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Financial engineering problems hold considerable significance in the academic realm, where there remains a continued demand for efficient methods to scrutinize and analyze these models. Within this investigation, we delved into a fractional nonlinear coupled system for option pricing and volatility. The model we examined can be conceptualized as a fractional nonlinear coupled wave alternative to the governing system of Black-Scholes option pricing. This introduced a leveraging effect, wherein stock volatility aligns with stock returns. To generate novel solitonic wave structures in the system, the present article introduced a generalized Ricatti mapping method and new Kudryashov method. Graphical representations, both in 3D and 2D formats, were employed to elucidate the system's response to pulse propagation. These visualizations enabled the anticipation of appropriate parameter values that align with the observed data. Furthermore, a comparative analysis of solutions was presented for different fractional order values. Additionally, the article showcases the comparison of wave profiles through 2D graphs. The results of this investigation suggested that the proposed method served as a highly reliable and flexible alternative for problem-solving, preserving the physical attributes inherent in realistic processes. To sum up, the main objective of our work was to conceptualize a fractional nonlinear coupled wave system as an alternative to the Black-Scholes option pricing model and investigate its implications on stock volatility and returns. Additionally, we aimed to apply and analyze methods for generating solitonic wave structures and compare their solutions for different fractional order values.

In this paper, the extended direct algebraic approach with the general fractional derivative is employed to attain various novel wave structures of the non-linear fractional Tzitzéica type non-linear evolution equations in the form of kink, singular, periodic singular, dark, bright and dark-bright combine solitons. For the purpose to illustrate the physical behavior of the acquired solutions, some of the extracted results are sketched in the pattern of 3-D and 2-D plots which show the efficiency and authenticity of the proposed method. The under consideration method can also utilize to any other non-linear model appears in optics and engineering.

The (4+1)-dimensional Davey-Stewartson-Kadomtsev-Petviashvili equation is explored in the present work, revealing its complex dynamics and solitary wave solutions. Modeling ocean and tidal waves, particularly tsunami and long water waves, depends significantly on this nonlinear equation. Additionally, these models can be used to simulate internal and external waves in rivers and oceans as well as wave packets in water with a finite depth. The Sardar subequation method, new Kudryashov's method, and (1ϑ(ζ),ϑ′(ζ)ϑ(ζ)) method are investigated to discover novel solitary wave solutions in the terms of hyperbolic, trigonometric and rational functions. A wide variety of solitons, as dark, bright, periodic, singular, combined dark-singular solitons and, combined dark-bright are obtained by these techniques. By taking accurate parameter values, certain three-dimensional and two-dimensional graphs are plotted to improve the physical description of solutions. The intriguing field of nonlinear waves and dynamic systems is signaled to readers by this work, which suggests a major advancement in understanding the intricate and unexpected behavior of this model.

Comprehending the temperature-dependent variable viscosity Phan Thien Tanner fluid film flow over a horizontal heated plate with surface tension is essential to enhancing coating and lubrication process prediction models. This paper accords with the flow analysis of a temperature-dependent variable viscosity Phan Thien Tanner fluid film over a horizontal heated plate. The flow over a heated plate occurs as a result of the plate’s motion and the surface tension gradient. The Adomian decomposition method is utilized to solve a system of linear and nonlinear ordinary differential equations, resulting in series-form solutions. The series-form expressions for flow variables such as velocity, temperature, volume flow rate, and surface tension are derived. Moreover, the positions of stationary points are computed using MATHEMATICA. The analysis delineated that when the inverse capillary number, variable viscosity parameter, Deborah number, elongation parameter, and Brinkman number increase, the stationary points move closer to the heated plate. The temperature also rises with an increase in these parameters. The temperature rises with increasing viscous dissipation while it lowers with increasing thermal diffusion. When the Deborah number is high, the Phan Thien Tanner fluid behaves like a solid and the flow is only driven by the motion of the plate. A comparison between Newtonian and Phan Thien Tanner fluids is made for velocity, temperature, and stationary points as a special case.

This study investigates the dynamic characteristics of the dual-mode resonant non-linear Schrodinger equation with a Bhom potential. Hydrodynamics, nonlinear optical fibre communication, elastic media, and plasma physics are just a few of the mathematical physics and engineering applications for this model. The study aims to achieve two main objectives: first, to discuss bifurcation analysis, and second, to extract optical soliton solutions using the extended hyperbolic function method. The study successfully derives various wave solutions, including bright, singular, periodic singular and dark solitons, based on the governing model. The findings conferred in this article show a crucial advancement in understanding the propagation of waves in non-linear media. Additionally, bifurcation of phase portraits of ordinary differential equation consistent with the partial differential equation under consideration is conducted. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. The existing literature shows that these methods are first time applied on this model.

