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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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