Basically, scalar equations have potential applications in various fields such as the transmission of nerve impulses between neurons through myelin substance and other disciplines. A particular model is well-known as the pantograph equation. The standard version of this scalar equation has been extensively investigated via different analytical and numerical techniques. This paper considers a variable version of the pantograph equation. Usually, constructing an exact or a closed form solution for a variable scalar equation is a challenge. However, this work proposes a developed hybrid approach to overcome such a difficulty. The solution of the current variable version is analytically obtained in different closed forms with addressing the convergence criteria. Under some conditions, such closed forms are successfully converted to different exact ones. Additionally, accurate approximations are provided and examined. Several comparisons with the available exact solutions are conducted as a validation of our approximations. Besides, the accuracy of our approximations is checked for some classes which have no exact solutions. Probably, the results demonstrate the elegance of the proposed approach to deal with a variable version of the Pantograph model.

This paper proposes different analytical approaches for solving a class of second-order scalar differential equations involving reflection of the argument. The first approach transforms the given model to a coupled system for which the series solution is obtained in an exact form. The second approach is based on the symmetry decomposition method which derives an equivalent system of two separate equations. Such two separate equations are then solved exactly utilizing well-known standard methods. The third approach converts the problem into a 4th-order ordinary differential equation, allowing the derivation of the same exact solution. The fourth approach applies a direct series method with easier computations of the series coefficients in comparison to the second approach. The obtained solution is expressed in terms of entire functions allowing the appearance of periodicity under a specific constraint of the involved parameters. The results provide a unified procedure for treating reflection-type scalar differential equations.

This study addresses a class of inhomogeneous delay differential equations characterized by variable coefficients and the presence of two proportional delays. An exponential-type transformation is employed to convert the inhomogeneous formulation into an equivalent homogeneous pantograph equation. This reformulation enables the derivation of an explicit analytical solution represented by a superposition of an exponential function and a power series with coefficients given in closed product form. A detailed investigation of the convergence properties of the associated infinite product and series is conducted, yielding clear criteria for global convergence. Furthermore, specific parameter configurations are identified under which the series solution reduces to a finite sum, leading to exact analytical expressions. As an important application, the inhomogeneous Ambartsumian equation involving two proportional delays is examined, and new closed-form and exact solutions are established. Numerical experiments are included to support the theoretical analysis and to demonstrate the rapid convergence and the practical effectiveness of the proposed solution approach.

This work investigated a scalar differential equation involving proportional arguments of the form & ccedil;b '(t) = ar & ccedil;b(gamma t) + j3 & ccedil;b(-gamma t). The terms & ccedil;b(gamma t) and & ccedil;b(-gamma t) represented proportional delay and reflection (time-reversal) effects, respectively, which arose in models exhibiting scaling and symmetry properties. A constructive analytical approach based on the Laplace transform was developed to derive a series representation of the solution. The method converted the original scalar equation into a recursive sequence of algebraic relations in the Laplace variable, allowing the systematic computation of successive terms. Explicit formulas for the series components were obtained, revealing a clear structure separating even and odd contributions. The convergence of the resulting series was rigorously established, showing that the solution existed globally and defined an entire function. Furthermore, the series representation was summed into a closed form expressed through hyperbolic functions, providing a compact analytical expression of the solution. Numerical illustrations confirmed the effectiveness of the derived formulation.

In this research, a phi(6)-model approximation method is employed to investigate the localized wave solutions for the Lakshmanan-Porsezian-Daniel equation with beta-derivative. This equation integrates the fundamental phenomena such as Space-Time Dispersion (STD), Group Velocity Dispersion (GVD) and parabolic-law-governed by nonlinear behavior. Avariety of optical soliton waves are obtained by applying the phi(6)-model expansion approach. These waves are expressed as Jacobi elliptic functions F(L, A) that based on the particular values of the parameter A, that can be converted into solutions of trigonometric or hyperbolic functions. This technique provides variety of solutions, including dark soliton solutions, hyperbolic solutions, periodic waves solutions, bright solitons, singular soliton solution and singular periodic waves solutions. To further explore the system's behavior, bifurcation analysis is done. For this analysis planar dynamical system is obtained by using Galilean transformation. This analysis offers deep understanding of the phase portraits, time series, chaotic behavior and sensitivity analysis of the equation to external perturbations. The sensitivity and dynamics of optical solitons are thoroughly investigated that offers significant insights into their behavior within fractional models.

