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Mariusz Iwaniuk

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These are answers submitted by Mariusz Iwaniuk

Hard for symbolic, easy for numerics....

Mariusz Iwaniuk 1476
April 07 2019
0 0

For symbolic solver is hard to find a solution. With numerics it's easy.See attached file.

_equations.mw

with(IntegrationTools):...

Mariusz Iwaniuk 1476
March 24 2019
2 0

More info in file:
 

Download integral.mw

 

Another ways:...

Mariusz Iwaniuk 1476
March 03 2019
1 0

Another ways:

1.Export to *.obj file and view in Online 3D Viewer.

Export("3dplot.obj", plot3d((x^2+y^2)/sqrt(x^2+y^2), x = -1 .. 1, y = -1 .. 1), base = homedir);

where homedir you can check by command:

kernelopts(homedir);

 

2.Export to *.stl file and view in Free online STL viewer:

Export("3dplot.stl", plot3d((x^2+y^2)/sqrt(x^2+y^2), x = -1 .. 1, y = -1 .. 1), base = homedir);

Not full answer....

Mariusz Iwaniuk 1476
January 13 2019
0 1

Almost is possible by Laplace Transform to find the solution of differenatial equation.

Problem is solving the nonlinear recurrence equation(See in worksheet at the end of page).
 

Download Laplace.mw

 

Adjustment...

Mariusz Iwaniuk 1476
January 11 2019
1 0

To insert Labels use CTRL+L on keyboard.


ode_problem.mw

Simpler way...

Mariusz Iwaniuk 1476
December 31 2018
1 1


 

> 

restart
> 

kernelopts(version)

`Maple 2018.2, X86 64 WINDOWS, Oct 23 2018, Build ID 1356656`

 (1)
> 

NULL
> 

`assuming`([simplify(ln(r)*sin(k1*Pi*ln(r/Rs)/ln(r4/Rs))/r)], [Rs > 0, r4 > 0, r4 > Rs, k1::posint])

ln(r)*sin(k1*Pi*(ln(Rs)-ln(r))/(-ln(r4)+ln(Rs)))/r

 (2)
> 

`assuming`([int(ln(r)*sin(k1*Pi*(ln(Rs)-ln(r))/(-ln(r4)+ln(Rs)))/r, r = Rs .. r4)], [Rs > 0, r4 > 0, r4 > Rs, k1::posint])

-(ln(r4)-ln(Rs))*(ln(r4)*(-1)^k1-ln(Rs))/(Pi*k1)

 (3)
> 

NULL
> 

evalf(eval(-(ln(r4)-ln(Rs))*(ln(r4)*(-1)^k1-ln(Rs))/(Pi*k1), [Rs = 2, r4 = 1, k1 = 3]))

-0.5097764806e-1

 (4)
> 

evalf(Int(eval(ln(r)*sin(k1*Pi*ln(r/Rs)/ln(r4/Rs))/r, [Rs = 2, r4 = 1, k1 = 3]), r = 2 .. 1))

-0.5097764806e-1

 (5)
> 

NULL
> 

`assuming`([simplify(sin(k1*Pi*ln(r/Rs)/ln(r4/Rs))*(r/r4)^(m1*Pi*`τds`/d)/r)], [Rs > 0, r4 > 0, r4 > Rs, k1::posint, m1::posint, d > 0, `τds` > 0])

sin(k1*Pi*(ln(Rs)-ln(r))/(-ln(r4)+ln(Rs)))*r4^(-m1*Pi*`τds`/d)*r^((`τds`*Pi*m1-d)/d)

 (6)
> 

`assuming`([int(sin(k1*Pi*(ln(Rs)-ln(r))/(-ln(r4)+ln(Rs)))*r4^(-m1*Pi*`τds`/d)*r^((`τds`*Pi*m1-d)/d), r = Rs .. r4)], [Rs > 0, r4 > 0, r4 > Rs, k1::posint, m1::posint, d > 0, `τds` > 0])

d*(2*ln(Rs)^2*sin((1/2)*Pi*k1)*cos((1/2)*Pi*k1)*m1*`τds`-4*ln(Rs)*ln(r4)*sin((1/2)*Pi*k1)*cos((1/2)*Pi*k1)*m1*`τds`+2*ln(r4)^2*sin((1/2)*Pi*k1)*cos((1/2)*Pi*k1)*m1*`τds`+2*ln(Rs)*cos((1/2)*Pi*k1)^2*d*k1-2*ln(r4)*cos((1/2)*Pi*k1)^2*d*k1-ln(Rs)*d*k1+ln(r4)*d*k1-r4^(-m1*Pi*`τds`/d)*Rs^(m1*Pi*`τds`/d)*d*k1*ln(Rs)+r4^(-m1*Pi*`τds`/d)*Rs^(m1*Pi*`τds`/d)*d*k1*ln(r4))/(Pi*(ln(Rs)^2*m1^2*`τds`^2-2*ln(r4)*m1^2*`τds`^2*ln(Rs)+ln(r4)^2*m1^2*`τds`^2+d^2*k1^2))

