MaplePrimes Questions

Dear Maple users,

I am progressing, but one last hitch, see below. A want the invariants of the PDE below. However, the final expression is too general to be useful. I would like to insert specific values for F1(R), F2(R), F3(R), and F4(R):

F3=F4=0; and F2=1, and F3(R)=aR, where a is a constant. R is one of my independent variables (the Reynolds number). 

 

I would like to do this at the step where the Infinitiesimals are generated by

infinies := Infinitesimals(PDE)

For example, the first entry would then be, infinies:=[ _xi[y](y, R, l, u) = y, ... ]. Then Invariants should give me much simplified expressions which I need.  How can i do this?

Thanks

Nadeem

 

with(PDEtools)

declare(u(y, R))

` u`(y, R)*`will now be displayed as`*u

(1)

declare(l(y, R))

` l`(y, R)*`will now be displayed as`*l

(2)

L := diff_table(l(y, R))

table( [(  ) = l(y, R) ] )

(3)

NULL``

U := diff_table(u(y, R))

table( [(  ) = u(y, R) ] )

(4)

DepVars := ([l, u])(y, R)

[l(y, R), u(y, R)]

(5)

PDE := U[y, y]+2*l(y, R)^2*U[y]*U[y, y]+2*l(y, R)*L[y]*U[y]^2+1/R = 0

diff(diff(u(y, R), y), y)+2*l(y, R)^2*(diff(u(y, R), y))*(diff(diff(u(y, R), y), y))+2*l(y, R)*(diff(l(y, R), y))*(diff(u(y, R), y))^2+1/R = 0

(6)

infinies := Infinitesimals(PDE)

[_xi[y](y, R, l, u) = _F2(R)*y+_F3(R), _xi[R](y, R, l, u) = _F1(R), _eta[l](y, R, l, u) = (1/2)*l*(-R*_F2(R)+_F1(R))/R, _eta[u](y, R, l, u) = (2*R*_F2(R)-_F1(R))*u/R+_F4(R)]

(7)

InfinitesimalGenerator(infinies, DepVars, prolongation = 1)

proc (f) options operator, arrow; add(xi[x[j]]*(diff(f, x[j])), j = 1 .. 2)+add(eta[u[m]]*(diff(f, u[m]))+eta[u[m], [y]]*(diff(f, u[m][y]))+eta[u[m], [R]]*(diff(f, u[m][R])), m = 1 .. 2) end proc

(8)

Phi := Invariants(infinies, DepVars)

