Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges

The articles published in the Special Issue “Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges” of the MDPI Mathematics journal are here collected [...]

A Student's Guide to Laplace Transforms

The Laplace transform is a useful mathematical tool encountered by students of physics, engineering, and applied mathematics, within a wide variety of important applications in mechanics, electronics, thermodynamics and more. However, students often struggle with the rationale behind these transforms, and the physical meaning of the transform results. Using the same approach that has proven highly popular in his other Student's Guides, Professor Fleisch addresses the topics that his students have found most troublesome; providing a detailed and accessible description of Laplace transforms and how they relate to Fourier and Z-transforms. Written in plain language and including numerous, fully worked examples. The book is accompanied by a website containing a rich set of freely available supporting materials, including interactive solutions for every problem in the text, and a series of podcasts in which the author explains the important concepts, equations, and graphs of every section of the book.

Simpson’s Second-Type Inequalities for Co-Ordinated Convex Functions and Applications for Cubature Formulas

Inequality theory has attracted considerable attention from scientists because it can be used in many fields. In particular, Hermite–Hadamard and Simpson inequalities based on convex functions have become a cornerstone in pure and applied mathematics. We deal with Simpson’s second-type inequalities based on coordinated convex functions in this work. In this paper, we first introduce Simpson’s second-type integral inequalities for two-variable functions whose second-order partial derivatives in modulus are convex on the coordinates. In addition, similar results are acquired by considering that powers of the absolute value of second-order partial derivatives of these two-variable functions are convex on the coordinates. Finally, some applications for Simpson’s 3/8 cubature formula are given.

Update and Acceleration of Health Care Using Artificial Intelligence in Medical Treatments and Diagnostics

Researchers support the growth of artificial intelligence and similar methods in health and medical care for the purpose of continuously improving processes. By focusing on the growth on data analytics, statistics, applied mathematics, and computer methods including machine learning, the future of health-care methods will change. The development of computerized methods and the growth of data systems produce ample materials for artificial intelligence to develop and to bring physician assistance programs to enable continuous improvement resulting in superior health and medical care. This includes applications in intensive care as well as diagnostic therapies. The focus is on examples in the use of the promising developments in data science methods, the accumulation of medical and research data. With quality and continuous improvement in process control applications where one determines the usefulness of data analytics, there are great possibilities of change in the improvement in medical applications as well as the management of medical and health-care treatment and diagnostic facilities.

A Critical Review on the Complex Potentials in Linear Elastic Fracture Mechanics

Introducing a crack in an elastic plate is challenging from the mathematical point of view and relevant within an engineering context of evaluating strength and reliability of structures. Accordingly, a multitude of associated works is available to date, emanating from both applied mathematics and mechanics communities. Although considering the same problem, the given complex potentials prove to be different, revealing various inconsistencies in terms of resulting stresses and displacements. Essential information on crack near-tip fields and crack opening displacements is nonetheless available, while intuitive adaption is required to obtain the full-field solutions. Investigating the cause of prevailing deficiencies inevitably leads to a critical review of classical works by Muskhelishvili or Westergaard. Complex potentials of the mixed-mode loaded Griffith crack, sparing restrictive assumptions or limitations of validity, are finally provided, allowing for rigorous mathematical treatment. The entity of stresses and displacements in the whole plate is finally illustrated and the distributions in the crack plane are given explicitly.

VIII International Conference Days of Applied Mathematics

These proceedings are the contributions made by the participants of the VIII International Conference / Days of Applied Mathematics (ICDAM). The main objective of the VIII ICDAM is to effectively share and transfer mathematical and physical knowledge for the purpose of problem-solving in different contexts of the sciences, engineering, and medicine. The VIII ICDAM was organized by the Facultad de Ciencias Ciencias Básicas y Biomédicas of the Universidad Simón Bolívar, was carried out in the city of San José de Cúcuta, Colombia, from October 20-22, 2021, in which dissertations and workshops developed by students, professors and researchers, those who worked in research lines related to the field of mathematics and physics from Francia, Mexico, República Dominicana, Chile, Argentina, Venezuela, and Colombia. List of Organizing Committee, National Scientific Committee, International Scientific Committee, Sponsor, Partners are available in this pdf.

Physics models in the course "The basics of computer simulation"

A review of the methodological literature on computer modeling shows the existence of different approaches to its teaching in secondary and higher education. It is a common approach in which the construction of models is carried out using the apparatus of higher mathematics, which is possessed mainly by senior students. This leads to the transfer of the course “Methods of mathematical modeling” for 7-8, and sometimes 9-10 semesters, which reduces its role in shaping the worldview of the future specialist, which takes place in high school and junior high school. This state of affairs forced us to create a propaedeutic course “The basics of computer simulation”, which was developed by the joint efforts of the Department of Informatics and Applied Mathematics of Kryvyi Rih State Pedagogical University and the Department of Informatics of the Kryvyi Rih Tsentralno-Miska gymnasium. The methodological support of the course is a textbook designed for high school students and junior high school students.

A different approach to b(an,bn)-hypermetric spaces

Introduction/purpose: The aim of this paper is to present the concept of b(an,bn)-hypermetric spaces. Methods: Conventional theoretical methods of functional analysis. Results: This study presents the initial results on the topic of b(an,bn)-hypermetric spaces. In the first part, we generalize an n-dimensional (n ≥ 2) hypermetric distance over an arbitrary non-empty set X. The b(an,bn)-hyperdistance function is defined in any way we like, the only constraint being the simultaneous satisfaction of the three properties, viz, non-negativity and positive-definiteness, symmetry and (an, bn)-triangle inequality. In the second part, we discuss the concept of (an, bn)-completeness, with respect to this b(an,bn)-hypermetric, and the fixed point theorem which plays an important role in applied mathematics in a variety of fields. Conclusion: With proper generalisations, it is possible to formulate well-known results of classical metric spaces to the case of b(an,bn)-hypermetric spaces.

Formal grammar methodology for digital visualization of Islamic geometric patterns

In the oriental practice of art and architecture, and among the regions under their influence, Islamic geometric patterns (IGPs) have been widely used, not only due to aesthetics and decoration but also to make it possible to cover wide flat surfaces, curved surface of domes, and perforated surfaces of window and partitions, with perfectly tessellated shapes. However, with advances in time and technology, these techniques could not connect to the new technologies and benefit from the capacities of digitalization. Recent progress in science and technology tends to open new doors to study geometrical patterns by digitalizing the old ones and developing new variations. This study looks at formal grammar and computer science to introduce a new approach to digital visualization of available IGPs, particularly, star patterns. We investigate the potentials of developing a re-writing system for simulation of IGPs to provide a flexible platform, which allows introducing IGP to CAD/CAM software without previous knowledge on their design or drawing techniques. This methodology allows designers to directly develop various scenarios of IGP applications and implement them on related CAD/CAM tools. Formal language and grammar theories, based on applied mathematics are contributing to the advancements of computer science and digital modeling. They can provide an opportunity to express relational definition and written equivalents of the geometries by using strings and symbols. It is supposed that by using the formal grammar frameworks, certain languages could be developed to visualize IGPs in a machine-friendly way, and consequently, this computational interpretation of IGPs facilitates their application and further developments, for example, regards to digital fabrication. The presented method of IGP visualization is developed as a C#-based add-on for Grasshopper in Rhino3D, one of the main modeling tools used by architects and product designers.