Direct method of solution for general boundary value problem of the Laplace equation |
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| Affiliation: | 1. Graduate School of Science and Engineering, Ibaraki University, Mito 310-8512, Japan;2. Kyushu Institute of Information Sciences, Dazaifu 818-0117, Japan;3. Department of Mathematical Sciences, Ibaraki University, Mito 310-8512, Japan;1. Department of Chemical Engineering, National Cheng Kung University, Tainan 701, Taiwan;2. National Synchrotron Radiation Research Center, Hsinchu 300, Taiwan;1. Département de Mathématiques, Université Libre de Bruxelles, Boulevard du Triomphe, 1050 Brussels, Belgium;2. Dipartimento di Matematica, Università di Roma La Sapienza, P. le A. Moro 2, 00185 Roma, Italy;3. Dipartimento di Matematica “Giuseppe Peano”, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy;1. Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda-shi, Chiba, 278-8510, Japan;2. Department of Creative Engineering, National Institute of Technology, Kushiro College, 2-32-1 Otanoshike-Nishi, Kushiro-Shi, Hokkaido 084-0916, Japan;1. College of Science, Hohai University, Nanjing 210098, PR China;2. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China |
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| Abstract: | A general boundary value problem for two-dimensional Laplace equation in the domain enclosed by a piecewise smooth curve is considered. The Dirichlet and the Neumann data are prescribed on respective parts of the boundary, while there is the second part of the boundary on which no boundary data are given. There is the third part of the boundary on which the Robin condition is prescribed. This problem of finding unknown values along the whole boundary is ill posed. In this sense we call our problem an inverse boundary value problem. In order for a solution to be identified the inverse problem is reformulated in terms of a variational problem, which is then recast into primary and adjoint boundary value problems of the Laplace equation in its conventional form. A direct method for numerical solution of the inverse boundary value problem using the boundary element method is presented. This method proposes a non-iterative and unified treatment of conventional boundary value problem, the Cauchy problem, and under- or over-determined problems. |
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