Maple Questions and Posts

These are Posts and Questions associated with the product, Maple

Dear, need help about Implicit finite difference method 

plz reply as soon as possible

Thanks

Hi,

Need help in plotting the function in the attached file. m, h and k are parameters.

 

Thanks

 

Bessel_Function.mw

> restart;
> u := -3*sqrt(mu)*tanh*(A+sqrt(-mu)*(x-(1/2)*mu*t))/sqrt(6*a)+3*sqrt(mu)*sech*(A+sqrt(-mu)*(x-(1/2)*mu*t))/sqrt(-6*a);

>
> mu := .5;
0.5
> lambda := 1;
1
> a := 0.5e-1;
0.05
> A := 1.5;
1.5
> plot3d([abs(u)], x = -3 .. 3, t = -3 .. 3);

Hello guys and gals!

I'm not strong enough with maple to get what the result I want.

It seems that it's because I'm asking for two lenths, and not a lenth and an angle, but I have no Idea how to tackle it diferently.

If you know a trick, please share it!

 

Here's an image:

http://imgur.com/xavAUoB

And here's the maple file attached (I think)

 complex_problem_from_the_internet.mw

Thanks,

Happy new year!

This post is my attempt to answer the question from here .  

The procedure  ContoursWithLabels  has 2 required parameters: Expr  is an expression in  x  and  y  variables,  Range1  and  Range2  are ranges for  x  and  y . In this case, the output is the list of floats for the contours and 8 black contours (with labels) (the axis of coordinates as a box). 

The optional parameters: Number is positive integer - the number of contours (by default Number=8),  S is a set of real numbers  C  for contours (for which Expr=C) (by default  S={}),  GraphicOptions  is a list of graphic options for plotting (by default  GraphicOptions=[color = black, axes = box]),  Coloring  is an equality  Coloring=list of color options for  plots[dencityplot]  command (by default Coloring=NULL). 

The code of the procedure:

restart;

ContoursWithLabels := proc (Expr, Range1::(range(realcons)), Range2::(range(realcons)), Number::posint := 8, S::(set(realcons)) := {}, GraphicOptions::list := [color = black, axes = box], Coloring::`=` := NULL)

local r1, r2, L, f, L1, h, S1, P, P1, r, M, C, T, p, p1, m, n, A, B, E;

uses plots, plottools;

f := unapply(Expr, x, y);

if S = {} then r1 := rand(convert(Range1, float)); r2 := rand(convert(Range2, float));

L := [seq([r1(), r2()], i = 1 .. 205)];

L1 := convert(sort(select(a->type(a, realcons), [seq(f(op(t)), t = L)]), (a, b) ->is(abs(a) < abs(b))), set);

h := (L1[-6]-L1[1])/Number;

S1 := [seq(L1[1]+(1/2)*h+h*(n-1), n = 1 .. Number)] else

S1 := convert(S, list)  fi;

print(Contours = evalf[2](S1));

r := k->rand(20 .. k-20); M := []; T := [];

for C in S1 do

P := implicitplot(Expr = C, x = Range1, y = Range2, op(GraphicOptions), gridrefine = 3);

P1 := [getdata(P)];

for p in P1 do

p1 := convert(p[3], listlist); n := nops(p1);

if n < 500 then m := `if`(40 < n, (r(n))(), round((1/2)*n)); M := `if`(40 < n, [op(M), p1[1 .. m-11], p1[m+11 .. n]], [op(M), p1]); T := [op(T), [op(p1[m]), evalf[2](C)]] else

if 500 <= n then h := floor((1/2)*n); m := (r(h))(); M := [op(M), p1[1 .. m-11], p1[m+11 .. m+h-11], p1[m+h+11 .. n]]; T := [op(T), [op(p1[m]), evalf[2](C)], [op(p1[m+h]), evalf[2](C)]]

fi; fi; od; od;

A := plot(M, op(GraphicOptions));

B := plots:-textplot(T);

if Coloring = NULL then E := NULL else E := ([plots:-densityplot])(Expr, x = Range1, y = Range2, op(rhs(Coloring)))  fi;

display(E, A, B);

end proc:

 

Examples of use:

ContoursWithLabels(x^2+y^2, -3 .. 3, -3 .. 3);

                             

 

 

ContoursWithLabels(x^2-y^2, -5 .. 5, -5 .. 5, {-20, -15, -10, -5, 0, 5, 10, 15, 20}, [color = black, thickness = 2, axes = box], Coloring = [colorstyle = HUE, colorscheme = ["White", "Red"], style = surface]);

                           

 

 

The next example, I took from here:

ContoursWithLabels(sin(1.3*x)*cos(.9*y)+cos(.8*x)*sin(1.9*y)+cos(.2*x*y), -5 .. 0, 2 .. 5, {seq(-2 .. 2, 0.5)}, [color = black, axes = box], Coloring = [colorstyle = HUE, colorscheme = ["Cyan", "Red"], style = surface]);

                                 

 

There are many more examples can be found in the attached file. 

