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Just extending seventh order PDE in four dimensions, to eight dimensions
Desired solution is U(x,y,z,t,u,v,w,p), given PDE:
∇4U + 2 ∇2U + 8 U = f(x,y,z,t,u,v,w,p)
∂4U(x,y,z,t,u,v,w,p)/∂x4 + ∂4U(x,y,z,t,u,v,w,p)/∂y4 +
∂4U(x,y,z,t,u,v,w,p)/∂z4 + ∂4U(x,y,z,t,u,v,w,p)/∂t4 +
∂4U(x,y,z,t,u,v,w,p)/∂u4 + ∂4U(x,y,z,t,u,v,w,p)/∂v4 +
∂4U(x,y,z,t,u,v,w,p)/∂w4 + ∂4U(x,y,z,t,u,v,w,p)/∂p4 +
2 ∂2U(x,y,z,t,u,v,w,p)/∂x2 + 2 ∂2U(x,y,z,t,u,v,w,p)/∂y2 +
2 ∂2U(x,y,z,t,u,v,w,p)/∂z2 + 2 ∂2U(x,y,z,t,u,v,w,p)/∂t2 +
2 ∂2U(x,y,z,t,u,v,w,p)/∂u2 + 2 ∂2U(x,y,z,t,u,v,w,p)/∂v2 +
2 ∂2U(x,y,z,t,u,v,w,p)/∂w2 + 2 ∂2U(x,y,z,t,u,v,w,p)/∂p2 +
8 U(x,y,z,t,u,v,w,p) = f(x,y,z,t,u,v,w,p)
Test a fourth order PDE in eight dimensions.
∇4U + 2 ∇2U + 8 U = f(x,y,z,t,u,v,w,p)
pde48hn_eq.java solver source code
pde48hn_eq_java.out verification output
∇4U + 2 ∇2U + 8 U = f(x,y,z,t,u,v,w,p)
pde48hn_eq.c solver source code
pde48hn_eq_c.out verification output
pde48hn_eq.adb solver source code
pde48hn_eq_ada.out verification output
Some programs above also need:
nuderiv.java basic non uniform grid derivative
rderiv.java basic uniform grid derivative
simeq.java basic simultaneous equation
deriv.h basic derivatives
deriv.c basic derivatives
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
Plotted output from pde48hn_eq.java execution
User can select any two variables for 3D view.
User can select values for other variables, option to run all cases.
Then, going to a spherical coordinate system in 8 dimensions
gen_8d_sphere.c source equations
gen_8d_sphere_c.out verification output
You won't find many open source or commercial 8D PDE packages
many lesser problems have many open source and commercial packages
en.wikipedia.org/wiki/list_of_finite_element_software_packages
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