l*exp(-(1/2)*(Int((-R*_F2(R)+_F1(R))/(R*_F1(R)), R))), -(Int(_F3(R)*exp(-(Int(_F2(R)/_F1(R), R)))/_F1(R), R))+y*exp(-(Int(_F2(R)/_F1(R), R))), u*exp(Int((-2*R*_F2(R)+_F1(R))/(R*_F1(R)), R))-(Int(_F4(R)*exp(Int((-2*R*_F2(R)+_F1(R))/(R*_F1(R)), R))/_F1(R), R)), u[y]*exp(Int((-R*_F2(R)+_F1(R))/(R*_F1(R)), R)), l[y]*exp(-(1/2)*(Int((-3*R*_F2(R)+_F1(R))/(R*_F1(R)), R))), (1/2)*Intat(-exp((1/2)*(Int((2*(diff(_F1(_j), _j))*_j+_j*_F2(_j)-_F1(_j))/(_j*_F1(_j)), _j)))*(-2*(diff(_F2(_j), _j))*_j^2*l[y]*(y*exp(-(Int(_F2(R)/_F1(R), R)))-(Int(_F3(R)*exp(-(Int(_F2(R)/_F1(R), R)))/_F1(R), R))+Int(_F3(_j)*exp(-(Int(_F2(_j)/_F1(_j), _j)))/_F1(_j), _j))*exp(Int(_F2(_j)/_F1(_j), _j)+(1/2)*(Int((-3*_j*_F2(_j)+_F1(_j))/(_j*_F1(_j)), _j))-(1/2)*(Int((-3*R*_F2(R)+_F1(R))/(R*_F1(R)), R)))+l*(-(diff(_F2(_j), _j))*_j^2+(diff(_F1(_j), _j))*_j-_F1(_j))*exp((1/2)*(Int((-_j*_F2(_j)+_F1(_j))/(_j*_F1(_j)), _j))-(1/2)*(Int((-R*_F2(R)+_F1(R))/(R*_F1(R)), R)))-2*(diff(_F3(_j), _j))*l[y]*_j^2*exp((1/2)*(Int((-3*_j*_F2(_j)+_F1(_j))/(_j*_F1(_j)), _j))-(1/2)*(Int((-3*R*_F2(R)+_F1(R))/(R*_F1(R)), R))))/(_j^2*_F1(_j)), _j = R)+l[R]*exp((1/2)*(Int((2*(diff(_F1(R), R))*R+R*_F2(R)-_F1(R))/(R*_F1(R)), R))), u[R]*exp(Int(((diff(_F1(R), R))*R-2*R*_F2(R)+_F1(R))/(R*_F1(R)), R))+Intat(-exp(Int(((diff(_F1(_k), _k))*_k-2*_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k)-(Int((-_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k))-(Int((-2*_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k)))*(-(diff(_F2(_k), _k))*_k^2*u[y]*(y*exp(-(Int(_F2(R)/_F1(R), R)))-(Int(_F3(R)*exp(-(Int(_F2(R)/_F1(R), R)))/_F1(R), R))+Int(_F3(_k)*exp(-(Int(_F2(_k)/_F1(_k), _k)))/_F1(_k), _k))*exp(Int(_F2(_k)/_F1(_k), _k)+Int((-2*_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k)+Int((-R*_F2(R)+_F1(R))/(R*_F1(R)), R))+(diff(_F4(_k), _k))*exp(Int((-_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k)+Int((-2*_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k))*_k^2-_k^2*u[y]*(diff(_F3(_k), _k))*exp(Int((-2*_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k)+Int((-R*_F2(R)+_F1(R))/(R*_F1(R)), R))+exp(Int((-_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k))*(u*exp(Int((-2*R*_F2(R)+_F1(R))/(R*_F1(R)), R))-(Int(_F4(R)*exp(Int((-2*R*_F2(R)+_F1(R))/(R*_F1(R)), R))/_F1(R), R))+Int(_F4(_k)*exp(-(Int((2*_k*_F2(_k)-_F1(_k))/(_k*_F1(_k)), _k)))/_F1(_k), _k))*(2*(diff(_F2(_k), _k))*_k^2-(diff(_F1(_k), _k))*_k+_F1(_k)))/(_k^2*_F1(_k)), _k = R)

In short, I want to create a density plot but with points. So I do a pointplot and I want to color it according to the value of a third function. 

So let's say I do:

>x1:=[seq(x,x=1..2,0.02)];

>y1:=[seq(y^3,y=1..2,0.02)];

>z1:=[seq(x1[i]/y1[i],i=1..51)];

And I want a plot like:

>pointplot([x1,y1])

but with a color corresponding to the gradient of z1.

What I can do is something like:

>pointplot([x1,y1], colorscheme=["valuesplit", z1, [0..0.5="Black",0.5..1="RoyalBlue"]])

but that's not as pretty as it could be. So is there a way to do something like colorscheme=["zgradient", z1]? This doesn't work because zgradient expects its first argument to be colors and then you can add markers, but they apply to the function you plot, not to a third function.

Any ideas?

colorscheme=["zgradient",["Orange","Red","NavyBlue"]]

I'm working towards creating a way to visualise real polynomial ideals! (or at least the solutions of the polynomials in the ideals) this code creates a plot showing the solutions to all the polynomials in the ideal generated by P1 and P2 (these are specified in the code)

with(plots);
P1 := x^2+2*y^2-3;
solve(P1, y);
Plot1 := plot([%], x = -2 .. 2);

P2 := -2*x^2+2*x*y+3*y^2+x-4;
solve(%, y);
Plot2 := plot([%], x = -4 .. 2);

P2*a+P1;
solve(%, y);
seq(plot([%], x = -4 .. 2), a = 0 .. 10, .1);
display(%, Plot1, Plot2)




This is because when you multiply two polynomials their set of solution curves is just the union of the sets of curves associated with the previous polynomials.

