Fourth Order Research Articles

Floquet theory and stability analysis for Hamiltonian PDEs

Abstract We analyze Floquet theory as it applies to the stability and instability of periodic traveling waves in Hamiltonian PDEs. Our investigation focuses on several examples of such PDEs, including the generalized KdV and BBM equations (third order), the nonlinear Schrödinger and Boussinesq equations (fourth order), and the Kawahara equation (fifth order). Our analysis reveals that the characteristic polynomial of the monodromy matrix inherits symmetry from the underlying PDE, enabling us to determine the essential spectrum along the imaginary axis and bifurcations of the spectrum away from the axis, employing the Floquet discriminant. We present numerical evidence to support our analytical findings.

Significance of heat transfer in a reversible esterification of Arrhenius activation energy with partial slip

The significance of heat transfer during a reversible esterification process in a magnetohydrodynamic boundary layer Casson fluid flow along a vertical stretching plate is examined. The multi-slip conditions are considered in a porous medium. The presence of chemical process requiring an activation energy is considered in the analysis. The study also investigates the hydromagnetic boundary layer Casson fluid flow alongwith partial slip conditions across a vertical stretching plate. The incorporation of multi-slip constraints in a porous medium, alongside magnetic fields and other parameters, highlights its relevance in diverse engineering fields such as thermal engineering, polymerization, and biodiesel industries. Understanding the characteristics of such fluids under complex conditions is vital for optimizing heat and mass transfer in industrial applications, making this investigation timely and valuable. The nonlinear differential set of equations are solved numerically involving Runge Kutta based shooting approach of fourth order and the results are verified with the bvp4c tool and the findings are explored using graphical plots. The predominance of significant factors on flow configurations are analyzed and presented in graphs and tables. A comprehensive analysis is provided on the effects on velocity, concentration, and temperature of diverse parameters such as reaction rate constant, magnetic parameter, suction parameter, mass Grashof number, Prandtl number, Casson parameter, thermal radiation parameter and slip parameters. The tabular representation of the adverse effects of drag coefficient, rate of mass transfer and Nusselt number on flow configurations for various significant parameters is presented. It is inferred that for the case of reversible and irreversible flows, the shear stress rate escalates by 29% when the magnetic parameter elevates from 0.5 to 1.5 and about 35% when the Casson parameter elevates from 0.5 to 1.5. For the suction parameter, the coefficient of drag increased by 27% and 26% for irreversible and reversible flows respectively. When the reaction rate increases from 0.5 to 1.5, the rate of shear stress elevates by 0.5% and 0.02% for irreversible and reversible flows in order. The Nusselt number decreased about 7% and 8% when the magnetic parameter and Casson parameter rises from 0.5 to 1.5 respectively, for irreversible and reversible flows. It is noteworthy that the previous studies are in precise agreement with the present investigation.

Experimental decomposition of nonlinear hydrodynamic loads and hydroelastic responses in wave–structure interactions of monopile wind turbines

This study explores the nonlinear effects of hydrodynamic loads and hydroelastic responses in monopile-supported offshore wind turbines, with a focus on their harmonic structures. High-quality experimental data were collected in a shallow water basin across various irregular waves. Their harmonic components up to the fourth order were successfully isolated using a novel four-phase decomposition technique, which effectively extracts harmonics from random time series. The results highlight significant contributions of higher-order harmonics across various sea states. Specifically, two physical models were employed: a flexible monopile to capture structural hydroelastic deformations during large wave events and a rigid monopile to accurately estimate hydrodynamic loads without hydroelastic effects. Additionally, numerical results from fully nonlinear potential-flow calculations were compared, revealing that while the potential-flow solver underestimates total wave loading, especially for higher harmonics. In terms of the free-surface elevations and hydrodynamic forces, this study confirms that their higher-order harmonics align well with the scaling of their corresponding linear components in both amplitude and phase. This enables the prediction of higher-order profiles based solely on their linear components. However, this efficient model is inadequate for accurately estimating the higher-order harmonics of monopile's hydroelastic responses during these extreme wave events, highlighting the need for further refinement.