Understanding the mechanisms involved in the movement and clearance of mucus within the human airways is important from a physiological perspective. A mathematical model of muco-ciliary two-layered fluid flow in a finite two-dimensional channel is proposed. Due to the presence of both the airway ciliary layer (ACL) and the periciliary liquid layer (PCL) in airways epithelial cells, the two-layered model is adopted. The third grade fluid models is used to characterize the viscoelastic nature of mucus secreted in the airways. The mathematical modeling of two-layer flow problem is simplified by using long wavelength and small Reynolds number approximation. A formulated set of partial differential equations (PDEs) is solved for series-form solutions of velocity, temperature, concentration, and pressure gradient by using the Adomian decomposition method (ADM). The analysis delineated that with an increase in pressure gradient at airways entrance ξ the velocity and temperature increases while concentration decreases. Deborah numbers impact ACL and PCL velocities differently, with De(A) influencing flow direction and De(P) causing fluid to flow faster in the forward direction. It also emphasizes the significance of the rate of thermal diffusion, viscous dissipation, and relaxation time. The results showed that the suggested two-layered muco-ciliary model provides a better description of mucus transport in the human airways. The pressure gradient at the airways entrance has a significant impact on mucus transport through the airways.

Abstract The analysis of peristaltic-ciliary transport in the human female fallopian tube, specifically in relation to the growing embryo, is a matter of considerable physiological importance. This paper proposes a biomechanical model that incorporates a finite permeable tube consisting of two layers, where the Jeffrey fluid model characterizes the viscoelastic properties of the growing embryo and continuously secreting fluid. Jeffrey fluid entering with some negative pressure gradient forms the core fluid layer while continuously secreting Jeffrey fluid forms the peripheral fluid layer. The resulting partial differential equations are solved for closed-form solutions after employing the assumption of long wavelength. The analysis delineated that increasing the constant secretion velocity, Darcy number, and Reynolds number leads to a decrease in the appropriate residue time of the core fluid layer and a reduction in the size of the secreting fluid bolus in the peripheral fluid layer. Eventually, the boluses completely disappear when the constant secretion velocity exceeds 3.0 Progesterone ( $$P_4$$ P 4 ) and estradiol ( $$E_2$$ E 2 ) directly regulate the transportation of the growing embryo, while luteinizing hormone (LH) and follicle-stimulating hormone (FSH), prolactin, anti-mullerian hormone (AMH), and thyroid-stimulating hormone (TSH) have an indirect effects. Based on the number and size of blastomeres, the percentage of fragmentation, and the presence of multinucleated blastomeres two groups were formed in an in vitro experiment. Out of 50 patients, 26 (76.5%) were pregnant in a group of the good quality embryos, and only 8 (23.5%) were in a group of the bad quality embryos. The transport of growing embryo in the human fallopian tube and preimplantation development of human embryos in in vitro are constraint by baseline hormones FSH, LH, prolactin, $$E_{2}$$ E 2 , AMH, and TSH.

The exploration of higher-order and multicomponent extensions of the nonlinear Schrödinger equation holds significant importance across diverse applications, notably within the field of optics. Among these equations, the integrable Sasa–Satsuma equation stands out for its captivating soliton solutions. The Sasa–Satsuma equation serves as a mathematical representation for describing the propagation of femtosecond light pulses through optical fibers. The breather, multiwave, combined dark–bright, singular, dark, bright and periodic singular optical soliton solutions are derived in this paper by using the 1φ(ϑ),φ′(ϑ)φ(ϑ) and multivariate generalized exponential rational integral function approach. Furthermore, the paper emphasizes the essential fiber parameters essential for the emergence of these structures and offers visual representations of chosen solutions to elucidate their physical characteristics. The novelty of this work lies in the inaugural application of these methods to the Sasa–Satsuma equation and gives a significant advancement in the understanding of optical phenomena. This study not only opens avenues for further research but also introduces a fresh perspective on the Sasa–Satsuma model in the field of nonlinear physics with its applications in optical systems.

In this study, we examine soliton solutions of extended (2+1)-dimensional Kadomtsev–Petviashvili equation arising in fluid mechanics and plasma physics. The research utilizes an improved modified extended tanh-function method to derive new soliton solutions. The diverse set of soliton solutions obtained in this study, featuring a combination of rational, trigonometric, and hyperbolic functions, enhances the model’s applicability for real-world fluid mechanics and plasma physics scenarios. The visual representations of the obtained solutions through contour, three-dimensional, and two-dimensional depictions in various simulations are shown in the figures. The results propose that the employed method is an efficient and powerful tool to be implemented for different differential equations in applied sciences and engineering.

