How can plot two  and bipolar figures as attached and extract their data.

Thank You.

I have reinstalled Maple 2019 several times, including its latest update. But when trying to graph the following:

plot(sin(x), x = -2*Pi .. 2*Pi)

But I get this:

Error, (in plot) expected a range but received x = -2*Pi .. 2*Pi

And if I enter:

sin(x)

The result is:

2.73949338633639*10^-116 + 2.73949338633639*10^-116*I

When trying this:

plot3d(x*exp(-x^2 - y^2), x = -2 .. 2, y = -2 .. 2, color = x)

I get this:

And if I try it with Graph Theory:

with(GraphTheory);
G := Graph({{a, b}, {a, c}, {b, c}});
G := Graph 1: an undirected unweighted graph with 3 vertices and 3 edge(s)

DrawGraph(G)

Error, (in GraphTheory:-DrawGraph) invalid input: modp received I, which is not valid for its 2nd argument, m
 

I do not know what is the reason for this anomalous behavior of Maple 2019, it will be some software bug or it will be an error caused by my computer...
I would like to know if this problem happens to other people or just to me. Any help or guidance on this problem will be greatly appreciated.

Best regards

We occasionally get asked questions about methods of Perturbation Theory in Maple, including the Lindstedt-Poincaré Method. Presented here is the most famous application of this method.

Introduction

During the dawn of the 20th Century, one problem that bothered astronomers and astrophysicists was the precession of the perihelion of Mercury. Even when considering the gravity from other planets and objects in the solar system, the equations from Newtonian Mechanics could not account for the discrepancy between the observed and predicted precession.

One of the early successes of Einstein's General Theory of Relativity was that the new model was able to capture the precession of Mercury, in addition to the orbits of all the other planets. The Einsteinian model, when applied to the orbit of Mercury, was in fact a non-negligible perturbation of the old model. In this post, we show how to use Maple to compute the perturbation, and derive the formula for calculating the precession.

In polar coordinates, the Einsteinian model can be written in the following form, where u(theta)=a(1-e^2)/r(theta), with theta being the polar angle, r(theta) being the radial distance, a being the semi-major axis length, and e being the eccentricity of the orbit:
 

# Original system.
f := (u,epsilon) -> -1 - epsilon * u^2;
omega := 1;
u0, du0 := 1 + e, 0;
de1 := diff( u(theta), theta, theta ) + omega^2 * u(theta) + f( u(theta), epsilon );
ic1 := u(0) = u0, D(u)(0) = du0;

The small parameter epsilon (along with the amount of precession) can be found in terms of the physical constants, but for now we leave it arbitrary:
 

# Parameters.
P := [
    a = 5.7909050e10 * Unit(m),
    c = 2.99792458e8 * Unit(m/s),
    e = 0.205630,
    G = 6.6740831e-11 * Unit(N*m^2/kg^2), 
    M = 1.9885e30 * Unit(kg), 
    alpha = 206264.8062, 
    beta = 415.2030758 
];
epsilon = simplify( eval( 3 * G * M / a / ( 1 - e^2 ) / c^2, P ) ); # approximately 7.987552635e-8

Note that c is the speed of light, G is the gravitational constant, M is the mass of the sun, alpha is the number of arcseconds per radian, and beta is the number of revolutions per century for Mercury.

We will show that the radial distance, predicted by Einstein's model, is close to that for an ellipse, as predicted by Newton's model, but the perturbation accounts for the precession (42.98 arcseconds/century). During one revolution, the precession can be determined to be approximately
 

sigma = simplify( eval( 6 * Pi * G * M / a / ( 1 - e^2 ) / c^2, P ) ); # approximately 5.018727337e-7

and so, per century, it is alpha*beta*sigma, which is approximately 42.98 arcseconds/century.
It is worth checking out this question on Stack Exchange, which includes an animation generated numerically using Maple for a similar problem that has a more pronounced precession.

Lindstedt-Poincaré Method in Maple

In order to obtain a perturbation solution to the perturbed differential equation u'+omega^2*u=1+epsilon*u^2 which is periodic, we need to write both u and omega as a series in the small parameter epsilon. This is because otherwise, the solution would have unbounded oscillatory terms ("secular terms"). Using this Lindstedt-Poincaré Method, we substitute arbitrary series in epsilon for u and omega into the initial value problem, and then choose the coefficient constants/functions so that both the initial value problem is satisfied and there are no secular terms. Note that a first-order approximation provides plenty of agreement with the measured precession, but higher-order approximations can be obtained.

