## Fractional Reduced Differential Transform Method...

Can anyone help me to frame the equations in Fractional Reduced Differential Transform Method

system of nonlinear ordinary diﬀerential equations
ds/ dt = b−γ s(t)− (δ s(t)(i(t) + βa(t)) /N − ε s(t) m(t)
de/ dt = δ (s(t)(i(t) + βa(t))/ N + ε s(t) m(t) − (1−ϑ) θ e(t) − ϑ α e(t) − γ e(t)
di/ dt = (1−ϑ) θ e(t) − (ρ + γ) i(t)
da/ dt = ϑ α e(t) − (σ + γ) a(t)
dr /dt = ρ i(t) + σ a(t) − γ r(t)
dm /dt = τ i(t) + κ a(t) − ω m(t)

## How can I select 2 variables from 3 to obtain a co...

Good day to all.

I have a function, z(x,y), that I am interested in exploring. The contour plot returns the z-contours on the x-y grid (see attached).

Can anyone suggest a way in which I can now assign z to the vertical axis (instead of y), x to the horizontal axis, and plot the contours for various y-values?

Contour_Plot_MaplePrimes.mw

## Maple coeff ignoring terms with a bar above them (...

I'm trying to extract coefficients from an expression that resulted from a matrix cross product operation. Maple inserts a bar above a number of variables (I'm assuming that this is a vinculum). When I use coeff it doesn't recognise the variables with a bar above them as the same as those without a bar above them.

Sample code attached with the problem manifesting.

## Problem of grid option in plot3d...

Hi MaplePrimes team,

I am struggling to display a correct projection of a sphere on a tangent plane disc of with Maple’s plot3d, but the resulting grid is not correct somewhere despite of the increasing of number of grids. See picture below:

Fig. 1: The sphere projection on a tangent plane shows a grid problem.

How can I do to fix the problem of undesired grid when the mathematical equation is correct?

Fig. 2: Default 49-grid for Lambert projection showing wrong grids.

To increase the number of grids such that grid = [1000, 1000], grid = [2000, 2000] or higher, or the idea of removing just one row grid row or one column grid like [299, 299] changes absolutely nothing. It still results some crossing meshes that hide desired projected map.

Even the use of the option numpoints fails.

 300-grid 1000-grid

Fig. 3: Higher grid for Lambert projection showing wrong grids.

Here, my parametric equations:

- The sphere equation is

S := (θ,φ)-> R * er(θ,φ)

Where,

The unit radial vector (US coordinates system) is:

er := (θ,φ)-> cos(θ) *cos(φ) * ex + cos(θ) *sin(φ)  * ey + sin(θ) * ez:

ex := <1,0,0>:          ey := <0,1,0>:           ez := <0,0,1>:

- θ: latitude or elevation variable;

- φ: longitude or azimuthal variable

Then,

- The projection on plane (Lambert projection) is

P := (θ,φ)-> X0 + R*α(θ,φ)*eρ(θ,φ):

Where

X0 := S(θ0, φ0)   Free tangent point belonging both to the sphere and the plane disc.

α := (θ,φ)-> arccos( cos(φ - φ0)*cos(θ)*cos(θ0) + sin(θ)*sin(θ0) ):

α is the arc angle of radius R between X0 and any point on the sphere.

eρ := (θ,φ)-> eN(θ,φ) &x er0:            Crossproduct

eρ is the polar direction from fixed point X0 to any point X = S(θ,φ)  on the sphere

eN := (θ,φ)-> if          evalf(abs(φ - φ0) - π) = 0 then

eθ0   #Arbitrary elevation unit vector, but unimportant now.

else      UNITVECTOR(er0 &x er(θ,φ))

fi:

Where,

- The unit tangential elevation vector

eθ0 :=-sin(θ0)*cos(φ0) * ex - sin(θ0)*sin(φ0) * ey + cos(θ0) * ez:

- The unit radial vector at X0

er0 := er0, φ0

Note that implicitplot3d-based method is not really preferred here because it is very high cost in grids unlike the parametric equation method which responds in few seconds at high grid number ~2000 with modern computer.

My Goal:

My final objective is to plot a world map projection of the Earth called Lambert Azimuthal Equal-Area (abbreviated LAEA) projection following any geographic coordinates involving the latitude = φ and the longitude = λ, all input angle can be in degree angle, but internally compiled in radian.

For that, I insert an initial equirectangular projection of world map in the Maple’s plot3d option.

Fig. 4: Initial equirectangular projection of world map (low 500-pixel image for test and large 1000-pixel or higher resolution for definitive image).

