A boundary integral approach to the eigenvalue problem for the anisotropic bidomain operator with perfect contact conditions
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| Format: | Preprint |
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2026
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| author | Felipe-Sosa, Raul García-Gómez, Yofre H. |
| author_facet | Felipe-Sosa, Raul García-Gómez, Yofre H. |
| contents | In this work, we study the eigenvalue problem associated with the bidomain operator in an anisotropic heterogeneous domain composed of three subregions representing the left ventricle, the septum, and the right ventricle. The anisotropic conductivity, together with the different orientations of the fiber directions in each subdomain, leads to an elliptic boundary value problem with discontinuous coefficients and transmission conditions across the interfaces.
Our main contribution consists in reformulating this problem using potential theory. By expressing the solution in terms of single- and double-layer potentials, we reduce the original boundary value problem to a system of Fredholm-type boundary integral equations. We derive explicit expressions for the fundamental solution of the associated anisotropic Helmholtz operator, as well as for the corresponding kernels, which are given in terms of Bessel functions.
Finally, we propose a numerical scheme for approximating the eigenvalues of the bidomain operator based on the discretization of the resulting integral system. This approach provides an efficient framework for the analysis of anisotropic boundary value problems with interface conditions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_03682 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A boundary integral approach to the eigenvalue problem for the anisotropic bidomain operator with perfect contact conditions Felipe-Sosa, Raul García-Gómez, Yofre H. Analysis of PDEs 35J05, 35P15, 45B05, 45L10, 92C50 In this work, we study the eigenvalue problem associated with the bidomain operator in an anisotropic heterogeneous domain composed of three subregions representing the left ventricle, the septum, and the right ventricle. The anisotropic conductivity, together with the different orientations of the fiber directions in each subdomain, leads to an elliptic boundary value problem with discontinuous coefficients and transmission conditions across the interfaces. Our main contribution consists in reformulating this problem using potential theory. By expressing the solution in terms of single- and double-layer potentials, we reduce the original boundary value problem to a system of Fredholm-type boundary integral equations. We derive explicit expressions for the fundamental solution of the associated anisotropic Helmholtz operator, as well as for the corresponding kernels, which are given in terms of Bessel functions. Finally, we propose a numerical scheme for approximating the eigenvalues of the bidomain operator based on the discretization of the resulting integral system. This approach provides an efficient framework for the analysis of anisotropic boundary value problems with interface conditions. |
| title | A boundary integral approach to the eigenvalue problem for the anisotropic bidomain operator with perfect contact conditions |
| topic | Analysis of PDEs 35J05, 35P15, 45B05, 45L10, 92C50 |
| url | https://arxiv.org/abs/2604.03682 |