To perform this in Maple, we can do the following:
 

# Transformed system, with the new independent variable being the original times a series in epsilon.
de2 := op( PDEtools:-dchange( { theta = phi/b }, { de1 }, { phi }, params = { b, epsilon }, simplify = true ) );
ic2 := ic1;

# Order and series for the perturbation solutions of u(phi) and b. Here, n = 1 is sufficient.
n := 1;
U := unapply( add( p[k](phi) * epsilon^k, k = 0 .. n ), phi );
B := omega + add( q[k] * epsilon^k, k = 1 .. n );

# DE in terms of the series.
de3 := series( eval( de2, [ u = U, b = B ] ), epsilon = 0, n + 1 );

# Successively determine the coefficients p[k](phi) and q[k].
for k from 0 to n do

    # Specify the initial conditions for the kth DE, which involves p[k](phi).
    # The original initial conditions appear only in the coefficient functions with index k = 0,
    # and those for k > 1 are all zero.
    if k = 0 then
        ic3 := op( expand( eval[recurse]( [ ic2 ], [ u = U, epsilon = 0 ] ) ) );
    else
        ic3 := p[k](0), D(p[k])(0);
    end if:

    # Solve kth DE, which can be found from the coefficients of the powers of epsilon in de3, for p[k](phi).
    # Then, update de3 with the new information.
    soln := dsolve( { simplify( coeff( de3, epsilon, k ) ), ic3 } );
    p[k] := unapply( rhs( soln ), phi );
    de3 := eval( de3 );

    # Choose q[k] to eliminate secular terms. To do this, use the frontend() command to keep only the terms in p[k](phi)
    # which have powers of t, and then solve for the value of q[k] which makes the expression zero. 
    # Note that frontend() masks the t-dependence within the sine and cosine terms.
    # Note also that this method may need to be amended, based on the form of the terms in p[k](phi).
    if k > 0 then
        q[1] := solve( frontend( select, [ has, p[k](phi), phi ] ) = 0, q[1] );
        de3 := eval( de3 );
    end if;

end do:

# Final perturbation solution.
'u(theta)' = eval( eval( U(phi), phi = B * theta ) ) + O( epsilon^(n+1) );

# Angular precession in one revolution.
sigma := convert( series( 2 * Pi * (1/B-1), epsilon = 0, n + 1 ), polynom ):
epsilon := 3 * G * M / a / ( 1 - e^2 ) / c^2;
'sigma' = sigma;

# Precession per century.
xi := simplify( eval( sigma * alpha * beta, P ) ); # returns approximately 42.98

Maple Worksheet: Lindstedt-Poincare_Method.mw

Hello everyone! 

I have a naive question... 
I'm traying to construct an array such entries are derivatives of a particular function. In my original problem, the elements of the array are numerous, so build it manual is not an option... A simplification of my task is :

So, as you can see, for some reason Maple doen't chanche the index "i" in my derivative while the cycle is runnin. So all i want to do  is construct an array of the form 

using a cycle. 

Thanks for yout help! 

Hello everybody

can anybody help me solve this differential equation please?

d^2y(t)/dt^2-dy(t)/dt*1/y(t)=a*y(t)^2

I am really not good at writing the code. The code was from the book (Stochastic process with maple).

It does not seem to generate the different plots when I run the code.

Also the last added code was from the book but it shows error.

Thank you for the help in advance. (My English also is not good.)
 

Download randomWalk.mw
 

Download randomWalk.mw

Could you help me?

Download randomWalk.mw
 

Download randomWalk.mw

restart; with(stats); with(plots); RandWalk :=proc(N,p,q,sa,sb,M) local A,B,r,PA,PC,PB, state,step,path: A:=-N: B:=N: r:=1-p-q: PA:=empirical[0.,sa,1-sa]: PC:= empirical[q,r,p]: PB := empirical[1-sb,sb,0.]: state:=0: path:=state: from 1 to M do if (A< state and state

with(LinearAlgebra):
assign(seq(U || i = `<,>`(x[i], y[i]), i = 1 .. 3));
U || 4 := U || 1; r := (1/6)*sqrt(3);
assign(seq(V || i = (U || i+U || (i+1))*(1/2)+(`<,>`(`<|>`(0, 1), `<|>`(-1, 0)) . (U || (i+1)-U || i))*r, i = 1 .. 3)); Norm(V || 3-V || 1, 2); is(Norm(V || 3-V || 1, 2) = Norm(V || 3-V || 2, 2)); is(Norm(V || 2-V || 1, 2) = Norm(V || 3-V || 2, 2));

Starting from 3 points of a base, then got 3 lines equations.

If having 3 lines equations of a cone surface, how to find the surface equation of a cone?

numbrix.mw

I find many tools about differential 

i had some differential ideal

how to use these tools to research what application can be applied with differential ideals I discovered?

Given a list of numeric value lists, calculate the average of each sub-data list

### I am writing a code for an Optimal control using pontryagins maximum principle. I have supplied all neccessary boundary conditions yet it fails to detect one of the conditions. It gives an error message whenever I run it.

