Advanced Search

vv

12453 Reputation

19 Badges

9 years, 285 days
Contact vv

MaplePrimes Activity


These are answers submitted by vv

convertions fail...

vv 12453
July 26 2019
1 0

For a solution given implicitely, as in this case, odetest may fail.

Actually the implicit solution is

> 

Int( 1/(y^3 + 1)^(2/3), y) + Int( 1/(x^3 + 1)^(2/3), x) + c = 0;

Int(1/(y^3+1)^(2/3), y)+Int(1/(x^3+1)^(2/3), x)+c = 0

 (1)

The problem is that the integrals are computed in terms of hypergeom, and Maple is not able to further manipulate.

> 

f:=1/(x^3 + 1)^(2/3);

1/(x^3+1)^(2/3)

 (2)
> 

int(f, x)

x*hypergeom([1/3, 2/3], [4/3], -x^3)

 (3)
> 

Z:=diff(% ,x) - f; # Z should be 0

hypergeom([1/3, 2/3], [4/3], -x^3)-(1/2)*x^3*hypergeom([4/3, 5/3], [7/3], -x^3)-1/(x^3+1)^(2/3)

 (4)
> 

is(Z=0) assuming x>0, x<1;

FAIL

 (5)
> 

# It seems that any usual convertion fails here.

An approximation...

vv 12453
July 24 2019
1 3

Salut Viorel, bine ai venit!

If you are interested in an approximate solution based on series, here it is:

restart;
ode:= diff(g(r),r$2)- r/R*g(r)=0;
ic:=g(2*R)=0, D[1](g)(0)=R;
dsolve([ode,ic],g(r)) assuming R>0;
G:=rhs(%);
Order:=6;
simplify(series(G, r=0));
convert(%, polynom):
g1:=convert(series(%, R=0), polynom);  
R:=1/3;  #############
dsol:=dsolve([ode,ic],g(r), numeric);
plots:-odeplot(dsol, r=0..2*R);
plot(g1, r=0..2*R);

So, for small R and r, an approximation is

allsolutions...

vv 12453
July 22 2019
1 1

RealDomains is not very reliable but it works in this case (Maple 2019).

eq:=(-2*cos(x)^2+2*sin(x+(1/4)*Pi)^2-1)/sqrt(-x^2+4*x) = 0:
RealDomain:-solve({-x^2+4*x > 0, eq}, {x}, allsolutions);
RealDomain:-solve(eq, {x}, allsolutions);

 

compact...

vv 12453
July 21 2019
2 0

If you want a compact solution:

9*x-(1/3)*Pi = (-1)^k*(7*x-(1/3)*Pi) + k*Pi:
solve(%,x); # k in Z

COMMAND...

vv 12453
July 20 2019
0 1

I think that in a package where almost everything is a module, such a COMMAND could exist only if it is provided by the package itself.
But we can use:

X := ''RandomVariable''(Normal(mu, sigma)):
Y := eval(X, mu=1);

 

Maybe...

vv 12453
July 19 2019
1 1

something like:

plots:-pointplot([x1,y1], color=[seq(COLOR(HUE,(t+1)/2),t=z1)]);

 

residue...

vv 12453
July 18 2019
4 4

It is not that difficult but unfortunately Maple 2019 gives 0 for it.

It can be computed using residues. This must be done by hand because Maple cannot compute residues for essential singularities.

But it can compute the resulting sum, and the result is   2*Pi*BesselJ(m, b)*exp(I/2*m*Pi)

F2_...

vv 12453
July 14 2019
0 5

 

> 

F1 := (x, y) -> 9/8*y + 9/8 - 21/8*y*exp(-2*x) - 21/8*exp(-x);

proc (x, y) options operator, arrow; (9/8)*y+9/8-(21/8)*y*exp(-2*x)-(21/8)*exp(-x) end proc

 (1)
> 

S1 := y -> fsolve(F1(x, y), x);

proc (y) options operator, arrow; fsolve(F1(x, y), x) end proc

 (2)
> 

S1(0.2);

.7548332751

 (3)
> 

g := (U,y) -> U + S1(2*y*exp(-U));

proc (U, y) options operator, arrow; U+S1(2*y*exp(-U)) end proc

 (4)
> 

g(0.6,0.2);

1.347716146

 (5)
> 

fsolve('g'(U,0.3) = 0.2, U);

-.4051840484

 (6)
> 

U1 := (x,y) -> fsolve('g(U,y) = x', U);

proc (x, y) options operator, arrow; fsolve('g(U, y) = x', U) end proc

 (7)
> 

U1(0.4,0.6);

-.1591368141

 (8)
> 

F2 := (x, y) -> 9/8*y + 9/8 - 21/8*y*exp(-2*x) - 21/8*exp(-x) + 3/4*int(F1(x - z, 2*y*exp(-z)), z = 0 .. U1(x,y));

proc (x, y) options operator, arrow; (9/8)*y+9/8-(21/8)*y*exp(-2*x)-(21/8)*exp(-x)+(3/4)*(int(F1(x-z, 2*y*exp(-z)), z = 0 .. U1(x, y))) end proc

