Mathematical Solution Research Articles

A study of the non-linear properties and wave generation of the multi-layer non-hydrostatic wave model SWASH

Despite the increasing use of non-hydrostatic models in the study of wave processes in coastal regions, there is still limited understanding of the non-linear properties of the governing equations and how they improve with increased vertical resolution. In this study, the governing equations of the non-hydrostatic wave model SWASH are analysed and the linear and non-linear solutions up to third-order of all dependent variables are derived, considering one to four vertical layers. The analysis concludes that the model can achieve excellent non-linear properties with respect to the Stokes theory over a large range of water depths using only a few vertical layers. Furthermore, deriving solutions for all variables enables the formulation of improved wave generation and absorption boundary conditions for non-hydrostatic models. A well-known issue of non-linear wave models is related to the generation and propagation of spurious free waves, resulting to non-homogeneous wave fields. In this study, it is proven that by imposing the derived exact mathematical solutions of the governing equations at the model’s boundaries, the target first- and second-order wave profiles can be generated with high accuracy, while the spurious waves can be entirely eliminated.

Soliton solutions to a nonlinear wave equation via modern methods

AbstractIn this pioneering study, we have systematically derived traveling wave solutions for the highly intricate Zoomeron equation, employing well-established mathematical frameworks, notably the modified (G′/G)-expansion technique. Twenty distinct mathematical solutions have been revealed, each distinguished by distinguishable characteristics in the domains of hyperbolic, trigonometric, and irrational expressions. Furthermore, we have used the formidable computational capabilities of Maple software to construct depictions of these solutions, both in two-dimensional and three-dimensional visualizations. The visual representations vividly capture the essence of our findings, showcasing a diverse spectrum of wave profiles, including the kink-type shape, soliton solutions, bell-shaped waveforms, and periodic traveling wave profiles, all of which are clarified with careful precision.

Team orienteering with possible multiple visits: Mathematical model and solution algorithms

This study addresses a new team orienteering problem in which each point can be visited possibly multiple times with decreasing scores. The problem is to determine a set of routes from a starting to a finishing point within an allowed travel time limit for the objective of maximizing the total collected score. A mixed integer programming model is developed to represent the problem mathematically. Then, due to the complexity of the problem, a basic iterated local search (ILS) algorithm is proposed that incorporates the ordinary perturbation and local search operators. Then, it is improved in two ways: (a) an extended ILS algorithm with multi-visit based local search operators and (b) a hybrid ILS algorithm with a simulated annealing technique to permit non-improving moves in the local search step. To test the performance of the ILS algorithms, computational experiments were done on modified benchmark instances under the cases of linear, convex and concave score decreases, and the results show that the hybrid ILS algorithm improves the others significantly and also gives near optimal solutions for small sized test instances. Finally, a case study was done for a naval warship reconnaissance operation, and the results are reported.

Explicit Runge-Kutta Method for Evaluating Ordinary Differential Equations of type v^vi = f(u, v, v′)

The initiative of this paper is to present the Runge Kutta Type technique for the development of mathematical solutions to the problems concerning to ordinary differential equation of order six of structure v vi = f(u, v, v′ ) denoted as RKSD with initial conditions. The three and four stage Runge-Kutta methods with order conditions up to order seven (RKSD7) have been designed to evaluate global and local truncated errors for the ordinary differential equation of order six. The framework and evaluation of equations with their results are well established to obtain the effectiveness of RK method towards implicit function satisfying the required initial conditions and for obtaining zero-stability of RKSD7 in terms of their accuracy with maximum precision under minimal processing.

