Maple Questions and Posts

Set Embedded Plot Component to Logarithmic...

Asked by: Lenny 30

October 07 2020

1 5

Hello Everyone,

So I am currently plotting some data within an embedded plot component using the following code:

SetProperty("DeterminantenRaumPlot1", value, dataplot(DetAnalysisNx2, DetAnalysisDeterm2));

Which is working just fine, DetAnalysisNx2 and DetAnalysisDeterm2 are single rows vectors with numbers in it.

However I need the plot to be in logarithmic mode for the vertical axis, but I do not know how to achieve this. I can set it manually, but when I save and reopen the document and re-do the calculation it turns back to linear vertical axis.

What kind of code segment am I missing? I have tried adding this like 'mode = log' but it is not working with dataplot. Maybe I need a different kind of plot? But then my Vectors will not work anymore, dataplot makes it so easy to use and the setting is available manually.

Any help would be fantastic!
Thanks and best regards,
Lennart

RE: Trogonomatric identity...

Asked by: pik1432 245

October 07 2020

1 2

Hello all,

Would you allow me to ask this question?

Is there a way to make the expression 'expression1', given below, to expression 'expression3'?

>

restart;

>	expression1 := -cos(theta)/2 + sin(theta)*sqrt(3)/2;

(1)

>	expression2 := expand(cos(theta-2/3*Pi));

(2)

>	is(expression1 = expression2);

(3)

>	expression3 := cos(theta-2/3*Pi);

(4)

>

Thank you,

Download Q20201007.mw

How do I fill the area between three equations...

Asked by: Thomas Dean 312

October 07 2020

0 3

I have four equations,

y = 1/x, y=3, y=1, x=0

restart;

P1 := plots:-implicitplot(y=1/x, x=-1..2, y=0..4, color=black, scaling=constrained):
P2 := plots:-implicitplot(y=3, x=-1..2, y=0..4, color=black):
P3 := plots:-implicitplot(y=1, x=-1..2, y=0..4, color=black):

plots:-display([P1,P2,P3]);

How do I fill the area between the first three equations and the y axis (x=0)?

Is it possible to do this with plottools:-transform?

How to obtain multiple solution...

Asked by: Madhukesh J K 140

October 07 2020

1 1

How to obtain the multiple solution and graph given in the paper.

Stefan Blowing and Slip Effects on Unsteady Nanofluid Transport Past a Shrinking Sheet: Multiple Solutions

https://doi.org/10.1002/htj.21470

Can anyone help to get solutions.

Maple helps in simulating multiscale-multidomain...

by: Dr. Venkat Subramanian 386 Product: Maple

October 06 2020

1 0

We recently published a paper on multiscale-multidomain simulation of battery models.

https://iopscience.iop.org/article/10.1149/1945-7111/abb37b

Some challenges are listed at

https://twitter.com/UT_MAPLE/status/1311356957941522438

Maple and symbolic math can play a critical role in solving many challenging problems. For example, consider a seemingly-trivial problem

uxx+uyy = 0

x = 0 and x = 1, ux = 0 for all y

y = 1, u = 0, for all x

y = 0, u =1, 0<=x<=0.5

y = 0, uy = 0, 0.5<x<=1

There is a singularity at (0.5,0) and most numerical methods will have trouble there. In these equations, uxx means the second derivative of u with respect to x.

Maple can help solve this problem with conformal mapping to achieve arbitrary precision. As of today, machine precision is not possible with any numerical method even with millions of Degrees of Freedom. The Maple code is given below. A FEM code is given below as well. Models like this can benefit from Maple adding parallel sparse direct and iterative solvers.

>

restart;

>	Digits:=15;

(1)

The domain is tranformed from Z = X+IY to w. The points tranformed are

>	Zdomain:=[[0,0],[1,0],[1,1],[0,1]];

(2)

>	wdomain:=[0,1,1+a,a+2];

(3)

>	eq1:=diff(Z(w),w)=-I*K1/sqrt(w)/sqrt(w-1)/sqrt(w-a-1)/sqrt(w-a-2);

(4)

a is not known apriori. The value of a should be found to make sure [1,1] in the Z coordinate is transformed to [1+a] in the w coordinate.

