Mathematical Methods for Wave Phenomena

Dear Colleagues,

Wave phenomena play a critical role in a wide range of scientific and engineering disciplines, encompassing everything from sound waves and seismic waves to electromagnetic waves. Understanding and accurately modeling these phenomena are essential for advancements in technology and scientific knowledge. In particular, the mathematical modeling of electromagnetic waves is foundational for numerous applications, such as communication systems, sensor development, and geophysical exploration. A precise solution to Maxwell's equations, for instance, is pivotal for the design of new electromagnetic sensors and the optimization of communication systems, including emerging technologies like 6G.

This modeling is also crucial in solving the so-called inverse problem, which involves determining the characteristics of a system or device based on its response to certain excitations. In electromagnetics, these responses are typically the electromagnetic fields or related quantities, and the characteristics to be determined might include the shape and electromagnetic properties of the materials involved. Solving the inverse problem has applications ranging from the optimization of microwave circuits and radar systems to remote sensing and imaging. Remote sensing, for example, involves determining the distribution and properties of materials (such as biological tissues in medical diagnostics) by analyzing their response to electromagnetic waves in the microwave frequency range and beyond.

Despite the availability of commercial software that suffices for many applications, there is a growing need for a deeper understanding of the mathematical foundations of these solutions, especially as we push the boundaries of current technology with higher frequency bands and more complex applications. This Special Issue, entitled "Mathematical Methods for Wave Phenomena", aims to gather seminal review papers by leading researchers, as well as recent advancements in the application of numerical methods like the Finite Element Method (FEM), Finite Difference Time Domain (FDTD), or Integral Equation (IE) to electromagnetic and other wave phenomena.

The scope of this Special Issue includes, but is not limited to, the following topics:

• Development of basis functions for numerical methods;
• Definition of analytical problems for verification and validation;
• Development of specific solvers for the training of AI models;
• Adaptive refinement and error estimation with methods and the impact of different parameters;
• Advantages of emerging and growing computing hardware and software infrastructure for numerical methods;
• Unconventional approaches to optimize the implementation of mathematical modeling software;
• Uncertainty quantification;
• New solvers for large, multiscale, or multiphysics modeling;
• Hybridization with other numerical tools;
• Applications of conventional and unconventional wave phenomena;
• Best practices for the development of new mathematical modeling software, including open-source projects, and reproducibility.

We invite researchers to submit their contributions to this Special Issue, which aims to provide a comprehensive overview of current trends and future directions in the mathematical modeling of wave phenomena, with a particular focus on electromagnetic waves but also on other types of wave phenomena.

Dr. Adrian Amor-Martin
Prof. Dr. Octavio Castillo-Reyes
Guest Editors

Manuscript Submission Information

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• mathematical modeling
• computational electromagnetics
• geophysics
• basis functions
• verification and validation
• artificial intelligence
• adaptivity
• high-performance computing
• heterogeneous computing
• applied mathematics