 (7)
> 

`assuming`([simplify(combine(d*(2*ln(Rs)^2*sin((1/2)*k1*Pi)*cos((1/2)*k1*Pi)*m1*`τds`-4*ln(Rs)*ln(r4)*sin((1/2)*k1*Pi)*cos((1/2)*k1*Pi)*m1*`τds`+2*ln(r4)^2*sin((1/2)*k1*Pi)*cos((1/2)*k1*Pi)*m1*`τds`+2*ln(Rs)*cos((1/2)*k1*Pi)^2*d*k1-2*ln(r4)*cos((1/2)*k1*Pi)^2*d*k1-ln(Rs)*d*k1+ln(r4)*d*k1-r4^(-m1*Pi*`τds`/d)*Rs^(m1*Pi*`τds`/d)*d*k1*ln(Rs)+r4^(-m1*Pi*`τds`/d)*Rs^(m1*Pi*`τds`/d)*d*k1*ln(r4))/(Pi*(ln(Rs)^2*m1^2*`τds`^2-2*ln(r4)*m1^2*`τds`^2*ln(Rs)+ln(r4)^2*m1^2*`τds`^2+d^2*k1^2))))], [Rs > 0, r4 > 0, r4 > Rs, k1::posint, m1::posint, d > 0, `τds` > 0])

(ln(r4)-ln(Rs))*d^2*k1*(Rs^(m1*Pi*`τds`/d)*r4^(-m1*Pi*`τds`/d)-(-1)^k1)/(Pi*(ln(Rs)^2*m1^2*`τds`^2-2*ln(r4)*m1^2*`τds`^2*ln(Rs)+ln(r4)^2*m1^2*`τds`^2+d^2*k1^2))

 (8)
> 

NULL
> 

evalf(eval((ln(r4)-ln(Rs))*d^2*k1*(Rs^(m1*Pi*`τds`/d)*r4^(-m1*Pi*`τds`/d)-(-1)^k1)/(Pi*(ln(Rs)^2*m1^2*`τds`^2-2*ln(r4)*m1^2*`τds`^2*ln(Rs)+ln(r4)^2*m1^2*`τds`^2+d^2*k1^2)), [Rs = 2, r4 = 1, k1 = 3, m1 = 2, d = 1/3, `τds` = 1/4]))

-1.786983541

 (9)
> 

evalf(Int(eval(sin(k1*Pi*ln(r/Rs)/ln(r4/Rs))*(r/r4)^(m1*Pi*`τds`/d)/r, [Rs = 2, r4 = 1, k1 = 3, m1 = 2, d = 1/3, `τds` = 1/4]), r = 2 .. 1))

-1.786983542

 (10)
> 

``


 

Download 2_integrals.mw

PDEtools...

Mariusz Iwaniuk 1476
December 28 2018
1 0

Probably pdsolve is designed to find only one solution not more.We can find more solution using packages PDEtools:

Download By_PDE-Tools.mw

 

Simpler code:...

Mariusz Iwaniuk 1476
December 13 2018
0 4 
int(convert(cos(2*x)/(1+2*sin(3*x)^2), exp), x = 0 .. Pi);

# 0

 

Solution by another math software....

Mariusz Iwaniuk 1476
December 02 2018
1 1

Using Rubi on Mathematica I have got a answer for case 3D.

See attached file for details:

Integral_3D.mw

....

Mariusz Iwaniuk 1476
November 12 2018
2 4

 To reproduce the computation below in Maple 2018.1 you need to install theMaplesoft Physics Updates (version 134 or higher)


 

> 

restart
> 

kernelopts(version)

`Maple 2018.2, X86 64 WINDOWS, Oct 23 2018, Build ID 1356656`

 (1)
> 

Physics:-Version()
> 

NULL
> 

PackageTools:-Install("5137472255164416", version = 192, overwrite)
> 

restart
> 

PDE := d*cp*(diff(T(t, z), t)) = k*(diff(T(t, z), z, z))+Q

d*cp*(diff(T(t, z), t)) = k*(diff(diff(T(t, z), z), z))+Q

 (2)
> 

sol := pdsolve({PDE, T(0, z) = Tamb, (D[2](T))(t, 0) = -h*(-Tamb+T(t, 0))/k, (D[2](T))(t, 8) = 0})