I need guidance. Thank you

restart;
with(plots);
r := 3; r[1] := 3; k := 10; a := 0.2e-1; b := 0.1e-1; c := 0.1e-1; beta := 0.3e-1; alpha := 0.3e-1; m := 0.5e-1;
z := 40; q := 5; p := 100; T := 3;
sigma := 0.1e-1; k[1] := 10; rho := 0.5e-1;

u[1] := min(max(0, z), 1); z := (a*m*k*lambda[2](t)*x(t)*y(t)-lambda[1](t)*r*(1+b*x(t)+c*y(t))*x(t)*x(t))/(z*k*(1+b*x(t)+c*y(t))); u[2] := min(max(0, q), 1); q := -lambda[1](t)*beta*x(t)*s(t)/q; u[3] := min(max(0, p), 1); p := -(r[1]*lambda[3](t)*s(t)*s(t))/(p*k[1]);
NULL;
sys := diff(x(t), t) = r*x(t)*(1-(1-u[1])*x(t)/k)-a*m*x(t)*y(t)/(1+b*x(t)+c*y(t))-beta*(1-u[2])*x(t)*s(t), diff(y(t), t) = -alpha*y(t)+a*m*x(t)*y(t)/(1+b*x(t)+c*y(t)), diff(s(t), t) = sigma*s(t)+r[1]*s(t)*(1-(1-u[3])*s(t)/k[1])-rho*s(t)*y(t), diff(lambda[1](t), t) = -lambda[1](t)*(r-2*r*(1-u[1])*x(t)/k-a*y(t)*(1+c*y(t))/((1+b*x(t)+c*y(t)) . (1+b*x(t)+c*y(t)))-beta*(1-u[2])*s(t))-lambda[2](t)*a*m*(1-u[1])*(1+c*y(t))*y(t)/((1+b*x(t)+c*y(t)) . (1+b*x(t)+c*y(t))), diff(lambda[2](t), t) = -lambda[1](t)*a*x(t)*(1+b*x(t))/((1+b*x(t)+c*y(t))*(1+b*x(t)+c*y(t)))+lambda[2](t) . (-alpha+(a*m*(1-u[1]) . (1+b*x(t)))*x(t)/((1+b*x(t)+c*y(t))*(1+b*x(t)+c*y(t))))+lambda[3](t)*rho*s(t), diff(lambda[3](t), t) = lambda[1](t)*beta*(1-u[2])*x(t)-lambda[1](t)*(r[1]-2*r[1]*(1-u[3])*s(t)/k[1]-sigma-rho*y(t)), x(0) = 10, y(0) = 20, s(0) = 10, lambda[1](T) = 0, lambda[2](T) = 0, lambda[3](T) = 0;
p1 := dsolve({sys}, type = numeric, method = bvp[midrich], abserr = .1);
Error, (in dsolve/numeric/bvp/convertsys) too few boundary conditions: expected 7, got 6
p2o := odeplot(p1, [t, y(t)], 0 .. 2, numpoints = 100, labels = ["Time (months)", " "*`badbiomass""spatina"""`], labeldirections = [horizontal, vertical], style = line, color = red, axes = boxed);
Error, (in plots/odeplot) input is not a valid dsolve/numeric solution
r := 3; r[1] := 3; k := 10; a := 0.2e-1; b := 0.1e-1; c := 0.1e-1; beta := 0.3e-1; alpha := 0.3e-1; m := 0.5e-1; sigma := 0.1e-1; k[1] := 10; rho := 0.5e-1;
z := 40; q := 5; p := 100; T := 3;

fun := diff(x(t), t) = r*x(t)*(1-x(t)/k)-a*m*x(t)*y(t)/(1+b*x(t)+c*y(t))-beta*x(t)*s(t), diff(y(t), t) = -alpha*y(t)+a*m*x(t)*y(t)/(1+b*x(t)+c*y(t)), diff(s(t), t) = sigma*s(t)+r[1]*s(t)*(1-s(t)/k[1])-rho*s(t)*y(t), x(0) = 10, y(0) = 20, s(0) = 10;
p2 := dsolve({fun}, type = numeric);
Error, (in f) unable to store 'HFloat(10.0)*r*(1-HFloat(10.0)/k)-HFloat(3.1538461538461537)' when datatype=float[8]
p2i := odeplot(p2, [t, y(t)], 0 .. 2, numpoints = 100, labels = ["Time(months)", " bad biomass"], labeldirections = [horizontal, vertical], style = line, axes = boxed, color = blue);
Error, (in plots/odeplot) input is not a valid dsolve/numeric solution
plots[display](p2i, p2o);
Error, (in plots:-display) expecting plot structure but received: p2i

We have just released an update to Maple, Maple 2019.1.

Maple 2019.1 includes corrections and improvement to the mathematics engine (including LPSolve, sum, statistics, and the Physics package),  visualization (including annotations and the Plot Builder), export to LaTeX (exporting output) and PDF (freezing on export), network licensing, MATLAB connectivity, and more.  We recommend that all Maple 2019 users install these updates.

This update is available through Tools>Check for Updates in Maple, and is also available from our website on the Maple 2019.1 download page, where you can also find more details.

do math regex solver exist in this world?

expect to output a set of regex for pattern

and then use regex to generate condition or rules equations and then use sorting algorithm to find whether match all data