>	a:=sqrt(2)-1;

(5)

Value of K1 is found using the transformation of [1,0] to 1 in the w coordinate

>	eq11:=1=int(rhs(eq1),w=0..1);

(6)

>	K1:=solve(eq11,K1);

K1 := -(1/2)*(2^(1/2)/EllipticK(2^(1/2)/2))

(7)

The height Y in the Z coordinate is found by integrating from 1 to 1+a in the w coordinate.

>	simplify(int(rhs(eq1),w=1..1+a));

(2*I)*2^(1/2)*EllipticK((2^(1/2)-1)/(2^(1/2)+1))/(EllipticK(2^(1/2)/2)*(2^(1/2)+1))

(8)

>

evalf(%);

(9)

The choice of a = sqrt(2)-1 gives the height of 1 for Z coordinate.

>	eval(simplify(int(rhs(eq1),w=0..1+a)));

(2^(1/2)*EllipticK(2^(1/2)/2)+EllipticK(2^(1/2)/2)+(2*I)*2^(1/2)*EllipticK((2^(1/2)-1)/(2^(1/2)+1)))/(EllipticK(2^(1/2)/2)*(2^(1/2)+1))

(10)

>

evalf(%);

(11)

integration from 0 to 1+a in the w coordinate gives 1,1 in the Z coordinate.

>	simplify(int(rhs(eq1),w=0..1/sqrt(2)));

EllipticF(2^(1/2)/(2+2^(1/2))^(1/2), 2^(1/2)/2)/EllipticK(2^(1/2)/2)

(12)

>

evalf(%);

(13)

Integrating w from 0 to wmid =1/sqrt(2) gives the point 0.5,0 in the Z coordinate

>	wmid:=1/sqrt(2);

(14)

Next w domain is transformed to Z2 domain Z2 = X2+IY2

>	Z2domain:=[[0,0],[1,0],[1,H],[0,H]];

(15)

>	wdomain2:=[0,1/sqrt(2),1+a,a+2];

(16)

>	eq2:=diff(Z2(w),w)=-I*K2/sqrt(w)/sqrt(w-wmid)/sqrt(w-1-a)/sqrt(w-a-2);

eq2 := diff(Z2(w), w) = -(2*I)*K2/(w^(1/2)*(4*w-2*2^(1/2))^(1/2)*(w-2^(1/2))^(1/2)*(w-2^(1/2)-1)^(1/2))

(17)

K2 is found based on the transformation of wmid to [1,0] in Z2 coordinate.

>	eq21:=1=int(rhs(eq2),w=0..wmid);

eq21 := 1 = -2*K2*EllipticK(1/(((2+2^(1/2))^(1/2))))/(2^(1/2)+1)^(1/2)

(18)

>	K2:=solve(eq21,K2);

K2 := -(1/2)*((2^(1/2)+1)^(1/2)/EllipticK(1/(((2+2^(1/2))^(1/2)))))

(19)

The corner 0,1 in the Z coordinate is mapped by integrating eq2 from 0 to 1 in the w coordinate

>	int(rhs(eq2),w=0..1);

1+EllipticF((2-2^(1/2))^(1/2), ((2^(1/2)+1)/(2+2^(1/2)))^(1/2))*I/EllipticK(1/(((2+2^(1/2))^(1/2))))

(20)

>	corner:=evalf(%);

(21)

This is the point 1,0 in the original coordinate.

The height in the Z2 coorinate is found by integrating eq2 from wmid to 1+a.

>	ytot:=int(rhs(eq2),w=wmid..1+a);

ytot := EllipticK(((2^(1/2)+1)/(2+2^(1/2)))^(1/2))*I/EllipticK(1/(((2+2^(1/2))^(1/2))))

(22)

>

evalf(%);

(23)

The magnitude in the Y direction is given by the coefficient of I, the imaginary number

>	Ytot:=coeff(ytot,I);

Ytot := EllipticK(((2^(1/2)+1)/(2+2^(1/2)))^(1/2))/EllipticK(1/(((2+2^(1/2))^(1/2))))

(24)

The analytial solution in the Z2 corordinate is a line in Y2 to satisfy simple zero flux conditions at X2 = 0 and X2 = 1.

