1. Ron Buckmire http://faculty.oxy.edu/ron
Application of Mickens finite differences to single-variable Bratu problems
Ron Buckmire
Occidental College
Los Angeles, CA
SIAM Annual Meeting Montreal, Canada

2. Outline Mickens finite differences Classic Bratu problem
Planar 1-dimensional Bratu problem
Numerical results
Cylindrical Bratu-Gelfand problem
Future work
Spherical Bratu-Gelfand problem

3. Mickens finite difference
Nonstandard discretization
Nonlinear denominator function
“Non-local discretization”
This is an exact Mickens discretization scheme.

4. Classic Bratu Problem
Appears in thermal combustion theory and elsewhere…
Quasi-elliptical boundary value problem
Nonlinear “eigenvalue” problem
Bifurcation problem in λ
No solution for λ > λc
Two solutions for 0< λ < λc

5. 1-D Planar Bratu Problem
Exact Solution
Note λc = coth(λc) =

6. 1-D Planar Bratu solution curves

7. Numerical Method for 1-D Bratu
Standard discretization
Mickens discretization

8. Numerical Results for 1D Bratu
Comparison of standard error (dashed) and Mickens error (solid)

9. Bratu-Gelfand Problem (in cylindrical coordinates)
Exact Solution
Note λc = 2

10. Bratu-Gelfand solution curves

11. Numerical Methods for cylindrical Bratu-Gelfand
Standard discretization
Mickens discretization

12. Numerical Results
Absolute error due to Mickens discretization for N=100, 200,400, 1000

13. Future work
Other singular boundary value problems… (w/ or w/o exact solutions)
Spherical Bratu-Gelfand problem
No known exact solution
Infinitely-valued solution at λ =2

14. SIAM Annual Meeting 2003 Montreal, Canada
Ron Buckmire
Occidental College Los Angeles, CA