Turbulent fluctuation and amplitude of heat transfer and magnetic nanoparticle motion around oscillating fuel sphere in fusion reactor systems is main objective of this analysis. Casson nanofluid, ohmic heating, viscous dissipation and mixed convection properties are applied. Oscillating thermo diffusion, nonlinear thermal radiation, magnetic field and Brownian diffusion aspects are used for fluctuating heat and mass improvement. The Buongiorno mathematical model of magnetic Casson nanofluid flow is developed. The model is transformed using dimensionless variables and oscillatory Stokes conditions over three pi & frasl;6-rad, pi & frasl;4-rad, and pi & frasl;3-rad angles. Steady, real and imaginary models are used to find fluid velocity, temperature and concentration fields using primitive variables. Asymptotic results of streamlines, isothermal lines, steady skin friction, steady heat flux and steady mass transfer are obtained through FORTRAN tool. The implicit finite difference method is used to calculate the stability in turbulent waves of heat and mass rate with Gaussian elimination approach. The ohmic heating Jh, radiation Rd, Lorentz force Mf, thermophoresis NT, Eckert number Ec, Brownian number NB, and buoyancy lambda T are used to find oscillations and amplitudes of heat and mass rates. Significant increment in temperature distribution is found with maximum choice of ohmic heating at angle pi & frasl;6-rad. Magnitude of streamlines and isothermal lines is increased as ohmic heating and Lorentz force enhances. Large velocity amplitude is noted for high ohmic heating. Large fluctuation and largeness in frictions and heat-mass is depicted for every choice of Lorentz force and ohmic heating. The measurement level of heating and mass flow is observed near 39 % and 31 % as Eckert number enhances. Large amplitude of heat oscillations and mass oscillations is detected as Ohmic heating and Lorentz force increases.

The delay differential equations (DDEs) are widely used to explore various engineering and physical applications. An example of DDEs with proportional delays is known as the pantograph model which governs the current collection in electric trains. DDEs with constant delays also have different applications. This paper introduces a unified approach to analyze a class of first order DDEs under arbitrary history functions (HFs). The proposed approach assumes that the arbitrary HF phi(t) can be represented as Maclaurin series with coefficients phi(m), m >= 0. Based on this assumption, the solution in each sub-interval of the problem's domain is obtained in explicit form in terms of the coefficients phi(m). Exact solutions are obtained for several examples subjected to history functions of different forms. Properties of the solution and its derivative are proved and examined theoretically. Existing results in the literature are derived from the current ones as special cases. In view of the obtained results, the exact solution of any first order linear delay differential equation can be directly determined once the coefficients phi(m) of the given history function is inserted into the standard solution. This reflects the advantage of the proposed approach over other techniques. Moreover, the suggested analysis can be easily extended to include higher order linear delay models.

Originally, the pantograph equation with one proportional delay is of practical applications in railways electrification. This paper introduces a powerful tool to extract the exact solutions for a class of the pantograph-type equations containing two different proportional delays. The fundamental solution is given in a compact series form. Moreover, the obtained compact series form is then implemented to establish exact solutions for the investigated class subject to different conditions on the involved parameters. In absence of these conditions, exact-form solutions are no longer attainable, and the considered class is treated through approximations. The convergence of the obtained series solution is theoretically examined and numerically confirmed through several graphs. The main advantage of the present approach is that it can be further generalized to delay differential equations with a finite number of proportional delays.

For the second-order conservative nonlinear oscillator and the Lienard equation we derive integral-type theoretical formulas to compute the periods through a generalized conservation law equipped with a weight function. Minimizing the error to satisfy periodicity conditions the optimal value of parameter is determined; hence, very accurate value of the period can be obtained for examples in testing. Three methods to solve periods and periodic solutions are developed for the Lienard equations without/with periodically-forcing terms. The periodicity conditions and nonlinear differential equation for the oscillatory motion constitute a special kind boundary value problem (BVP), rather than the initial value problem (IVP). This study is concerned with two possible cases: (a) given initial values, and (b) unknown initial values. For case (a) the boundary values at two end points of a time interval are known but with an unknown period, while for case (b) the resulting BVP is more difficult with both unknown period and boundary values on an unknown time interval. By means of boundary shape function method (BSFM) we can transform the BVP to an IVP for an easy computation of periodic problem, where the initial values for the new variable are given and derived explicitly; however, the period and terminal values of the new variable are unknown to be determined iteratively. BSFM's periodic solutions automatically satisfy the periodicity conditions. Numerical examples disclose the merit of BSFM, which is convergent fast to offer very accurate period and periodic solution.

