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This paper considers the movement of an incompressible non-Newtonian Walters' B liquid within a rotating annulus. The flow lies within porous media and is magnetized by a uniform normal magnetic field. Mass transmission is included in the arrangements of homogeneous and heterogeneous chemical reactions (HHCR) with a thermal diffusion effect. Heat diffusion is exposed with a temperature- dependent heat source (THS) and an exponential radial-dependent heat source (ERHS), besides Joule heating and viscous dissipations. The mathematical construction is shown by momentum, energy and concentration of chemical materials equations along with time-periodic boundary conditions. The innovation of this study stems from considering mass transfer utilizing HHCR, with thermal diffusion effects in the flow within a rotating annulus, trying to understand the effect of these reactions on the flow and related applications. The principal regulating the structure of nonlinear partial differential equations is transferred by ordinary ones utilizing appropriate similarity analysis. These equations are analyzed by the homotopy perturbation method. Subsequently, the analytical solutions of the main profiles are obtained, and a graphical construction is established to clarify the effects of the relevant physical elements. It is observed that the velocity waves decrease with annulus width enlargement and increase with the wave phase amplitude. With an increase in all significant parameters, heat transmission gets better. Furthermore, as the strength of HHCR is increased, the concentration of molecules decreases. These outcomes are significant in several applications like cancer therapy, biochemical manufacturing purposes through reactors, where biochemical interactions can be regulated by accurate control of various parameters. [ABSTRACT FROM AUTHOR]

AbstractThis article investigates the thermal, variable viscosity axial Couette flow between two con- centric circular cylinders, taking into account both the Navier and the dynamic slip boundary conditions. In order to put in evidence the effect of dynamic slip on the behavior of the solution, we have kept the flowing fluid to a simple Newtonian viscous fluid. The nonlinear governing equations for momentum and energy balance are solved under startup condition using the Laplace transform (LT) technique, in conjunction with the Homotopy Perturbation Method (HPM). The first two orders of approximation for temperature and velocity are obtained, as well as the entropy generation in the space occupied by the flow and the volumetric flux. Numerical results and graphical representations of the solution are provided and discussed to depict the effects of the slip parameters on the velocity and the temperature profiles, on the entropy generation rate and on the volumetric flux. [ABSTRACT FROM AUTHOR]

In this paper, inscribed non-Newtonian Williamson fluid relaxation and contraction phenomena due to the effect of magnetohydrodynamics (MHD) is taken into account. This issue is modeled, and precise solutions are handled using a low Reynolds number and long wavelength approximation. Graphical depiction of the consequences of various parameters on temperature and velocity are provided. It becomes clear that the temperature reactions to increases in Biot numbers are decreased. Considering a cylindrical coordinate system, the impact of various physical parameters on the temperature distribution, velocity, the pressure gradient, and frictional force are secured from the flow for non-Newtonian Williamson fluid exposed from MATLAB. Additionally, it has been discovered that the velocity profile at the tube's center increases for shear-thinning a fluid and decreases for thickening a fluid. Furthermore, the peristaltic walls show the opposite behavior. [ABSTRACT FROM AUTHOR]

The current study aims to utilize the homotopy perturbation method (HPM) to solve nonlinear dynamical models, with a particular focus on models related to predicting and controlling pandemics, such as the SIR model. Specifically, we apply this method to solve a six-compartment model for the novel coronavirus (COVID-19), which includes susceptible, exposed, asymptomatic infected, symptomatic infected, and recovered individuals, and the concentration of COVID-19 in the environment is indicated by S (t) , E (t) , A (t) , I (t) , R (t) , and B (t) , respectively. We present the series solution of this model by varying the controlling parameters and representing them graphically. Additionally, we verify the accuracy of the series solution (up to the (n − 1) t h -degree polynomial) that satisfies both the initial conditions and the model, with all coefficients correct at 18 decimal places. Furthermore, we have compared our results with the Runge–Kutta fourth-order method. Based on our findings, we conclude that the homotopy perturbation method is a promising approach to solve nonlinear dynamical models, particularly those associated with pandemics. This method provides valuable insight into how the control of various parameters can affect the model. We suggest that future studies can expand on our work by exploring additional models and assessing the applicability of other analytical methods. [ABSTRACT FROM AUTHOR]

AbstractCarbon nanotubes have gained significant attention due to their unique properties like small dimensions, high thermal conductivity, and unique architecture. These nanotubes have enormous potential in various fields such as nanomedicine, mechanical compound materials, sensor devices, bio-sensors, and many other applications. The present study investigates the impact of adding nanoparticles such as iron oxide (Fe3O4), single-walled carbon nanotubes (SWCNT), and multi-walled carbon nanotubes (MWCNT) on magnetohydrodynamics (MHD) oscillatory water-based flow over a porous medium. The aim is to examine the effect of heat and mass transfer rate and radiative viscous dissipation that interacts with Fe3O4 nanoparticles and carbon nanotubes. The governing equations of the dynamic nanofluid systems are transformed into a set of coupled nonlinear partial differential equations through suitable dimensionless transformations and further converted into ordinary differential equations by taking appropriate solutions for oscillatory-type nanofluid flow. The homotopy perturbation method is applied to solve the governing equations with oscillatory boundary conditions and the solution is also compared with the numerical solver bvp4c. The rate of increase in the temperature and Heat Transfer are computed by the linear slope regression method and results are computed for nanoparticles SWCNT, MWCNT, and Ferro. Also, Skin friction and rate of heat transfer are computed and compared for the nanoparticles with varying Peclet and Eckert numbers. The velocity profile and temperature profile of SWCNT/MWCNT/Ferro nanofluids are being compared. The heat transfer rate is observed to be high for Ferro nanoparticles compared to SWCNT and MWCNT. [ABSTRACT FROM AUTHOR]

