Fourth-order Runge-Kutta Research Articles

Global Analysis of Meningitis Disease With Optimal Control

The meningitis epidemic has impacted lives negatively, especially in Sub-Sahara Africa, dubbed the ‘Meningitis Belt’. The epidemic has been a public health concern due to an improper understanding of the disease’s dynamics. To implement a control measure that will help minimize the epidemic, we introduce a non-linear Meningitis model that describes the dynamic behaviour of the disease and explains the transmission trend. The model explores the condition that leads to local or global asymptomatic stability of the equilibria. The model is subjected to a sensitivity analysis to find the parameters that influence the R0. The model is modified into an optimal control by adding timedependent controls. The control model is solved qualitatively using Pontryagin’s maximum principle and numerically using MATLAB and the fourth-order Runge-Kutta method. We provide a control strategy that can be relied on for management decision-making based on the results.

Reversible esterification process in Casson nanofluids: A numerical analysis of non-newtonian hydromagnetic flow

A computational mechanism has been formulated to scientifically analyze the phenomenon of combined convection in the occurrence of magnetohydrodynamic flow behavior of Casson nanofluid. This study emphasizes the trajectory of the nanofluid along a stretching sheet with nonlinear permeability, while considering additional physical effects such as reversible chemical reaction, thermal radiation, energy source/sink, suction and viscous dissipation. In this study, we have additionally integrated the nanofluid paradigm proposed by Buongiorno, which encompasses the repercussion of thermophoresis along with Brownian motion. The utilization of appropriate similarity renovations serves the purpose of recasting the dominant multivariate differential equations to a collection of nonlinear differential equations in single variable. The Runge–Kutta shooting mechanism is implemented for the intention of numerically determining the unknown boundary conditions. The solution to the dominant equations was obtained by employing the fourth-order Runge–Kutta technique and to ensure the dependability of the outcomes, the bvp4c method was employed. The graphical and tabulated observations are presented herein to facilitate a comprehensive analysis of the underlying physical characteristics inherent in the problem at hand. The utilization of the Casson parameter has been found to be advantageous in the reduction of shear stress rate, along with the enhancement of mass and energy transfer rates. Additionally, the application of suction has been observed to be beneficial in the enhancement of Sherwood number and energy transfer rate. Within the scope of the context of the esterification process, the purpose of this proposal is to evaluate the impact that numerous arithmetical values exert on the temperature field, the velocity profile and the volumetric concentration. An in-depth graphical analysis that takes into account the relevant physical consequences is used to evaluate the most important factors that relate to the physical quantities that describe the contours of temperature, velocity and concentration. In addition to being accompanied by their respective explanations, the tabular portrayal of the local Sherwood number, shear stress rate along with local Nusselt number all feature in this paper. There is a clear distinction that can be made between irreversible and reversible flows in the evaluation of the local Sherwood number, rate of shear stress and local Nusselt number. This differentiation is brought about by taking into analysis thermophoresis parameter, Brownian motion parameter and suction parameter. The results that were produced from the theoretical simulations have significant repercussions for a variety of fields that are related to energy engineering. The findings derived from the analysis indicate a negative correlation between the Brownian motion parameter and both irreversible and reversible flow in terms of rate of energy transfer and shear stress rate. It is imperative to highlight that reversible flow holds greater significance in comparison to irreversible flow.

Transforming chaotic and stiff systems to improve numerical accuracy

AbstractSystems of differential equations can exhibit chaotic or stiff behavior under specific conditions, posing challenges for time-stepping numerical methods. For explicit methods, the required time step resolution significantly exceeds the resolution associated with the smoothness of the exact solution for specified accuracy. In order to improve efficiency, the question arises whether transformation to asymptotically stable solutions can be performed, for which neighbouring solutions converge towards each other at a controlled rate. Employing the concept of local Lyapunov exponents, it is shown that chaotic differential equations can indeed be transformed to obtain high accuracy, whereas stiff equations cannot. For instance, the accuracy of explicit fourth order Runge–Kutta solution of the Lorenz chaotic equations can be increased by several orders of magnitude. Alternatively, the time step can be significantly extended with retained accuracy. The transform method applies broadly to chaotic systems, including weather prediction and turbulence.