Generalized Linear Differential Equation using Hyers - Ulam Stability Approach

In this paper, We demonstrate the Hyers - Ulam stability of linear differential equation of fourth order. We interact with the differential equation\begin{align*}\gamma^{iv} (\omega) + \rho_1 \gamma{'''} (\omega)+ \rho_2 \gamma{''} (\omega) + \rho_3 \gamma' (\omega) + \rho_4 \gamma(\omega) = \chi(\omega),\end{align*}where $\gamma \in c^4 [\alpha,\beta], \chi \in [\alpha,\beta]$. Hyers-Ulam stability concerns the robustness of solutions of functional equations under small perturbations, ensuring that a solution approximately satisfying the equation is close to an exact solution. We extend this concept to fourth-order linear differential equations and continuous functions. Using fixed-point methods and various norms, we establish conditions under which such equations exhibit Hyers-Ulam stability. Several illustrative examples are provided to demonstrate the application of these results in specific cases, contributing to the growing understanding of stability in higher-order differential equations. Our findings have implications in both theoretical research and practical applications in physics and engineering.

Analysis of the third class mechanism using the modeling method in the Mathcad software environment

Purpose. To carry out a kinematic analysis of a flat twelve-link hinged mechanism with three leading links, the basis of which is a structural group of links of the third class of the fourth order with rotational kinematic pairs, which is used in technological equipment. Methodology. The kinematic study of a flat complex mechanism of the third class with three leading links was performed using the mathematical modeling method in the Mathcad software environment. Findings. Using the method of mathematical modeling, the mechanism was presented in the form of free vectors built on its links, for which, taking into account the presence of three leading links, three vector contours were selected, which made it possible to draw up a system of vector equations of their closedness with further solving by a numerical method in the Mathcad software environment. The functions of the rotation angles of the mechanism links, their angular velocities and accelerations were obtained in analytical and graphical form depending on the rotation angle of the first driving crank of the mechanism, the calculation of the angular velocities and accelerations of all the driven links of the mechanism was performed. Originality. A sequence was developed and a kinematic study of a complex planar mechanism of the third class with three leading links was done, the basis of which is a structural group of links of the third class of the fourth order. Schematic modeling of the mechanism with three leading links was performed, a visualization schedule of its kinematic scheme was built, and a valid version of its assembly was obtained from many possible ones, for which the values of kinematic parameters of all its links were determined. Practical value. An analysis of the effect of the kinematic parameters of the three leading links of the third-class mechanism on the movement of the working body of the technological equipment was carried out. From the analysis of the obtained research results for the cycle of the mechanism the reason was found for the incomplete technological stoppage of the link, which should provide the working body of the machine for the required period of time. Recommendations of the elimination were made to identify the deficiency and proposals for the need to improve the drive of this equipment.

A dynamical optimal control theory and cost-effectiveness analyses of the HBV and HIV/AIDS co-infection model

Studies have shown that the co-infection of Human Immunodeficiency Virus (HIV) and Hepatitis B Virus (HBV) poses a major threat to the public health due to their combined negative impacts on health and increased risk of complications. Even though, some scholars formulated and analyzed the HBV and HIV co-infection model they did not consider the compartment that contains protected individuals against both HBV and HIV infections. They incorporated the optimal control theory and cost-effectiveness analysis simultaneously. With this in mind, we are motivated to formulate and analyze the HBV and HIV co-infection model, considering the protected group and incorporating optimal control theory and cost-effectiveness. In this study, we have theoretically computed all of the models disease-free equilibrium points, all the models effective reproduction numbers and unique endemic equilibrium points. The two sub-models disease-free equilibrium points are locally as well as globally asymptotically stable whenever their associated effective reproduction numbers are less than one. We reformulated the optimal control problem by incorporating five time-dependent control measures and conducted its theoretical analysis by utilizing the Pontryagin's maximum principle. Using the fourth order Runge–Kutta numerical method and MATLAB ODE45, we performed the numerical simulations with various combinations of control efforts to verify the theoretical results and investigate the impacts of the suggested protection and treatment control strategies for both the HBV and HIV diseases. Also, we carried out a cost-effectiveness analysis of the proposed control strategies. Eventually, we compared our model results with other researcher similar model results whenever cost-effectiveness analysis is not carried out the findings of this particular study suggest that implementing each of the proposed control strategies simultaneously has a high potential to reduce and control the spread of HBV and HIV co-infections in the community. According to the cost-effectiveness analysis, implementing the HBV treatment and the HIV and HBV co-infection treatment measures has a high potential effect on reducing and controlling the HBV and HIV co-infection transmission problem in the community.