A local fractional modification of the Duffing equation is considered, and the homotopy perturbation method is employed to reveal its frequency-amplitude relationship, which is of paramount importance in the optimal design of the energy harvesting devices and chatter detection. Effects of the initial conditions on the periodic property is also discussed. [ABSTRACT FROM AUTHOR]

Purpose: Financial mathematics is one of the most rapidly evolving fields in today's banking and cooperative industries. In the current study, a new fractional differentiation operator with a nonsingular kernel based on the Robotnov fractional exponential function (RFEF) is considered for the Black–Scholes model, which is the most important model in finance. For simulations, homotopy perturbation and the Laplace transform are used and the obtained solutions are expressed in terms of the generalized Mittag-Leffler function (MLF). Design/methodology/approach: The homotopy perturbation method (HPM) with the help of the Laplace transform is presented here to check the behaviours of the solutions of the Black–Scholes model. HPM is well known for its accuracy and simplicity. Findings: In this attempt, the exact solutions to a famous financial market problem, namely, the BS option pricing model, are obtained using homotopy perturbation and the LT method, where the fractional derivative is taken in a new YAC sense. We obtained solutions for each financial market problem in terms of the generalized Mittag-Leffler function. Originality/value: The Black–Scholes model is presented using a new kind of operator, the Yang-Abdel-Aty-Cattani (YAC) operator. That is a new concept. The revised model is solved using a well-known semi-analytic technique, the homotopy perturbation method (HPM), with the help of the Laplace transform. Also, the obtained solutions are compared with the exact solutions to prove the effectiveness of the proposed work. The different characteristics of the solutions are investigated for different values of fractional-order derivatives. [ABSTRACT FROM AUTHOR]

The analysis of MHD UCM fluid flow between a pair of rectangular plates with the top plate moving either towards or apart from the stationary bottom porous plate is presented in this work. By employing similarity transformations, the governing equations of momentum are transformed into non-linear ordinary differential equations. An approximate analytical solution is achieved by adopting the homotopy perturbation technique. The aim of this work is to determine the velocity profile and coefficients of skin friction for various values of different physical parameters. It has been found that as the magnetic parameter enhances, the squeezing flow retards in injection and suction cases. It is also observed that the velocity field declines in the core region with a rise in the porosity parameter. This theoretical study is helpful in the processing of visco-elastic polymers in industry. [ABSTRACT FROM AUTHOR]

Homotopy perturbation method (HPM) is applied to approximate and simulate the cell-cell adhesion model. [ABSTRACT FROM AUTHOR]

This study examines Titania nanofluids that conduct electricity and are combined with various base fluids, focusing on various aspects. A magnetic field analyzes a continuous flow of nanofluid over a rotating disk that is positioned at an angle with a three-dimensional geometry assumption. The continuity, momentum, and energy balance equations are established, transformed using similarity variables, and solved through Akbari–Ganji Method (AGM) and Homotopy Perturbation Method (HPM) semi-analytical methods. The effects of several parameters, such as Magnetic parameter (M), Hall parameter (m), Porosity parameter (γ), Radiation parameter (Rd), and Thickness of liquid (δ), are examined through a graphical representation of state variables, including skin friction and Nusselt number. Additionally, the Response Surface Method (RSM) method has been utilized to simultaneously display the effects of porosity parameters and various factors across the disk. The findings showed that the magnetic, porosity, Hall, and thickness parameters had a significant impact on the behavior of the nanofluid. Specifically, the magnetic field had a strong influence on the velocity and temperature distribution of the fluid as it flowed over the rotating disk, while changes in porosity, Hall, and thickness parameters also affected these state variables. [ABSTRACT FROM AUTHOR]

The non-Newtonian characteristic of the fluid flow phenomena has several applications in industries and the production of engineering gadgets. Based upon the role, the present work deals with the flow of Williamson nanofluid past two inclined parallel sheets. Further, the influence of radiating heat due to thermal radiation, Ohmic heating because of the magnetization, and inclined angle on Williamson fluid are presented, which are vital for the heat transfer enhancement. The proposed study became novel due to the scrutiny of entropy for the irreversibility processes within the system that is obtained due to heat transfer. The designed model with dimensional form is developed by converting to dimensionless forms with the help of similarity transformations. The finalized transformed equations are handled analytically using the Homotopy perturbation method (HPM). Heat irreversibility is calculated by inserting the observed values into entropy generation and Bejan number. The suggested method of solution for the momentum profile is verified by comparing it to a previously published numerical result in some limited circumstances, and great agreement is found. The result reveals that the entropy generation is enhanced near the upper wall of the channel for the magnetized particle but thermal radiation attenuates it, however, the fluid temperature boosts up for the enhanced coupling constraint i.e. for the existence of Eckert number. [ABSTRACT FROM AUTHOR]

