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Int(piecewise(t < T1, exp((1/2)*t*(1+2*I-I*sqrt(3))), t < T2, -1000*exp((1/2)*t*(1+2*I-I*sqrt(3)))*(-1/1000+T1-t), T2 <= t, -1000*exp((1/2)*t*(1+2*I-I*sqrt(3)))*(-1/1000-T2+T1)), t)

 

 

Let us consider the definite integral

J:=int(abs(x-(-x^5+1)^(1/5)), x = 0 .. 1);

Maple fails with it, Mathematica 10.1 finds it in terms of  special functions. Let us look at the integrand:
plot(x-(-x^5+1)^(1/5), x = 0 .. 1);

We see the expression under the modulus changes its sign at the unique point of RealRange(0,1). Therefore

solve(x-(-x^5+1)^(1/5));


Then

J:= int(-x+(-x^5+1)^(1/5), x = 0 .. (1/2)*2^(4/5))+int(x-(-x^5+1)^(1/5), x = (1/2)*2^(4/5) .. 1);

which outputs a complicated expression

(1/8)*2^(4/5)*(4*hypergeom([-1/5, 1/5], [6/5], 1/2)-2^(4/5))+(1/2)*2^(4/5)*((1/2)*2^(1/5)-(1/4)*2^(4/5))-(1/25)*Pi*csc((1/5)*Pi)*(-(25/2)*sin((1/5)*Pi)*GAMMA(4/5)*2^(4/5)*hypergeom([-1/5, 1/5], [6/5], 1/2)/Pi+(5/4)*sec((3/10)*Pi)*cos((1/10)*Pi)*2^(3/5)*Pi^(1/2)*csc((3/10)*Pi)/GAMMA(7/10))/GAMMA(4/5).

At the same time we have

int(abs(x-(-x^5+1)^(1/5)), x = 0 .. 1, numeric);

                          0.5000000000

How to obtain 1/2 symbolically?






according to help on timelimit

"Note: For efficiency reasons, the timelimit bound is ignored while in built-in routines."

Which is not very useful, since I want to limit  int() to some CPU time.

There are some integrals that can hang Maple easily. I'd like to set some CPU time on an int() and
have it terminate with error, but I am not able to find how to do that.

For example this

int((a+a*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(5/2),x)

Will hangs Maple.
Is there a trick some expert here can show to limit the CPU time on a build in operation?
May be some package or other command can do this?

I am only interested in int() now, but if it can work also on dsolve, that will be good.

thank you
ps. Mathematica supports putting time constraint on build-in commands. So I do
not see why Maple can't also do the same.

of the cut-off sphere

Изображение

Изображение?

Of course, with Maple.

Any suggestions (or perhaps related examples?) illustrating how I might numerically solve for f(t) in the following non-linear integral equation?  In Fortran, I would start with a guess f(t)=T0, and then search in the neighborhood for a minimum (in the error), but I am not familiar with numerical searches and methods in Maple.  Thank you for any suggestions or leads.

(a,b,... etc are all real)


T__0 := 298.

`&Delta;T` := 25.

0 < beta and beta <= 1

``

f*t = T[0]+`&Delta;T`*[1-exp(-a(int(exp(-b/f(y)), y = y[1] .. t))^beta)]

NULL


Download Integraleqn.mw

 

 

 

I use Maple 15 to calculate some (nasty) integrals at my university. Because our university also offers a server on which I can run my Maple program, I would like to do that. (instead of occupying a workspace). But at the computer on my workspace the integrals are evaluated fine, but on the server the integrals are just returned with no numerical evaluation.