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.

Local Convergence Study for an Iterative Scheme with a High Order of Convergence

In this paper, we address a key issue in Numerical Functional Analysis: to perform the local convergence analysis of a fourth order of convergence iterative method in Banach spaces, examining conditions on the operator and its derivatives near the solution to ensure convergence. Moreover, this approach provides a local convergence ball, within which initial estimates lead to guaranteed convergence with details about the radii of domain of convergence and estimates on error bounds. Next, we perform a comparative study of the Computational Efficiency Index (CEI) between the analyzed scheme and some known iterative methods of fourth order of convergence. Our ultimate goal is to use these theoretical findings to address practical problems in engineering and technology.

Sobolev regularity of bivariate isogeometric finite element spaces in case of a geometry map with degenerate corner

We investigate Sobolev regularity of bivariate functions obtained in Isogeometric Analysis when using geometry maps that are degenerate in the sense that the first partial derivatives vanish at isolated points. In particular, we show how the known C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^1$$\\end{document}-conditions for D-patches have to be tightened to guarantee square integrability of second partial derivatives, as required when computing finite element approximations of elliptic fourth order PDEs like the biharmonic equation.

Dispersion relation from Lorentzian inversion in 1d CFT

Starting from the Lorentzian inversion formula, we derive a dispersion relation which computes a four-point function in 1d CFTs as an integral over its double discontinuity. The crossing symmetric kernel of the integral is given explicitly for the case of identical operators with integer or half-integer scaling dimension. This derivation complements the one that uses analytic functionals. We use the dispersion relation to evaluate holographic correlators defined on the half-BPS Wilson line of planar N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills, reproducing results up to fourth order in an expansion at large t’Hooft coupling.

Shock wave propagation in a real gas with or without gravitational field in the presence of magnetic field and monochromatic radiation via group invariance method

PurposeThis article aims to find the similarity solutions for the one-dimensional motion of spherical symmetric shock wave in non-ideal gas influenced by the azimuthal magnetic field and monochromatic radiation in the presence or absence of gravitational field. This paper also aims to study the effects of physical parameters on the strength of shock wave, and on the flow variables in the flow-field region behind the shock front.Design/methodology/approachThe Roche model is used to describe the gravitational field effects due to a massive nucleus at the point of symmetry. To derive the similarity solutions, the Lie group symmetry method has been used. Also, the numerical solutions to the present problem are obtained by using Rung–Kutta method of the fourth order with the use of Mathematica software. The effects of variation in the parameter of non-idealness of the gas, the gravitation parameter, the strength of the ambient magnetic field and the adiabatic index of the gas on the shock wave, and on the flow variables is discussed. A comparative study between with and without gravitational field is also, made.FindingsFor different choices of the arbitrary constants that appeared in the solution of infinitesimal generators, we have obtained seven distinct cases of similarity solutions. In the absence of the gravitational field, the similarity solution exists to the power and exponential law shock paths, but in the presence of gravitational field, the similarity solution exists to the power law shock path case only. In the absence of gravitational field, the shock strength is enhanced in the exponential law shock path case in comparison to the power law shock path case. It is found that the shock wave decays with an increase in the value of the adiabatic exponent, the strength of magnetic field, non-idealness of the gas or gravitational parameter.Research limitations/implicationsThe consideration of medium under the influence of gravitational field due to a heavy nucleus at the center and presence of magnetic field decrease the shock strength. This result may be helpful in designing space vehicle and jet engine.Practical implicationsThe result of the present study may be used in the analysis of data from the measurements by space craft in the solar wind and in neighborhood of the Earth’s magnetosphere.Social implicationsThe obtained results may be used for mankind.Originality/valueThe study of spherical shock wave propagation influenced by monochromatic radiation and azimuthal magnetic field in a non-ideal gas with or without gravitational field has yet to be discussed by any authors by using the Lie group symmetry method. In this article, we have discussed all possible cases of similarity solutions using the Lie group symmetry method, which is not studied by anyone as known to us.

