Mathematical Modeling
Vol. 7 (2023), Issue 3

Table of Contents

  • THEORETICAL FOUNDATIONS AND SPECIFICITY OF MATHEMATICAL MODELLING

    • New concepts about the infinite numbers and functions and their application in modeling technological and mathematical problems

      pg(s) 69-73

      In this paper more theory and examples about the infinite numbers and functions that were introduced in a previous paper is presented. These numbers and functions apply in modeling physical and mathematical systems and processes, where infinity somehow appears, i.e. series of numbers, limits calculation, kinematics problems, infinite expansion of certain universe geometric shapes. By strictly applying the theory of limits of functions, more properties of the infinite numbers which are limits of complex functions tending to infinity, are defined. The mirror infinite numbers and some properties of them are also demonstrated and presented. Furthermore, infinite geometric shapes whose sides are infinite numbers, as triangles, circles, spheres and ellipsoids are defined and used in modeling theoretical and technological problems.

    • The parametric model of reflected solar radiation in a cloudless atmosphere

      pg(s) 74-75

      Inverse problems in atmospheric optics, for example, the reconstruction of the optical and microphysical properties of aerosols and clouds, operational algorithms based on measurements or calculations of solar radiation are used. In this work the patterns of formation of the field of reflected solar radiation in a cloudless atmosphere that were obtained during statistical modeling for various opticalgeometric parameters of the observation scheme are discussed. In addition, statistical data processing, the main goal of which was to construct a basic parametric model of solar haze brightness, was performed and analyzed.

    • Application of Sturm Liouville Problem in the Wave Equation

      pg(s) 76-79

      Partial differential equations (PDEs) are differential equations in which there is more than one independent variable. They arise in the modelling of a wide-range of physical phenomena including electromagnetism, fluid flow, elasticity, quantum mechanics and heat conduction. The wave equation serves as a fundamental model for understanding various wave phenomena in physics and engineering. In this paper, we explore the application of Sturm-Liouville problems to solve the wave equation. The results of our investigations not only showcase the accuracy and computational advantages of the Sturm-Liouville method but also shed light on the physical interpretations of the obtained eigenfunctions and eigenvalues. In conclusion, this paper contributes to the body of knowledge regarding the application of SturmLiouville problems in wave equation modeling and analysis. It offers a valuable perspective for researchers, scientists, and engineers seeking efficient and insightful solutions to wave-related challenges. The versatility and effectiveness of the Sturm-Liouville approach make it a compelling tool for gaining deeper insights into wave phenomena and their practical applications.

  • MATHEMATICAL MODELLING OF TECHNOLOGICAL PROCESSES AND SYSTEMS

    • Numerical solution of a system of equations arising in telecommunication system modeling

      pg(s) 80-82

      An overall telecommunication system including users and a queuing system with FIFO discipline of services of the requests at the switching stage is considered. A detailed conceptual model has been developed including BPP (Bernoulli-Poisson-Pascal) input flow; repeated calls; limited number of homogeneous terminals; losses due to abandoned and interrupted dialing, blocked and interrupted switching, not available intent terminal, blocked and abandoned ringing and abandoned communication. Previously, on the basis of this conceptual model, a system of algebraic equations has been derived characterizing the overall state of the telecommunication system. In the present paper, a method for solving the system of equations with respect to certain dynamic parameters is proposed.

    • Evaluation of Various Cooling Conditions of Vertical Commercial Refrigerator Using Mathematical Modeling

      pg(s) 83-86

      Commercial kitchen products have a larger capacity than household products and are required to withstand harsh conditions because they are used more intensively. In addition, commercial refrigerators, among these products, are critical for the long-term preservation of food and beverages for many commercial businesses. Commercial refrigerators are generally expected to provide operating conditions between -2ºC and +8ºC. Variations in cooling conditions are observed due to the intensive use of commercial refrigerators in businesses (refrigerator doors being opened too much, irregular placement of different food products, etc.). Therefore, this situation does not allow the commercial refrigerator to be used efficiently. It can also cause food waste. It was first evaluated on a vertical commercial refrigerator using mathematical modeling to eliminate these drawbacks. This study aimed to obtain data to provide the most suitable cooling conditions by evaluating the temperature and air circulation at different points of the vertical commercial refrigerator through simulation studies.

  • MATHEMATICAL MODELLING OF SOCIO-ECONOMIC PROCESSES AND SYSTEMS

    • Bivariate Composite distributions with Pareto tail for modeling bivariate data

      pg(s) 87-89

      Financial data or insurance claim data often exhibit skewness to the right and extreme values, so that classical right skewed distributions like Exponential, Gamma, Weibull or Lognormal fail to capture their behavior. However, built from different distributions on distinct contiguous intervals, two-component spliced (or composite) models often provide a better fit on the right tail, especially since the right tail distribution is considered to be of heavy-tailed type (usually Pareto). In this work, we introduce two bivariate composite distributions defined from a bivariate type I Pareto distribution for values larger than some thresholds, and a bivariate distribution less heavy-tailed on the complementary domain. We present some properties of the new distributions and discuss an estimation method, illustrated on a real data set from insurance.

  • MATHEMATICAL MODELLING OF MEDICAL-BIOLOGICAL PROCESSES AND SYSTEMS

    • Examine of stress-strain state of a spongy bone of an implanted jaw

      pg(s) 90-93

      A spongy bone can be considered a multi-porous area with its fissures and pores as the most evident components of a double porous system. The work studies the stress-strain state of a spongy jawbone near the implant under occlusal loading. A mathematical model of the problem is the contact problem of the theory of elasticity between the implant and the jawbone. The problem is solved by using the boundary element methods, which are based on the solutions of Flamant’s (BEMF) and Boussinesq’s (BEMB) problems. The cases of various lengths of implant diameter are considered. Stressed contours (isolines) in the jawbone are drafted and the results obtained by BEMF and BEMB for the different diameter implants are compared.

    • Modelling, Simulation, and Prototyping of Hollow Microfluidic Channel for Investigation of Blood Cells

      pg(s) 94-97

      The current publication presents an approach for the elaboration of a disposable microfluidic hollow micro-channel using 2 photon polymerization technology and Photonic Professional GT2 (Nanoscribe, Germany) equipment. The design of a 3D model of a microchannel is realized by the CAD analysis software – SOLIDWORKS. A suitable laminar flow is generated by using computational fluid dynamics (CFD) software. As a result, the critical points of the pressure, velocity, and wall shear stress into the microfluidic channel are obtained. A real prototype of the hollow microfluidic device is created, using a highly innovative technology of 3D nanoprinting by twophoton polymerization. Experimental studies with dilute erythrocyte suspensions are conducted to test the functionality of the developed prototype of nano 3D printed microchannel.

    • A Machine Learning approach in 3D object reconstruction using spherical harmonics functions

      pg(s) 98-100

      Artificial intelligence (AI) and machine learning techniques have revolutionized various fields, including 3D modelling of anatomical structures. One such area of research involves the use of AI and machine learning algorithms for approximating spherical harmonics functions in the realm of anatomical structure modelling.
      Spherical harmonics functions are mathematical tools that describe functions on the surface of a sphere. In 3D modelling of anatomical structures, these functions are employed to represent complex surface details with accurate precision. However, calculating values of these functions for complex anatomical structures is time-consuming and prone to errors. This is where AI and machine learning come into play.
      Using AI and machine learning algorithms, we have developed models that can automatically learn the inherent patterns and complexities of anatomical structures from vast amounts of training data. These models can then approximate the spherical harmonics functions that accurately represent the surface details of these structures. This automation significantly reduces the time and effort required in the 3D
      modelling process.