Differential quadrature lattice Boltzmann method for simulating fluid flows

In this study, a novel method for simulating incompressible flows using the lattice Boltzmann method is presented, aiming to improve robustness. This method is enhanced by the Qadyan numerical method. In the Qadyan method, Partial differential equations are solved using semi-discrete schemes combined with the differential quadrature method. To discretize the spatial derivatives of the lattice Boltzmann equation, the upwind differential quadrature method is employed, while the first-order forward difference scheme and/or fourth-order Runge-Kutta method handle the temporal term. The decision to exclude more complex methods stems from their requirement for refined computational grids. This work focuses on achievable results with similar types of uniform grids. The proposed scheme introduces and evaluates a novel method for solving both steady and unsteady problems. The present numerical method is validated by solving five benchmark problems, including heat diffusion on a slab, Stokes’ first problem, Taylor-Green vortex flow, flow around a flat plate, and flow in a lid-driven cavity. Reasonable agreements are obtained between the solutions obtained using this method and those from analytical/other numerical approaches. The Qadyan formulation delivers more accurate results compared to the finite-difference lattice Boltzmann method, (FDLBM), without requiring an increase in the number of grids. Notably, the novel Qadyan method achieves this increased accuracy without introducing additional complexity to the standard lattice Boltzmann method or resorting to non-uniform grids. This is possible due to the nature of the differential quadrature method, which utilizes more combinations of candidate stencils. The Qadyan method has a higher computational cost in comparison with the FDLBM.

Chaotic behavior and multistability of a tree trunk structure driven by self-and parametric hydrodynamic forces

Abstract This study investigates the chaotic behavior of a tree trunk under dynamic
wind loads, focusing on control strategies and multistability. We consider time varying wind speeds and analyze a specific case where hydrodynamic drag forces align
with the flow velocity. The stability of the model’s equilibrium points is analyzed
theoretically and numerically. Melnikov’s method is employed to identify conditions
for homoclinic bifurcation. Numerical simulations employing basin of attraction
confirm the analytical predictions. Our findings show a decrease in the threshold
for chaos with increasing amplitudes of external excitation, damping coefficient,
and parametric damping. The global dynamics are explored numerically using a
fourth-order Runge-Kutta method. When solely subjected to external excitation, the
system exhibits period doubling bifurcations, multiperiodic oscillations, mixed-mode
oscillations, and chaos. Conversely, with self- and parametric drag forces, the system
displays reverse periodic bifurcations, periodic bubbling oscillations, antimonotonicity,
transient chaos, and chaos. Poincar´e maps analyze the geometric structure of chaotic
attractors, revealing a strong influence of dimensionless drag parameters. These
parameters can be manipulated to control or eliminate chaos. Furthermore, the system
exhibits multistability, with coexisting attractors. Beyond its application in protecting
infrastructure from wind damage, this research can contribute to ecological balance by
improving our understanding of tree wind resistance.

Computational analysis of the Covid-19 model using the continuous Galerkin–Petrov scheme

Abstract Epidemiological models feature reliable and valuable insights into the prevention and transmission of life-threatening illnesses. In this study, a novel SIR mathematical model for COVID-19 is formulated and examined. The newly developed model has been thoroughly explored through theoretical analysis and computational methods, specifically the continuous Galerkin–Petrov (cGP) scheme. The next-generation matrix approach was used to calculate the reproduction number ( R 0 ) ({R}_{0}) . Both disease-free equilibrium (DFE) ( E 0 ) ({E}^{0}) and the endemic equilibrium ( E ⁎ ) ({E}^{\ast }) points are derived for the proposed model. The stability analysis of the equilibrium points reveals that ( E 0 ) ({E}^{0}) is locally asymptotically stable when R 0 < 1 {R}_{0}\lt 1 , while E ⁎ {E}^{\ast } is globally asymptotically stable when R 0 > 1 {R}_{0}\gt 1 . We have examined the model’s local stability (LS) and global stability (GS) for endemic equilibrium \text{ } and DFE based on the number ( R 0 ) ({R}_{0}) . To ascertain the dominance of the parameters, we examined the sensitivity of the number ( R 0 ) ({R}_{0}) to parameters and computed sensitivity indices. Additionally, using the fourth-order Runge–Kutta (RK4) and Runge–Kutta–Fehlberg (RK45) techniques implemented in MATLAB, we determined the numerical solutions. Furthermore, the model was solved using the continuous cGP time discretization technique. We implemented a variety of schemes like cGP(2), RK4, and RK45 for the COVID-19 model and presented the numerical and graphical solutions of the model. Furthermore, we compared the results obtained using the above-mentioned schemes and observed that all results overlap with each other. The significant properties of several physical parameters under consideration were discussed. In the end, the computational analysis shows a clear image of the rise and fall in the spread of this disease over time in a specific location.