This study's primary objective is to investigate the Jeffery–Hamel model and entropy generation in the magnetohydrodynamic (MHD) flow of hybrid nanofluid across stretching, shrinking, converging, and diverging channels. Aluminum oxide (Al2O3) and silver (Ag) are nanoparticles, using blood as the base fluid. The controlling nonlinear coupled partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) with similarity transformations and then solved using the Homotopy Perturbation Method (HPM) and shooting technique (Bvp4c). The HPM is compared to the numerical method (NM), and the results are more accurate and reliable. The effects of velocity, temperature, entropy generation, and the Bejan number on physical parameters like a magnetic field, the Reynolds number, magnetic field, and the Brinkman number are discussed through graphs. The heat transfer and skin friction coefficients are also studied and portrayed as graphs and tables. The influence of a magnetic field on the velocity profile of stretching and shrinking through converging and diverging channels. When increasing the magnetic field, the velocity profile increases. Physically, an increase in the magnetic field produces a higher Lorentz force, which improves fluid resistance and restricts fluid particle flow within the given topology. The temperature profile decreases in the stretching and shrinking of converging channels as magnetic field values rise. As well as the opposite behavior observed in stretching and shrinking for diverging channels. In this model, biomedical applications include studying physiological systems, surgical procedures, nano-pharmacological delivery systems, and nano-mediated treatment of atherosclerosis. [ABSTRACT FROM AUTHOR]

The research aims to scrutinize the significance of MHD, radiation, Joule heating, porous medium, and heat generation impacts on the convective boundary flow. It contains blood as the base fluid, silver (Ag), and aluminum oxide (Al2O3) as the nanoparticles over a curved stretching and shrinking sheet. To investigate the rheology of blood, we used a Williamson fluid model with comparisons for both curved stretching/shrinking sheets for velocity and a comparison between linear and nonlinear thermal radiation for temperature, entropy production, and Bejan number. This model is used to study physiological systems, surgical procedures, and nano-pharmacological delivery systems. The governing nonlinear coupled PDEs are transformed into nonlinear associated ODEs via the similarity transformation in the Maple software solver using the Homotopy Perturbation Method. The Homotopy Perturbation Method (HPM) results are compared with the Numerical Method (NM) results, where HPM gives reliable results. The results and discussion provide graphical results with several parameters for velocity, temperature, entropy production, Bejan number, Nusselt number, and skin friction. As the magnetic field parameter (M) increases, the velocity profile decreases for curved stretching sheet. We observe the opposite behavior of the velocity profile for a curved shrinking sheet. As the volume fractions increase, the temperature profile increases for linear and non-linear thermal radiation. The temperature profile rises due to the thermal conductivity of nanoparticles, which is amplified by estimations of the nanoparticle volume fractions. [ABSTRACT FROM AUTHOR]

This paper advances fundamental knowledge of how environmental conditions and physical phenomena at different scales can be included in the differential equation that models the flight dynamics of dipteran insects. The insect's anatomical capability of modifying their mass inertia and flapping-wing damping properties during flight are included by modeling inertia and damping forces with fractal derivatives. An expression for calculating fractal dimension linked to the temporal distribution of non-geometric quantities related to atmospheric processes such as turbulence flow is introduced using, for the first time ever, the two-scale fractal dimension definition and adopting the flow energy spectrum of eddies that occur at large and small scales. The applicability of the derived expression is illustrated with the prediction of the fractal dimension observed in turbulent flows. Then, the two-scale fractal dimension transform is used to re-write the dipteran flight equation of motion in equivalent form to derive its approximate solution using harmonic balance and homotopy perturbation methods. Numerical predictions computed from the derived approximate solutions allow to elucidate how insects and animals could adapt to flight under different environmental conditions. [ABSTRACT FROM AUTHOR]

The main goal of this work is to focus on using He's two-scale fractal dimension transform, the Caputo–Fabrizio fractional-order derivative, and the harmonic balance and the homotopy methods are applied for deriving the approximate solution of the fractal, damped cubic–quintic Duffing's equation when the fractional derivative order of the inertia term is not twice of that of the damping term. Numerical results obtained from the derived expressions and the numerical integration solution show good agreement, especially at small values of the nonlinear parameters. Furthermore, when the fractal order of the damping term decreases, the damping oscillation frequency values increase with a decrease in the system wavelength values, which indicates a slower decay in the system oscillation amplitudes. Our solution procedures elucidate the applicability of He's two-scale fractal dimension transform for solving nonlinear dynamic systems with inertia and damping fractal terms. [ABSTRACT FROM AUTHOR]

Diabetes is referred to a chronic metabolic disease signalized by elevated levels of blood glucose (also known as blood sugar level), which results over time in serious damage to the heart, blood vessels, eyes, kidneys, and nerves in the body. A mathematical assessment of the diabetes model using the Caputo fractional order derivative operator is given in this research paper. The concept of a Caputo fractional order derivative is a novel class of noninteger order derivative that has many applications in real-life scenarios. The proposed model is represented by a set of fractional ordinary differential equations. The authors employed the Sumudu Transform Homotopy Perturbation Method (STHPM) for finding the series solutions of the model being studied. By giving various numerical values to the respective model parameters, graphical analysis is also performed. It is observed in the numerical discussion that a decrease in both fractional order and leads to decrease in the number of diabetic people. [ABSTRACT FROM AUTHOR]

In order to unveil different features of gravitational waves due to massive objects like black hole etc., researchers primarily attempt to find the corresponding quasi normal modes from the relevant wave equations. In the present article, homotopy perturbation method (HPM) is employed to obtain the wave functions by solving wave equations for one‐dimensional potential barriers. Under this scheme (i) for constant potential barrier a generalized solution which reduces to the known solution as done by Chandrasekhar and Detweiler [Proc. R. Soc. Lond. A 344 (1975) 441] is produced, (ii) the wave equation for the Price potential [Phys. Rev. D 5 (1972) 2419; Phys. Rev. D 5 (1972) 2439] by HPM is solved. The wave functions (graphically) as well as quasi‐normal modes to those of the Press solution (1973) are compared. The present investigation is important in connection to further study related to the Zerilli potential and Regge‐Wheeler potential in case of black hole. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to inspect the unsteady MHD mixed convective Couette flow of viscous dissipative fluid in a vertical channel filled with porous material. The steady state analytical solutions of temperature, velocity, frictional force and heat transfer rate are obtained using the Homotopy perturbation method (HPM). The implicit finite difference technique is used to solve the transient leading equations numerically. Plots and tables with the dynamic flow parameters and dimensionless variables are displayed. During this computation, it was found that raising the Darcy parameter (Da) and the mixed convection (Gre) parameters both push the fluid flow upwards, but increasing the magnetic field's values had the opposite effect (M). [ABSTRACT FROM AUTHOR]