I constructed a MWE to look where it goes wrong. I set the printlevel to 25 so I could see what was going on. The MWE was suprisingly simple, on both machines (via ssh) I executed within maple:

evalf(Int(1/sqrt(x), x=0..2))

This of course would normally just give 2*sqrt(2). On my workplace-pc it worked fine and it found 2.828427125. The server just returned the integral. After looking at the steps, they where both exactly the same until the following part:

Workplace-PC:

         General_flags := {_NoNAG, _DEFAULT, _NoMultiple}

            NAG_methods := {_d01ajc, _d01akc, _d01amc}

                        Method := _DEFAULT

                          HFDigits := 15

                                       -12
                        HFeps := 0.1 10

                                            -9
                    HFeps := 0.5000000000 10

   oldEvents := overflow = default, division_by_zero = default

                         callNAG := true

                            fcns := {}

                  result := 2.82842712474618807

Server:

        General_flags := {_NoNAG, _DEFAULT, _NoMultiple}

           NAG_methods := {_d01ajc, _d01akc, _d01amc}

                       Method := _DEFAULT

                         HFDigits := 15

                                      -12
                       HFeps := 0.1 10

                                           -9
                   HFeps := 0.5000000000 10

   oldEvents := overflow = default, division_by_zero = default

                         callNAG := true

                           fcns := {}

       overflow = exception, division_by_zero = exception

It seems that the server has a problem with the singularity and thus throwing an exception, but I just don't get why. The Maple-versions are both the same.

Does somebody know what this could be?

Dear community,

The VectorCalculus package in Maple 18 exhibits some rather odd behaviour when calculating an integral over a 3D-domain (an elipsoid centered at (0,0,1) with length of semi-axes 2, 2, and 1) given by a 'Region'. It also reports an incorrect result when calculating the flux of a vector field through the surface of this ellipsoid.

See below for a minimal working example:

 

// integrate several functions over ellipsoid

int(2*x+2*y, [x, y, z] = Region(-2 .. 2, -sqrt(-x^2+4) .. sqrt(-x^2+4), -sqrt(1-(1/4)*x^2-(1/4)*y^2)+1 .. sqrt(1-(1/4)*x^2-(1/4)*y^2)+1));    ---- output: 0

int(2*z, [x, y, z] = Region(-2 .. 2, -sqrt(-x^2+4) .. sqrt(-x^2+4), -sqrt(1-(1/4)*x^2-(1/4)*y^2)+1 .. sqrt(1-(1/4)*x^2-(1/4)*y^2)+1));    ---- output: 32*Pi/3

int(2*x+2*y+2*z, [x, y, z] = Region(-2 .. 2, -sqrt(-x^2+4) .. sqrt(-x^2+4), -sqrt(1-(1/4)*x^2-(1/4)*y^2)+1 .. sqrt(1-(1/4)*x^2-(1/4)*y^2)+1));    ---- output: -32*Pi/3

 

Why does the integral change sign?

 

// Calculate flux of vector field over surface of ellipsoid

F2 := VectorField(`<,>`(x^2, y^2, z^2));

S2 := Surface(`<,>`(2*sin(t)*cos(s), 2*sin(t)*sin(s), 1+cos(t)), s = 0 .. 2*Pi, t = 0 .. Pi);

Flux(F2, S2, 'outward');   ---- output: -32*Pi/3

 

However, since 'z' is positive over the entire ellipsoid, by the Divergence Theorem we know the result should be positive (in fact, equal to +32*Pi/3). Changing 'outward' to 'inward' does not change the sign, by the way.

 

Is this a bug inside the VectorCalculus package which appears more often, or have I done something wrong?

Thank you for responding and apologies if a similar question has already been answered in another thread.

Best wishes,

QQ

of the improper integral of exp((1-x)/((1-x)^2+y^2)) over the unit disk x^2+y^2 <= 1 with Maple? For purists the function is assumed to be undefined at (1,0). It is not so difficult to verify that statement  by hand. It is not easy to prove that with Maple.

My try was

f := evalc(exp(Re(1/(1-x-I*y))));


VectorCalculus:-int(f, [x, y] = Circle(`<,>`(0, 0), 1), numeric);

and

evalf(Int(f, [y = -sqrt(-x^2+1) .. sqrt(-x^2+1), x = -1 .. 1]));
.

Edit. The formula for f.                  

PS.

 

 

 

Hello everyone,

 

I am working on a program in Maple and got stuck in exchanging limits for integrals. For example, if I have an expression of following type.