 (9)
> 

F2(0.2,0.3);

-1.008550780

 (10)
> 

####################################################################
> 

X:=[seq(k*0.1, k=0..10)];
Y:=[seq(k*0.1 ,k=0..10)];

[0., .1, .2, .3, .4, .5, .6, .7, .8, .9, 1.0]

  

[0., .1, .2, .3, .4, .5, .6, .7, .8, .9, 1.0]

 (11)
> 

F1_:= Interpolation:-SplineInterpolation( [X, Y],  [seq([seq(F1(x,y),y=Y)],x=X)] );

Interpolation:-SplineInterpolation([_rtable[18446744074399121150], _rtable[18446744074399121270]], _rtable[18446744074399113214], verify = false)

 (12)
> 

plot3d([F1(x,y), '1+F1_(x,y)'], x=0..1, y=0..1);

 
> 

S1_:=Interpolation:-SplineInterpolation( X,  [seq(fsolve(F1_(x, y), x), y=Y)] );

Interpolation:-SplineInterpolation([_rtable[18446744074449580390]], Vector[row](11, {(1) = .8471962164, (2) = .7961144432, (3) = .7548618542000001, (4) = .7211061446, (5) = .6931443837, (6) = .6697004904, (7) = .6498166028, (8) = .6327736053999999, (9) = .6180256179, (10) = .6051531019, (11) = .593829713}), verify = false)

 (13)
> 

S1_(1/2);

HFloat(0.6697004904)

 (14)
> 

solu:=(x,y)->fsolve(U + 'S1_'(2*y*exp(-U))=x, U);

proc (x, y) options operator, arrow; fsolve(U+('S1_')(2*y*exp(-U)) = x, U) end proc

 (15)
> 

solu(0.3,0.4);

-.2865364247

 (16)
> 

U1_:= Interpolation:-SplineInterpolation( [X, Y],  [seq([seq(solu(x,y),y=Y)],x=X)] );

Interpolation:-SplineInterpolation([_rtable[18446744074439092814], _rtable[18446744074439092934]], _rtable[18446744074439093054], verify = false)

 (17)
> 

U1(0.1,0.2); U1_(0.1,0.2);

-.5354543130

  

HFloat(-0.5354485486)

 (18)
> 

F2_:=proc(x,y)
  local z, u:=evalf(U1_(x,y));
  evalf(9/8*y + 9/8 - 21/8*y*exp(-2*x) - 21/8*exp(-x)) + 3/4*int(''F1_''(x - z, 2*y*exp(-z)), z = 0 .. u, numeric)
end:
 
> 

F2(0.1,0.2), F2_(0.1,0.2);

-1.139973599, HFloat(-1.1400076769496776)

 (19)
> 

 


 

Download int_iters.mw

 

On my Windows 7&10, the Maple comman...

vv 12453
July 14 2019
0 0

On my Windows 7&10, the Maple commands work as expected:

!cmd
                               -1

!calc # ==> opens a console and launch calc.exe (the Windows calculator)
                               0
system(calc);
                               0
system("calc.exe");
                               0
!"cmd /c dir";
                               0
!"cmd /c calc";
ssystem("dir"); # ==>  contents of the current directory (Maple)

 

depends...

vv 12453
July 11 2019
0 5 
restart;
vars := [x,y,z]:
m1,m2,m3 := (a-b)*x*y*z^2, c*x^3, 3*b*x*y*z^2:
EqMon := (m1,m2,vars::list(name)) -> not depends(m1/m2, vars):
EqMon(m1,m2,vars), EqMon(m2,m3,vars), EqMon(m1,m3,vars);
#                       false, false, true

 

simple workaround...

vv 12453
July 11 2019
2 0 
SOL:= u(x, t) = limit(op(1,rhs(sol)),n=0) + subs('n=0..infinity' = 'n=1..infinity', rhs(sol)):
plot3d(subs(infinity=100, rhs(SOL)), x=0..3, t=0..2);

Int...

vv 12453
July 10 2019
1 20

You should inspect visually the integrals.
It's easy to see that the integral wrt phi can be computed symbolically (and actually ==> some integrals are 0).
Then compute numerically wrt theta and r. It will be much faster.

Also, for other 3 dimensional int, try a fast (low precision)  e.g.
Int( ...,  epsilon=1e-2, method=_CubaDivonne)

 

 

series...

vv 12453
July 10 2019
1 1 
s4:=series(ex, epsilon, 4): # ex is your expression
coeff(s4, epsilon, 2); # coeff of epsilon^2

 

int~...

vv 12453
July 03 2019
0 3 
A := Matrix(2,3,()->randpoly(t,degree=1));
int~(A,t=-1..1);

 

subs...

vv 12453
June 25 2019
1 0

That is what subs must do:

subs(a=b, {c=d}, {e=f}, {g=h} ) ==> successive substitutions  a=b, c=d, e=f  in  {g=h}.
In your case sol := {c=d}, {e=f}, {g=h}

First 41 42 43 44 45 46 47 Last Page 43 of 111