Dynamical behavior of soliton solutions to the fractional phi-four model via two analytical techniques

This study explores solutions for a mathematical equation called the time-space fractional phi-four equation using two methods: the Sardar-subequation method and the modified extended auxiliary equation method. The phi-four equation is connected to the Klein–Gordon model and is important in different scientific areas like biology and nuclear physics. Understanding its solutions is crucial. By using a specific wave transformation, the equation is changed into a simpler form for analysis. The methods proposed give a variety of solutions, such as Kink, bright singular, dark, combo dark bright, periodic, and singular periodic solutions. Each solution we find using these methods has specific rules that determine when it’s correct. We carefully choose specific values for the parameters to help us understand more about the solutions. This helps us see the detailed features of the solutions and improves our understanding of how the model behaves in the real world. These methods create a strong framework for studying solitons, which are specific types of mathematical solutions. The study compares the outcomes of these methods with earlier ones to get a complete understanding. Graphical illustrations are used to visually represent some of these solutions, helping us grasp their characteristics. Visual representations in two- and three-dimensional figures add originality to the findings. Importantly, these methods can be applied to solve similar problems with fractional derivatives in various scientific contexts. In summary, this research not only deepens our understanding of the phi-four equation but also introduces powerful methods with broad applications in fractional differential equations.

Numerical illustration of diffusive flow of blood-based tri-hybrid nanofluid generated by a curved stretching sheet using law of porosity

The study of flow of blood through the blood vessels carries nanoparticles which opens new ways in the research. This study explores one such flow of trihybrid nanoparticles suspended in blood over a curved stretching surface which undergoes the Darcy-Forchheimer porosity model. The analysis employs Cattaneo-Christov heat flux model along with chemical reaction, linear heat source allowing the surface to slip with heat and mass convection. The trihybrid nanofluids consist of a mixture of titanium dioxide, iron oxide, and silica nanoparticles mixed with base fluid (blood) to generate TiO 2 - Fe 3 O 4 - SiO 2 /blood hybrid nanofluids. Runge-Kutta-Fehlberg 4th–5th order technique, a robust numerical technique, was employed to obtain mathematical solutions. Throughout the numerical process, all parameters, except those under examination, were kept at their default values. Additionally, various plots were generated based on numerical results to illustrate the obtained data. The findings indicate that an increase in the Forchheimer number and porosity results in a reduction in the velocity profile. Furthermore, an increase in the radiation parameter and thermal Biot number leads to an increase in the temperature profile, while the temperature profile decreases with an increase in the thermal relaxation parameter. Additionally, an increase in the chemical reaction parameter and concentration Biot number results in an increase in the concentration profile. Moreover, higher Brinkman and Prandtl numbers enhance the Nusselt number.

Technological limitations of the Markov method for dependability analysis

Aim. To identify and classify the technical limitations of the Markov method of system dependability analysis, to define their boundaries taking into account the actual feasibility of the method, to show the opportunities of expanding the applicability of the Markov method. The paper examined the limitations associated with the automation of state graph construction and the mathematical solution of the corresponding systems of algebraic and differential equations using software. Methods. An approach is proposed to automatically constructing state graphs of technical systems defined by a structural diagram of dependability, as well as communication networks defined by a structural diagram. The paper examines approaches to reducing the dimension of a state graph that do not reduce the accuracy of dependability indicator calculation. The author also considers ways of reducing the dimensionality of a state graph by means of truncation or merging inoperative states that cause biased calculated dependability values. Results. The actual limits of applicability of the Markov method are shown along with their relevance in the context of modern information technologies. Approaches are shown that expand the applicability of the Markov method for analysing the dependability of large systems. The existing approach to state graph truncation were examined. Approaches to merging inoperative graph states were proposed and investigated (for bias). Conclusion. The approaches discussed in the paper allow extending the applicability of the Markov method of system dependability analysis to systems with a larger number of subsystems.

