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This paper conducts an extensive comparative analysis of numerical methods employed in modeling blood flow through arteries with Magnetohydrodynamics (MHD) and hybrid nanofluids. The study investigates the effectiveness and precision of distinct numerical approaches: Akbari Ganji's Method (AGM), Fourth-Order Runge–Kutta (RK4), Finite Volume Method (FVM), and the Finite Element Method (FEM). These methods are essential for comprehending the intricate fluid dynamics that arise in the presence of magnetic fields and hybrid nanofluids a phenomenon relevant in numerous medical applications. Blood flow is subjected to a homogeneous magnetic field in a radial direction while the magneto-hemodynamics effect is taken into account. A variety of medical, physiological, and surgical procedures, as well as the regulation of blood pressure, heat distribution, wound healing, diagnostic imaging, and drug discovery, depend on blood flow through arteries to carry out vital functions such as oxygen and nutrition delivery, organ maintenance, and wound healing. Our findings highlight that while each method has strengths, their applicability varies based on the problem's characteristics and computational resource constraints. This analysis aids researchers and practitioners in selecting the most suitable method for their modeling requirements, advancing numerical techniques for complex fluid dynamics involving MHD and hybrid nanofluids. [ABSTRACT FROM AUTHOR]

Under the consideration of computational software Mathematica on the basis of extended simple equation method, constructed the periodic wave soliton solutions for nonlinear Zakharov–Kuznetsov modified equal width equation. In this study, constructed soliton solutions included periodic singular solitons, bright solitons, kink solitons, anti-kink solitons, travelling waves, dark solitons and solitary waves are more interesting, prices and some are novel. The graphical demonstration of some investigated results shown by contour, 2-dim and 3-dim by computing software Mathematica under numerical simulation to visualizing the physical phenomena of obtained results. The demonstrated results will be helpful in the study of nonlinear phenomena in various kinds of domains including nonlinear optics, nonlinear dynamics, solitons wave theory, optical fiber, nonlinear acoustic wave, communications system, networking system and transmission of signals. The extended simple equation technique is a concise, straightforward, powerful and reliable approach for the investigation of different kinds of nonlinear partial evolution equations. [ABSTRACT FROM AUTHOR]

The Sharma–Tasso–Olver–Burgers (STOB) equation is a nonlinear partial differential equation that appears in many branches of science, engineering and describes significant phenomena including wave propagation and fluid dynamics. The STOB equation characteristics and solutions for the nonlinear situation are thoroughly examined in this research work. The analytical techniques, namely the extended (G ′ G 2) -expansion method, stability analysis, and sensitivity analysis, are used to determine the solitons solution of STOB equation. The study begins with an overview of the equation, highlighting its significance in modeling. The nonlinear nature of the equation leads to interesting dynamics and challenges in its analysis and solution. The research paper focuses on understanding the behavior of solutions and identifying the solution with the help of graphical interpretation. Numerous of the identified solutions are depicted in figures to provide a physical comprehension. Because it is critical, we use appropriate values of parameters to highlight the physical aspects of the supplied data using 3D, 2D, and contour charts. The proposed techniques are valuable and contribute to the field of nonlinear sciences. Various nonlinear evolutionary equations are employed to represent models of nonlinear physical phenomena [ABSTRACT FROM AUTHOR]

This study investigates the exact solutions of the time-fractional (3+1)-dimensional combined Korteweg–de Vries Benjamin–Bona–Mahony (KdV-BBM) equation. The considered model describes the long surface gravity waves of small amplitude, which portrays the two-way propagation of waves. The modified generalized Kudryashov method and the exp(- φ (ξ ))-expansion methods are employed to resolve the aforementioned issue because of their effectiveness and simplicity. For generality, the time fractional version is studied; more advanced solutions that do not exist in the literature were obtained. As a result, a variety of the exact wave solutions of the conformable (3+1)-dimensional KdV-BBM equation are obtained. The dynamical behaviors of some obtained solutions are represented with the proper parameter values. The used methods yield noteworthy results in obtaining the analytical solutions of fractional differential equations under various conditions. Besides, the sensitivity of regarding dynamical system is assessed to show the numerical stability effects. [ABSTRACT FROM AUTHOR]

