International Symposium on Electromagnetic Theory (EMTS 2016)

The fourth edition of the *URSI Commission B School for Young Scientists* is arranged on the occasion of the EMTS 2016. This one-day School is sponsored jointly by URSI Commission B and the EMTS 2016 Local Scientific Committee. The School offers a short, intensive course, where a series of lectures will be delivered by a leading scientist in the Commission B community. Young scientists are encouraged to learn the fundamentals and future directions in the area of electromagnetic theory from these lectures.

Mathematical Models and Numerical Methods

Date: Sunday, August 14, 2016

Venue: Finnish Nature Centre Haltia (program includes excursion to Nuuksio National Park)

Schedule (Coffee breaks are also included):

Lecture 1 | 9:00-12:00 |

Lunch | 12:00-13:00 |

Lecture 2 | 13:00-15:30 |

Outside activity and group discussions | 16:00-18:00 |

Departure at 8.15 Hostel Domus Academica (Helsinki) and at 8.30 Radisson Blu (Espoo)

Return at 18.45 (Haltia – Radisson Blu, Espoo – Hostel Domus Academica, Helsinki)

Mathematical problems arising in electromagnetics and acoustics always attracted attention of mathematicians. The traditional (physical) diffraction theory was created by Huygens (who formulated in 1660s the famous Huygens principle, according to which the wave propagation is caused by secondary sources), Fresnel (1818), Maxwell (1850s), Helmholtz (1859), Kirchhof (1882), and others. The modern level was achieved owing to the studies of Poincare (1892) and Sommerfeld (1896) when it became clear that the mathematical theory of diffraction is connected with certain nonselfadjoint boundary value problems (BVPs) for partial differential equations (PDEs) of mathematical physics in unbounded domains. Recently, this theory has been developed on a completely new mathematical level involving the theory of distributions (Colton, Kress, Kleinmann, Werner, Vineberg, Costabel, Stephan, and others). The corresponding aspects of the theory of pseudodifferential operators were developed by Kohn and Nirenberg (1965), Eskin (1973), Shubin (1978), Taylor (1981), Rempel and Schulze (1982), and Mazja (1970-80s) who considered the problems on manifolds with sharp edges. The typical BVPs associated with the wave diffraction are stated in domains that have noncompact boundaries stretching to infinity and contain inclusions large in comparison with a characteristic parameter of the problem (e.g. wavelength) or dimensions of the boundary inhomogeneities. Recent remarkable progress of computational resources has opened new possibilities for solving such problems based on the use of huge computer clusters employing parallel computations. These circumstances dictate the necessity of deeper studies of mathematical foundations of the electromagnetic field theory that would enable further development and creation of specifically oriented mathematical and numerical methods and techniques.

The course focuses on foundations of the mathematical theory of electromagnetic fields and waves and includes (i) review of differential operations and theorems of the vector analysis; (ii) basic notions of the electromagnetic theory, Maxwell’s equations; (iii) statements and analysis of the boundary value problems (BVPs) for Maxwell’s and Helmholtz equations in unbounded domains associated with the wave diffraction, plane waves, conditions at infinity; (iv) statements and analysis of the BVPs for Maxwell’s and Helmholtz equations associated with the wave propagation in guides, the mathematical nature of electromagnetic waves; (v) introduction to the integral equation method with application to the solution of the BVPs for the Helmholtz equation; and (vi) introduction to the theory of numerical solution of PDEs by finite difference, finite element, and Galerkin methods. The course offers a possibility of solving practical exercises and problems based on the considered basic theoretical items.

Course compendium Course slides

**List of topics for Part 1 (Lecture 1)**

- Differential operations and theorems of the vector analysis.
- Basic electromagnetic theory. Maxwell’s and Helmholtz equations.
- Statements and analysis of the boundary value problems (BVPs) for Maxwell’s and Helmholtz equations in unbounded domains associated with the wave diffraction. Plane waves. Conditions at infinity

**List of topics for Part 2 (Lecture 2)**

- Statements and analysis of the BVPs for Maxwell’s and Helmholtz equations associated with the wave propagation in guides. The mathematical nature of waves.
- Introduction to the integral equation (IE) method with application to the solution of the BVPs for the Helmholtz equation.
- Introduction to the theory of numerical solution of ordinary and partial differential equations by finite difference, finite element, and Galerkin methods.

**List of topics for Part 3 (Outside activity and group discussions)**

A review of the statements and methods concerning basic proofs of the existence and uniqueness of the BVPs for Maxwell’s and Helmholtz in electromagnetic field theory. Solution to some course problems included to the course Compendium.

Dr. Yury Shestopalov, Professor of Mathematics, University of Gävle, Gävle, Sweden

Course tutor: Dr Eugen Smolkin, University of Gävle, Gävle, Sweden

Yury Shestopalov is a Professor of Mathematics at the University of Gävle, Gävle, Sweden. He received the M.S., Ph.D., and D.Sc. (Doctor of Science) degrees from the M. V. Lomonosov Moscow State University, Russia, in 1975, 1978, and 1988, respectively. From 1978 to 2000, he has been working as Assistant Professor, Associate Professor, Professor, and Department Head at the Department of Computational Mathematics and Cybernetics and Kolmogorov School of the M. V. Lomonosov Moscow State University. From 2000 to 2013, he has been occupying the positions of Associate Professor (Docent) and Professor at the Department of Mathematics of the Karlstad University, Sweden. In 1989 he performed postdoctoral research at the Royal Institute of Technology and Stockholm University under the supervision of Lennart Carleson and Staffan Ström.

Professor Shestopalov is the co-author of five books on mathematical methods in electromagnetics, over 60 journal papers, over 120 conference papers and 30 book chapters. He has organized and chaired several big international events including three PIER Symposia and various special sessions in international conferences. His current research interests span over a broad range of areas, including many topics in pure and applied mathematics: partial differential equations, integral equations, numerical methods, spectral theory, nonlinear analysis, inverse problems, and mathematical methods in electromagnetics with applications in optics, scattering and propagation of waves, analysis of nonlinear media. He has been principal investigator of a number of research projects and supervised five Ph.D. works. Dr. Shestopalov is currently on the Editorial Board of Contemporary Analysis and Applied Mathematics, Electromagnetic Waves and Electronic Systems, Advances in Radio Science. In 1995-2001 he was scientific editor of Journal of Communications, Technology, and Electronics. Dr. Shestopalov is IEEE Fellow and Electromagnetic Academy member.

The registration fees for the School are as follows:

- EMTS 2016 participants (except YSA recipients): 80 Euros
- Non EMTS 2016 participants: 100 Euros
- YSA recipients: free

It is strongly recommended that the recipients of the EMTS 2016 Young Scientist Award (YSA) participate in the School.

For any inquiries on the School, please contact:

Professor Ari Sihvola, Chair, URSI Commission B

Aalto University, Espoo, Finland

Email: ari.sihvola@aalto.fi

Professor Kazuya Kobayashi, Vice-Chair, URSI Commission B

Chuo University, Tokyo, Japan

Email: kazuya@tamacc.chuo-u.ac.jp