Investigating preservice teachers’ feedback to students’ mathematical solutions

ABSTRACT Teachers require support in developing the pedagogical skill of interpreting students’ mathematical thinking and providing effective feedback that promotes learning. However, there is a lack of research assessing the quality of feedback based on teacher noticing in mathematics classes. This study analysed 20 preservice teachers’ (PSTs) feedback approaches through two perspectives: mathematical alignment and the responsiveness of feedback to students’ mathematical approaches. The findings indicated alignment of the focused mathematical concepts/processes in noticing and feedback. PSTs paid more attention to mathematical accuracy than the strength or potential of students’ strategies. Furthermore, there was a relationship between the complexity of student solutions and PSTs’ depth of noticing and the cognitive demands of feedback.

Creation of fermions in a two-dimensional de Sitter space via a quantum gravity approach

We discuss the production process of Dirac particles in a two-dimensional de Sitter geometry via the framework of rainbow gravity from the general relativity and the teleparallel theory perspectives. On this purpose, we find out the exact analytical solutions for the selected space-time model and then conclude that the general relativity and the teleparallel theory versions of the covariant Dirac equation in the (1+1) -dimensional deformed de Sitter space-time have the same exact solutions. In the subsequent step, making use of the explicit form of the Dirac spinor components and the Bogoliubov coefficients, we turn our attention to investigating the density of produced half-spin particles. Finally, we obtain two mathematical solutions for the fermionic energy spectrum.

Penerapan Kelas Virtual Sevima Edlink dengan Pendekatan Realistik untuk Memecahkan Masalah Matematika Kelas XI MAN 3 Jombang

ABSTRACT Education is the most important aspect for the progress of a nation. In the last year, Indonesian education is being tested by the COVID-19 pandemic, which forces all those involved in education to carry out online learning routines. This is a new challenge for both teachers and students. The teacher must think of the most effective way for learning so that it does not seem monotonous and boring. One alternative is with semiva edlink virtual class. The purpose of this study is whether there is an increase in mathematical problem solving skills by using a virtual class Sevima Edlink with a realistic approach. This research is a quantitative research with pre-experiment that applies the Sevima edlink application as a learning medium using a realistic approach to improve problem solving skills. The sample in this study was class XI IPS 6 MAN 3 Jombang, amounting to 47 students. The data were processed using inferential statistical techniques. The results of this study indicate that there is an increase in the value of students' mathematical solutions before the treatment (pretest) and after the treatment (posttest). Judging from the results of the statistical price = 20,207, with db = 46 and sig. = 0.00 < 0.05 or H0 is rejected. Thus H1 is accepted, which means there is an increase in mathematical problem solving ability by using a virtual class Sevima Edlink with a realistic approach

Neurocomputational mechanisms underlying perception and sentience in the neocortex.

The basis for computation in the brain is the quantum threshold of "soliton," which accompanies the ion changes of the action potential, and the refractory membrane at convergences. Here, we provide a logical explanation from the action potential to a neuronal model of the coding and computation of the retina. We also explain how the visual cortex operates through quantum-phase processing. In the small-world network, parallel frequencies collide into definable patterns of distinct objects. Elsewhere, we have shown how many sensory cells are meanly sampled from a single neuron and that convergences of neurons are common. We also demonstrate, using the threshold and refractory period of a quantum-phase pulse, that action potentials diffract across a neural network due to the annulment of parallel collisions in the phase ternary computation (PTC). Thus, PTC applied to neuron convergences results in a collective mean sampled frequency and is the only mathematical solution within the constraints of the brain neural networks (BNN). In the retina and other sensory areas, we discuss how this information is initially coded and then understood in terms of network abstracts within the lateral geniculate nucleus (LGN) and visual cortex. First, by defining neural patterning within a neural network, and then in terms of contextual networks, we demonstrate that the output of frequencies from the visual cortex contains information amounting to abstract representations of objects in increasing detail. We show that nerve tracts from the LGN provide time synchronization to the neocortex (defined as the location of the combination of connections of the visual cortex, motor cortex, auditory cortex, etc.). The full image is therefore combined in the neocortex with other sensory modalities so that it receives information about the object from the eye and all the abstracts that make up the object. Spatial patterns in the visual cortex are formed from individual patterns illuminating the retina, and memory is encoded by reverberatory loops of computational action potentials (CAPs). We demonstrate that a similar process of PTC may take place in the cochlea and associated ganglia, as well as ascending information from the spinal cord, and that this function should be considered universal where convergences of neurons occur.

