Rolling without slipping on a non-orientable surfa...

by: Maple 15

The Möbius strip  Mobius_strip_rolling.mw

Variants :

The line and the curve on the surface.

How do I solve a system with more equations than v...

Hi everyone.

I'm using Maple 15 to try to give random values to a matrix with some columns fixed, the problem is that it has to be unitary, so I managed to create a system of equations (with 36 linear equations and 24 variables) which should give me the information i require.

I have been trying with fsolve, etc. but i haven't been able to solve it yet, as I usually get the error: "Error, (in fsolve) number of equations, 36, does not match number of variables, 24"

Is there any way to find out which equations are neccesary and get rid of the rest of them? Or can Maple give ramdon values to some of the variables and calculate the others?

Convert Peano curve into a function...

Hi!

I have seen th following procedure to compute the image of the points of [0,1] under the so called Peano space-filling curve (sorry, I have to pasted the code in "text plane" mode):

P[0] := (x, y) -> ((1/3)*y, (1/3)*x);

P[1] := (x, y) -> (-(1/3)*x+1/3, (1/3)*y+1/3);

P[2] := (x, y) -> ((1/3)*x, (1/3)*y+2/3);

P[3] := (x, y) -> ((1/3)*x+1/3, -(1/3)*y+1);

P[4] := (x, y) -> (2/3-(1/3)*y, 2/3-(1/3)*x);

P[5] := (x, y) -> ( (1/3)*x+1/3, 1/3-(1/3)*y));

P[6] := (x, y) -> (1/3)*x+2/3, (1/3)*y);

P[7] := (x, y) -> (-(1/3)*x+1, (1/3)*y+1/3);

P[8] := (x, y) -> ((1/3)*x+2/3, (1/3)*y+2/3);

peano := proc (t::numeric, depth::integer)

local q, r; global P;

if depth = 0 then return 0, 0 end if;

q := floor(9*t); r := 9*t-q;

return P[q](peano(r, depth-1))

end proc;

Now, I need to use the procedure "peanofun" as a function. For instance, if we define f:=(x,y)->x+y, I need to use (plot, compute, etc) for instance, the function f(peanofun(t,5))

Can you help me with this issue, please?

run code for plot the region...

can anyone run my code?
please do it and send it for me.
thanks.

restart;
with(plottools): with(LinearAlgebra): with(plots):
ode := `assuming`([diff(Y(X), `\$`(X, 2))+2*alpha*(diff(Y(X), X))+beta^2*Y(X) = 0], [alpha >= 0, beta >= 0, alpha+beta > 0]):
F := unapply(solve(subs({X = x, Y(X) = y, diff(Y(X), X) = yp, diff(Y(X), `\$`(X, 2)) = yz}, ode), yz), x, y, yp):
Fp := unapply(solve(subs({X = x, Y(X) = y, diff(Y(X), X) = yp, diff(Y(X), `\$`(X, 2)) = yz, diff(Y(X), `\$`(X, 3)) = yt}, diff(ode, X)), yt), x, y, yp, yz):
Ni := seq(i, i = 0 .. 9), 15, 20:
for ni in Ni do
print(ni);
st := time[real]();
f[0, ni] := F(x[0], y[0, ni], yp[0, ni]);
fp[0, ni] := Fp(x[0], y[0, ni], yp[0, ni], f[0, ni]);
f[1, ni] := F(x[1], y[1, ni], yp[1, ni]);
fp[1, ni] := Fp(x[1], y[1, ni], yp[1, ni], f[1, ni]);
y[2, 0] := y[0, ni]+2*h*yp[0, ni]+(6/5)*f[0, ni]*h^2+(4/15)*fp[0, ni]*h^3+(4/5)*f[1, ni]*h^2+(4/15)*fp[1, ni]*h^3;
yp[2, 0] := yp[0, ni]+2*f[0, ni]*h+(2/3)*fp[0, ni]*h^2+(4/3)*fp[1, ni]*h^2;
for j to ni do
f[2, j-1] := F(x[2], y[2, j-1], yp[2, j-1]);
fp[2, j-1] := Fp(x[2], y[2, j-1], yp[2, j-1], f[2, j-1]);
y[2, j] := y[1, ni]+h*yp[1, ni]+(7/20)*f[1, ni]*h^2+(1/20)*fp[1, ni]*h^3+(3/20)*f[2, j-1]*h^2-(1/30)*fp[2, j-1]*h^3;
yp[2, j] := yp[1, ni]+(1/2)*f[1, ni]*h+(1/12)*fp[1, ni]*h^2+(1/2)*f[2, j-1]*h-(1/12)*fp[2, j-1]*h^2;
end do:
Ms := Matrix(4, 4); Ms[1, 3] := 1; Ms[2, 4] := 1;
y[2, ni] := collect(algsubs(h*alpha = H1, expand(algsubs(h*beta = H2, expand(y[2, ni])))), {y[0, ni], y[1, ni], yp[0, ni], yp[1, ni]});
Ms[3, 1] := coeff(y[2, ni], y[0, ni]);
Ms[3, 2] := coeff(y[2, ni], yp[0, ni])/h;
Ms[3, 3] := coeff(y[2, ni], y[1, ni]);
Ms[3, 4] := coeff(y[2, ni], yp[1, ni])/h;
hyp[2, ni] := collect(algsubs(h*alpha = H1, expand(algsubs(h*beta = H2, expand(h*yp[2, ni])))), {y[0, ni], y[1, ni], yp[0, ni], yp[1, ni]});
Ms[4, 1] := coeff(hyp[2, ni], y[0, ni]);
Ms[4, 2] := coeff(hyp[2, ni], yp[0, ni])/h;
Ms[4, 3] := coeff(hyp[2, ni], y[1, ni]);
Ms[4, 4] := coeff(hyp[2, ni], yp[1, ni])/h;
sol := Eigenvalues(Ms);
print(time[real]()-st);
st := time[real]();
SR[ni, 1] := implicitplot(max(seq(abs(sol[ii]), ii = 1 .. numelems(sol))) <= 1, H1 = 0 .. 3, H2 = 0 .. 3, filledregions, gridrefine = 3, axes = Boxed, view = [-2 .. 3, -3 .. 3], labels = [H[1], H[2]], labeldirections = ["horizontal", "vertical"]);
SR[ni, 2] := implicitplot(max(seq(abs(sol[ii]), ii = 1 .. numelems(sol))) <= 1, H1 = -2 .. 3, H2 = -2 .. 3, gridrefine = 3, axes = Boxed, view = [-2 .. 3, -3 .. 3], labels = [H[1], H[2]], labeldirections = ["horizontal", "vertical"]);
print(time[real]()-st);
end do;
for i in Ni do
i;
display({SR[i, 1], SR[i, 2], line([-1, 0], [3, 0], color = red, linestyle = dash), line([0, -3], [0, 3], color = red, linestyle = dash)});
end do;
display({seq(SR[i, 2], i = 0 .. Ni)});