>	phianal:=1+b*Y2;

(25)

The value of phi is zero at Y2 = Ytot (originally the cathode domain in the Z domain)

>	bc:=subs(Y2=Ytot,phianal)=0;

bc := 1+b*EllipticK(((2^(1/2)+1)/(2+2^(1/2)))^(1/2))/EllipticK(1/(((2+2^(1/2))^(1/2)))) = 0

(26)

>	b:=solve(bc,b);

b := -EllipticK(1/(((2+2^(1/2))^(1/2))))/EllipticK(((2^(1/2)+1)/(2+2^(1/2)))^(1/2))

(27)

The analytical solution is given by

>

phianal;

1-EllipticK(1/(((2+2^(1/2))^(1/2))))*Y2/EllipticK(((2^(1/2)+1)/(2+2^(1/2)))^(1/2))

(28)

>	evalf(phianal);

(29)

The potential at the corner is given by substituting the imaginary value of corner for Y2 in phinanal)

>	phicorner:=subs(Y2=Im(corner),phianal);

phicorner := 1-.559419351518322*EllipticK(1/(((2+2^(1/2))^(1/2))))/EllipticK(((2^(1/2)+1)/(2+2^(1/2)))^(1/2))

(30)

>	evalf(phicorner);

(31)

local flux/current density calculation, written in terms of w is

>	curr:=b*rhs(eq2)/rhs(eq1);

curr := -(2^(1/2)+1)^(1/2)*2^(1/2)*EllipticK(2^(1/2)/2)*(w-1)^(1/2)/(EllipticK(((2^(1/2)+1)/(2+2^(1/2)))^(1/2))*(4*w-2*2^(1/2))^(1/2))

(32)

average flux/current density calculation for the anode

>	currave:=int((curr),w=0..wmid)/wmid;

currave := -(1/2)*((2^(1/2)+1)^(1/2)*(2^(1/2)-1)*EllipticK(2^(1/2)/2)*(2^(3/4)+2^(1/4)+arctanh(2^(3/4)/2))*2^(1/2)/EllipticK(2^(3/4)/2))

(33)

>	Digits:=25:

The average current density at Y =0, local current density at X = 0,Y=0 and potential at X=1,Y=0 (Corner) can be used to study convergence of FEM and other numerical methods

>	evalf(currave),evalf(subs(w=0,curr)),evalf(phicorner);

(34)

>

Download Conformalmapping.mws

This FEM code is for solving Laplace's equation with primary current distribution considered in Model 1.
This code is based on FEM weak-form. Biquadratic Lagrange shape functions (9nodes in an element) are used.

>

restart;

>	with(LinearAlgebra):

>	Lx:=1: #length in X

>	Ly:=1: #length in Y

>	nx:=10: #number of elements in X (even numbers only)

>	ny:=10: #number of elements in Y, to be kept same as nx in this version

>	hx:=Lx/nx: #element size x

>	hy:=Ly/ny: #element size y

Procedure to perform numerical integration on shape functions to obtain local matrices (can be replaced with analytical integration for this particular problem)
-Shape functions are also used as weight functions in applying weak formulation. Numerical integration is done using Simpson's rule.
-Local cartesian coordinates x,y are converted to natural coordinates zeta and eta. This transformation is not required for this simple geomerty but useful in general. zeta and eta are obtained by scaling x and y with hx/2 and hy/2, respectively, in this code.

>	localmatrices:=proc(a1,a2,a3,b1,b2,b3,q1,q2) global Kx,Ky,Nx,Ny,zeta,eta,c; local A,dAdzeta,dAdeta,y,x,J,terms,i,j,k,l,dx,dy,fx,fy,fxy,fyy,dzeta,deta,J1,J2;

>	A:=[(1-zeta)zeta(1-eta)eta/4,-(1-(zeta)^2)(1-eta)eta/2,-(1+zeta)zeta(1-eta)eta/4,(1-(eta)^2)(1+zeta)zeta/2,(1+zeta)zeta(1+eta)eta/4,(1-(zeta)^2)(1+eta)eta/2,-(1-zeta)zeta(1+eta)eta/4,-(1-(eta)^2)(1-zeta)(zeta)/2,(1-(zeta)^2)*(1-(eta)^2)]; #bi quadratic langrange shape functions

