Dirichlet principle using computers

Abstract

In this article we shall give practical and numerical solutions of the Laplace equation on multidimensional spaces and show the numerical experiments by using computers. Our method is based on the Dirichlet principle by combinations with generalized inverses, Tikhonov's regularization and the theory of reproducing kernels.

Keywords:

• Laplace equation
• Dirichlet problem
• Inverse problem
• Approximation of functions
• Reproducing kernel
• Tikhonov regularization
• Sobolev space
• Generalized inverse
• Approximate inverse
• AMS Subject Classifications: Primary 30C40
• 31A05
• 31B05
• 35J05
• 45B05

Acknowledgements

T. Matsuura is supported in part by the Gunma University Foundation for the Promotion of Science and Engineering. S. Saitoh is supported in part by the Grant-in-Aid for Scientific Research (C)(2) (No. 16540137) from the Japan Society for the Promotion of Science.

Notes

Email: [email protected]

Additional information

Notes on contributors

Saburou Saitoh §

Email: [email protected]