ContoursWithLabels1.mw

 

Edit. The attached file has been corrected.

Hi,

 

  I think similar question has been asked by several people, but I did not find a suitable thread. My question is, suppose I have a probablity distirubtion function like

  p(x,y) = exp(-alpha (x+y) ) x^2 y^2 / |x-y|  , alpha>0

 x,y goes from - \infty to + \infty. This function is normalizable but unbounded, which makes the rejection algorithm a bit difficult(?).

 

  How to generate samping points from this type of probability distribution function?

 

Thank you very much!

 

hi. i am tottaly new to the maple and i have a problem.

consider function f with variables x & y which are not independent. x & y are functions of t and the relation is unknown.

for example i wanna to the below job:

define f as f=x^2+y^2

differentiate it with respect to t : diff(f,t) which should give me 2(dx/dt)x+2(dy/dt)y

i've googled it alot and i couldn't find anything usefull.

(the problem is how to set x as a function of t with unknown relation and use it in another function and then differentiate it with respect to t)

thanks alot :)

 

has anybody experienced issues with Maple connectiong to Solidworks?  There doen't seem to be much info ion trouble shooting a failed "OpenConnection" call.

 

thanks,

Bill

From a Maple Primes answer two years ago:

f(x,y) is the equation of a line through point [m,n]. The solve command finds values of a and b for which f(x,y) are lines through [m,n] and tangent to x^2 + y^2 = r^2.

f := proc (x, y) options operator, arrow; a*(x-m)+b*(y-n) end proc

solve([f(0, 0) = r, a^2+b^2 = 1], [a, b])

These commands are far from the conventional solution. Why do they provide the correct answers?

Hello,

I would like to plot gait diagrams (the lines you can see on the picture belowà from the solutions obtained with a NL oscillator (composed with 8 coupled odes). Here the result that I would like to obtain.

Initial plot:

 

Desired plot

 

 

I would like to obtain 4 lines corresponding to the 4 elliptic trajectories obtained with the NL oscillator. The four lines should be done like this. When the trajectory is above 0, the line should be colored in green. When the trajectory is below 0, the line should be colored in black. 

May you help me to define this kind of graph called gait diagrams from the solution of the NL oscillator ?

Here you can find my maple code:

K:=Matrix([<0, -1, 1, -1>,<-1, 0, -1, 1>,<-1, 1, 0,-1>,<1, -1, -1,0>]);

for i to 4
do
r[i]:=sqrt((u[i](t))^2+(v[i](t))^2):
omega[i]:=omega[sw]/(1+exp(b*v[i](t)))+omega[st]/(1+exp(-b*v[i](t))):
Equ[i]:=diff(u[i](t),t)=Au*(1-r[i]^2)*u[i](t)-omega[i]*v[i](t):
Eqv[i]:=diff(v[i](t),t)=Av*(1-r[i]^2)*v[i](t)+omega[i]*u[i](t)+MatrixVectorMultiply(K,<seq(v[i](t),i=1..4)>)[i]:
EqSys[i]:=[Equ[i],Eqv[i]]:
end do:

paramsCycle:=omega[st]=4*2*Pi,omega[sw]=2*Pi,Au=5,Av=50,b=100;
params:=paramsCycle;