The Burger equation has been widely used to study nonlinear acoustic plane waves in gas-filled tubes, waves in shallow water, and shock waves in gas. Recently, a more comprehensive version of the equation, known as the fractional-order Burger equation, has emerged. However, finding a closed-form for approximate analytic solution using analytical methods for this type of equation is challenging. This paper focuses on researching the multidimensional fractional-order time and space Burger equation based on the Caputo-Katugampola derivative. An approximate analytic solution is obtained using the generalized Laplace homotopy perturbation method. The coefficients of the approximate analytic solution have a recurrence relation similar to the Catalan number in number theory, and the closed form of the approximate analytic solution can be obtained using number theory knowledge. It is worth noting that the Caputo-Katugampola derivative can be reduced to the Caputo derivative, and hence, the closed form for the approximate analytic solution of the multidimensional fractional-order time and space Burger equation based on the Caputo derivative is also obtained. [ABSTRACT FROM AUTHOR]

The primary goal of the present study is to investigate how to obtain a periodic solution for a fractal Mathieu–Duffing oscillator. To achieve this, the fractal oscillator in the fractal space has been transformed into a damping Mathieu–Duffing equation in the continuous space by employing a new modification of He's definition of the fractal derivative. The required analytical periodic solution has been based on the rank upgrade technique (RUT) presented. The RUT successfully generates a periodic solution without sacrificing the damping coefficient by creating an alternate equation, aside from any challenges in managing the impact of the linear damping component. The homotopy perturbation method (HPM) has been used to find the required periodic solution for the alternate equation. A comparison of the numerical solutions of the original equation and the alternative equation showed good agreement. The stability behavior in the non-resonance case as well as in the sub-harmonic resonance case has also been discussed. Further, another method, "the non-perturbative approach", that deals with the obtained equation has been introduced. [ABSTRACT FROM AUTHOR]

The exploration of collisional fragmentation pheno-mena remains largely unexplored, yet it holds considerable importance in numerous engineering and physical processes. Given the nonlinear nature of the governing equation, only a limited number of analytical solutions for the number density function corresponding to empirical kernels are available in the literature. This article introduces a semi-analytical approach using the homotopy perturbation method to obtain series solutions for the nonlinear collisional fragmentation equation. The method presented here can be readily adapted to solve both linear and nonlinear integral equations, eliminating the need for domain discretization. To gain deeper insights intothe accuracy of the proposed method, a convergence analysis is conducted. This analysis employs the concept of contractive mapping within the Banach space, a well-established technique universally acknowledged for ensuring convergence. Various collisional kernels (product and polymerization kernels), breakage distribution functions (binary and multiple breakage) and various initial particle distributions are considered to obtain the new series solutions. The obtained results are successfully compared against finite volume method [26] solutions in terms of number density functions and their moments. The error between the exact and obtained series solutions is shown in plots and tables to confirm the applicability and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we study a generalization of a Lotka–Volterra system with Holling type III functional response, where the Caputo fractional derivative is considered. Applying a multistage homotopy perturbation method, we obtain an analytical solution for the system. Moreover, analyzing the eigenvalues of the Jacobian matrix around the equilibria, we find sufficient conditions in order to guarantee the local stability and we present several examples to illustrate the behavior of solutions. [ABSTRACT FROM AUTHOR]

The main features of scientific efforts in physics and engineering are the development of models for various physical issues and the development of solutions. In order to solve the time-fractional coupled Korteweg–De Vries (KdV) equation, we combine the novel Yang transform, the homotopy perturbation approach, and the Adomian decomposition method in the present investigation. KdV models are crucial because they can accurately represent a variety of physical problems, including thin-film flows and waves on shallow water surfaces. The fractional derivative is regarded in the Caputo meaning. These approaches apply straightforward steps through symbolic computation to provide a convergent series solution. Different nonlinear time-fractional KdV systems are used to test the effectiveness of the suggested techniques. The symmetry pattern is a fundamental feature of the KdV equations and the symmetrical aspect of the solution can be seen from the graphical representations. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. Additionally, the system's approximative solution is illustrated graphically. The results show that these techniques are extremely effective, practically applicable for usage in such issues, and adaptable to other nonlinear issues. [ABSTRACT FROM AUTHOR]

In this note, the hybrid method (combination of the homotopy perturbation method (HPM) and the Gauss elimination method (GEM)) is developed as a semi-analytical solution for the first kind system of Cauchy-type singular integral equations (CSIEs) with constant coefficients. Before applying the HPM, we have to first reduce the system of CSIEs into a triangle system of algebraic equations using GEM, which is then carried out using the HPM. Using the theory of the bounded, unbounded and semi-bounded solutions of CSIEs, we are able to find inverse operators for the system of CSIEs of the first kind. A stability analysis and convergent of the proposed method has been conducted in the weighted Lp space. Moreover, the proposed method is proven to be exact in the Holder class of functions for the system of characteristic SIEs for any type of initial guess. For each of the four cases, several examples are provided and examined to demonstrate the proposed method's validity and accuracy. Obtained results are compared with the Chebyshev collocation method and modified HPM (MHPM). Example 3 reveals that the error term of the MHPM is slightly superior to that of the HPM. One of the features of the proposed method is that it can be solved as a complex-valued system of CSIEs. Numerical results revealed that the hybrid method dominates others. [ABSTRACT FROM AUTHOR]

