@sand15 

Thank you. It is indeed not a question, but more of a small investigation into how to deal with a PDE.

Now only for u(x,t) solutions surfaces

The plot legend now attached makes it much clearer, because the start for this PDE solution is when C(t) = 0 is an initial surface, but I do not see this initial surface in the plot?

An Explore plot would also be a good idea to see the solution surfaces for different boundary conditions.

@Rouben Rostamian  

It's all done via AI, because I'm not a experienced Maple user, as I don't have the knowledge at my fingertips at the moment.

PlotCilinderInBol(0., 1, true)

Rotating spherical surface  containing a moving cylinder 
The cylinder diameter can be adjusted and the animation can be turned on or off.

bol_en_cilinder_animatie_mprimes_10-10-2025.mw

@Rouben Rostamian  
I do nothing with transparency.
The second new code does not affect transparency, but simply draws now a hole in the spherical surface.

For a surface other than a sphere, you could determine the 3D intersection curve, turn it into a white surface, and position it.

animatie_halve_bol_met_cilinder_mprimes_recreatief_9-10-2025.mw

@salim-barzani 
t_plot_domein_definition_for_functie_K_mprimes_8-10-2025.mw

@sand15 
Thanks for the oversight of this subject.

BesselI and BesselK are the modified Bessel functions of the first and second kinds, respectively.  They satisfy the modified Bessel equation:
            "x^2*`y''` + x*`y'` - (v^2 + x^2)*y = 0"
Seems that here is no explicit third kind modified bessel function in Maple ?

There is a singularity at (0,0)  ?

@Kitonum 
Thanks 

with(Optimization);
[ImportMPS, Interactive, LPSolve, LSSolve, Maximize, Minimize, 

  NLPSolve, QPSolve]

Minimize(419*x^2 + 116*x*y - 426*x*z + 78*y^2 - 142*y*z + 133*z^2 - 1604*x - 682*y + 1086*z + 2306);
       [0., [x = 7.00000000000007, y = 11.0000000000001, 

         z = 13.0000000000002]]

Seems to be not strict symbolic ?

@salim-barzani 
Is this function for the 3 plots on different times ?
"can you plot by Ai you have lets see how many shape you have ?"   what are shapes ?

2*((t*(-alpha*conjugate(lambda[1] + lambda[2]*I)^3 + b*conjugate(lambda[1] + lambda[2]*I) + c*r[2] + a) + 2*beta*(y*conjugate(lambda[1] + lambda[2]*I) + z*r[2] + x))/(2*beta) + (t*(-alpha*(lambda[1] + lambda[2]*I)^3 + b*(lambda[1] + lambda[2]*I) + c*r[1] + a) + 2*beta*(y*(lambda[1] + lambda[2]*I) + z*r[1] + x))/(2*beta))/(((t*(-alpha*(lambda[1] + lambda[2]*I)^3 + b*(lambda[1] + lambda[2]*I) + c*r[1] + a) + 2*beta*(y*(lambda[1] + lambda[2]*I) + z*r[1] + x))*(t*(-alpha*conjugate(lambda[1] + lambda[2]*I)^3 + b*conjugate(lambda[1] + lambda[2]*I) + c*r[2] + a) + 2*beta*(y*conjugate(lambda[1] + lambda[2]*I) + z*r[2] + x)))/(4*beta^2) - 4/((lambda[1] + lambda[2]*I - conjugate(lambda[1] + lambda[2]*I))^2*alpha))

@salim-barzani 

That resembles a normal 3D plot at u= 0 , where you see peaks and a density plot at level -1.

Is that possible?

@salim-barzani 
conjugate_functie_voorschriftmprimes25-9-2025.mw

@salim-barzani 

It seems to be incorrect in Maple to use direct index variables (such as Lambda) in an arrow operator function.(Line -1-Done)

It is possible to do so indirectly, as shown in the plot example.

"how many graph and hsape of design you can make it?"      hsape ?

The z= 0 is only for creating a contour plot over the x-y plane of the lumps. ?

@salim-barzani 
? 
trajectory_lump_vraag_mprimes_25-9-2025.mw

@acer Thanks

This seems the shortest solution. 

eq := B12 = -6*(p1 + p2)/(p1 - p2)^2;
F2 := (theta1*theta2*p1^2 + (-2*p2*theta1*theta2 - 6)*p1 + theta1*theta2*p2^2 - 6*p2)/(p1 - p2)^2;


simplify(F2, {eq}, [p1, p2]);