Differential system 
sys:=map(op,eval([seq(EqSys[i],i=1..4)],[params]));
ic:=[u[1](0)=0, v[1](0)=0,u[2](0)=0, v[2](0)=-0.1,u[3](0)=0, v[3](0)=0.1,u[4](0)=0, v[4](0)=0.1];
Résolution1
res:=dsolve([sys[],ic[]],numeric):
Initial boundaries
tcalc:=4;
ic2:=[seq(u[i](0)=eval(u[i](t), res(tcalc)),i=1..4),seq(v[i](0)=eval(v[i](t), res(tcalc)),i=1..4)];
Résolution2
res:=dsolve([sys[],ic2[]],numeric):

tmax:= 40:
numpts:=100*tmax:
plots:-odeplot(res,[t,v[1](t)],0..tmax,thickness=2, view=[0..5, -1.5..1.5],numpoints = numpts);
plots:-odeplot(res,[t,v[2](t)],0..tmax,thickness=2, view=[0..5, -1.5..1.5],numpoints = numpts);
plots:-odeplot(res,[t,v[3](t)],0..tmax,thickness=2, view=[0..5, -1.5..1.5],numpoints = numpts);
plots:-odeplot(res,[t,v[4](t)],0..tmax,thickness=2, view=[0..5, -1.5..1.5],numpoints = numpts);
plots:-odeplot(res,[seq([t,v[i](t)+i*5],i=1..4)],0..tmax,thickness=2,view=[0..5,0..25], numpoints = numpts);

Thanks a lot for your help

I'm not really an expert in maple. I'm stuck with a problem that could be related to the precision of the computational engine...

I defined a function which is relatively complicated as follows:

m := proc (x, l, T) options operator, arrow; (exp(l*(x-T)/(1-exp(-l*T)))*exp(l*T/(1-exp(-l*T)))*(1-exp(-l*T))+(2-exp(-l*T)-exp(l*T))*exp(l*(x-T)/(1-exp(-l*T))))/(exp(l*T)-1)-(l*(x-T)*exp(l*(x-T)/(1-exp(-l*T)))-2*exp(-l*T)+4-2*exp(l*T))/(exp(l*T)-1) end proc

whenever i simplify the equation or factor some of the terms, I get completely different results.

As an example, you can define this function as follows

m := proc (x, l, T) options operator, arrow; (exp(x*l/(1-exp(-l*T)))*(1-exp(-l*T))+(2-exp(-l*T)-exp(l*T))*exp(l*(x-T)/(1-exp(-l*T))))/(exp(l*T)-1)-(l*(x-T)*exp(l*(x-T)/(1-exp(-l*T)))-2*exp(-l*T)+4-2*exp(l*T))/(exp(l*T)-1) end proc

 

By drawing the function, you can see how things get messy.

plot(m(450, (1/250)*x, 250), x = 0 .. 100)

 

I know the problem occurs as a results of computation round-off. How could I ameliorate that? I have already increase the precision in the option menu, but that didn't help. By the way, how can I be sure that the anwers maple gives me is correct, except using my intuition?

 


The year 2015 has been one with interesting and relevant developments in the MathematicalFunctions  and FunctionAdvisor projects.

• 

Gaps were filled regarding mathematical formulas, with more identities for all of BesselI, BesselK, BesselY, ChebyshevT, ChebyshevU, Chi, Ci, FresnelC, FresnelS, GAMMA(z), HankelH1, HankelH2, InverseJacobiAM, the twelve InverseJacobiPQ for P, Q in [C,D,N,S], KelvinBei, KelvinBer, KelvinKei, KelvinKer, LerchPhi, arcsin, arcsinh, arctan, ln;

• 

Developments happened in the Mathematical function package, to both compute with symbolic sequences and symbolic nth order derivatives of algebraic expressions and functions;

• 

The input FunctionAdvisor(differentiate_rule, mathematical_function) now returns both the first derivative (old behavior) and the nth symbolic derivative (new behavior) of a mathematical function;

• 

A new topic, plot, used as FunctionAdvisor(plot, mathematical_function), now returns 2D and 3D plots for each mathematical function, following the NIST Digital Library of Mathematical Functions;

• 

The previously existing FunctionAdvisor(display, mathematical_function) got redesigned, so that it now displays more information about any mathematical function, and organized into a Section with subsections for each of the different topics, making it simpler to find the information one needs without getting distracted by a myriad of formulas that are not related to what one is looking for.

More mathematics

 

More mathematical knowledge is in place, more identities, differentiation rules of special functions with respect to their parameters, differentiation of functions whose arguments involve symbolic sequences with an indeterminate number of operands, and sum representations for special functions under different conditions on the functions' parameters.

Examples

   

More powerful symbolic differentiation (nth order derivative)

 

Significative developments happened in the computation of the nth order derivative of mathematical functions and algebraic expressions involving them.