Bifurcation analysis of Van der Pol–Mathieu equation for the dust grain charge with regularized (κ)-distributed electron-ion

This paper presents a novel viewpoint on nonlinear dust-acoustic waves (DAWs) in an unmagnetized collisionless plasma containing regularized [Formula: see text] distributed electrons and ions (ei) and negative dust grain. The nonlinear oscillatory system based on hybridization of the Van der Pol–Mathieu equation (VdPME) is derived by a new technique. By bifurcation analysis of the planar dynamical system (DS), the effects of parameters with the assistance of phase planes and time series of VdPME are studied. After analyzing the equation to identify the resonance region, a fourth-order Runge–Kutta method is used to solve it numerically. We explained the behavior of DA periodic, stable limit cycle, and chaotic limit cycle wave solutions with different parameters. These types of numerical solutions are illustrated in two-dimensional and three-dimensional graphics by changing the rate at which charged dust grain is produced [Formula: see text], as well as waste [Formula: see text] and comparing the results with those of earlier research. A novel bifurcation analysis of VdPE and VdPME is obtained with the effects of the cut-off factor [Formula: see text] of regularized [Formula: see text]-distribution (RKD) distributed ei, the superthermality of ei particles [Formula: see text], the ratio of ion to electron temperature [Formula: see text], and the ratio of dust to electron density [Formula: see text] illustrated. It is noticed that the DAW shows promotion in width and amplitude as the frequency [Formula: see text] increases and it becomes rarefaction as the cut-off parameter [Formula: see text] increases. On the contrary, it becomes compressive under the impact of superthermality [Formula: see text] and [Formula: see text]. The obtained conclusions may help explain and comprehend a variety of applications in experimental plasma containing highly energy regularized [Formula: see text] dispersed ei, as well as in the interstellar medium.

Nonlinear deterministic dynamic analysis of a GPL micropipe conveying pulsating laminar flow

In several industries, the handling of fluid-filled micropipes is common. New-brand structures are increasingly being manufactured using advanced materials. This article tackles one of the crucial dynamic analyses that an advanced fluid-filled micropipe encounters. The dynamics of a graphene nanoplatelets-reinforced micropipe (GPL micropipe) under pulsating laminar flow for the first time is examined. The Euler-Bernoulli beam model follows the modified couple stress theory (MCST) and von-Karman’s nonlinear strain to formulate the problem. The impression of the flow profile exerted by a real flow as a consequence of fluid viscosity, is taken into account. The plug flow model results are compared to those regarding real flow consideration. Using the method of multiple scales (MMS), we determine the instability area and the steady state response of the GPL micropipe subjected to the principal parametric resonance of one of its modes due to pulsating laminar flow by applying this method to the discretized couple nonlinear gyroscopic governing equations. The Floquet theory treats the linear equations and fourth-order Runge–Kutta (RK4) method attacks nonlinear ones to validate MMS findings. It is confirmed that the 0.3 % addition of the GPL to matrix phase significantly enhances its advanced attributes, making it a highly effective material. The GPL reinforcement phase in the FGO pattern increases the critical flow velocity that exhibits the micropipe to static instability by 37.98 % , and critical flow stimulation amplitude fraction by 152.19 % , while declines instability area bandwidth by 36.95 % , and steady state response by 67.17 % relative to a non-reinforced micropipe. A denser flow degrades the corresponding values by 18.40 % , 42.59 % , 41.67 % , and 227.28 % in comparison to a lighter fluid. The declared findings provide crucial insights into the key design elements affecting significantly the functioning of fluid-filled GPL micropipes system, and regulate future research and expand openings for industry applications.