Nematicons upgrade the recognition of light localization in the reorientation of non-local media. the current research employs a powerful integral scheme using a different procedure, namely, the modified simple equation method (MSEM), to analyze nematicons in liquid crystals from the controlling model. The expanded MSEM is investigated to enlarge the applicability of the standard one. The suggested expansion depends on merging the MSEM and the ansatz method. The new generalized nonlinear n-times quadruple power law is included. With the aid of the symbolic computational package Mathematica, new explicit complex hyperbolic, periodic, and more exact spatial soliton solutions are derived. Moreover, the related existence constraints are obtained. To show the dynamical properties of some of the obtained nematicons, three-dimensional profiles with corresponding contours are depicted with the choice of appropriate values of arbitrary parameters. The fractional impacts in various applicable senses are analyzed to investigate the generality of the considered model. [ABSTRACT FROM AUTHOR]

The combined Korteweg-de Vries (KdV)-modified KdV (mKdV) equation widely appears as an integrable model in a wide range of interacting physical phenomena to explore their nonlinear dynamical features with geometrical structures. In this work, we present a comprehensive physical analysis of a bunch of solitary waves associated with a combined KdV-mKdV equation, aiming to elucidate the properties and behavior of these waves in the context of propagation in the background of interactions. The study is specifically concerned with analytically examining of combined KdV-mKdV equation, with a focus on algebraic and geometrical properties of solitary wave solutions with their stability analysis. The nature and stability of solitary waves are studied using mathematical methods such as the Kudrayashov methodology and the exp (- Φ (ζ)) -expansion method. This investigation tries to shed light on the behavior of solitary waves in various beginning situations and parameter regimes. Plotting 3D surface graphs, contour plots, and line graphs for different parameter values enables the observation of the graphical properties of the developed solutions. The obtained solutions exhibit kink, singular periodic, and periodic behaviors. The answers generated using these two methodologies also show the graphical representations of the soliton propagation. [ABSTRACT FROM AUTHOR]

This analysis examines the time-fractional mixed hyperbolic-elliptic p-system of conservation laws by applying the new extended direct algebraic method. The p-system with generalized cubic van der Waals flux, and potential applications in the field of compressible isothermal viscosity-capillarity fluids, is investigated. In particular, this issue describes the longitudinal isothermal motion in elastic bars or fluids. A diverse periodic, kink, and singular soliton structures are extracted. The 3D dynamical behaviors and corresponding contour profiles of some obtained solitons are displayed. The fractional effects in the sense of Beta, M-truncated, and modified Riemann–Liouville, are discussed and illustrated. The method shows the straightforward, reliability, and efficiency for solving complex physical phenomena that is modeled by nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

The use of renewable energy sources is leading the charge to solve the world's energy problems, and non-Newtonian nanofluid dynamics play a significant role in applications such as expanding solar sheets, which are examined in this paper, along with the impacts of activation energy and solar radiation. We solve physical flow issues using partial differential equations and models like Casson, Williamson, and Prandtl. To get numerical solutions, we first apply a transformation to make these equations ordinary differential equations, and then we use the MATLAB-integrated bvp4c methodology. Through the examination of dimensionless velocity, concentration, and temperature functions under varied parameters, our work explores the physical properties of nanofluids. In addition to numerical and tabular studies of the skin friction coefficient, Sherwood number, and local Nusselt number, important components of the flow field are graphically shown and analyzed. Consistent with previous research, this work adds important new information to the continuing conversation in this area. Through the examination of dimensionless velocity, concentration, and temperature functions under varied parameters, our work explores the physical properties of nanofluids. Comparing the Casson nanofluid to the Williamson and Prandtl nanofluids, it is found that the former has a lower velocity. Compared to Casson and Williamson nanofluid, Prandtl nanofluid advanced in heat flux more quickly. The transfer of heat rates are 25.87 % , 33.61 % and 40.52 % at R d = 0.5 , R d = 1.0 , and R d = 1.5 , respectively. The heat transfer rate is increased by 6.91 % as the value of Rd rises from 1.0 to 1.5. This study is further strengthened by a comparative analysis with previous research, which is complemented by an extensive table of comparisons for a full evaluation. [ABSTRACT FROM AUTHOR]

