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Lecture 28e, 5D five dimensions, independent variables

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

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

.5  ∂4U(w,x,y,z,t)/∂w4 + 1.5 ∂4U(w,x,y,z,t)/∂w∂x∂y∂z + 2.5 ∂4U(w,x,y,z,t)/∂w2∂t2 +
∂4U(w,x,y,z,t)/∂x4 + 2 ∂4U(w,x,y,z,t)/∂y4 + 3 ∂4U(w,x,y,z,t)/∂z4 + 4 ∂4U(w,x,y,z,t)/∂t4 +
5 ∂3U(w,x,y,z,t)/∂x∂y∂t + 6 ∂4U(w,x,y,z,t)/∂y2∂z2 +
7 ∂3U(w,x,y,z,t)/∂z∂t2 + 8 ∂3U(w,x,y,z,t)/∂t3 + 9 ∂3U(w,x,y,z,t)/∂y2∂t + 
10 ∂2U(w,x,y,z,t)/∂z∂t + 11 ∂U(w,x,y,z,t)/∂t + 12 U(w,x,y,z,t) = f(w,x,y,z,t)

With f(w,x,y,z,t) given in pde45.txt, Dirichlet boundary values from U(w,x,y,z,t)

Development requires test cases to verify correctness of code.
Check computing first through fourth order derivative of five variables:
test_5d.ads specification of test functions 
test_5d.adb test functions  
check_test_5d.adb check test functions
check_test_5d_ada.out check output

Now, test a fourth order PDE in five dimensions.
pde45_eq.adb extended pde44_eq.adb
pde45_eq_ada.out verification output

The test case used in pde45_eq was created with the help of Maple
pde45.txt text of Maple output

Plotting solution against 5D independent variables
Designed for interactive changing of variables plotted and variables values.
pot5d_gl.c plot program




∇4U + 2 ∇2U + 5 U = 0

Fourth order Biharmonic PDE solution in 5 dimensions in "C"
pde45h_eq.c extended pde44h_eq.c
pde45h_eq_c.out output

Fourth order Biharmonic PDE solution in 5 dimensions in Fortran 90
pde45h_eq.f90 extended pde44h_eq.f90
pde45h_eq_f90.out output

Fourth order Biharmonic PDE solution in 5 dimensions in Java
pde45h_eq.java extended pde44h_eq.java
pde45h_eq_java.out output

 

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Other links

• CMSC 455 home page
• Syllabus - class dates and subjects, homework dates, reading assignments
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• Projects -
• Partial Lecture Notes, one per WEB page
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• Some brief notes on Ada 95
• An Ada math library (gnatmath95)
• Finite difference approximations for derivatives
• MATLAB examples, some ODE, some PDE
• parallel threads examples
• Reference pages on Taylor series, identities,
  coordinate systems, differential operators
• selected news related to numerical computation

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