Eq:=4*Int(f(x), x=0..1/3)+Int(x*f(x), x=0..2/3);

I want to convert it in to an expression of form

4*Int(f(x),x=0..x)-4*Int(y,x=1/3..x)+ Int(x*f(x), x=0..x)- Int(x*f(x), x=2/3..x)


In short, I want to split both the integral at x but flip in limits in the second integral. I tried as follows which did not work

applyrule(Int(f::anything,y=c::numeric..d::numeric)=Int(f,y=c..x)-Int(f,y=c..x), Eq)

Please, help me!

Thank you for your time.

hello i need plot integrale of siampson thank you

> Simpson := proc(f, a, b, n)
> local h, S1, S2, S, i;
> h := (b-a)/n;
> S1 := 0.0;
> for i from 0 to n-1 do
> S1 := S1 + f(a + (2*i+1)*h);
> end do;
> S2 := 0.0;
> for i from 1 to n-1 do
> S2 := S2 + f(a + (2*i)*h);
> end do;
> S := (h/3) * ( f(a)+f(b) + 4*S1 + 2*S2 );
> return S;
> end proc:
> Digits := 5;

                             Digits := 5

> f := x -> 1/sqrt(39.24*x-44.65*(x*arccos(x)-sqrt(1-x^2)-13.88*(1-x^2)^1.5));

  f := x -> 1/sqrt(39.24 x

                                          2                2 1.5
         - 44.65 (x arccos(x) - sqrt(1 - x ) - 13.88 (1 - x )   ))

> Simpson(f, 0, 1, 100):
>
> p:=int(f(x), x=0..0.1);

                            p := 0.0038931

> w:=int(f(x), x=0.1..0.2);

                            w := 0.0039570

> m:=int(f(x), x=0.2..0.3);

                            m := 0.0040826

> l:=int(f(x), x=0.3..0.4);

                            l := 0.0042836

> kohv:=int(f(x), x=0.4..0.5);

                          kohv := 0.0045860

> q:=int(f(x), x=0.5..0.6);

                            q := 0.0050373

> s:=int(f(x), x=0.6..0.7);

                            s := 0.0057306

> d:=int(f(x), x=0.8..0.9);

                            d := 0.0089874

> f:=int(f(x), x=0.9..1);

                            f := 0.013349

I am trying to numerically double integrate x^2+sqrt(y), with the bounds y=0..x and x=1..1.5.

Then I tried the following code:

 

int(int(x^2+sqrt(y),method=trapezoid,y=0..x),method=trapezoid,x=1..1.5);

 

I know how to write the code if instead of a 'x' in my upper limit for my integral, I had a real number, but I'm not sure how to remedy to code in order make it work. Any help would be appreciated. Thanks!

 

I am trying to solve the Morrison equation for a normal force acting on a cylinder in a viscous fluid by using the potentional theory. With my limited experience in Maple I do not get why my dubble integral cannot be computed. Any help would be appreciated, tips are also very welkom as I am trying to expand my knowledge of Maple. 

I am facing a kind of strange problem. Whenever I enter Int(exp(-s t) t^2,t) and try to see full solution using Student[Calculus1]:-ShowSolution(), it gives empty square brackets [ ] as superscripts of e. If I restart Maple engine and perform the same, sometimes it produces right solution. Kindly help, what is this? Same integral does not give problem if done with parameter 'r' instead of 's'.

At the internet site of The Heun Project, a strong declaration is made that only Maple incorporates Heun functions, which arise in the solution of differential equations that are extremely important in physics, such as the solution of Schroedinger's equation for the hydrogen atom.  Indeed solutions appear in Heun functions, which are highly obscure and complicated to use because of their five or six arguments, but when one tries to convert them to another function, nothing seems to work.  For instance, if one inquires of FunctionAdvisor(display, HeunG), the resulting list contains

"The location of the "branch cuts" for HeunG are [sic, is] unknown ..." followed by several other "unknown" and an "unable". Such a solution of a differential equation is hollow.

Incidentally, Maple's treatment of integral equations is very weak -- only linear equations with simple solutions, although procedures by David Stoutemyer from 40 years ago are available to enhance this capability.

When can we expect these aspects of Maple to work properly, for applications in physics?

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