In this research, a [Formula: see text]-model approximation method is employed to investigate the localized wave solutions for the Lakshmanan-Porsezian-Daniel equation with beta-derivative. This equation integrates the fundamental phenomena such as Space-Time Dispersion (STD), Group Velocity Dispersion (GVD) and parabolic-law-governed by nonlinear behavior. A variety of optical soliton waves are obtained by applying the [Formula: see text]-model expansion approach. These waves are expressed as Jacobi elliptic functions [Formula: see text] that based on the particular values of the parameter A, that can be converted into solutions of trigonometric or hyperbolic functions. This technique provides variety of solutions, including dark soliton solutions, hyperbolic solutions, periodic waves solutions, bright solitons, singular soliton solution and singular periodic waves solutions. To further explore the system's behavior, bifurcation analysis is done. For this analysis planar dynamical system is obtained by using Galilean transformation. This analysis offers deep understanding of the phase portraits, time series, chaotic behavior and sensitivity analysis of the equation to external perturbations. The sensitivity and dynamics of optical solitons are thoroughly investigated that offers significant insights into their behavior within fractional models.

Obtaining accurate solutions for mathematical models of neutron diffusion systems may lead to a deeper understanding of processes in reactor physics. The present paper applies the Laplace transform to the time-dependent neutron diffusion equation (together with the delayed neutron precursor equation) under a reflective boundary condition at one edge. The residue theorem is employed to obtain the inverse transform, leading to a series solution structured as a modal expansion associated with the eigenvalues of a transcendental equation. Moreover, the obtained series solution is theoretically proven to converge. The numerical results show acceptable accuracy based on residual errors. Physically, the neutron flux exhibits oscillatory behavior within the spatial domain, resulting in a wave-like alternating surface. Additionally, the delayed neutron precursor concentration stabilizes over time, gradually approaching a stationary profile, which is consistent with the physical expectations. The results also support the effectiveness of the Laplace transform technique in capturing the early-time behavior of the system. Differences between the present results and those reported in the relevant literature are explained.

Fluctuating and turbulent energy dissipation effect on plate heat exchanger presents essential applications in marine turbines, power generation systems, large refrigeration systems, chemical processing industry, automotive industries, and food processing industries. This analysis presents thermophoretic convection and heat dissipation effects into radiative cooling performance and heating efficiency of plate heat exchanger in marine turbine under Darcy Forchheimer medium and vibration conditions. Dimensionless form signifies the balanced Maxwell model for prominent computational results of heat and mass flow rates. Turbulent framework is altered into primitive form of steady, real and imaginary models and solved through Gaussian elimination and finite difference methods in FORTRAN software. The velocity contours, temperature contours, flow dynamics, nanoparticle concentration, steady-turbulent heating efficiency, steady-turbulent mass flow and skinfriction are depicted. The enhancing rate of velocity contours and temperature contours is deduced as Maxwell factor and Darcy/Forchheimer-medium decreases. Amplitude in flow velocity, surface temperature and nanoparticle motion increases as the parametric Darcy porous medium is enhanced. Increasing largeness and amplitude in turbulent heating efficiency and turbulent mass flow is depicted under strong Darcy porous medium, heat dissipation and Maxwell fluid factor. The increasing percentage results of heating efficiency and mass flow are noticed under strong radiative cooling.