Examples

   

Mathematical handling of symbolic sequences

 

Symbolic sequences enter various formulations in mathematics. Their computerized mathematical handling, however, was never implemented - only a representation for them existed in the Maple system. In connection with this, a new subpackage, Sequences , within the MathematicalFunctions package, has been developed.

Examples

   

Visualization of mathematical functions

 

When working with mathematical functions, it is frequently desired to have a rapid glimpse of the shape of the function for some sampled values of their parameters. Following the NIST Digital Library of Mathematical Functions, a new option, plot, has now been implemented.

Examples

   

Section and subsections displaying properties of mathematical functions

 

Until recently, the display of a whole set of mathematical information regarding a function was somehow cumbersome, appearing all together on the screen. That display was and is still available via entering, for instance for the sin function, FunctionAdvisor(sin) . That returns a table of information that can be used programmatically.

With time however, the FunctionAdvisor evolved into a consultation tool, where a better organization of the information being displayed is required, making it simpler to find the information we need without being distracted by a screen full of complicated formulas.

To address this requirement, the FunctionAdvisor now returns the information organized into a Section with subsections, built using the DocumentTools package. This enhances the presentation significantly.

Examples

   

These developments can be installed in Maple 2015 as usual, by downloading the updates (bundled with the Physics and Differential Equations updates) from the Maplesoft R&D webpage for Mathematical Functions and Differential Equations


Download MathematicalFunctionsAndFunctionAdvisor.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Hello,

I've tried to find the solution to my problem, but none of my attempts was succesful.

I have a function which is one-to-one in a particular domain which I am interested in. I would like to get the inverse function of it only in this domain. Here is my function and plot for xp=0..10000:

x := xp-> (-1)*720.5668720*sinh(0.2043018094e-3*xp-0.8532729286)+84952.59423+4.014460003*10^5*arcsinh(0.1219272144e-1*sinh(0.2043018094e-3*xp-0.8532729286)-0.2032498888)

I would appreciate any help,

Iza

I am tryng to write and solve within Maple the equations of movement for a single body subject to a gravitatinal field, F_=-GMm/r2

But I get an error message when trying to define the Angular Momentum which doesn't make sense for me.

Thank you for any help on this topíc.


restart; with(Physics[Vectors]); conventions, Setup(mathematicalnotation = true);
conventions, [mathematicalnotation = true]
r_ := rho*_rho;
r_ := rho _rho
_rho(t);
_rho(t)
rho(t);
rho(t)
v_ := diff(rho(t)*_rho(t), t);
/ d \ / d \
v_ := |--- rho(t)| _rho(t) + rho(t) |--- phi(t)| _phi(t)
\ dt / \ dt /
a_ := diff(%, t);
/ 2 \
| d | / d \ / d \
a_ := |---- rho(t)| _rho(t) + 2 |--- rho(t)| |--- phi(t)| _phi(t)
| 2 | \ dt / \ dt /
\ dt /

/ 2 \ 2
| d | / d \
+ rho(t) |---- phi(t)| _phi(t) - rho(t) |--- phi(t)| _rho(t)
| 2 | \ dt /
\ dt /

eq[1] := -G*M*_rho(t)/r^2-a_ = 0;
/ / 2 \ 2\
| G M | d | / d \ |
eq[1] := _rho(t) |- --- - |---- rho(t)| + rho(t) |--- phi(t)| |
| 2 | 2 | \ dt / |
\ r \ dt / /

/ / 2 \
| / d \ / d \ | d |
+ _phi(t) |-2 |--- rho(t)| |--- phi(t)| - rho(t) |---- phi(t)|
| \ dt / \ dt / | 2 |
\ \ dt /

\
|
| = 0
|
/
Eq[1, 2] := seq(Component(lhs(eq[1]), n) = 0, n = 1 .. 2);
/ 2 \ 2
G M | d | / d \
Eq[1, 2] := - --- - |---- rho(t)| + rho(t) |--- phi(t)| = 0,
2 | 2 | \ dt /
r \ dt /

/ 2 \
/ d \ / d \ | d |
-2 |--- rho(t)| |--- phi(t)| - rho(t) |---- phi(t)| = 0
\ dt / \ dt / | 2 |
\ dt /
NULL;

L_ := `&x`(r_, m*v_);
Error, (in Physics:-Vectors:-&x) found the unit vector _rho present also as a function _rho(t); either one form or the other - not both - can be present in an algebraic expression

 

HI, I am trying to solve two PDEs but in boundry conditions there is arising an error plz help.
Nazi.mw

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