The astute now is to arrange the ranges such that the surface equation is defined on

ϕ = -π/2...π/2     and    λ = 0..2π

And taking account the image scanning (Width x Height) have to be swept however in the range:

λ = 0..2π    then    ϕ = 0..π,

Then, I get this instruction without to detail all options:

plot3d(S(ϕ - Pi/2, g(λ - Pi)), λ = 0..2*Pi, ϕ = 0..Pi, image = “C:\....\World_map.jpg”, …)

where g is a longitudinal λ-periodic function on 0..2π in the objective to keep the continuity of the map at any longitude.

Here, the first result of LAEA projection at φ = 0 and λ = 0

(a)

 (b) (c)

Fig. 5: LAEA projection near latitude = 0 and longitude = 0 shows negligible grid failure at circular edge.

So now, the problem begins worse when input begins far from (0°,0°), for instance (60°, -90°).

In fact, the overall resulting map seems to be correct, but it is unfortunately hidden by an unnecessary secondary grid layer trending an undesirable shortcut grid lines explained at the beginning of this report.

Fig. 6: Wrong grids seem to be the source of mask problem for LAEA projection at any φ, λ.

Best.

Guy.

Here the full PDF report:

Problem_of_grid_option_in_plot3d_-_by_Guy_April_27th_2020.pdf

## 2D Polar Coordinate Plot...

How can I  get the "phase portrait including vector fields" and several trajectories of the following dynamical system in the polar coordinates (Note that the system is also in polar coordinate)??

## How can we remove c1 from the expression of c2 and...

(-(2*I)*c1*eta+c1*sqrt(-e0^2-4*eta^2))/e0 = c2, ((2*I)*c2*eta+c2*sqrt(-e0^2-4*eta^2))/e0 = -c1

How can we remove c1 from the expression of c2 and similarly c2 from c1?

## SolidOfRevolution ...

Hi

I try to visualize and animate the hollow volume between two functions, but I do not have an optimal rendering

Ideas? Thanks

## Why are the Gauss-Bonnet figures incorrect in my w...

The worksheet below displays dual cones inside the unit sphere, but the Gauss-Bonnet formula (area of upper cone's spherical cap plus length of dual cone's intersection with unit sphere) does not equal 2*Pi.

Dual_cones.mw

## How do I turn a system of parametric equations int...

In Maple 2020, I clicked on "Math Apps" and then clicked on "Algebra & Geometry", followed by "Boy's Surface".

It is a set of 3 parametric equations in terms of {u,v}. I would like to eliminate u and v and turn this into a single "error" function in terms of (x,y,z) such that anywhere this function is zero, the original set of equations are satisfied for some value of {u,v}. (Ideally, the values of u and v could be determined as well, but that would be a bonus.)

I tried this:

alpha = 1;
u in [-Pi/2, Pi/2];
v in [0, Pi];
eq1 := x(u, v) = (sqrt(2)*cos(v)^2*cos(2*u) + cos(u)*sin(2*v))/(2 - alpha*sqrt(2)*sin(3*u)*sin(2*v));
eq2 := y(u, v) = (sqrt(2)*cos(v)^2*sin(2*u) - sin(u)*sin(2*v))/(2 - alpha*sqrt(2)*sin(3*u)*sin(2*v));
eq3 := z(u, v) = 3*cos(v)^2/(2 - alpha*sqrt(2)*sin(3*u)*sin(2*v));
eliminate({eq1, eq2, eq3}, {u, v});

but got no response and no error message.

Additionally, I tried explaining that I want a function f such that f(x,y,z)=0, but again got no response and no error message:

eq4 := f(x, y, z) = 0;
solve({eq1, eq2, eq3, eq4}, f);

Any ideas what I'm doing wrong?

Thank you!

-- Glenn

## Finding the intersection point...

Hellow everyone,

Consider the following diffrential equation:

(dA(x)/dx )^2+V(A)=0,
1- How to plot dA/dx vs A?

2- How to find the coordinate of intersection point of f(x)=e-b*exp(x)-c*(1-2*x/c)^(1/2) with df/dx? e,b,c are constants.

## Can anyone tell me what is causing this error and ...

Hi everyone! I was writing this code for an interactive component, when I stumbled into a problem that I couldn't solve...