Abstract The objective of this work is to investigate the perturbed Gerdjikov–Ivanov (GI) equation along spatio-temporal dispersion which explains the dynamics of soliton dispersion and evolution of propagation distance in optical fibers, photonic crystal fibers (PCF), and metamaterials. The algorithms, namely hyperbolic extended function method and generalized Kudryashov’s method, are constructed to obtain the new soliton solutions. The dark, bright, periodic, and singular solitons are derived of the considered equation with the appropriate choice of parameters. These results are novel, confirm the stability of optical solitons, and have not been studied earlier. The explanation of evaluated results is given by sketching the various graphs in 3D, contour and 2D plots by using Maple 18. Graphical simulations divulge that varying the wave velocity affects the dynamical behaviors of the model. In summary, this research adds to our knowledge on how the perturbed GI equation with spatio-temporal dispersion behaves. The obtained soliton solutions and the methods offer computational tools for further analysis in this field. This work represents an advancement in our understanding of soliton dynamics and their applications in photonic systems.

In many nonlinear partial differential equations, noise or random fluctuation is an inherent part of the system being modeled and have vast applications in different areas of engineering and sciences. This objective of this paper is to construct stochastic solitons of Biswas-Arshed equation (BAE) under the influence of multiplicative white noise in the terms of the Itô calculus. Bright, singular, dark, periodic, singular and combined singular-dark stochastic solitons are attained by using the Sardar subequation method. The results prove that the suggested approach is a very straightforward, concise and dynamic addition in literature. By using Mathematica 11, some 3D and 2D plots are illustrated to check the influence of multiplicative noise on solutions. The presence of multiplicative noise leads the fluctuations and have significant effects on the long-term behavior of the system. So, it is observed that multiplicative noise stabilizes the solutions of BAE around zero.

The (2 + 1)-dimensional Konopelchenko-Dubrovsky system contributes to the field of atmospheric science by investigating the behaviour of nonlinear waves, revealing subtle scattering effects and extended-range interactions within the tropical and mid-latitude troposphere. This equation provides insights into the interplay between equatorial and mid-latitude Rossby waves, capturing their complex interactions and dynamics. Nonlinear waves are significant in atmospheric processes and understanding their dynamics is important for comprehending weather patterns. This study centres around the utilisation of the simplest equation method and extended hyperbolic function method to analyse these waves and derives many solutions that illustrate different wave patterns and behaviours intrinsic to the governed system. The simplest equation method enables the derivation of kink and multi-soliton solutions. This method allows for the extraction of solutions characterised by multiple solitary waves propagating without distortion. On the other hand, the extended hyperbolic function method provides anti-bell-shaped, periodic, and singular solutions. These diverse solution types portray various wave patterns and behaviours intrinsic to the equation under study. Additionally, 3D and 2D graphical representations are generated to visually depict the obtained solutions. Hence, this study not only contributes to the comprehension of non-linear wave dynamics in atmospheric science but also imparts knowledge on the broader applicability of mathematical methods in uncovering the underlying intricacies of natural phenomena in different fields of non-linear sciences.

In this scholarly exploration, we employ new mapping method to unveil new soliton solutions to the nonlinear fractional Kudryashov’s equation, using β-derivative and M-Truncated fractional derivatives. Soliton phenomena, invaluable for enhancing computational capabilities in computer systems, find particular utility in tasks like data analysis, image processing, and simulations across various computer science domains. Our research reveals diverse solution forms, encompassing dark, periodic, kink, singular, dark-bright, and bright solitons. These versatile soliton types offer adaptable tools for both modelling and simulation. To enhance understanding, we provide 3D and 2D graphical representations, which facilitate a comparative analysis of solutions derived from the two fractional derivatives. Additionally we compare the effects of different fractional derivative operators (β-derivative and M-Truncated derivative) involve in the nonlinear fractional Kudryashov’s equation. The incorporation of both derivatives allows for a broader exploration of solution forms and enhance the understanding of the equation’s behaviour and its solution. Our work bridges soliton physics and computer science, underscoring the significance of soliton phenomena in advancing computational modelling and simulations. This research contributes to the intersection of theoretical mathematics and practical computer science, highlighting the vital role of solitons in scientific fields.