Mathematical definition of the fine-structure constant: A clue for fundamental couplings in astrophysics

Astrophysical tests of the stability—or not—of fundamental couplings (e.g., can the numerical value ∼1/137 of the fine-structure constant α = e2/ℏc vary with astronomical time?) are a very active area of observational research. Using a specific α-free non-relativistic and nonlinear isotropic quantum model compatible with its quantum electrodynamics (QED) counterpart yields the 99% accurate solution α = 7.364 × 10−3 vs its experimental value 7.297 × 10−3. The ∼1% error is due to the deliberate use of mean-field Hartree approximation involving lowest-order QED in the calculations. The present theory has been checked by changing the geometry of the model. Moreover, it fits the mathematical solution of the original nonlinear integro-differential Hartree system by use of a rapidly convergent series of nonlinear eigenstates [G. Reinisch, Phys. Lett. A 498, 129347 (2024)]. These results strongly suggest the mathematical transcendental nature—e.g., like for π or e—of α’s numerical value of ∼1/137 and, hence, its astrophysical as well as its cosmological stability.

Modeling of reaction–desorption process by core–shell particles dispersed in continuously stirred tank reactor (CSTR)

Abstract Transient responses of reaction–desorption process were predicted from mathematical solutions of modeling equations for CSTR (continuously stirred tank reactor) containing core–shell adsorbent particles. Analytical solutions on the core–shell particles were derived for core–shell spherical, cylindrical, and slab-type morphologies, assuming inert-cores. Unlike continuous adsorber, CSTRs for reaction–desorption process containing spherical particles exhibited the slowest reduction rate of concentration of adsorbate, because the amount of adsorbed component on the particles is the largest among three kinds of particle shapes. Factors affecting the transient concentration in bulk medium of reaction–desorption process were investigated by adjusting inert-core thickness, inlet flow rate, initial concentration of reactant in inflow stream, amount of adsorbent, and Thiele modulus. Concentration profile inside the particles as well as average intra-particle concentration could be also predicted for comparison with bulk concentration of CSTR. For non-linear isotherm and non-linear reaction kinetics, concentration of active component could be predicted by solving non-linear coupled differential equation using finite element method. For connected CSTRs in series, systems of reaction-diffusion equations were solved by finite element method to study the effect of number of connected reactors. When the number of reactors was sufficiently large, the reactor system could be approximated to fixed bed reactor for reaction–desorption process.

Noise Impact on Optimum Parameters for Signal Deconvolution

In many areas, such as speech processing, communication, and measurement systems, where the actual signal is not directly available, Wiener deconvolution is a widely used signal restoration technique. The accuracy of the estimated signal in Wiener deconvolution is determined by the unknown filter parameter. To compute this unknown parameter, complex and time-consuming numerical as well as iterative computation approaches were proposed in the literature, which also often compute an unoptimized unknown filter parameter. To address the computational problem and estimate a more accurate unknown filter parameter, the authors reported an analytical solution in a prior work for a given signal to noise ratio. However, it was revealed that the actual optimum parameter value also varies with noise also. In this paper, the authors propose a new analytical solution for optimum parameter estimation that includes noise as well. The authors compared the modified analytical solution to existing numerical computation methods for different values of Signal to Noise Ratio and demonstrated that the proposed methods’ optimum parameter estimation is always close to the actual optimum value, which resulted in more accurate signal deconvolution. Proposed mathematical solutions are also verified with experimental signals, acquired in a laboratory.