The stretching, magnetic coating and thermal radiating heat impact on conducting elastic-polymer surface has noticeable applications in drug-eluting polymer stents, biodegradable stents, tracheal stents, absorbable stents, vascular stents, coronary stents, and fabrication process of stents for enhanced heat and mass transport. Utilizing Powell-Eyring nanofluid, the amplitude, oscillation and phase change behavior in heating distribution and mass characteristics over elastic-polymer surface are deduced numerically. The polymer surface model is solved using dimensionless units, primitive-formulation, oscillatory stokes transformation, Gaussian-elimination technique and implicit finite difference method. The computational and tabular outcomes are deduced by using FORTAN tool and data is presented through Tecplot-360 with significant asymptotic values of unknown quantities. The steady values are examined first and then utilized in main domain to find the amplitude and oscillation in momentum-thermal transport of non-Newtonian fluid. The influence of thermal buoyancy force, radiation, magnetic field, thermophoresis and Brownian motion on the physical quantities such as velocity, streamlines, temperature, isotherms and concentration is deduced. It is noticed that the high amplitude in fluid velocity factor is depicted for high values of magnetic field, radiation and Powell-Eyring fluid parameter. Using Brownian motion and thermal buoyancy, the steady heating variation and steady nanoparticle concentration flow is better with noticeable difference. The fluctuation, oscillating frequency and phase angle of heating distribution and concentration/mass rate is increased as Schmidt factor and Prandtl coefficient increases. The concept of Powell-Eyring nanoparticles with improved theoretical and innovative mechanism is very useful in artificial heart surgery, heating-cooling of medical devices, stents coating, biodegradable magnesium stents, polymer cardio stents and other bio-medical equipment.

The generalized Riccati equation mapping method and the modified extended tanh-function are the two analytical techniques that are used to solve the nonlinear fractional-order differential equations. The Riemann Liouville sense is used to define the fractional derivative in Jumaries. Through saturated ferromagnetic materials with negligible conductivity, a nonlinear ultrashort wave pulse moves according to the fractional Kraenkel-Manna-Merle system. A number of families of analytical solutions to the fractional Kraenkel-Manna Merle model are produced by applying the proposed methods. When the proper values are given to the parameters, these methods successfully recover both hyperbolic and trigonometric solutions. The contour, 3D, and 2D graphs are given to illustrate how the parameters affect these solutions. In addition, phase portrait characterization is performed and the system is converted into a planar dynamical structure. Furthermore, the dynamical system’s sensitivity examination verifies that even small changes to the starting circumstances will not significantly affect the solution’s stability.

The investigation of delay differential equations (DDEs) spans a diverse array of practical applications. In the realm of applied sciences, DDEs typically involve either constant/pure delays or proportional delays, each posing significant analytical challenges. The task of deriving exact solutions becomes increasingly intricate when both types of delays are integrated within a single model. This paper derives an explicit unified analytical solution for a DDE with combined delays using the method of steps (MoS). A unified solution formula is presented, applicable across any sub-interval of the problem's domain. Additionally, the theoretical properties of the solution and its derivative—such as continuity and the presence of discontinuities at specific points—are meticulously examined. The proposed methodology also encompasses existing findings in the literature, with applications extending to fields including astronomy and railway electrification.

Obtaining a solution of a given SDE is essential in neuroscience, especially, in modeling transmission of nerve impulses between neurons through myelin substance. This paper analyzes a particular scalar differential equation (SDE). The current scalar model involves two categories of differential equations–advanced and delayed–based on the domain of the independent variable. The results are consistent with existing literature as the advance/delay parameter approaches unity. Theoretical and graphical analyses of the solution’s properties are presented. To the best of our knowledge, this is the first study to analyze this form of SDE.

Wave oscillations of periodic boundary layers and enhancement of fluctuating heat and mass distribution along vertical cone using Powell-Eyring nanofluid aspects is the novelty of current analysis. The significance of entropy generation, thermophoresis, nonlinear radiation and buoyancy force is applied for oscillating heat transfer enhancement. The unsteady partial differential formulation is developed and reduced into simple equations using unit-less variables. The oscillation behavior of heat and mass distribution, streamlines, isothermal lines, fluid velocity, fluid temperature and concentration are explored using steady and periodic conditions. To obtain steady and fluctuating outcomes, the Stokes oscillations and primitive factors are used to make similar relation of energy, momentum and mass equations. The algorithm is generated in FORTRAN tool using the implicit scheme of finite difference approach. The unknown thermal and flow quantities are obtained using Gaussian elimination scheme. The amplitude and phase angles are explored using oscillation formula to calculate the oscillatory and periodical quantities of heating stability and mass/concentration circulation. It is noticed that higher amplitude in nanofluid-velocity variation and wall-temperature is observed for large radiations. The uniform heating rate and concentration distribution increases as Brownian motion and thermophoresis force increases. Large oscillations and amplitudes in heat and mass transfer are observed for each value of thermal radiation. It is depicted that the streamline variation increases as Powell-Eyring parameter decreases and mixed convection parameter increases. The maximum scale of isotherm contour is found as thermal radiation rises and Powell-Eyring parameter drops. The outstanding improvement in mass and heat transmission is deduced as mixed convection parameter enhances. The rate of skin friction is enhanced as Powell-Eyring material factor increases.