This paper deals with theoretical analysis of heat transfer in n-immiscible layers of a horizontal Jeffrey fluid film, driven by a heated plate and surface tension gradient. The formulation of the horizontal n-immiscible layers is based on mass, momentum, and energy conservation laws. The resulting systems of coupled linear ordinary differential equations are solved to obtain exact solutions for velocity, temperature, volume flow rate, shear stress, and rate of heat transfer. The effects of flow control parameters of interest, such as the inverse capillary number C, thermal conductivity of fluid χ, and Brinkmann number Br, on the velocity and temperature of the 5-immiscible layers, are investigated through graphical analysis. It is observed that increasing C leads to an increase in the velocity of all 5-immiscible layers. The temperature of all 5-immiscible layers also rises with increasing values of C, χ, and Br. The analysis of these 5-immiscible layers provides significant knowledge and understanding that may be applied to systems with a higher or lower number of immiscible layers utilizing the expressions derived for n-immiscible layers of a fluid film. Moreover, a comparison between immiscible layers of Newtonian fluid and Jeffrey fluid films is made in terms of velocity and temperature.

Understanding the behaviour of temperature dependent viscosity thin film flows on a vertically upward moving heated belt is crucial for improving predictive models in various engineering applications, such as in coating and lubrication processes. This paper presents the heat transfer analysis of temperature-dependent viscosity fluid flow on a vertical upward moving heated belt. The formulated coupled system of nonlinear ordinary differential equations is solved using the Adomian decomposition method. The study investigates the impact of flow controlling parameters: Stokes number St, Brinkmann number Br, variable viscosity parameter β, Weissenberg number We, effective viscosity ϕ and slip parameter e is observed on velocity, temperature, and stationary points. It is perceived that the temperature of the fluid decreases with the increase of St, Br, β, and ϕ, while it increases with the increase of We and e. Positions of stationary points relocate towards the heated belt surface by the increment of St, Br, β, and ϕ. On the other hand, their positions relocate away from the heated belt surface by the increment of We and e. The analysis also highlights the influence of heat generated by viscous dissipation and heat transported by thermal diffusion on the velocity, temperature, and stationary points. Moreover, the flow variables discussed for the temperature dependent viscosity Johnson–Segalman fluid are also discussed for the temperature dependent viscosity Newtonian fluid, and an analogy between both is provided.

This paper explores the double-chain model of DNA known as deoxyribonucleic acid, which has a significant role in the preservation and conversion of genetic data in biological fields. The setup consists of two rods that represent polynucleotide chains found in the DNA and are interconnected by an elastic membrane, showing the hydrogen bonds existing between the base pairs of the chains. The extended hyperbolic function method (EHFM) is analyzed to extract the solutions, including singular, bright, periodic, and kink soliton. The 3D, 2D, and contour graphs provide information on longitudinal and transverse displacements within the DNA structure. The visual exploration of some results reveals the presence of solitary waves in the DNA strands, and these findings have the potential to evaluate various applications. This study represents a significant advancement in our understanding of DNA dynamics, and illustrates the fundamental nature of genetic information transmission and processing.

The coupled Drinfeld–Sokolov–Wilson (DSW) equation is a system of nonlinear partial differential equations (NLPDEs) in (1+1)-dimensions that play a fundamental role in soliton theory and integrable systems. Due to its ability to accurately describe wave phenomena in dispersive water waves, coupled DSW equation has wide-ranging applications in fluid dynamics, plasma physics, and mathematical physics. This study investigates the solitary wave solution of coupled DSW equation by using ϕ6-model expansion. This technique reveals the diverse types of solutions, including bright, dark, singular, and periodic singular solitons. The 3D, contour, and 2D plots are also illustrated to demonstrate the physical behavior of the obtained solutions. The findings of this study can contribute to the development of new analytical and numerical tools for solving other nonlinear equations in the future. The inclusion of constraint conditions in the ϕ6-model expansion technique enhances its applicability and reliability, rendering it a valuable approach for investigating nonlinear systems in various scientific domains.

In this research paper, the improved generalized Riccati equation mapping (IGREM) method is employed to get analytical soliton solutions of (3+1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV-ZK) equation. By giving appropriate parametric values, we obtain soliton solutions of various types, such as kink soliton, singular soliton, and periodic-singular soliton. The description provided in physical variables allows for the study of genuine multispecies plasmas, plasma models, and frequency regimes. These solutions are essential in illuminating several physical phenomena in engineering and other applied sciences. In addition, 3D and 2D graphs of some selected solutions are sketched for the best physical characterization of the obtained results. This method can also be implemented in many non-linear equations in contemporary research areas. Comparison with previous studies shows that the discovery of singular soliton is novel. Therefore, our research represents a significant advancement in the field and contributes to expanding the knowledge regarding soliton behavior. The applications of this study include electron and ion-acoustic modes in regular plasmas with hot Boltzmann electron species, as well as ion and dust-acoustic modes, depending on the specific modeling of the heavier components.

