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Using the Complex Polynomial Method with Mathematica to Model Problems Involving the Laplace and Poisson Equations

A. C. Poler
A. W. Bohannon
S. J. Schowalter
T. V. Hromadka II
Journal / Anthology

Year: 2009
Volume: 25
Issue: 3
Page range: 657-667

The complex polynomial method variant of the well-known complex variable boundary element method (CVBEM) is reexamined in its utility in solving Partial Differential Equations (PDE) of the Poisson and Laplace type. Because the CVBEM was recently extended to three and higher dimensions, the use of complex polynomials to solve higher dimension PDE becomes apparent and therefore the advantages afforded by the use of complex polynomials can be brought to focus on higher dimension problems. Because complex polynomials involve use of computational algorithms that require high accuracy in numerical precision, including the solution of fully populated nonsymmetric matrices, the computer program Mathematica is evaluated for use as the underlying computational engine. Furthermore, Mathematica is evaluated for its internal high-accuracy computational features and algorithms, including ease of program setup. In this research, the new program is found to provide at least a 5-fold increase in complex polynomial degree utilization (from degree 10 to degree 50), with computational speed less than was involved in the original degree 10 approximation of Hromadka and Guymon [ASCE J Hydraulic Eng 110 (1984), 329-339], and with exceptional computational accuracy and reporting features. The Mathematica program is quite small and is provided to the reader as freeware and can be obtained from the first author.


complex polynomials, Laplace's equation, numerical methods, Dirichlet problem