<- previous    index    next ->

Lecture 28f, 6D six dimensions, Biharmonic 

Just extending fourth order PDE in four dimensions, to six dimensions


Desired solution is U(v,w,x,y,z,t), given PDE:

∇4U + 2 ∇2U + 6 U = 0

∂4U(v,w,x,y,z,t)/∂v4 + ∂4U(v,w,x,y,z,t)/∂w4 +
∂4U(v,w,x,y,z,t)/∂x4 + ∂4U(v,w,x,y,z,t)/∂y4 +
∂4U(v,w,x,y,z,t)/∂z4 + ∂4U(v,w,x,y,z,t)/∂t4 +
2 ∂2U(v,w,x,y,z,t)/∂v2 + 2 ∂2U(v,w,x,y,z,t)/∂w2 +
2 ∂2U(v,w,x,y,z,t)/∂x2 + 2 ∂2U(v,w,x,y,z,t)/∂y2 +
2 ∂2U(v,w,x,y,z,t)/∂z2 + 2 ∂2U(v,w,x,y,z,t)/∂t2 +
6 U(v,w,x,y,z,t) = 0



6D cubes and spheres

faces.c program
faces.out Data on 2D to 6D
test_faces.c program
test_faces.out sphere data

faces.java program
faces_java.out Data on 2D to 6D
test_faces.java program
test_faces_java.out sphere equations test




Test a fourth order PDE in six dimensions.

∇4U + 2 ∇2U  + 6 U = 0
pde46h_eq.adb extended pde44h_eq.adb

pde46h_eq_ada.out verification output

∇4U + 2 ∇2U  + 6 U = 0
pde46h_eq.c

pde46h_eq_c.out verification output

∇4U + 2 ∇2U  + 6 U = 0
pde46h_eq.f90

pde46h_eq_f90.out verification output

∇4U + 2 ∇2U  + 6 U = 0
pde46h_eq.java

pde46h_eq_java.out verification output


Plotted with plot6d_gl.c, stopped on one of the moving output
pde46h_eq_java.dat data written for plot

∇4P + 2 ∇2P  = f(x,y,z,t,u,v)
Solving for P = exp(x+y+t*z+u+v) six independent variables, 4th order
pde64eb_eq.c testing source code
pde64eb_eq_c.out verification output
d6p4eb.mw Maple test generation

∇4P + 2 ∇2P  = f(x,y,z,t,u,v)

Solving for P = sin(x*u+y*v+z+t) six independent variables, 4th order
pde64sb_eq.c testing source code
pde64sb_eq_c.out verification output
d6p4sb.mw Maple test generation
d6p4sb_rhs.jpg RHS




Plotting solution against 6D independent variables

Designed for interactive changing of variables plotted and variables values.
pot6d_gl.c plot program




With a small change, we obtain a nonuniform refinement

Now we use "nuderiv" rather than "rderiv" that can use
nonuniform and different grids in each dimension.



Test a fourth order PDE in six dimensions with nonuniform refinement.

∇4U + 2 ∇2U  + 6 U = 0
pde46h_nu.adb extended pde46h_eq.adb

pde46h_nu_ada.out verification output

∇4U + 2 ∇2U  + 6 U = 0
pde46h_nu.java

pde46h_nu_java.out verification output

∇4U + 2 ∇2U  + 6 U = 0
pde46h_nu.c

pde46h_nu_c.out verification output
 


Plotting solution against 6D independent variables

Designed for interactive changing of variables plotted and variables values.
pot6d_gl.c plot program


pde46h_nu_ada.jpg


Least Square Fit 6 independent variables up to sixth power

lsfit.ads Least Square Fit 6D 6P
lsfit.adb Least Square Fit 6D 6P
test_lsfit6.adb test program
test_lsfit6_ada.out test results


Some programs above also need

simeq.f90 solve simultaneous equations
deriv.f90 compute numerical derivatives
array3d.ads 3D arrays
array4d.ads 4D arrays
array5d.ads 5D arrays
array6d.ads 6D arrays
real_arrays.ads 2D arrays and operations
real_arrays.adb 2D arrays and operations
integer_arrays.ads 2D arrays and operations
integer_arrays.adb 2D arrays and operations


You won't find many free or commercial 5D and 6D PDE packages


many lesser problems have many opensource and commercial packages

en.wikipedia.org/wiki/list_of_finite_element_software_packages


    <- previous    index    next ->

Other links

Go to top