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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 plot4P + 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 generation4P + 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
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