Analysis of heat generation and viscous dissipation with thermal radiation on unsteady hybrid nanofluid flow over a sphere with double-stratification: Case of modified Buongiorno's model

PurposeThis work uses entropy generation analysis to numerically analyze magnetohydrodynamics (MHD) unsteady flow, heat and mass transfer. The study considers changing viscosity, thermal radiation, viscous dissipation, mass suction, heat generation, and stratification processes in a hybrid nanofluid around a spinning sphere. A two-phase nanofluid flow model (Buongiorno model) is used to tackle the current challenge. Both the free stream velocity and the sphere's angular velocity changed over time. Design/methodology/approachThe case study's complicated partial differential equations are transformed into simpler ordinary differential equations utilizing similarity transformation. Implementing the fourth-order Runge-Kutta Fehlberg method with a shooting scheme in MATHEMATICA has allowed us to get numerical solutions for ordinary differential structures. FindingsThe many aspects of these regulated physical characteristics have been elucidated and thoroughly examined via the use of charts and tables. For increasing values of unsteadiness parameter, the velocity profiles in the x-direction grow while they decrease in the z-direction. On the other hand, the temperature profile exhibits a dual pattern, and the concentration profile decreases. As the chemical reaction parameter climbs from 0.25 to 1.0, the Sherwood number for the nanofluid rises by 6721.39% and for the hybrid nanofluid, 3818.9%. When Nr increases from 0.2 to 0.8, nanofluid Nusselt number climbs 22.5% and hybrid nanofluid's Nusselt number rises 21.9%. The Brinkmann number and nanoparticle volume percentage are strongly associated with entropy production. Entropy generation is dual for the temperature difference, magnetic, radiation, and changeable viscosity parameters. The findings are also compared to those from the existing literature and are in excellent agreement with them.

SACR EPIDEMIC MODEL FOR THE SPREAD OF HEPATITIS B DISEASE BY CONSIDERING VERTICAL TRANSMISSION

Hepatitis B is an infectious disease that causes inflammation of the liver due to infection with the Hepatitis B virus. Hepatitis B is divided into two phases: the acute phase and the chronic phase. Hepatitis B virus (HBV) can be prevented through vaccination and treatment of susceptible and infected individuals. The spread of the virus can be modeled using mathematical modeling of epidemics. In this study, the model used consists of four classes, namely vulnerable individuals (S), acute individuals (A), chronic individuals (C), and recovered individuals (R). The purpose of this study is to explain the formation of the Hepatitis B disease epidemic model, analyze the stability of the model, perform simulations, and conduct parameter sensitivity analysis on the basic reproductive number. The result of this study is the construction of an epidemic model of the spread of hepatitis B disease in the form of a SACR model. This model takes into account the transmission that occurs not only through interactions between susceptible individuals and chronic individuals but also through the birth process, which occurs in chronic subpopulations because babies born can be chronically infected (vertical transmission from mother to baby). The model produces two equilibrium points, the disease-free equilibrium and the endemic equilibrium. Both points were analyzed for stability using the linearization method and were found to be asymptotically stable. Furthermore, the model simulation was carried out using the fourth-order Runge-Kutta method and sensitivity analysis of the basic reproduction number. From the results obtained, it can be concluded that the spread of hepatitis B disease can be minimized by reducing contact between susceptible and chronic individuals, increasing treatment of chronic individuals, and increasing the number of vaccinated individuals in susceptible populations.

Prediction of wellbore gas hydrate generation based on temperature-pressure coupling model in deepwater testing

With the advancement of oil and gas exploration and development technology, the development of the world's oil and gas resources is gradually advancing from land and shallow waters to deep waters. During the development of gas wells in deepwater regions, the temperature of the wellbore decreases from the bottom of the well to the wellhead, and therefore, common well conditions of high pressure, low temperature, and the presence of free water are encountered, which can lead to gas hydrate plugging of the pipeline and affect the transportation of natural gas. Simulation of temperature and pressure along the wellbore and identification of gas hydrate-generating zones in the wellbore can be used to take timely preventive measures and minimize losses. In this study, a wellbore gas hydrate generation and prediction model is developed by combining the mass, momentum, and energy conservation equations, and the model is solved by the fourth-order Runge-Kutta method, taking into account the temperature-pressure coupling in the wellbore. The effects of gas production, specific heat capacity, thermal conductivity, tubing size, test duration, and ground temperature gradient on the location of gas hydrate generation and the amount of inhibitor used were analyzed in a case study well in the South China Sea. Some suggestions for practical engineering operations, such as increasing the gas production in moderation and choosing smaller tubing sizes within the acceptable range, were given. The study's results can provide a theoretical basis for flow assurance for hydrocarbon production in the wellbore of deepwater gas wells.