Three-Dimensional Model for Bioventing: Mathematical Solution, Calibration and Validation

Bioventing is an established technique extensively employed in the remediation of soil contaminated with petroleum hydrocarbons. In this study, the objective was to develop an improved foundational bioventing model that characterizes gas flow in vadose zones where aqueous and non-aqueous phase liquid (NAPL) are present and immobile, accounting for interphase mass transfer and first order biodegradation kinetics. By incorporating a correlation for the biodegradation rate constant, which is a function of soil properties including initial population of petroleum degrader microorganisms in soil, sand content, clay content, water content, and soil organic matter content, this model offers the ability to integrate a specific biodegradation rate constant tailored to the soil properties for each site. The governing equations were solved using the finite volume method in OpenFOAM employing the “porousMultiphaseFoam v2107” (PMF) toolbox. The equation describing gas flow in unsaturated soil was solved using a mixed pressure-saturation method, where calculated values were employed to solve the component transport equations. Calibration was done against a set of experimental data for a meso-scale reactor considering contaminant volatilization rate as the pre-calibration parameter and the mass transfer coefficient between aqueous and NAPL phase as the main calibration parameter. The calibrated model then was validated by simulating a large-scale reactor. The modelling results showed an error of 2.9% for calibrated case and 4.7% error for validation case which present the fitness to the experimental data, proving that the enhanced bioventing model holds the potential to improve predictions of bioventing and facilitate the development of efficient strategies to remediate soil contaminated with petroleum hydrocarbons.

Non-magnetic four-port electronic circulators based on 90∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$90^\\circ $$\\end{document} non-reciprocal phase-shifters

This paper presents a family of four-port electronic circulators adhering to a new topology symmetry that enables linear, low-loss transistor-based circuit implementations. The underlying principle of operation employs a property of the 90∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$90^\\circ $$\\end{document} non-reciprocal phase shifter (NRPS) derived in this article. Under quadrature excitation, the NRPS transfers or reflects exciting signals depending on their respective phase lead. The fundamental topology consists of two back-to-back quadrature hybrid couplers with a 90∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$90^\\circ $$\\end{document} NRPS connected in parallel over the line of symmetry, interrupting the circuit’s reciprocity to achieve circular propagation by bypassing or reflecting at the NRPS but not through. We break down the circuit into three fundamental four-port sub-circuits. The transfer function of the cascaded sub-circuits enables an analysis with specific hybrid couplers. It also allows a synthesis of other four-port passive sub-circuits that, with an NRPS, achieve a four-port circulator transfer function by solving a matrix equation. Some of the mathematical solutions have circuit realizations, which are adjusted quadrature hybrid structures that differ from each other by the characteristic impedance of their arms. Two familiar solutions, including the standard quadrature hybrid and a modified design with equal Z0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Z_0$$\\end{document}, λ/4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda /4$$\\end{document} arms, are simulated utilizing lossless lumped element arms and a 4-Path, 65-nm NMOS 90∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$90^\\circ $$\\end{document} NRPS. The simulation results verify the theoretical analysis and enable a comparison between the performance of the two circuit solutions around 1 GHz. The four-port circulator with equal arms is implemented on a PCB and measured, yielding better than 1.5 dB insertion loss between the circulator ports, over 17 dB port-to-port reverse isolation, and better than 20 dBr port matching around 1 GHz.

Carbon Reduction Incentives under Multi-Market Interactions: Supply Chain Vertical Cooperation Perspective

The greening trend in consumer markets and the marketization and financialization of carbon emission rights have begun to revitalize carbon assets. However, solitary efforts and the spillover of environmental protection effects still hamper enterprises’ enthusiasm for carbon emission reduction. To tackle this challenge, two vertical cooperation mechanisms, cost cooperation and alliance cooperation, are proposed. The mathematical models and solutions are developed for both of the two mechanisms, and their values and applicability are explored, respectively. In addition, the impact of fluctuations in consumer markets, financial markets, and carbon markets on cooperation is examined. The results show that both cooperation models effectively motivate enterprises to enhance carbon reduction and boost market demand. However, cost cooperation may result in inflated product prices and even weaken the profitability of the supply chain. In contrast, alliance cooperation can enhance product price performance and effectively increase supply chain profits. Concerning environmental performance, the initial market is better suited for alliance cooperation, whereas cost cooperation fits the mid-to-late market. The higher financing costs of the financial market and the trading price of the carbon market will strengthen the applicability of cost cooperation. This study offers managerial insights for collaborative decision-making in the context of a multi-market cross-section.