Ordinary differential equations (ODEs) are fundamental tools for modeling and understanding a wide range of chemistry, physics, and biological phenomena. However, solving complex ODEs often presents significant challenges, necessitating advanced numerical approaches beyond traditional analytical techniques. Thus, a novel machine learning (ML)-based method for solving and analyzing ODEs is proposed in the current investigation. In this study, we utilize a feed-forward neural network (FNN) with five fully connected layers trained on data samples generated from the exact solutions of specific ODEs. To show the efficacy of our suggested method, we will conduct a thorough evaluation by comparing the anticipated solutions of the FNN with the exact solutions for some ODEs. Furthermore, we analyze the absolute error and present the loss functions for some ODE examples, providing valuable insights into the model's performance and potential areas for further development. [ABSTRACT FROM AUTHOR]

In fluid dynamics, mixed-type systems of conservation laws model a wide range of phase transition problems in compressible media. This analysis studies analytically the time-fractional mixed-type hyperbolic-elliptic van der Waals p-system with generalized cubic flux function for the first time. For this purpose, the expanded ansatz method is introduced and employed to derive new closed and approximate topological, non-topological, singular, and periodic solitons solutions for the considered model. The 2D, 3D, and contour plots of the dynamical behaviors of some obtained results with fractional effects , in conformable sense, are illustrated. Furthermore, many other solution profiles can be obtained from our results with the open choices of parameters. The numerical sensitivity analysis of the regarding dynamical system is also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we study linear and nonlinear mixed convection, activation energy, and heat radiation effects caused by nanoparticles. This study aims to improve the understanding of how nanofluids behave in the presence of rotating disks and develop more efficient and effective cooling technologies. The flow problem consisted of partial differential equations (PDE). It is challenging to calculate these equations as a result of these nonlinear PDEs. Consequently, we use appropriate similarities to transform them into ordinary differential equations (ODEs). The bvp4c Matlab built-in technique is then used to resolve these ODEs. The velocities, temperature, and concentration outcomes with the various factors are examined graphically. Additionally, tables are employed to analyze the skin friction and Nusselt number values. It is analyzed that increasing the linear and linear mixed convection parameters enhances the velocity profiles of nanofluid. Enhancements in heat are analyzed by increasing nonlinear thermal radiation and enhancement in concentration is examined by increasing activation energy. Furthermore, as the variables for thermophoresis and Brownian motion are increased, the Nusselt number falls. The heat transfer rate is 27.16% for R d = 0.6 and 39.28% for R d = 1.4 . Thus, the heat transfer rate is enhanced 12.12%. This study’s practical applications include improving the behavior of fluids and the transfer of heat in rotating frameworks, which may affect energy systems, heat exchangers, and cooling advances in technology. [ABSTRACT FROM AUTHOR]

The purpose of this manuscript is to investigate the precise soliton solution of the (1 + 1) -dimensional Sharma–Tasso–Olver–Burgers equation. This study thoroughly explores the characteristics and solutions of this nonlinear equation. To pinpoint the exact soliton solution, we employ analytical techniques, specifically the exp (- ϕ (ξ)) -expansion method, as well as stability and sensitivity analysis. We begin by introducing the equation and discussing its role in mathematical modeling. It's worth noting that the equation's inherent nonlinearity adds complexity and poses challenges for analysis and solution. Our research objective is to identify solutions through graphical interpretation, allowing us to gain insights into their behavior. To achieve this, we utilize relevant parameter values to emphasize the physical properties of the provided data. We use the Mathematica and Maple software to demonstrate the solutions that have been discovered in 3D, 2D, and contour plots for the purpose of physical expression and graphical representation. The investigated system is shown to be stable since a little change in the initial conditions does not result in an immense shift in solutions. The sensitivity and stability analyses are also provided at various initial conditions. Importantly, these methods have broader applications and are valuable for conveying nonlinear physical models in the field of nonlinear sciences, not limited to the Sharma–Tasso–Olver–Burgers equation alone. [ABSTRACT FROM AUTHOR]