Thermal study of MHD hybrid nano fluids confined between two parallel sheets: Shape factors analysis

This article discusses the heat and air transfer characteristics of stable magnetohydrodynamic nanofluid flow between two interconnected sheets under the influence of a magnetic field. The nanofluid is a mixture of equal proportions of ethylene glycol and water. This study examined hybrid nanoparticles containing multi-walled carbon nanotubes (MWCNT) and silver (Ag). This research presents, for the first time, a new method for solving nonlinear equations using HPM Python and AGM Python. In addition, the symbolic solution to HPM and AGM has been attained by employing SymPy and SciPy libraries in Python. The results are presented graphically by comparing them with the fourth-order Runge-Kutta number. The final results reflect a high level of agreement between the analytical and numerical methods on the one hand and HPM Python and AGM Python on the other hand. This examination also investigates the effect of various parameters, including magnetic properties, viscosity coefficients, thermophoretic parameters, Brownian parameters, and nanofluid parameters such as velocity, temperature, and concentration. The results prove that velocity and concentration increase as the magnetic field decreases, whereas the temperature displays an opposite trend. As the Schmidt number increases, both the Nusselt number and concentration decrease. The relationship between concentration and temperature with respect to the Prandtl number indicates that when the Prandtl number decreases, the temperature increases while the concentration declines. It is important to note that the employment of hybrid nanofluids leads to an increase in velocity, temperature, and concentration.

Using Short Time Series of Monofractal Synthetic Fluctuations to Estimate the Foreign Exchange Rate: The Case of the US Dollar and the Chilean Peso (USD–CLP)

Short time series are fundamental in the foreign exchange market due to their ability to provide real-time information, allowing traders to react quickly to market movements, thus optimizing profits and mitigating risks. Economic transactions show a strong connection to foreign currencies, making exchange rate prediction challenging. In this study, the exchange rate estimation between the US dollar (USD) and the Chilean peso (CLP) for a short period, from 2 August 2021 to 31 August 2022, is modeled using the nonlinear Schrödinger equation (NLSE) and calculated with the fourth-order Runge–Kutta method, respectively. Additionally, the daily fluctuations of the current exchange rate are characterized using the Hurst exponent, H, and later used to generate short synthetic fluctuations to predict the USD–CLP exchange rate. The results show that the USD–CLP exchange rate can be estimated with an error of less than 5%, while when using short synthetic fluctuations, the exchange rate shows an error of less than 10%.

Stochastic Optimal Control Analysis for HBV Epidemic Model with Vaccination

In this study, we explore the concept of symmetry as it applies to the dynamics of the Hepatitis B Virus (HBV) epidemic model. By incorporating symmetric principles in the stochastic model, we ensure that the control strategies derived are not only effective but also consistent across varying conditions, and ensure the reliability of our predictions. This paper presents a stochastic optimal control analysis of an HBV epidemic model, incorporating vaccination as a pivotal control measure. We formulate a stochastic model to capture the complex dynamics of HBV transmission and its progression to acute and chronic stages. By leveraging stochastic differential equations, we examine the model’s stationary distribution and asymptotic behavior, elucidating the impact of random perturbations on disease dynamics. Optimal control theory is employed to derive control strategies aimed at minimizing the disease burden and vaccination costs. Through rigorous numerical simulations using the fourth-order Runge–Kutta method, we demonstrate the efficacy of the proposed control measures. Our findings highlight the critical role of vaccination in controlling HBV spread and provide insights into the optimization of vaccination strategies under stochastic conditions. The symmetry within the proposed model equations allows for a balanced approach to analyzing both acute and chronic stages of HBV.