Critical buckling load analysis of Euler-Bernoulli beam on two-parameter foundations using Galerkin method

The critical buckling load determination of Euler-Bernoulli beams on two-parameter elastic foundations (EBBo2PFs) is important to avert buckling failures. The governing equation for buckling of thin beam on two-parameter elastic foundation is a homogeneous ordinary differential equation (HODE) of fourth order and constant parameters when the beam is prismatic and homogeneous. The HODE has been solved in this work by Galerkin method for simply supported, clamped and clamped-simply supported ends. One-parameter algebraic shape function formulation was used to reduce the problem to an algebraic eigenvalue problem, which is solved to find the critical buckling load for each studied case. The critical buckling load for EBBo2PF for simply supported boundary conditions was found to be closely identical to the exact solutions. The solutions for clamped-clamped edges and clamped-simple supports were found to be accurate. The merit of the Galerkin method is the simplicity and the accuracy even when one-parameter shape function has been used.

Three-dimensional stagnation point motion of bioconvection nanofluid via moving stretching sheet with convective and anisotropic slip condition

The current article deals with the SM of a NFs on a 3D moving surface containing bioconvection, microorganisms, and convective conditions over anisotropic slip. This work reports describes the characteristics of bioconvection aspects of nanofluid containing microorganisms with anisotropic slip condition. The study may be important from the application point of view in microfluidic devices, antibiotics, enhanced oil recovery and many other areas. Applying the similarity variables, the basic gov. Eqs are translated to ordinary ones, and such Eqs are calculate by R-K-F fourth order programming with shooting scheme by MATLAB software. The physical parameters on the moving surface are explained in detail via graphs. We found that the temperature increases with a higher revolution and larger numerical numbers (convection parameter), while the HTR declined for various values of the SFP. The HTR enhances with surface convection, and therefore, the convection condition is helpful in reducing the viscous drag on the moving SS.

Sodium alginate-based unsteady nanofluid flow over a porous stretching surface with endothermic and exothermic reaction and bioconvection effects

The impact of nanofluid (NF) circulation under gyrotactic microorganisms and endo/exothermic chemical reactions across a permeable medium is the primary emphasis of this investigation. Sodium Alginate and Silicon dioxide are the base fluid and nanoparticle utilised here. The governing equations are reduced to a set of ODEs by employing the appropriate similarity variables. To get the numerical results, the shooting method and Runge–Kutta Fehlberg's fourth fifth order (RKF-45) are implemented. The numerical outcomes are conveyed graphically across numerous parameters. In addition, the engineering factors are analyzed and explained. It was found that the velocity profile decreased with the enhanced Casson and porous parameter values. An increment in temperature profile is experienced during exothermic reaction and is reduced for endothermic reaction and also the concentration profile got accelerated as the impact of chemical reaction parameter is upsurged. There is a significant growth in surface drag force from NF to base liquid of about 2%, rate of thermal and mass distribution of about 0.1% in context to various constraints.

Stability Analysis and Optimal Control for Spreading Typhoid Fever Model with Direct and Indirect Transmissions

This article discusses a model of spreading typhoid fever with direct and indirect transmissions. There are five compartments included in the model, namely susceptible individuals, infected individuals, chronic carrier individuals, recovered individuals, and salmonella typhi bacteria in the environment. As an effort to control and minimize the numbers of infected individuals and chronic carrier individuals, we introduce simultaneously campaign for the susceptible individuals and treatment for the infected individuals and for chronic carrier individuals. The resulting model is a system of non-linear differential equations and quite challenging to analysis globally. Existence of both disease-free equilibrium point and endemic equilibrium point are analysed analytically. Stability of the equilibrium point is analysed locally by determining the eigenvalues of the associated Jacobian matrix, Routh-Hurwitz stability criteria, and basic reproduction number () via next generation matrix. The minimum Pontryagin principle and Hamiltonian equation are referred to minimize the numbers of infected and chronic carrier individuals. Simulation is carried out using suitable values of parameters. We found that the eigenvalues for endemic equilibrium point are all negative real numbers and . Optimal paths for state and constate variables are plotted using fourth order forward-backward Runge-Kutta method. In case there are no campaign and treatments, we found and endemic occurs. When campaign and treatments are included together in the model, we found optimal paths for all compartments that minimize the number of infected and chronic carrier individuals. Giving campaign and treatments increase significantly the susceptible and recovered individuals. At the same time, it reduces significantly the infected, chronic carrier individuals, and salmonella typhi bacteria in the environment. Involving campaign and treatment in the model of spreading typhoid fever can be considered as an effective strategy to minimize the numbers of infected and chronic carrier individuals.