T(t, z) = `casesplit/ans`(Int(Sum(-128*exp(-(1/64)*lambda[n]^2*k*(t-tau)/(cp*d))*(-(1/64)*lambda[n]*(d^(1/2)*cp^(1/2)*k^(5/2)*(k/(d*cp))^(1/2)*sin(d^(1/2)*cp^(1/2)*lambda[n]*(k/(d*cp))^(1/2)/k^(1/2))*lambda[n]+8*cos(d^(1/2)*cp^(1/2)*lambda[n]*(k/(d*cp))^(1/2)/k^(1/2))*h*k^2-8*k^2*h)*cos((1/8)*d^(1/2)*cp^(1/2)*lambda[n]*(k/(d*cp))^(1/2)*z/k^(1/2))+sin((1/8)*d^(1/2)*cp^(1/2)*lambda[n]*(k/(d*cp))^(1/2)*z/k^(1/2))*h*(d^(1/2)*cp^(1/2)*k^(1/2)*(k/(d*cp))^(1/2)*cos(d^(1/2)*cp^(1/2)*lambda[n]*(k/(d*cp))^(1/2)/k^(1/2))*h-d^(1/2)*cp^(1/2)*k^(1/2)*(k/(d*cp))^(1/2)*h+(1/8)*sin(d^(1/2)*cp^(1/2)*lambda[n]*(k/(d*cp))^(1/2)/k^(1/2))*k^2*lambda[n]))*Q*k^(1/2)/(d^(1/2)*cp^(1/2)*(k/(d*cp))^(1/2)*(16*d^(1/2)*cp^(1/2)*k^(3/2)*(k/(d*cp))^(1/2)*cos(d^(1/2)*cp^(1/2)*lambda[n]*(k/(d*cp))^(1/2)/k^(1/2))^2*h*lambda[n]+(k^3*lambda[n]^2-64*h^2*k)*sin(d^(1/2)*cp^(1/2)*lambda[n]*(k/(d*cp))^(1/2)/k^(1/2))*cos(d^(1/2)*cp^(1/2)*lambda[n]*(k/(d*cp))^(1/2)/k^(1/2))+64*(k/(d*cp))^(1/2)*lambda[n]*cp^(1/2)*d^(1/2)*(h^2*k^(1/2)+(1/64)*k^(5/2)*lambda[n]^2-(1/4)*k^(3/2)*h))), n = 0 .. infinity), tau = 0 .. t)+Tamb, {And(tan(lambda[n])*k^(1/2)*d^(1/2)*cp^(1/2)*(lambda[n]^2*k/(cp*d))^(1/2)+8*h = 0, -infinity <= lambda[n] and lambda[n] <= infinity), And(tan(lambda[n])*(k^2)^(1/4)*d^(1/2)*cp^(1/2)*(lambda[n]^2*(k^2)^(1/2)/(cp*d))^(1/2)+8*h = 0, -infinity <= lambda[n] and lambda[n] <= infinity)})

 (3)
> 

``


 

Download PDE.mw

UPDATED:

Physics package was fixed and now with version 195 works fine.

UPDATED 15.11.2018:

Solution with Physics:-Version 200:

PDE_with_Physics-version_200.mw

Thumb if You like.

Bug!...

Mariusz Iwaniuk 1476
November 11 2018
1 2

Yes it's a Bug in KroneckerDelta function for m=4 it should be 0 not 1.

with(Physics, KroneckerDelta);

[seq(KroneckerDelta[m, 0], m = 0 .. 10)];

#[1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]

but It should be: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

See workaround in attached file:

Download bug.mw

 

Thumb if You like.

 

Workaround....

Mariusz Iwaniuk 1476
November 11 2018
1 0

infolevel[IntegrationTools] := 3;

`assuming`([int(GAMMA(a, s)*exp(-b*s), s = 0 .. infinity)], [a > 0, b > 0]);

#Definite Integration:   Integrating expression on s=0..infinity
Definite Integration:   Using the integrators [distribution, piecewise, series, o, polynomial, ln, lookup, cook, ratpoly, elliptic, elliptictrig, meijergspecial, improper, asymptotic, ftoc, ftocms, meijerg, contour]
IntegralTransform LookUp Integrator:   Integral might be a laplace transform with expr=GAMMA(a,s) and s=b.
LookUp Integrator:   unable to find the specified integral in the table
Definite Integration:   Returning integral unevaluated.
 

Then we see Maple can't find in Table a Laplace Transform(?) ,but:

inttrans:-laplace(GAMMA(a, s), s, b);#Works fine

#GAMMA(a)*(1-(1+b)^(-a))/b

 

Thumb if You like.

 

....

Mariusz Iwaniuk 1476
November 09 2018
1 0

If is indefine integral then is not possible.

More info in attached file.

integral.mw

 

Only workaround....

Mariusz Iwaniuk 1476
October 30 2018
1 2

INT := int(exp(-a^2-b^2-c^2), a = b^2/(4*c) .. infinity);

int(INT, b = -infinity .. infinity, c = 0 .. infinity, numeric);

#0.9786008283

 

Add......

Mariusz Iwaniuk 1476
October 27 2018
1 0

Add option to roots: sqrt(2).

roots(x^3-x^2-8*x+8, sqrt(2));

#[[-2*sqrt(2), 1], [1, 1], [2*sqrt(2), 1]]

 

Thumb if You like.

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