Here is the code.

use DocumentTools in
if type(Do(%NIniziale), posint) then
if Do(%NIterazioni)<=100 then
if type(Do(%NIterazioni), posint) then
Do(%E="Valid Values!");
Even:=Do(%Pari);
Odd:=Do(%Dispari);
Iterazioni:=proc (N, i)
local m, n;
n:=N;
for m from 1 to i do
if type(n, even) = true then
n := Even;
else
n := Odd;
end if;
end do;
return (SetCellFormula("Risultati", i, 1, eval(n)));
end proc;
ix:=Do(eval(%NIterazioni));
Nx:=Do(eval(%NIniziale));
seq(Iterazioni(Nx,y),y=1..eval(ix));
if Do(%NIterazioni)<100 then
seq(SetCellFormula("Risultati", x, 1, "//"), x=1+ix..100);
end if;
else
Do(%E="Inavalid Values");
seq(SetCellFormula("Risultati", x, 1, "ERROR"), x=1..100);
end if;
else
Do(%E="Invalid Values");
seq(SetCellFormula("Risultati", x, 1, "ERROR"), x=1..100);
end if;
else
Do(%E="Invalid Values");
seq(SetCellFormula("Risultati", x, 1, "ERROR"), x=1..100);
end if;
end use;

Now this are what the names stands for:

%Pari is a math conainer in which there is a function of n that is to be used when n is even;

%Dispari is a math conainer in which there is a function of n that is to be used when n is odd;

"Risultati" is the name of a spreadsheet on which the results will be displayed;

%NIterazioni is a math conainer in which there is a posint that indicates how many times the procedure is to be iterated;

%NIniziale is a math conainer in which there is the starting number (another posint) to which on which the procedure will operate;

%E is a text area that will signal the validity of the input.

What I wanted to do was a procedure that reiterated two function on a given number a given amount of times and returned the result of every passage, however, when computing, it returns the function in n and not the final numbers.

I have opened the Spread package in the startup-code, and I really want to use the spreadsheet even though it is deprecated in newer versions. When you'll open, the function will be the ones of the Collatz Conjecture, however I want it to be able to iterate any function of n, which will be acquired via input, and will be to the user's choice.

I tried everything that came to my mind: using text area, assigning n outside of the preocedure, using evalf, eval, putting the return inside the if system, opening and closing a do inside "then" and one isndide "else" , changing names and everything... But I couldn't solve it. on the internet and on the MapleHelp I didn't find anything.

So can anyone tell me ( most importantly) why it doens't compute the formulas and just leaves them as input and (less important then the why, but still important and would appreciate) how can it be solved? Thanks to everyone even if you just read this, I know it is amybe a little too much.

## How to search and remove elements in a list of var...

Hello

Unfortunately I run out of ideas on how to solve the problem of searching and removing elements in a variable size list.  Below you can find an example.

```parms:=[seq(seq(alpha[i,j],j=0..9),i=1..3)];
abc:=combinat:-choose(parms,3):```

the condition to remove the element of the list is given by the following procedure.

```conds:= proc(varCoef::list,vars::name:=alpha)
local T1,T2,res;
#
T1 := table([2=3,3=2,5=6,6=5,7=9,9=7]):
T2 := table([2=3,3=2]):
#
res:=map(u->vars[`if`(assigned(T2[op(1,u)]),T2[op(1,u)],op(1,u)),
`if`(assigned(T1[op(2,u)]),T1[op(2,u)],op(2,u))],
varCoef);
res:=convert(convert(res,set),list):
end proc:```

Something like

`member(conds(abc[1]),abc,'k')`

if it is a member, the element indexed by k should be removed from abc and then the search continues using the next element in abc (that now has one less element).   Otherwise, the element is kept and the search moves to next one.  The process ends when the last element in a short version of abc is tested.

Thanks all for the help and patience.

Ed

## numerical solutions of PDE-integral equation...

Dear experts

I am interested to solve the following equation numerically by Maple. I would appreciate it if you let me how I can do and what the boundary conditions and initial values are needed

eq:= diff(2*diff(eta(x,y,t),t)+3*eta(x,y,t)*diff(eta(x,y,t),x)+(1/3-1/epsilon/B)*diff(eta(x,y,t),x,x,x),x)+diff(eta(x,y,t),y,y)-1/sqrt(Pi*R)*int(diff(eta(x+zeta,y,t),x,x)/sqrt(zeta),zeta=0..t/epsilon)=0;
where

1) epsilon, B and r are constant

2) 1/epsilon/B is not equal to 1/3 at all

## shading, style and lightmodel oddities...

choosing lightmodel=none and shading=none, produces a dark grey grided surface
plot3d(x*y, x = 0 .. 10, y = 0 .. 10, lightmodel = none, shading = none)

adding the style=wireframe option gives a blank plot.  Grid probably white?  Changing style to patchnogrid the surface is indeed white. However chosing both shading and style options to none regardless of the lightmodel will produce a plot that appears empty.  Is this to be expected?

However, just the style=wireframe option produces a colored grided wireframe as to expect
plot3d(x*y, x = 0 .. 10, y = 0 .. 10, lightmodel = none, style = wireframe)