This paper investigates the mathematical result of the nonlinear fractional scheme of equations representing the singular and non-singular thermoelastic system. The suggested method is a graceful combination of Shehu transformation with homotopy perturbation method and fractional derivative described with the Caputo operator. To show and confirm the effectiveness of the future method, I examined three cases and evaluated in terms of fractional-order the projected model. Furthermore, the physical behavior of the fractional-order solution achieved has been shown in graphs, and the mathematical model is proved to confirm the accuracy. The solutions achieved clarify that the suggested methodology is simple to apply, highly organized, and accurate to study the attitude of nonlinear system differential equations of arbitrary order arising in the related fields of engineering and science. [ABSTRACT FROM AUTHOR]

In this paper, two well-known analytic approaches for solving diffusion equations are implemented. We propose a modified version of the homotopy perturbation method and Adomian decomposition methods utilizing the ZZ transform. In this sense, the fractional derivative of the Atangana–Baleanu derivative is provided. In addition, concrete examples are as well provided to demonstrate the precision of the proposed methodologies. The proposed solution is observed to have the desired rate of convergence towards the exact answer. The key advantage of the proposed method is the small amount of calculations. To demonstrate the validity of the suggested methods, we give graphical representation of analytical and exact solutions that are in close contact. [ABSTRACT FROM AUTHOR]

Thermal analysis of serrated fins which are consist of annular and plain sections are investigated. Serrated fin's thermal conductivity is assumed to change linearly with temperature. Nonlinear differential equations are obtained by applying the energy balance equation for both sections of the serrated fin and these equations are solved by applying homotopy perturbation method. Insulated fin tip, constant fin base temperature and common boundary conditions between the interface of two sections are considered. Serrated fin radii ratio (𝜀), segment height ratio (𝛿), thermo-geometric fin parameter (𝜓) and thermal conductivity parameter (𝛽) effecting the thermal performance and temperature distribution are investigated. The results showed that the homotopy perturbation is a reliable method for the solutions of such nonlinear differential equations. A very good agreement with the homotopy perturbation method and the numerical finite difference method are obtained. It is seen that, serrated fin efficiency lays between annular and rectangular fins and increases with the increase of segment height ratio and thermal conductivity parameter. Such as, fin efficiency values under the condition of 𝜀 = 2, 𝜓1 = 1.0 and 𝛽 = 0 for 𝛿 = 0, 0.5, and 1 are 0.692, 0.718, and 0.762, respectively. [ABSTRACT FROM AUTHOR]

In this paper, the He–Elzaki transform method (HEM) is proposed. The method is formulated by combining He's variation iteration method and the modified Laplace transform, known as the Elzaki integral transform. This method is designed to solve the time-fractional telegraph equation that arises in electromagnetics. The Caputo sense is used to describe fractional derivatives. One of the advantages of this method is that the computation of the Lagrange multiplier is not necessarily required through the convolution theorem or integration in recurrence relations. Additionally, to reduce nonlinear computational terms, He's polynomial is determined using the homotopy perturbation method. The proposed method is applied to several examples of nonlinear fractional telegraph equations. The results obtained from these examples demonstrate that the proposed method is an efficient technique that facilitates the process of solving time-fractional differential equations. [ABSTRACT FROM AUTHOR]

This paper estimated the problem of one-dimensional generalized thermoelastic half-space in medium considering two models: the Lord and Shulman (LS) model and the Dual-Phase-Lag (DPL) model. We assumed that the surface of the half-space was free from traction force and under an exponentially varying external heat source at the boundary with time. The technique of homotopy perturbation has been applied to find the approximate solution for the interactions of thermoelasticity with the applied boundary condition. The effect of a heat source that varies with the time and the free traction force are investigated for the temperature, displacement, and stress. The numerical results obtained are presented graphically to show the influence of the new external parameters. The results obtained illustrate the strong impacts on the displacement, temperature, and stress with the variations in the two models as well as the relaxation time parameter. The results show the agreement between the present results and the previous obtained results of the phenomenon and applicable, especially in biology, acoustics, engineering, and geophysics. [ABSTRACT FROM AUTHOR]

Background: Tuberculosis (TB), caused by Mycobacterium tuberculosis, is a contagious infectious disease that primarily targets the lungs but can also impact other critical systems such as the bones, joints, and neurological system. Despite significant efforts to combat TB, it remains a major global health concern. To address this challenge, this study aims to explore and evaluate various tuberculosis control approaches using a mathematical modeling framework. Results: The study utilized a novel SEITR mathematical model to investigate the impact of treatment on physical limitations in tuberculosis. The model underwent qualitative analysis to validate key aspects, including positivity, existence, uniqueness, and boundedness. Disease-free and endemic equilibria were identified, and both local and global stability of the model was thoroughly examined using the derived reproduction number. To estimate the impact of each parameter on each compartment, sensitivity analysis was conducted, and numerical simulations were performed using Maple 18 software with the homotopy perturbation method. The obtained results are promising and highlight the potential of the proposed interventions to significantly reduce tuberculosis virus prevalence. The findings emphasize the significance of fractional-order analysis in understanding the effectiveness of treatment strategies for mitigating tuberculosis prevalence. The study suggests that the time fractional dynamics of TB treatment correspond to the treatment's efficacy, as the conceptual results showed that non-local interactions between the disease and the treatment may lead to more accurate ways of eradicating tuberculosis in real-world scenarios. These insights contribute to a better understanding of effective treatment strategies and their potential impact on tuberculosis control and public health. Conclusions: In conclusion, scientists, researchers, and healthcare personnel are urged to take action and utilize the discoveries from this research to facilitate the eradication of the hazardous tuberculosis bacteria. [ABSTRACT FROM AUTHOR]