Dynamics of heat transfer in complex fluid systems: Comparative analysis of Jeffrey, Williamson and Maxwell fluids with chemical reactions and mixed convection

This study addresses the complex dynamics of heat transfer in magnetohydrodynamic (MHD) systems involving Jeffrey, Williamson, Maxwell, and Newtonian fluids, focusing on how chemical reactions, activation energy, porosity, and mixed convection impact fluid behavior. The problem is critical due to the significant influence these factors have on industrial processes and applications involving non-Newtonian fluids. The developed a mathematical model represented by partial differential equations (PDEs), which were solved using similarity transformations and the fourth order Runge-Kutta (R-K) method combined with shooting technique, with MATLAB software facilitating the solution process. The results reveal that variations in magnetic field strength, porosity, and buoyancy force significantly affect fluid velocities, while radiation, Brownian motion, and thermophoresis alter temperature profiles. Furthermore, chemical reaction rates, Schmidt number, relaxation constant, and activation energy influence fluid concentrations. Key findings include that increasing porosity and magnetic field strength generally decreases fluid velocity, while higher radiation and Prandtl numbers reduce temperature. Chemical reactions and activation energy decrease fluid concentrations, with non-Newtonian fluids showing more pronounced effects compared to Newtonian fluids. The novelty of this work lies in its comprehensive analysis of multiple interacting parameters and their combined effects on heat transfer in MHD systems, providing insights that extend beyond previous studies in the literature. This research offers valuable implications for optimizing fluid dynamics in various industrial applications, including food processing, ink formulation, and friction reduction.

A study on the fractional-order COVID-19 SEIQR model and parameter analysis using homotopy perturbation method

In this article, we present a fractional-order susceptible-exposed-infected-quarantine-recovered (SEIQR) model to analyze the dynamics of the COVID-19 pandemic. The model includes susceptible (S), exposed (E), infected (I), quarantined (Q), and recovered (R) populations and uses a fractional-order differential equation to provide a further accurate representation of the disease's progression. We employ the homotopy perturbation method (HPM) to derive analytical solutions and the Runge-Kutta fourth-order (RK4) method to obtain numerical solutions. The results indicate that the fractional-order model, particularly for a fractional parameter α = 0.40, provides better accuracy and stability compared to the classical integer-order model. This study highlights the importance of fractional-order modeling in understanding the spread of COVID-19 and suggests its potential application in predicting and controlling future epidemics.

Effect of side ratio on the two-degrees-of-freedom vortex-induced vibrations of the rectangular cross section model

The effect of the mutative side ratio (D/B) on the vortex-induced vibration (VIV) characteristic, aerodynamic characteristics, and the surrounding time-averaged and transient flow field of a rectangular cross section model were simulated numerically. Based on Fluent 19.0 platform, overset grid technology and the fourth-order Runge–Kutta method were used at Re 22 000. First, a rectangular cross section model with D/B = 0.25 was selected, and the simulation method and parameter settings were validated against previous literatures. The subsequent analysis compared and evaluated the effect of side ratio on the VIV response by focusing on statistical values of aerodynamic force coefficients, self-spectra, amplitude ratio, motion trajectory, and phase transition changes for stream-wise and cross-flow directions. Moreover, the study examined the influence of different models at different reduced velocities (Ur) on wake vortex-shedding. The findings suggest that, within a fixed cross-sectional area, a smaller side ratio leads to a weaker VIV characteristic and notably lower aerodynamic performance compared to a larger side ratio. The vortex-shedding mode of the rectangular cross section, particularly with a large side ratio, is less sensitive to changes in Ur compared to the standard square cylinder. An examination of the Reynolds number (Re) effect on the minimum and maximum side ratio models reveals that it has a more pronounced impact on the aerodynamic performance and VIV of the cross-flow when compared to in-line flow. In general, it is noted that larger side ratio model exhibits a stronger sensitivity to the variation of Re.

