Integral Transforms in Applied Mathematics and Statistics

Integral transforms play a foundational role in applied mathematics, statistics, and theoretical physics, serving as powerful tools to tackle complex analytical and computational problems. Over the past decades, these transforms—such as the Mellin-Barnes and Laplace transforms—have become central in exploring Feynman integrals in quantum field theory, studying Green’s functions, and analyzing correlation structures in both physical and statistical systems. Despite their extensive utility in solving integro-differential equations arising in disciplines like fluid mechanics, quantum mechanics, and particle physics, critical challenges remain in handling multi-fold contour integrals, ensuring mathematical rigor, and bridging exact analytical results with modern computational approaches.

Recent years have witnessed the growing importance of integral transforms not only in theoretical frameworks but also as benchmarks for contemporary methods, such as neural networks and numerical solvers, yet there remains a pressing need for more accessible exact analytical solutions and systematic methods to obtain such solutions. The existence, uniqueness and stability of the solutions to these equations in fluid dynamics may be proved by using appropriate estimates of the integrals that appear at intermediate steps. However, it is always better to obtain, via integral transformations, an exact solution which may be used to confirm results obtained via neural networks. In theoretical physics, integral transformations may be used to find exact solutions to Schrödinger equations.

This Research Topic aims to underscore the indispensable nature of integral transformations and highlight their ability to produce exact results in applied mathematics, statistics, and theoretical physics. The objective is to foster a comprehensive dialogue around the methodological advances, innovative applications, and theoretical developments related to integral transforms. Contributions are encouraged that address both the classic usage of these tools in solving canonical equations and the exploration of novel techniques for evaluating intricate contour integrals. Specific questions include: How can integral transforms yield new insights into the existence, uniqueness, and stability of solutions? What modern computational strategies can complement or extend analytical methods? Which open challenges remain in practical applications?

This Research Topic is dedicated to applications of integral transformations for tasks in fluid mechanics, theoretical physics and mathematical statistics. Its scope encompasses any issues which may appear when integral transforms are applied to solve integro-differential equations in theoretical physics. This includes the Bethe-Salpeter equation in particle physics, Schroedinger equations in quantum mechanics, Navier-Stokes equations in fluid dynamics, and correlation functions in quantum field theory and mathematical statistics, when the solutions are given in terms of contour integrals of Barnes type or in terms of more complicated contour integrals which may be transformed into Barnes integrals by complex mapping in the complex planes of Mellin variables. These contour integrals are frequently multi-fold, requiring new and advanced methods of calculations.

To gather further insights into the advancements of integral transforms in analytical and computational sciences, we welcome articles addressing, but not limited to, the following themes:

o Application of Mellin-Barnes and related transforms in Feynman diagram calculations;

o Exact solutions and estimation methods for integro-differential equations in fluid dynamics, quantum field theory, and quantum mechanics;

o Advances in contour integral evaluation, including multi-fold Barnes integrals;

o Role of integral transforms in mathematical statistics;

o Development of novel analytical and computational techniques, including hybrid approaches involving machine learning and neural networks.

We accept original research articles, comprehensive reviews, methodological papers, and perspective pieces that contribute to the themes of this Research Topic.

Article types and fees

This Research Topic accepts the following article types, unless otherwise specified in the Research Topic description:

• Brief Research Report
• Curriculum, Instruction, and Pedagogy
• Data Report
• Editorial
• FAIR² Data
• FAIR² DATA Direct Submission
• General Commentary
• Hypothesis and Theory
• Methods
• ... View all formats

Articles that are accepted for publication by our external editors following rigorous peer review incur a publishing fee charged to Authors, institutions, or funders.

Article types

This Research Topic accepts the following article types, unless otherwise specified in the Research Topic description:

• Brief Research Report
• Curriculum, Instruction, and Pedagogy
• Data Report
• Editorial
• FAIR² Data
• FAIR² DATA Direct Submission
• General Commentary
• Hypothesis and Theory
• Methods
• Mini Review
• Opinion
• Original Research
• Perspective
• Review
• Technology and Code

Keywords: integral transforms, Feynman diagrams, fluid dynamics, neural networks, integro-differential equations, partial differential equations

Important note: All contributions to this Research Topic must be within the scope of the section and journal to which they are submitted, as defined in their mission statements. Frontiers reserves the right to guide an out-of-scope manuscript to a more suitable section or journal at any stage of peer review.