In this article, we study the fractional SIR epidemic model with the Atangana–Baleanu–Caputo fractional operator. We explore the properties and applicability of the ZZ transformation on the Atangana–Baleanu–Caputo fractional operator as the ZZ transform of the Atangana–Baleanu–Caputo fractional derivative. This study is an application of two power methods. We obtain a special solution with the homotopy perturbation method (HPM) combined with the ZZ transformation scheme; then we present the problem and study the existence of the solution, and also we apply this new method to solving the fractional SIR epidemic with the ABC operator. The solutions show up as infinite series. The behavior of the numerical solutions of this model, represented by series of the evolution in the time fractional epidemic, is compared with the Adomian decomposition method and the Laplace–Adomian decomposition method. The results showed an increase in the number of immunized persons compared to the results obtained via those two methods. [ABSTRACT FROM AUTHOR]

In this paper, we examined the approximations to the time-fractional Kawahara equation and modified Kawahara equation, which model the creation of nonlinear water waves in the long wavelength area and the transmission of signals. We implemented two novel techniques, namely the homotopy perturbation transform method and the Elzaki transform decomposition method. The derivative having fractional-order is taken in Caputo sense. The Adomian and He's polynomials make it simple to handle the nonlinear terms. To illustrate the adaptability and effectiveness of derivatives with fractional order to represent the water waves in long wavelength regions, numerical data have been given graphically. A key component of the Kawahara equation is the symmetry pattern, and the symmetrical nature of the solution may be observed in the graphs. The importance of our suggested methods is illustrated by the convergence of analytical solutions to the precise solutions. The techniques currently in use are straightforward and effective for solving fractional-order issues. The offered methods reduced computational time is their main advantage. It will be possible to solve fractional partial differential equations using the study's findings as a tool. [ABSTRACT FROM AUTHOR]

This study presents the modified form of the homotopy perturbation method (HPM), and the Yang transform is adopted to simplify the solving process for the Kuramoto–Sivashinsky (KS) problem with fractal derivatives. This scheme is established by combining the two-scale fractal scheme and Yang transform, which is very helpful to evaluate the approximate solution of the fractal KS problem. Initially, we transfer the fractal problem into its partners using the two-scale fractal approach, and then we use the Yang transform ( T) to obtain the recurrent relation. Second, the HPM is then introduced to deal with the nonlinear elements of the fractal model. The numerical example demonstrates how the suggested technique is incredibly straightforward and precise for nonlinear fractal models. In addition, the graphical error of the proposed fractal model is compared with the calculated results of our suggested approach and the exact results. This graphical error displays the strength and authenticity of our proposed scheme. [ABSTRACT FROM AUTHOR]

This study investigates the hemodynamics of nanofluid flow through modelled stenosis-aneurysm models. The models were created using mathematical functions to increase their realism. This study aims to explore how temperature-sensitive drugs coated on nanoparticles could be delivered to diseased areas, with the mathematical model aiding in the treatment of vascular stenosis. To effectively treat stenosis, medication-coated nanoparticles should be applied to the exterior surface of a catheter. The blood flow was modelled as a micropolar fluid flow, which led to the development of highly nonlinear coupled equations for momentum, temperature, and concentration. The dispersion of nanoparticles resulted in changing viscosity effects, making the fluid flow equations even more complex. The model considered the porous nature of the stenosis, no-slip at the catheter surface, and free slip at the blood vessel surface. The homotopy perturbation method was used to solve the formulated mathematical model. The study investigated the convergence of perturbed solutions for temperature and concentration and showed the degree of deformation. Drug delivery to a targeted region is faster in a converging tapered blood vessel than in a diverging and non-tapered artery. Concentration dispersion is more significant in the stenotic region, while temperature dispersion is more significant in the aneurysm region. The results of the study can be used to understand the improvement in mass dispersion and heat transfer in unhealthy blood arteries, which may be useful in delivering drugs to treat stenotic diseases. [ABSTRACT FROM AUTHOR]

The effect of thermal dissipation on a steady, fully developed, mixed convection viscous, incompressible fluid in an annulus partially filled with porous materials has been thoroughly examined in this work. Fluid flow begins within the annulus when a pressure gradient is applied abruptly in the flow direction. The fluid flow in the porous zone is characterized by the Brinkmann‐extended Darcy model. The fluid is divided into transparent and porous parts by a minimal interface. By matching their velocities and considering the shear stress jump conditions at the interface, the clear fluid and the porous region are connected. Additionally, the viscous dissipation effect is considered while determining the energy equation in the clear fluid zone. However, in the porous area, the Darcy dissipation effects are considered in addition to the viscous dissipation influence. In the model, the results of various fluid parameters in the problem were addressed using line graphs and the homotopy perturbation method. The study found that when the porous region's thickness grows, heat transmission on the annular surface enclosing the clear fluid region increases while it decreases on the border surface close to the porous region. In addition, a thicker porous region requires a greater pressure gradient to propel the flow. [ABSTRACT FROM AUTHOR]