Chaotic dynamics of flexible graphene electronic membranes with variable density in motion

With advancements in artificial intelligence and wearable technology, flexible electronic devices characterized by their flexibility and extensibility have found widespread applications in fields such as information technology, healthcare, and military. Printing technology can accurately print a circuit diagram onto a flexible membrane substrate by the pressure transfer of a conductive ink, which makes the large-scale printing of flexible graphene electronic membranes possible. However, during the roll-to-roll printing process used to prepare flexible graphene electron membranes, the density of electron membranes is variable due to the uneven distribution of inkjet-printed circuits, which limits the printing speed of flexible graphene electron membranes. Hence, investigating the dynamic properties of flexible graphene electron materials with different densities is of paramount importance to improve the production efficiency and quality of flexible graphene electron membranes. This paper takes roll-to-roll intelligent graphene electronic membranes as the research object. According to Hamilton’s principle, nonlinear vibration partial differential equations for the motion of flexible graphene electron membranes with varying densities were established and subsequently discretized using the assumed displacement function and the Bubnov–Galerkin method. Through numerical calculations, the simulation results obtained based on the fourth-order Runge–Kutta method and the multiscale algorithm were compared, and the multiscale algorithm was verified to be more correct and effective. The primary resonance amplitude–parameter characteristic curve, along with phase-plane portraits, Poincaré maps, power spectrum, time history plots, and bifurcation diagrams, for the nonlinear behavior of the membrane was obtained. The impacts of the density coefficient, velocity, damping ratio, excitation force, and detuning parameters on the nonlinear primary resonance and chaotic behavior of the moving graphene electron membrane were determined, and the stable operational region was identified, laying a theoretical foundation for the development of flexible graphene electronic membranes.

Advancing superconductivity research: Insights from numerical simulations of potassium fullerenide-60 and gold and with Ginzburg-Landau theory

There are many types of superconductors, including gold ormus and some fullerene derivatives. Gold can become a superconductor at extremely low temperatures (<1 K), allowing it to conduct electricity without resistance. While not as commonly used as materials like niobium or lead, gold superconductors are valuable for research and development in superconductivity. Fullerene derivatives like potassium fullerenide-60 also exhibit high superconductivity. Limited studies have been conducted on both gold ormus and superconducting fullerene derivatives. Our study of numerical simulations of the Ginzburg-Landau theory in superconductors for gold ormus and potassium fullerenide-60 has yielded important results. We have successfully simulated class-I and class-II superconducting gold ormus, as well as potassium fullerenide-60, using the Runge-Kutta fourth order method. Our analysis demonstrates the convergence of our simulation outcomes and highlights the importance of considering truncation error and selecting appropriate step sizes for accurate results. The periodic factor of penetration (PFP) for each superconductor has been determined, with class-I superconducting gold having a PFP of 250 nm, class-II superconducting gold having a PFP of 566.2 nm, and potassium fullerenide-60 having a PFP of 1.374 nm. Additionally, our study reveals the relationship between the periodic penetration factor and the length of the penetration depth, showing that the PFP reaches a minimum value at a penetration depth length of 130 nm. Overall, our findings contribute to a better understanding of superconductivity in gold ormus and potassium fullerenide-60, emphasizing the importance of accurate numerical simulations for studying complex physical phenomena. Our study confirmed the accuracy of the Runge-Kutta fourth-order method in simulating superconductors. By examining the PFP for various superconducting materials, we identified trends in penetration depth, shedding light on superconductivity. Our simulations give valuable insights for advancing research in this field, with the Runge-Kutta fourth-order method striking a balance between accuracy and efficiency. Careful parameter adjustment ensures reliable simulations and contributes to progress in superconductivity research.

Studies on drug delivery from dissolving microneedle and its uptake in tissue

Abstract In this study, a mathematical model for transdermal drug administration using a dissolving conical microneedle (MN) is developed. The drug is absorbed in the blood compartment after passing through the skin’s viable epidermis. Reversible reaction kinetics then allow the drug to enter the tissue compartment. An unsteady reaction equation is used to predict the drug’s distribution into the skin following MN’s dissolution. An unsteady reaction mechanism is used to determine the concentrations in the blood and tissue compartments. Using the fourth-order Runge–Kutta method, the governing equations with their initial conditions are solved numerically. The MN is expected to dissolve completely in 108.1 s, according to the predicted results, while the concentration in the skin peaks at 94.4 s. The concentrations in the tissue and blood compartments, however, reach their respective maximal amounts later on. In this investigation, the impacts of the drug’s mass fraction, volume fraction, and penetration rate cannot be completely ruled out. A thorough sensitivity analysis has been carried out for some of the parameters involved. Our findings closely align with the findings reported in the literature.