Thermal transport and mass/concentration transfer through heat exchanger vertical plate is important component of nuclear power reactor for heat transfer in primary coolant loop to secondary loop. These loops generate stream in turbine to generate electricity energy. In nuclear power reactors, the heat exchanger plate provides enhanced cooling. The fluctuations and turbulence behavior of heat-mass transfer of oscillatory Williamson nanofluid over the heated surface with viscous dissipation, thermophoresis, and nonlinear radiating heat effects. Mathematical model is developed for unsteady flow to discuss the steady profiles and shifting amplitude of periodical heat/thermal and mass-concentration transport. The fluctuating Stokes conditions are applied to change the steady and fluctuating model into real/imaginary equations. Real, steady and imaginary models are transformed through primitive transformations to create the similarity in all equations under FORTRAN language. For asymptotic and fluctuating results of heat and mass transfer, the Gaussian elimination approach is used through implicit finite difference technique for programming algorithm. The velocity streamlines and isothermal lines are plotted along the heat exchanger plate for Eckert number (Ec), radiation parameter (Rd), thermophoresis (NT) and Weissenberg number (W). The steady profiles of skin friction, heat transfer and mass rate are examined and then used in oscillatory formula to draw the periodical-skin friction, periodical heat transfer and periodical-mass rate with amplitude and turbulent effects. It is found that magnitude streamlines and isothermal lines increases as radiation parameter increases. The high fluctuations and turbulence in heat and mass transfer is noted for maximum thermophoresis and radiation parameter with greater amplitude.

In the realm of fluid mechanics, the behavior of materials often defies simplistic characterization, requiring sophisticated models to capture their intricate dynamics, especially when involving non-Newtonian fluids. This research endeavors to elucidate the bioconvection flow of Reiner-Rivlin nanofluid incorporating gyrotactic microorganisms in the presence of magnetohydrodynamics (MHD), multiple slip, and thermal radiation over a rotating surface. Thermal and solutal convective boundary conditions have also been considered. The Buongiorno model is added together with the governing equations in partial differential equations (PDEs). By the adoption of transformation, the complex PDEs are transformed into ordinary differential equations (ODEs). The physical quantities of interest in this study are drag friction, heat, mass, and local motile microorganism density. Numerical solutions are calculated by the bvp4c solver in MATLAB, and the impacts of the governing parameters are visualized through graphical representations and tabulated data. Visual representations are used to inspect the significant effects of the changing parameters on the involved fields. The results demonstrate that radial velocity and temperature are improved for a strong Reiner-Rivlin material parameter. Enhancing slip variables lessens the shear stress by up to 30%, considerably. Tables show the relation between various estimations of emergent parameters and the behavior of microorganism density, friction coefficients, rate of heat and mass are provided. By comparing the reported results to an existing published study, we can verify that the intended model is authentic. We get a proper correlation between the two sets of results. These findings emphasize the possibility for enhancing refrigeration procedures for spinning systems like windmills and disk-based thermal exchangers by combining viscous nanofluids and microbial growth. This study sheds light on the development of innovative bio-nanofluidic technologies and effective heat control techniques for commercial and medical uses.

This paper introduced a new aftertreatment technique for solving a fractional nonlinear Susceptible-Infectious-Recovered (SIR)-epidemic model. The proposed approach reformulated the power series solution via incorporating the Laplace transform and its inverse. Basically, it applied the Laplace transform to convert the series solution into different Pade-approximants involving Laplace's parameter with arbitrary order. This procedure facilitated the method of deriving the inverse Laplace transform explicitly, as a final step, using basic special functions in fractional calculus. Our analysis was capable of obtaining a sequence of closed-form approximations in terms of the Mittag-Leffler functions. The current results may be provided for the first time regarding the Mittag-Leffler solution of the fractional SIR-model. Additionally, the outcome of this analysis revealed that our approach was not only effective but also applicable to a wide range of fractional differential equations and systems.