>	dAdzeta:=diff(A,zeta);

>	dAdeta:=diff(A,eta);

>	y:=[-a1,-a2,-a3,0,a3,a2,a1,0,0];x:=[-b1,0,b1,b2,b3,0,-b3,-b2,0];

>	J:=simplify(add(dAdzeta[i]x[i],i=1..9)add(dAdeta[i]y[i],i=1..9)-add(dAdzeta[i]y[i],i=1..9)add(dAdeta[i]x[i],i=1..9));

>	Nx:=[seq((simplify(add(dAdeta[i]y[i],i=1..9))dAdzeta[j]-simplify(add(dAdzeta[i]y[i],i=1..9))dAdeta[j])/simplify(J),j=1..9)];

>	Ny:=[seq((-dAdzeta[j]simplify(add(dAdeta[i]x[i],i=1..9))+dAdeta[j]simplify(add(dAdzeta[i]x[i],i=1..9)))/simplify(J),j=1..9)];

>	Kx:=Matrix(nops(A),nops(A),datatype=float[8]):

>	Ky:=Matrix(nops(A),nops(A),datatype=float[8]): c:=Vector(nops(A),datatype=float[8]):

>	terms:=20:#number of terms for numerical integration dzeta:=2/terms: deta:=2/terms:

>

for i from 1 to nops(Nx) do #loop to obtain local matrices

for j from 1 to nops(Ny) do
Kx[i,j]:=0;
Ky[i,j]:=0;

for k from 0 to terms do #outer loop double integration, integration in zeta

if k = 0 then fx[k]:= subs(zeta=-1,Nx[i]*Nx[j]*J); fy[k]:= subs(zeta=-1,Ny[i]*Ny[j]*J);
elif k = terms then
fx[k]:= subs(zeta=-1+(k*dzeta),Nx[i]*Nx[j]*J);
fy[k]:= subs(zeta=-1+(k*dzeta),Ny[i]*Ny[j]*J);
elif irem(k,2) = 0 then
fx[k]:= 2*subs(zeta=-1+(k*dzeta),Nx[i]*Nx[j]*J);
fy[k]:=     2*subs(zeta=-1+(k*dzeta),Ny[i]*Ny[j]*J);
else fx[k]:= 4*subs(zeta=-1+(k*dzeta),Nx[i]*Nx[j]*J);
fy[k]:=     4*subs(zeta=-1+(k*dzeta),Ny[i]*Ny[j]*J); end if;

for l from 0 to terms do #inner loop double integration, integration in eta

if l = 0 then fxy[l]:= subs(eta=-1,fx[k]); fyy[l]:= subs(eta=-1,fy[k]);
elif l = terms then fxy[l]:= subs(eta=-1+(l*deta),fx[k]); fyy[l]:= subs(eta=-1+(l*deta),fy[k]);
elif irem(l,2) = 0 then fxy[l]:= 2*subs(eta=-1+(l*deta),fx[k]); fyy[l]:= 2*subs(eta=-1+(l*deta),fy[k]);
else fxy[l]:=4*subs(eta=-1+(l*deta),fx[k]); fyy[l]:=4*subs(eta=-1+(l*deta),fy[k]); end if;
Kx[i,j]:=Kx[i,j]+fxy[l];
Ky[i,j]:=Ky[i,j]+fyy[l];

end do;

end do;
Kx[i,j]:=Kx[i,j]*dzeta*deta/9;
Ky[i,j]:=Ky[i,j]*dzeta*deta/9;
end do;
end do:

>

end proc:

>	n:=nx*ny; #total number of elements

(1)

>	Nx1:=nx*2+1: #number of nodes in x in one row

>	N:=Nx1(2ny+1); # total number of nodes/equations

(2)

>	K:=Matrix(N,N,storage=sparse): # global K matrix

>	C:=Vector(N,storage=sparse): # global c matrix

>	L2G:=Matrix(n,9): #mapping matrix - each row has node numbers for each element

>	l:=1:k:=1: localmatrices(hy/2,hy/2,hy/2,hx/2,hx/2,hx/2,0,0): kx:=copy(Kx):ky:=copy(Ky):c0:=copy(c):