To paint a triangle on a sphere...

My profile picture was formerly animation and looked like this:

It would be interesting to paint a triangle on a sphere. How can I do that?

Second-order curve in 3d

by: Maple 15

Parametric equation of second-order curve in 3d. Draghilev method.
PLAN_CURVE_3d_1.mw
Examples:
x1^2+x1*x3+13*x2^2+x3-1=0;
x1+x2+x3=0;

x1^2+0.1*x2^2+x3^2-9=0;
x1+3*x3+1=0;

x1^2-0.1*x2^2+x3^2-9=0;
x1+3*x3+1=0;

Circle in 3d

by: Maple 15

Parametric equation of a circle in 3d by three points. Draghilev method.

CIRCLE_3_POINTS_geom3d_2.mw

Isolation of sides of the surface on the graph

by: Maple 15

One way is coloring a surface on both sides. We build equidistant surface with very small radius and stain the equidistant surface in color different from the color of the original surface.
Examples coloring of surfaces on both sides.  Radius equal to abs (0.0001).
x3-0.5*exp(sin(x1+2.5*x2+x3))=0;
(x1^2+x2^2-0.4)^2+(x3+sin(x1*x2+x3))^4-0.1=0;

2_COLORS.mw

To find the numerical solutions of system of nonli...

Dears

Hope everything fine with you. I want to solve the attached problem by numarically and want to plot it but failed. Please see the attachement and correct it. I am waiting your positive respone.

System_of_ODEs.mw

With my best regards and sincerely.

School of Mathematical Sciences
Peking University, Beijing, China

Simplify the expression ...

Hello dear! Hope you will be fine. I want to simplify the equation (4) like equation (3) in the attached file please correct it. I shall be thankful to you.

Help.mw

With my best regards and sincerely.

School of Mathematical Sciences
Peking University, Beijing, China

Numeric solution of three-point differential equat...

Hello! Hope everything fine with you. I am try to solve the three-point differential by numerical method in attached file but failed. Please see the attachement and solve my problem. I am very thankful your kind effort. Please take care.

three-point.mw

With my best regards and sincerely.

School of Mathematical Sciences
Peking University, Beijing, China

Runge Kutta method...

Hi guys, i'm working on a problem using the runge kutta method (4). The problem is, when I run this program, with the given starting values, it perfectly calculates every value of x[w-i] except x[0] and i have no idea why. Can anyone help? Thanks!

Error with Analytic()...

Hi!

I am an error with the use of the function "Analytic" of the packpage RootFinding. These are the procedures:

CreaCos := proc (C, n, m, t) local k, F; F := C[1][1]+(C[1][2]-C[1][1])*t; for k to n-1 do F := F, C[k+1][1]+((1/2)*C[k+1][2]-(1/2)*C[k+1][1])*(1-cos(Pi*m^k*t)) end do; return F end proc;

Then, for k=50, 100, 150... the instruction

works correctly. However, for higher values of k (for instance, k=250) returns the below error. Some idea or suggets about occurs this error?

Error, (in RootFinding:-Analytic) unable to evaluate `@`(evalf, proc (x) option remember; table( [( 0.524900000000000000000000000000e-1+Float(undefined)*I ) = Float(undefined)+Float(undefined)*I ] ) 31250*Pi*sin(62500*Pi*x)/(7/18-(1/2)*cos(62500*Pi*x)) end proc) at the value 0.524900000000000000000000000000e-1+Float(undefined)*I. The expression to be solved was probably not analytic.

Problem to solve system of linear PDEs with BSc an...

Hello

Hope everything going fine with you. I am facing problem to fine the exact (numerical) solution of the attached system of linear PDEs associated with BSc and ICs. I tried to solve it without BCs and ICs, with BCs and with ICs also all the times I failed. Please solve it either general, with ICs or BCs. You can try to solve it numrically. In attached file H(t) represent the unit step function. I am waiting your positive response.

With my best regards and sincerely.

School of Mathematical Sciences
Peking University, Beijing, China

Problem to create a loop...

Hello!

Hope everyone good. I am facing problem to define a loop. Actually I want to solve the attached file for different values of H like H=[100,125,140,160] for this I don't know how to tackle it. My program do working on single value of H. Please solve my problem as soon as possible. I am waiting your positive response.

Help.mw

With my best regards and sincerely.