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Lecture 28m, extending to 9 dimensions 

Just extending eighth order PDE in four dimensions, to nine dimensions


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

∇4U + 2 ∇2U + 10 U = f(x,y,z,t,u,v,w,p,q)

∂4U(x,y,z,t,u,v,w,p,q)/∂x4 + ∂4U(x,y,z,t,u,v,w,p,q)/∂y4 +
∂4U(x,y,z,t,u,v,w,p,q)/∂z4 + ∂4U(x,y,z,t,u,v,w,p,q)/∂t4 +
∂4U(x,y,z,t,u,v,w,p,q)/∂u4 + ∂4U(x,y,z,t,u,v,w,p,q)/∂v4 +
∂4U(x,y,z,t,u,v,w,p,q)/∂w4 + ∂4U(x,y,z,t,u,v,w,p,q)/∂p4 +
∂4U(x,y,z,t,u,v,w,p,q)/∂q4 +
2 ∂2U(x,y,z,t,u,v,w,p,q)/∂x2 + 2 ∂2U(x,y,z,t,u,v,w,p,q)/∂y2 +
2 ∂2U(x,y,z,t,u,v,w,p,q)/∂z2 + 2 ∂2U(x,y,z,t,u,v,w,p,q)/∂t2 +
2 ∂2U(x,y,z,t,u,v,w,p,q)/∂u2 + 2 ∂2U(x,y,z,t,u,v,w,p,q)/∂v2 +
2 ∂2U(x,y,z,t,u,v,w,p,q)/∂w2 + 2 ∂2U(x,y,z,t,u,v,w,p,q)/∂p2 +
2 ∂2U(x,y,z,t,u,v,w,p,q)/∂q2 +
10 U(x,y,z,t,u,v,w,p,q) = f(x,y,z,t,u,v,w,p,q)


Maple check on solution

pde49hn_mws.out analytic solution



Test a fourth order PDE in nine dimensions.


∇4U + 2 ∇2U  + 10 U = f(x,y,z,t,u,v,w,p,q)
pde49hn_eq.c solver source code
pde49hn_eq_c.out verification output

pde49h_eq.adb solver source code
pde49h_eq_ada.out verification output

pde49hn_eq.java solver source code
pde49hn_eq_java.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 pde49hn_eq.c execution
plot9d.java plotter source code



User can select any two variables for 3D view.
User can select values for other variables, option to run all cases.




You won't find many open source or commercial 9D 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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