Complex fuzzy sets (CFSs) have recently emerged as a potent tool for expanding the scope of fuzzy sets to encompass wider ranges within the unit disk in the complex plane. This study explores complex fuzzy numbers and introduces their application for the first time in the literature to address a complex fuzzy partial differential equation that involves a complex fuzzy heat equation under Hukuhara differentiability. The researchers utilize an implicit finite difference scheme, namely the Crank–Nicolson method, to tackle complex fuzzy heat equations. The problem's fuzziness arises from the coefficients in both amplitude and phase terms, as well as in the initial and boundary conditions, with the Convex normalized triangular fuzzy numbers extended to the unit disk in the complex plane. The researchers take advantage of the properties and benefits of CFS theory in the proposed numerical methods and subsequently provide a new proof of the stability under CFS theory. A numerical example is presented to demonstrate the proposed approach's reliability and feasibility, with the results showing good agreement with the exact solution and relevant theoretical aspects. [ABSTRACT FROM AUTHOR]

The current research aims to implement the numerical results for the Holling third kind of functional response delay differential model utilizing a stochastic framework based on Levenberg-Marquardt backpropagation neural networks (LVMBPNNs). The nonlinear model depends upon three dynamics, prey, predator, and the impact of the recent past. Three different cases based on the delay differential system with the Holling 3rd type of the functional response have been used to solve through the proposed LVMBPNNs solver. The statistic computing framework is provided by selecting 12%, 11%, and 77% for training, testing, and verification. Thirteen numbers of neurons have been used based on the input, hidden, and output layers structure for solving the delay differential model with the Holling 3rd type of functional response. The correctness of the proposed stochastic scheme is observed by using the comparison performances of the proposed and reference data-based Adam numerical results. The authentication and precision of the proposed solver are approved by analyzing the state transitions, regression performances, correlation actions, mean square error, and error histograms. [ABSTRACT FROM AUTHOR]

The solution of partial differential equations has generally been one of the most-vital mathematical tools for describing physical phenomena in the different scientific disciplines. The previous studies performed with the classical derivative on this model cannot express the propagating behavior at heavy infinite tails. In order to address this problem, this study addressed the fractional regularized long-wave Burgers problem by using two different fractional operators, Beta and M-truncated, which are capable of predicting the behavior where the classical derivative is unable to show dynamical characteristics. This fractional equation is first transformed into an ordinary differential equation using the fractional traveling wave transformation. A new auxiliary equation approach was employed in order to discover new soliton solutions. As a result, bright, periodic, singular, mixed periodic, rational, combined dark–bright, and dark soliton solutions were found based on the constraint relation imposed on the auxiliary equation parameters. The graphical visualization of the obtained results is displayed by taking the suitable parametric values and predicting that the fractional order parameter is responsible for controlling the behavior of propagating solitary waves and also providing the comparison between fractional operators and the classical derivative. We are confident about the vital applications of this study in many scientific fields. [ABSTRACT FROM AUTHOR]

We define the notions of weakly μ -countably compactness and nearly μ -countably compactness denoted by W μ -CC and N μ -CC as generalizations of μ -compact spaces in the sense of Csaśzaŕ generalized topological spaces. To obtain a more general setting, we define W μ -CC and N μ -CC via hereditary classes. Using μ θ -open sets, μ -regular open sets, and μ -regular spaces, many results and characterizations have been presented. Moreover, we use the properties of functions to investigate the effects of some types of continuities on W μ -CC and N μ -CC. Finally, we define soft W μ -CC and N μ -CC as generalizations of soft μ -compactness in soft generalized topological spaces. [ABSTRACT FROM AUTHOR]

The current study suggests a new generalisation of highly dispersive nonlinear Schrödinger-type equation (NLSE) with perturbation terms. With polynomial refractive index, known by cubic–quintic–septic (CQS) law and Hamiltonian-type cubic perturbation terms, the new model includes eighth-order dispersion term. The generalised Riccati simplest equation method (RSEM) and the modified simplest equation method (MSEM) are successfully utilised to analytically process the fractional version of the considered NLSE. A diverse collection of bright, dark and singular optical solitons under some constraints, in hyperbolic, periodic and rational-exponential forms are derived. Graphical interpretations of some obtained solutions are displayed. The two considered schemes, with different algorithms, show an influential mathematical tool for processing nonlinear fractional evolution equations. [ABSTRACT FROM AUTHOR]