>

for i from 1 to n do #modifying,adding and assembling matrices to get global matrix

if i<=nx/2 then
a1:=copy(kx); a2:=copy(ky); a3:=0; a1[1..3,1..9]:=IdentityMatrix(3,9); a2[1..3,1..9]:=Matrix(3,9,shape=zero); a4:=a1+a2; c:=copy(c0); c[1..3]:=1.0;

elif i=nx/2+1 then a1:=copy(kx); a2:=copy(ky); a3:=0; a1[1,1..9]:=IdentityMatrix(1,9); a2[1,1..9]:=Matrix(1,9,shape=zero); a4:=a1+a2; c:=copy(c0); c[1]:=1.0;

elif i>nx*(ny-1) then a1:=copy(kx); a2:=copy(ky); a3:=0; a1[5..7,5..7]:=IdentityMatrix(3,3);
a1[5..7,1..4]:=ZeroMatrix(3,4);
a1[5..7,8..9]:=ZeroMatrix(3,2); a2[5..7,1..9]:=Matrix(3,9,shape=zero); a4:=a1+a2; c:=copy(c0); c[5..7]:=0;

else a1:=kx; a2:=ky; a3:=0;a4:=a1+a2; c:=c0; end if;

L2G[i,1..9]:=Matrix([l,l+1,l+2,l+2+Nx1,l+2+Nx1*2,l+1+Nx1*2,l+Nx1*2,l+Nx1,l+1+Nx1]):

k:=k+1:

if k>nx then k:=1; l:=l+Nx1+3;

else l:=l+2; end if:

indx2:=L2G[i,1..9]:
indx2:=convert(indx2,list):

C[indx2]:=C[indx2]+c[1..9];
c[1..9];

for i1 from 1 to 9 do
indx1:=L2G[i,i1]:
K[indx1,indx2]:=K[indx1,indx2]+a4[i1,1..9]:
end do:

end do:

>	phi:=LinearSolve(K,C,method=SparseLU): #linear set of equations solved using Sparse LU solver

>	phi_at(1,0):=phi[Nx1];

(3)

>	dNdy:=copy(Ny):

>	dNdy_bottom_left:=subs(eta=-1,zeta=-1,dNdy):

>	current_at(0,0):=add(dNdy_bottom_left[i]*phi[L2G[1,i]],i=1..nops(Ny));

(4)

>	if irem(nx/2,2)=0 then current_at(0,0.25):=add(dNdy_bottom_left[i]phi[L2G[nx/4+1,i]],i=1..nops(Ny)); else dNdy_bottom_center:=subs(eta=-1,zeta=0,dNdy): current_at(0,0.25):=add(dNdy_bottom_center[i]phi[L2G[(nx/2+1)/2,i]],i=1..nops(Ny)); end if;

(5)

>

Download FEM_2D.mws

Combining algebraic expression...

Asked by: sunit 105

October 06 2020

0 4

Hi,

It might be a really silly question, but I am wondering is it possible to simplify expression like this

a^2+b^2+2*a*b+c^2 . Just by looking it we know that we can write it in the form of (a+b)^2+c^2. This is the basic exmaple I come up with. I have very lengthy expressions in maple, which can be factorize like this, but factor command will not work as it will try to factorize entire expression. So I am wondering is it doable in maple or I have to do it manually by collecting terms and check whether they can be factorize or not.

Thanks in Advance.

With Regards

sunit

how to use Jordan form inside a proc?...

Asked by: nm 8552

October 06 2020

2 2

I know this has to do with name scoping issue. But I do not see how to fix it now.

Calling J,Q:=LinearAlgebra:-JordanForm(A,output=['J','Q']); works in global, but not inside a proc.

What is the correct way to use this inside a proc?

Download jordan_issue.mw

how much costs maple 9.5 licencel ?...

Asked by: carrijo 5

October 06 2020

1 2

Hi,

I would like to instal Maple 9.5 in my laptop, once I already have some few programs for his version. I don't know how much cost Maple 9.5, and how do download it.

