float(undefined).mw

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