This analysis utilizes the generalized Riccati simple equation method to construct nematicons in liquid crystals from its governing system. A new type of nonlinearity is studied for the first time in the context of liquid crystals. It is the nonlinear quadruple power law. The fractional version of the governed model, with conformable sense, is considered. With the aid of Mathematica, Bright, dark, and singular types of solutions are derived with the constraints guaranteeing their existence. Moreover, some obtained results are depicted to show the real physical characteristics of nematicons. The used method provides a powerful tool for extracting exact solitary wave solutions. By simple calculations, we show that the generalized Riccati simple equation method can be treated as a general case of the well-known simple equation, exp - φ (η) -expansion, and G ′ / G -expansion methods. [ABSTRACT FROM AUTHOR]

The telecommunications revolution finds its foundation thanks to new service offerings including voice, images, security of computer data and much more. Thereby, telecommunications can be defined as the transmission of information at a distance using electronic, computer, wired, optical or electromagnetic technologies. At an expense of these characteristics, this paper explores the dynamics of various travelingwaves, through telecommunications systems. In this perspective, we use a (2+1)-dimensional nonlinear transmission line equation, by solving it analytically using a novel developed techniques: The improved Sardar-subequation. As a result, we discover several novel voltage characteristics such as bright, trigonometric, exponential and dark function solutions. Using certain appropriate values, we provide several different surfaces of the solutions produced in this work. These findings will be essential in the telecommunications system used to represent wave propagations. [ABSTRACT FROM AUTHOR]

In this paper, we examined four different forms of generalized (2+1)-dimensional Boussinesq–Kadomtsev–Petviashvili (B-KP)-like equations. In this connection, an accurate computational method based on the Riccati equation called sub-equation method and its Bäcklund transformation is employed. Using this method, numerous exact solutions that do not exist in the literature have been obtained in the form of trigonometric, hyperbolic, and rational. These solutions are of considerable importance in applied sciences, coastal, and ocean engineering, where the B–KP-like equations modeled for some significant physical phenomenon. The graph of the bright and dark solitons is presented in order to demonstrate the influence of different physical parameters on the solutions. All of the findings prove the stability, effectiveness, and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

This work provides a technical applied description of the residual power series method (RPSM) to develop a fast and accurate algorithm for mixed hyperbolic–elliptic systems of conservation laws with Riemann initial datum. The RPSM does not require discretization, reduces the system to an explicit system of algebraic equations and consequently of massive and complex computations, and provides the solution in a form of Taylor power series expansion of closed-form exact solution (if exists). Theoretically, convergence hypotheses are discussed, and error bounds of exponential rates are derived. Numerically, the convergence and stability of approximate solutions are achieved for systems of mixed type. The reported results, with application to general Cauchy problems, which rise in diverse branches of physics and engineering, reveal the reliability, efficiency, and economical implementation of the proposed algorithm for handling nonlinear partial differential equations in applied mathematics. [ABSTRACT FROM AUTHOR]

In the current analysis, the conformable generalized Kudryashov equation of pulses propagation with power non-linearity is processed. The considered higher order equation represents a generalized mathematical model of many well-known ones in nonlinear media. A variety of multiple kinks, bi-symmetry, periodic, singular, bright and dark optical solitons are extracted via the generalized Riccati equation mapping method. Basing on the Riccati differential equation, the theoretical algorithm extracts a number of empirical solutions that do not exist in the literature. The obtained results showed that the present technique is an effective and strong tool for solving nonlinear fractional partial differential equations and produces a very large number of solutions. [ABSTRACT FROM AUTHOR]

The conformable derivative and adequate fractional complex transform are implemented to discuss the fractional higher-dimensional Ito equation analytically. The Jacobi elliptic function method and Riccati equation mapping method are successfully used for this purpose. New exact solutions in terms of linear, rational, periodic and hyperbolic functions for the wave amplitude are derived. The obtained solutions are entirely new and can be considered as a generalization of the existing results in the ordinary derivative case. Numerical simulations of some obtained solutions with special choices of free constants and various fractional orders are displayed. [ABSTRACT FROM AUTHOR]