The present article explores the effect of viscous dissipation on steady natural convection Couette flow subject to convective boundary condition. Due to the nonlinearity and coupling of the governing equations in the present situation, the homotopy perturbation method was employed to obtain the solutions of the energy and momentum equations. The impacts of the controlling parameters were investigated and discussed graphically. In the course of investigation, it was found that fluid temperature increases with an increase in viscous dissipation while the reverse trend was observed in fluid velocity. However, it was also discovered that heat generation leads to a decrease in the rate of heat transfer on the heated plate and it increases on the cold plate. Finally, it was concluded that the velocity boundary layer thickness increases with an increase in Biot number. [ABSTRACT FROM AUTHOR]

Some strongly nonlinear oscillators have been investigated with the help of combined modified Lindstedt–Poincare and homotopy perturbation methods. The solutions are more accurate than those obtained by the modified Lindstedt–Poincare method. Moreover, the determination of solutions is simpler than usual homotopy perturbation methods. Some similarities and dissimilarities between the present method and other existing methods have also been discussed. [ABSTRACT FROM AUTHOR]

This study proposes a new method, called the Fractal Yang transform method (F TM), for obtaining the fractal solution of the modified Camassa–Holm (mCH) and Degasperis–Procesi (mDP) models with fractal derivatives. The authors use the two-scale fractal approach to convert the fractal problem into its differential components and implement the Yang transform ( T) to achieve the recurrence iteration. We then apply the homotopy perturbation method (HPM) to overcome the difficulty of nonlinear elements in the recurrence iteration, which makes it simple to acquire further iterations. The most advantage of this fractal approach is that it has no restriction on variables and provides successive iterations. The fractal results are presented in the sense of a series that converges to the exact solution only after a few iteration. Graphical behavior demonstrates that this fractal approach is a very fast and remarkable solution, particularly with fractal derivatives. [ABSTRACT FROM AUTHOR]

Rotavirus infection is also a major cause of death in babies and children. Rotavirus causes severe diarrhea in babies and infants in developed and underdeveloped countries. It has multiple types of transmission: human to human and human to the environment. This paper investigates the time‐fractional order rotavirus epidemic model in an uncertain environment defined in the Caputo–Fabrizio (CF) derivative sense. In the titled model, the initial conditions are considered as fuzzy numbers. A double parametric form (DPF) of fuzzy numbers has been used in the fuzzy fractional rotavirus model. A semi‐analytical approach called the homotopy perturbation Elzaki transform method (HPETM) has been employed to solve the present model with the fuzzy initial condition. In order to derive the existence and uniqueness of the solution of the model, the concept of fixed‐point theory is utilized. Various fuzzy and interval solutions have been computed by considering different values of fractional orders, fuzzy parameters, and parameters involved in the given model. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the fractional-order arms race model. The model has emerged as an important tool for the investigation of international conflict and arms races. The variational iteration method, the homotopy perturbation method, and the adomian decomposition method are used to solve the mathematical model with Caputo’s fractional derivative. Several numerical computations have been provided to establish the validity and accuracy of the acquired results. It is shown that the fractional-order model can be solved easily using semi-analytical methods. The results obtained by all methods are compared. [ABSTRACT FROM AUTHOR]

The current study investigates the stability structure of the base periodic motion of an inverted pendulum (IP). A uniform magnetic field affects the motion in the direction of the plane configuration. Furthermore, a non-conservative force as one that dampens air is considered. Its underlying equation of motion is derived from traditional analytical mechanics. The mathematical analysis is made simpler by substituting the Taylor theory in order to expand the restoring forces. The modified Homotopy perturbation method (HPM) is employed to achieve a roughly adequate regular result. To support the prior result, a numerical method based on the fourth-order Runge-Kutta method (RK4) is employed. The graphs for both the analytic and numerical solutions are highly consistent with one another, which indicates that the perturbation strategy is accurate. The solution time history curve exhibits a decaying performance and indicates that it is steady and without chaos. The resonance and non-resonance cases are found through the stability study by using the time scale method. In all perturbation approaches, the methodology of multiple time scales is actually regarded as a further standard approach. The time history is used to create a collection of graphs. Some graphical representations are used to illustrate how the typical physical values affect the behavior of the discovered solution. It has been discovered that the statically unstable IP can have its instability reduced by raising the spring torsional constant stiffness as well as the damped coefficient. Moreover, the magnetic field has a significant role in the stability configuration, which explains that at higher values of this field, the decaying waves take much more time than the smaller values of this field. Accordingly, it can be employed in various engineering devices that need a certain period of time to be more stable. [ABSTRACT FROM AUTHOR]

The Black - Scholes Model is commonly used for option pricing, which is one of the most important applications in ?nance. In the absence of a transaction cost, the value of the option is determined by using B-S model. In the view of Caputo sense, this study proposes a result for the fractional Black-Scholes equation (FBSE) problem. The main goal of this paper is to show how to solve the FBSE by a semi-analytical method called the homotopy analysis shehu transform method (HASTM) and compare that homotopy analysis method (HAM), homotopy perturbation method (HPM), elzaki transform homotopy perturbation method (ETHPM). The HASTM result is quite similar to the HAM, HPM, ETHPM solution. The HASTM and other methods analytical solutions are also represented. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to study the non-linear vibration of a delaminated cross-ply composite beam with two overlapping delaminations. The effect of geometrical non-linearity and large deflection on the axial forces due to the existence of two overlapping delaminations are considered. The unknown axial forces created due to the delaminations are obtained by changing the length of the delaminated segments during the vibration behavior. The homotopy perturbation method (HPM) is utilized to obtain the frequency ratio of the delaminated cross-ply composite beam with two overlapping delaminations. This study describes the effects of different parameters such as two delaminations total length, overlap distance, and the distance between center points of two delaminations in axial and thickness directions on the frequency ratio of the delaminated composite beam in detail. [ABSTRACT FROM AUTHOR]