I live in Brazil.

Thanks in advance.

Carrijo.

jose.carrijo@gmail.com

To eliminate some entries of a list that are not ...

Asked by: Magma 70

October 06 2020

0 2

Let L=[a₁,a₂,...,a_n] be a list of positive real numbers.

My Question:

How to write a procedure to find a minimal interval such as [b,c]

provided that this interval covers entries of L which are close together.

Example:

let L:=[8.1 , 2.03 , 3.5 , 0.05 , 4.1]. Then the output of the procedure is the interval [2.03 , 4.1].

In fact we want to eliminate some entries of L that are not close to other entries of L.

Thanks in advance

Why do the summation index not count as a god matr...

Asked by: Eric Ragnarsson 50

October 06 2020

1 3

Hi

I am on my way to construct a matrix for centripetal and Coriolis forces for a robot arm. However this requeers that I use a sum where the matrix index is the sumations index.

So I started testing things out before I constructed the full expresion, this is what I did.

According to the error page, this is du to that k in M[1,k] is an bad index, however in the sum this would be k=1 och k=2 whitch are vallid.

Do anybody know a way around this problem or can tell me what I did wrong.

Many thanks

Eric Ragnarsson

How to add a legend (color bar with number) on the...

Asked by: winfredgg 25

October 06 2020

4 3

plots:-contourplot(16.70196911*2^(1/2)*((x^2 + 0.1*y)/((1 - x)*(3*x^2 + 0.2*y)))^(1/2)/(4.373839156*(x^2 + 0.1*y)/((1 - x)*(3*x^2 + 0.2*y)) + 1)^(1/2), x = 0.001 .. 1, y = 0.001 .. 1, contours = 20, thickness = 0, coloring = ["blue", "yellow"], axes = "boxed", filledregions = true)

I have tried many times, such as adding legendstyle = [position=correct],legend = true command, but it always report errors.

Can anyone help me solve this problem? Thank you.

find Databases for the NASA Seven-Coefficient Poly...

Asked by: 9009134 110

October 06 2020

0 7

Hi.

in the ThermophysicalData[Chemicals] package that compute the coefficients for different species how I can find that coefficients for seven coefficients not nine of them

in other words, I am seeking to find Databases for the NASA Seven-Coefficient Polynomial Fits for Calculating Thermodynamic Properties of Individual Species.

Best

Disappointing results regarding complex expr simpl...

Asked by: zphaze 20

October 06 2020

0 5

Hi,

I'm trying to make Maple reduce an extensively simple complex equation, something so simple anybody with basic knowledge of complex numbers would take about 2 mins to solve manually.

Yet, I can't seem to get Maple to do it, which is disappointing. I can only hope the issue is on my side (I would be glad to be the one in the wrong).

example.mw

I expect at the last line a reduction to s^2 + 2*z*omega__n * s + omega__n^2

I tried different combinations of simplification functions without success. Simplify doesn't work much better.

Basically looking at something like this, but on Maple (this comes from Maxima), if possible *WITHOUT* the need for assume() on any scale (just like Maxima and Mathematica) :

Thanks!

why the Elzaki transform is not working for deriva...

Asked by: saher 35

October 06 2020

2 4

I made a customised procedure of Elzaki transform which is working for functions but it is not for derivatives.Elziki_costume.mw

does Maple have option for fraction free eigenvect...

Asked by: nm 8552

October 05 2020

0 5

I remember seeing sometime ago an option called something like "fraction free" in LinearAlgebra. But may be I was looking at something else or different package. I can't remember now. I searched the help pages now and googled and can't find it.

In Maple, when asking for eigenvectors of matrix, I'd like the vectors to come out fraction free, like with Mathematica.

It is ofcourse easy to write code to post process this and remove the fractions.

But before I do this, I thought to ask. Here is an example

restart;
A:=Matrix([[48,-30,-14,1],[65,-41,-19,0],[17,-10,-5,3],[-35,22,10,0]]);
(e,v):=LinearAlgebra:-Eigenvectors(A);

In Mathematica:

Anyone knows if such option exists somewhere?

Maple 2020.1

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