On Bäcklund transformations to stationary equations in hierarchy of the second Painlevé equation

The analytical properties of solutions to stationary equations of the second and fourth orders in the hierarchy of the second Painlevé equation are considered. For the second-order equation, it is shown that the Bäcklund transformation in the general case determines the formula of the addition theorem for the Weierstrass elliptic function. For the fourth and sixth-order equations, Bäcklund transformations and special classes of solutions are constructed. It has been established that, for a certain relationship between the parameters, the set of solutions to the first term of the stationary hierarchy is a subset of solutions to the second term and the set of solutions to the second term of the hierarchy is a subset of solutions of the six-order equation of the stationary hierarchy of the second Painlevé equation.

Prevalence of early initiation of breastfeeding and its associated factors among women in Mauritania: evidence from a national survey

BackgroundTimely initiation of breastfeeding is crucial for positive health outcomes for babies and mothers. Understanding the factors influencing timely initiation of breastfeeding is vital for reducing child morbidities and mortalities in Mauritania. This study, therefore, assessed the prevalence of early initiation of breastfeeding and its associated factors among women in Mauritania, providing significant insights for improving maternal and child health in the country.MethodsWe performed a secondary analysis of the 2019–2021 Mauritania Demographic and Health Survey data. A weighted sample of 4,114 mother-child pairs was included in the study. We used percentage to present the prevalence of early initiation of breastfeeding. A four-modelled multilevel binary logistic regression was used to examine the factors associated with early initiation of breastfeeding. The regression results were presented using adjusted odds ratio (aOR) with their respective 95% confidence interval (CI). Stata software version 17.0 was used to perform all the analyses.ResultsThe prevalence of early initiation of breastfeeding was 57.3% (95% CI 54.5, 60.00). Birth order was associated with early initiation of breastfeeding with the highest odds among those in the fourth birth order (aOR 1.61; 95% CI 1.08, 2.39). Mothers who practiced skin-to-skin contact were more likely to initiate breastfeeding early than those who did not (aOR 1.46; 95% CI 1.14, 1.87). There were regional disparities in the early initiation of breastfeeding. The odds of timely initiation of breastfeeding was lower among women who were delivered by caesarean section (aOR 0.22; 95%CI 0.14, 0.36), those who were working (aOR 0.57; 95% CI 0.45, 0.73), those who had four or more antenatal care visits (aOR 0.67; 95%CI 0.47, 0.94)], and those in the richest wealth quintile (aOR 0.61; 95% CI 0.38, 0.98) compared to those who had normal delivery, those who were not working, those who had zero antenatal care visits, and those in the poorest wealth quintile households, respectively.ConclusionOur study found a relatively low prevalence of early initiation of breastfeeding among women in Mauritania. Factor such as birth order, region of residence, mother and newborn skin-to-skin contact after birth, antenatal care visits, caesarean delivery, employment status, and wealth index were associated with early initiation of breastfeeding. Improving optimal breastfeeding practices, such as early initiation of breastfeeding in Mauritania, should be given adequate attention. There is a need for interventions such as baby-friendly facilities, providing an enabling environment for mothers to breastfeed their newborns early. Addressing regional health access disparities is important to improve early initiation of breastfeeding and other maternal, newborn, and child health interventions.

Effect of ambisonic order on spatial release from masking.

In two experiments, spatial release from masking (SRM) was measured using an ambisonic reproduction system for a range of different ambisonic orders. The first experiment used ambisonic panning while the second experiment used impulse responses recorded from a sixth-order ambisonic microphone. Both experiments found a progressive increase in SRM with increasing ambisonic order for speech presented against a single speech-shaped-noise interferer. SRM increased progressively up to at least fourth order and continued to asymptotically improve at higher-ambisonic orders toward the level achieved with point sources. The second experiment found that this effect was robust for different acoustic environments.

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.

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.