Cognitive Improvement in Thermodynamic Functions With Calculus Chain Rule

This research was motivated by the low ability of students to apply mathematical solution methods to thermodynamic functions. Thermodynamic functions are functions that consist of pressure, temperature, volume and enthalpy functions where changes in one function can occur due to changes in other functions. The aim of this research was to show the implementation of chain rules in thermodynamic function problems by relying on a process technique to improve students' cognitive awareness of thermodynamic functions. The place of this research was at the TD Pardede Institute of Science and Technology. The process of understanding chain rules is very impactful for understanding thermodynamic state functions. Differential decomposition of functions within functions is also described in the chain rule method. Thermodynamic functions which consist of pressure, temperature, volume and enthalpy functions can depend on each other so that the chain rule analysis method can be clearly used. With this method students will also be given the opportunity to connect one concept with other thermodynamic concepts. Apart from that, this method can be used anywhere. This shows a cognitive increase in terms of cognitive elements, cognitive structure and cognitive function.

Transient elastodynamic responses of the concrete ceiling reinforced with nano-materials: A new strategy to improve the efficiency of building construction

Concrete is a durable and resilient material that can withstand substantial damage and decay. The durability of a concrete ceiling is often extensive, requiring little maintenance or repairs. Concrete has the capacity to be molded and reshaped into a wide variety of shapes, allowing architects and designers the liberty to create unique and attractive ceiling designs. This research provides practical suggestions for improving the effectiveness and durability of transient elastodynamics in concrete ceilings that are reinforced with nano-materials. Nano-materials are often used in building construction because of their many benefits. To accurately model the concrete structure reinforced by nanocomposites, a new mathematical simulation is introduced in this study. The simulation utilizes the Halpin-Tsai method, stress-strain relation in linear poroelasticity for the composite layer, considers the function of mixing, and incorporates Hamilton’s principle. The objective is to achieve an accurate representation of the concrete structure. The governing equations of the time-dependent mathematical simulation are solved using Newmark’s time integration method, an isogeometric approach (IGA), and NURBS functions as the basis functions. The current study’s results compared to the published article in the literature showcase the accuracy of the mathematical simulation and solution approach. Incorporating nanoparticles into the concrete ceiling may ultimately improve the efficiency of this kind of construction, especially when exposed to high external transient stresses.

Philosophy of Mathematics in Primary Education Mathematics Learning: Ontological, Epistemological, and Methodological

The purpose of this study is to examine the relationship, position, and role of the philosophy of mathematics in learning mathematics in basic education through the perspective of the scope of philosophy, namely ontological mathematics, epistemological mathematics, and methodological mathematics. This research is a qualitative descriptive research with a literature study approach. Data collection techniques are carried out through the documentation method of articles originating from the Google Scholar database, as well as books related to the philosophy of mathematics. Data analysis techniques consist of data reduction, data presentation, and conclusion. The results highlight three main aspects within the scope of the philosophy of mathematics: ontological, epistemological, and methodological. Ontological mathematics addresses the ontological foundations of mathematics for teaching, including the role of mathematics in various contexts. The epistemology of mathematics emphasizes the importance of understanding the discipline from multiple perspectives to enhance learning and curriculum design. Meanwhile, the methodology of mathematics highlights the role of specific methods in determining mathematical solutions. In conclusion, understanding the philosophy of mathematics in these three aspects is important for educators to improve the learning process of mathematics at the primary level comprehensively and optimally.