This paper investigates the flow and heat transfer of special third-grade fluid with a viscous dissipation effect over a stretching sheet. This model, adequate for many non-Newtonian fluids, is used to characterize the behavior of the fluids domain. The governing momentum and energy equation are reduced to ordinary nonlinear differential (self-similar) equations via the Lie group transformation method. The Homotopy Perturbation Method (HPM) is applied to solve these obtaining results. For validation, current results have been compared with the fourth-order Runga method (RK4) and shooting technique. The effects of physical parameters on fluid velocity and temperature profile were investigated with the aid of figures and tables by simply altering a single parameter while keeping the others constant. It is observed that both the non-Newtonian parameter and the Prandtl number have the effect of decreasing the temperature of the stretching surface, while the opposite behavior was found for the Eckert number. [ABSTRACT FROM AUTHOR]

In this paper, a promising technique based on the homotopy perturbation method for solving singular nonlinear system of boundary value problems (BVPs) is employed. The original singular system of BVPs is transformed into an equivalent system of integral equations to overcome the singular behavior at x=0$$ x=0 $$, and then, the recursive series for the solution is established. The mathematical formulation is further supported by conducting the convergence analysis of the recursive schemes. The accuracy of the proposed method is tested by comparing against the current variational iteration method using many standard examples. It is shown that the proposed method is more accurate and efficient than the existing method such as Chebyshev operational matrix method and variational iteration method. [ABSTRACT FROM AUTHOR]

In this research, we successfully demonstrated the use of the homotopy perturbation method with Green's function to find approximate solutions for the fuzzy system of boundary value problems. Our results showcase the effectiveness of this method in providing accurate and reliable solutions. The consistent way to reduce the size of the computation gives to reach the exact solution is one of the best methods adopted to determine the behavior of the solution directly in order to determine the approximate solution analytically, Finally, the problems that have been addressed confirmed the validity of the method applied analytically in this research using comparison with some numerical problems. [ABSTRACT FROM AUTHOR]

This article uses the Yang transform decomposition method and the homotopy perturbation transform method to study the seventh-order time-fractional Sawada–Kotera–Ito equation. The fractional derivative is taken into account in the Caputo sense. We used the Yang transform with the Adomian decomposition process and homotopy perturbation procedure on the time-fractional Sawada–Kotera–Ito problem to obtain the solution. We looked at a single case and contrasted it with the actual result to validate the methodologies. These techniques create recurrence relations representing the proposed problem's solution. We then produced graphical representations that allowed us to visually check all of the outcomes in the proposed case for various fractional order values. The results of applying the current methodologies revealed strong connections to the precise resolution of the problem under investigation. The present study also illustrates error analysis. The numerical results obtained using the suggested techniques show that the methods are both simple and have excellent computational merit. [ABSTRACT FROM AUTHOR]

COVID-19 pandemic is now increasing concern to authorities and public health officials. The study is aimed to investigate the impact of vaccination in the absence of media induced fear. In this research, analysis and simulation of a mathematical model of COVID-19 incorporated with media induced fear and vaccination was made, the total population is divided into 5 sub-population classes; Susceptible, Exposed, Infected, Quarantine and Recovery. The disease free and the endemic equilibrium of the model were carried out and the basic reproduction number was obtained using the next generation matrix. The stability analysis of the model was done and it was ascertained that the disease free equilibrium of the biological model is stable. Due to its efficiency and accuracy in handling nonlinear coupled ordinary differential equations, the homotopy perturbation method is applied to obtain the approximate solution of the mathematical model and the obtained results was simulated using the computation software Maple 18 to study the impact of vaccination in each compartment of the model when the media induced fear. The outcome of the simulation process were presented graphically and interpreted accordingly and it was discovered that in eradicating the spread of COVID-19 in a society where there is no fear, vaccination is an alternative and better measure. [ABSTRACT FROM AUTHOR]

A good initial guess and an appropriate homotopy equation are two main factors in applications of the homotopy perturbation method. For a nonlinear oscillator, a cosine function is used in an initial guess. This article recommends a general approach to construction of the initial guess and the homotopy equation. Duffing oscillator is adopted as an example to elucidate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

This article investigates the seventh-order Lax's Korteweg–de Vries equation using the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM). The physical phenomena that emerge in physics, engineering and chemistry are mathematically expressed by this equation. For instance, the KdV equation was constructed to represent a wide range of physical processes involving the evolution and interaction of nonlinear waves. In the Caputo sense, the fractional derivative is considered. We employed the Yang transform, the Adomian decomposition method and the homotopy perturbation method to obtain the solution to the time-fractional Lax's Korteweg–de Vries problem. We examined and compared a particular example with the actual result to verify the approaches. By utilizing these methods, we can construct recurrence relations that represent the solution to the problem that is being proposed, and we are then able to present graphical representations that enable us to visually examine all of the results in the proposed case for different fractional order values. Furthermore, the results of the current approach exhibit a good correlation with the precise solution to the problem being studied. Furthermore, the present study offers an example of error analysis. The numerical outcomes obtained by applying the provided approaches demonstrate that the techniques are easy to use and have superior computational performance. [ABSTRACT FROM AUTHOR]