Ron Buckmire http://faculty.oxy.edu/ron Application of Mickens finite differences to single-variable Bratu problems Ron Buckmire http://faculty.oxy.edu/ron.

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Presentation transcript:

Ron Buckmire http://faculty.oxy.edu/ron Application of Mickens finite differences to single-variable Bratu problems Ron Buckmire http://faculty.oxy.edu/ron ron@oxy.edu Occidental College Los Angeles, CA SIAM Annual Meeting 2003 Montreal, Canada

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

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

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

1-D Planar Bratu Problem Exact Solution Note λc = coth(λc) = 3.51380719

1-D Planar Bratu solution curves

Numerical Method for 1-D Bratu Standard discretization Mickens discretization

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

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

Bratu-Gelfand solution curves

Numerical Methods for cylindrical Bratu-Gelfand Standard discretization Mickens discretization

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

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

SIAM Annual Meeting 2003 Montreal, Canada Ron Buckmire http://faculty.oxy.edu/ron ron@oxy.edu Occidental College Los Angeles, CA