For the next step I'd like to create a graph of the solutions associated with an ideal with three generators. To stop this from being excessively messy I'd like to do it with the RGB value of the colour of a curve is determined by  a and b where the formula for a generic polynomial that we are solving and graphing is given by:

P1+a*P2+b*P3;

where P3 is given by

P3 := x*y-3

I've tried various ways to use cury to make this work (my intuition is cury is the right function to use here)  but got no where. Any ideas how to procede?

I wanted to remove entry from a list that contain y=y or x=x in it. Here is an example

f:= (x-1)*y^4/(x^2*(2*y^2-1));
S:=[singular(f)]

Where I wanted to remove those entries highlighted above to obtain

This is below how I ended up doing it. I'd like to ask if there is a better or more elegent way. I had to use map, since could not get remove() to work on the original list in one shot. 

foo:= z->remove(has,z,{y = y,x = x});
map(foo,[singular(f)])

Which gives the output above.

Is there a better way to do this? I always learn when I find how to do something better.

Maple 2019.1

 

 

Why does the iscont( ) function declare that the square root function is continous over Riscont_error.mw
 

iscont(sqrt(x), x = -infinity .. infinity)

true

(1)

``


 

Download iscont_error.mw

 

Dear Users!

Hoped you will be fine. I want to define a block matrix A for any value of M like this way

A = [A[0], A[1], A[2], ..., A[M]]

A[i] = [A[i,0], A[i,1], A[i,2], ..., A[i,M]]

and

A[i,j] = Transpose([a[i,j,0], a[i,j,1], a[i,j,2], ..., a[i,j,M]]);

Kindly help me in this matter. Thanks in advance.

A research paper published in 1929 claimed that this integral was solved in 1896.

int(exp(I*m*omega + I*b*cos(omega) ),omega=0..2*Pi)           (m integer,  b positive constant)

but it defeats Maple (and other).  Can anybody suggest a way to solve this integral with Maple?

If f is a procedure e.g.  f := x -> x^2;  then f[a](b)  evaluates to f(b)  (= b^2).
Do you know where is this feature documented?
(I have used this in the past, but I need now a reference and I cannot find it).

 

Hello. There is a complex function U11[n]. Let's say I take its derivative and want to extract the real part. That is all right. Now let's say I want to get the derivative at a point and extract the real part. I can't. Are there any solutions?


HELP.mw

 

is the reason for Image result?


 

R+r__i*cos(`θ__j`)

R+(1.-0.750482401378084e-231*I)*r__i

(1)

``


 

Download cos

 

I would like to change the color and font type on the Maple-2019 inputs and outputs, so that these formatting would be applied every time I started a new document. I tried doing this through the Styles option on the Format menu, but I could not. Can anyone help? Thank you.

Which sorting related with famous sequence

for example 

sorting differential equation in a list

then access the list with famous sequence as index such as using https://oeis.org/

after access with sequence as index, use choose function to get combinations then most result are isomorphism differential ideals?

is there methods about this sorting in Richard Stanley Combinatiric book? which page of it?

Are there commands in Maple to find the order and degree of an ODE?  Searching help I could not find anything so far.

For an example, given 

restart;
ode:=(1+diff(y(x),x)^2)^(3/2)=diff(y(x),x$2)

I want the command to return 2 for the order of the ODE and degree is also 2 in this case.

I looked at DEtools package and googled. I am sure Maple have build in commands to do this without me having to parse the ODE myself to find out.

 

 

Folks,

I am new to maple and looking to solve for all value of a list and return alist as the result 

fI := [12.5, 16, 20, 25, 31.5, 40, 50, 63, 80, 100, 125, 160, 200, 250, 315, 400, 500, 630, 800, 1000, 1250, 1600, 2000, 2500, 3150, 4000, 5000, 6300, 8000, 10000, 12500, 16000, 20000];

x:= 23

y:= 36

Lp:= 8-10*log[10]((1+(f1/(2*x))^2.5)*(1+(y/(2.*f1))^1.7))

I am looking to solve Lp to give a result in the form: [a,b,c,d....] i.e solve for  all values of f1 and return a list.

Many Thanks

Last month I still can read file

by 

read “c://Users//hello//Documents//h.m”

but

now it return error

no read access c://Users//hello//Documents/

and 

in security I add the m file into readable 

I saw open file at c drive has many shell folders 

i just add m file

but still the same error

i unencrypted m file by window properties

still the same error

i save file into maple roaming directory under 12 folder , still the same error

i save into maple installation directory maple 12 , still the same error

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