In this paper, with the aid of the Mathematica package, several classes of exact analytical solutions for the time-fractional (2 + 1) -dimensional Ito equation are obtained. To analytically tackle the above equation, the Kudryashov simple equation approach and its modified form are applied. Rational, exponential-rational, periodic, and hyperbolic functions with a number of free parameters were represented by the obtained soliton solutions. Graphical illustrations with special choices of free constants and different fractional orders are included for certain acquired solutions. Both approaches include the efficiency, applicability and easy handling of the solution mechanism for nonlinear evolution equations that occur in the various real-life problems. [ABSTRACT FROM AUTHOR]

The main objective of our analysis is to modify the reduced differential transform method (RDTM) to be applicable for a wide range of nonlinear partial differential equations. The Adomian polynomials are employed to overcome the huge size and complexity of computing reduced differential transforms, and new generalizations of transformed formulas are established. Our new approach provides a series solution with effortless components to compute. A new sufficiency convergence proof of series solution is introduced. The (1 + m)-dimensional telegraph and Burgers equations are tested to illustrate robustness and reliability of our modification. [ABSTRACT FROM AUTHOR]

The simple equation method and modified simple equation method are employed to seek exact traveling wave solutions to the (1 + 1)‐dimensional van der Waals gas system in the viscosity‐capillarity regularization form. Under the help of Mathematica, new classes of kink solutions are derived. Numerical simulations with special choices of the free parameters are displayed by three‐ and two‐dimensional plots. The two methods demonstrate simplicity, reliability, and efficiency. [ABSTRACT FROM AUTHOR]

In this paper, the simplest equation method is implemented to construct exact traveling-wave solutions to the modified Fornberg-Whitham equation. With the help of Mathematica symbolic computa- tion software, some new classes of explicit exact solutions are derived. Graphical representations of some obtained solutions are displayed. The method demonstrates the applicability, reliability and efficiency in treating a wide range of nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

Periodic and soliton solutions are derived for the (1+1)-dimensional van der Waals gas system in the viscosity-capillarity regularization form. The system is handled via the e -ϕ(ξ) -expansion method. The obtained solutions have been articulated by the hyperbolic, trigonometric, exponential and rational functions with arbitrary constants. Mathematical analysis and numerical graphs are provided for some solitons, periodic and kink solitary wave solutions to visualize the dynamics of equations. Obtained results reveal that the method is very influential and effective tool for solving nonlinear partial differential equations in applied mathematics. [ABSTRACT FROM AUTHOR]

The aim of the present analysis is to generalize the recently devised technique, known as the residual power series method (RPSM), for analytic treatment of higher-order non-linear partial differential equations (NPDEs). This method based on constructing series solutions in a form of rapidly convergent series with easily computable components and without need of linearization, discretization, perturbation or unrealistic assumptions. Convergence and error analysis are discussed in details. To illustrate the efficiency, reliability and simplicity of proposed method, exact solutions for some diffusion processes of power, rational and exponential law diffusivities are obtained. [ABSTRACT FROM AUTHOR]

This paper studies the propagation of the short pulse optics model governed by the higher-order nonlinear Schrödinger equation (NLSE) with non-Kerr nonlinearity. Exact one-soliton solutions are derived for a generalized case of the NLSE with the aid of software symbolic computations. The modified Kudryashov simple equation method (MSEM) is employed for this purpose under some parametric constraints. The computational work shows the difference, effectiveness, reliability, and power of the considered scheme. This method can treat several complex higher-order NLSEs that arise in mathematical physics. Graphical illustrations of some obtained solitons are presented. [ABSTRACT FROM AUTHOR]

This work deals with a new modified version of the Adomian-Rach decomposition method (MDM). The MDM is based on combining a series solution and decomposition method for solving nonlinear differential equations with Adomian polynomials for nonlinearities. With application to a class of nonlinear oscillators known as the Lienard-type equations, convergence and error analysis are discussed. Several physical problems modeled by Lienard-type equations are considered to illustrate the effectiveness, performance and reliability of the method. In comparison to the 4th Runge-Kutta method (RK4), highly accurate solutions on a large domain are obtained. [ABSTRACT FROM AUTHOR]