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Lecture 28g, extending to 7 dimensions 

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


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

∇4U + 2 ∇2U + 8 U = f(x,y,z,t,u,v,w)

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



Test a fourth order PDE in seven dimensions.

∇4U + 2 ∇2U  + 8 U = f(x,y,z,t,u,v,w)
pde47h_nu.adb extended pde46h_nu.adb

pde47h_nu_ada.out verification output
pde47h_eq.adb solver source code
pde47h_eq_ada.out verification output

pde47h_eq.f90 solver source code
pde47h_eq_f90.out verification output



Test a fourth order PDE in seven dimensions.

∇4U + 2 ∇2U  + 8 U = f(x,y,z,t,u,v,w)
pde47h_nu.java extended pde46h_nu.java

pde47h_nu_java.out verification output
pde47h_eq.java solver source code
pde47h_eq_java.out verification output
with i7, j7 in place of iiiiiii, jjjjjjj
pde47hn_eq.java solver source code
pde47hn_eq_java.out verification output


∇4U + 2 ∇2U  + 8 U = f(x,y,z,t,u,v,w)
pde47h_nu.c

pde47h_nu_c.out verification output
pde47h_eq.c solver source code
pde47h_eq_c.out verification output


Plotted with plot7d_gl.c, stopped on one of the moving output
pde47h_nu_c1.dat data written for plot



Boltzmann Equation in seven dimensions, using Galerkin FEM.

∇U(x,y,z,t,p,q,r)  = Sf(x,y,z,t,p,q,r)
boltzmann_7la.java extended fem_check44_la.java
boltzmann_7la_java.out verification output
boltzmann_7la.dat for plot7d_gl and lsfit_file



least square fit of 7 dimensional data

lsfit.adb Least square fit, up to 7 dimensions, 4th order
test_lsfit7.adb Test least square fit, up to 7 dimensions, 4th order
test_lsfit7_ada.out Test results
lsfit.java Least square fit, up to 7 dimensions, 4th order
lsfit_file.java fit polynomial solution to PDE numerical solution
lsfit_file_java7.out Test results


Some programs above also need:


simeq.f90 solver
deriv.f90 numerical derivatives

array3d.ads 3D arrays
array4d.ads 4D arrays
array5d.ads 5D arrays
array6d.ads 6D arrays
array7d.ads 7D 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
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
plot7d_gl.c plot solution to PDE



You won't find many open source or commercial 7D 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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