Big data is defined as a large set of data that could be structured or unstructured. In manufacturing shop-floor, big data incorporates data collected at every stage of the production process. This includes data from machines, connecting devices, and even manufacturing operators. The large size of the data available on the manufacturing shop-floor presents a need for the establishment of tools and techniques along with associated best practices to leverage the advantage of data-driven performance improvement and optimization. There also exists a need for a better understanding of the approaches and techniques at various stages of the data life cycle. In the work carried out, the data life-cycle in shop-floor is studied with a focus on each of the components - Data sources, collection, transmission, storage, processing, and visualization. A narrative literature review driven by two research questions is provided to study trends and challenges in the field. The selection of papers is supported by an analysis of n-grams. Those are used to comprehensively characterize the main technological and methodological aspects and as starting point to discuss potential future research directions. A detailed review of the current trends in different data life cycle stages is provided. In the end, the discussion of the existing challenges is also presented.

In this study, our focus is on the construction of novel soliton solutions of non-linear Schrödinger equation with parabolic law and non-local law non-linearities via new extended hyperbolic function method. The acquired solutions are original and could be helpful in the field of nonlinear optics, fluid dynamics, and plasma physics. From these outcomes, it is believed that this method is a promising technique to handle a wide variety of such type of equations. For physical understanding and better realization of our constructed solutions, some of the obtained solutions under appropriate parameters are demonstrated by 2D and 3D plots. Analysis discloses that the proposed scheme is consistent and can be applied to more advanced models in engineering and physics.

In this paper, we study the well known Zakhrov equation (ZE) which states to the Langmuir waves propagation in the ionized plasma. The exact soliton solutions of ZE are extracted using new extended direct algebraic method (EDAM). The obtained solutions are given in the form of trigonometric, rational and hyperbolic functions. Moreover, some of the obtained solutions are illustrated graphically to establish the physical nature of the selected solutions. The existence of these exact solutions is guaranteed by the constraints on the used parameters. It is noted that the new EDAM is easy and efficient technique and it can be useful to many other non-linear partial differential equations (PDEs).

Looking for the exact solutions in the form of optical solitons of nonlinear partial differential equations has become very famous to analyze the core structures of physical phenomena. In this paper, we have constructed some various type of optical solitons solutions for the Kaup-Newell equation (KNE) and Biswas-Arshad equation (BAE) via the generalized Kudryashov method (GKM). The conquered solutions help to understand the dynamic behavior of different physical phenomena. These solutions are specific, novel, correct and may be beneficial for edifying precise nonlinear physical phenomena in nonlinear dynamical schemes. Graphical recreations for some of the acquired solutions are offered.

This paper discusses optical soliton solutions for the paraxial wave model (PWM) in Kerr law media. Mathematically, the PWM for a monochromatic beam is equivalent to the Schrödinger equation of free quantum particle. The ϕ6 model expansion technique is used to obtain dark, bright, singular, bright-dark combined and periodic solitons. The solutions obtained by this method are in the form of trigonometric, hyperbolic, and exponential functions. To prompt the important propagated structures, some scrutinized solutions are demonstrated in the form of 3D, contour, and 2D plots using specific values to the parameters with constrained conditions. It is observed that this method is an efficient, secure, and stable tool for new precise solitons for various NLPDEs in engineering and applied sciences.

By employing the extended hyperbolic function method (EHFM), we extract the exact solutions of the (2+1)-dimensional nonlinear soliton equation (SE). A soliton equation is used for investigation of the dynamics of nonlinear waves in plasma physics and fluid dynamics. A various new techniques for finding exact solutions of the (2+1)-dimensional nonlinear SE are satisfactorily acquired with the help of EHFM. The EHFM presents various types of new solutions in the form of dark, singular, periodic, bright solitons and some rational function solutions. In addition, for the physical characterization of the acquired solutions of (2+1)-dimensional SE, some 2-dim and 3-dim plots are drawn. The attained results are novel for the considered equation, and results reveal that the method is concise, direct and competent which can be assembled in other complex phenomena.

In this article, a strain wave equation (SWE) is studied, which is used to model wave propagation in microstructured materials that earn a noteworthy place in solid-state physics. This equation also signifies the dynamics of various physical phenomena. The Sardar-subequation method (SSM) is utilized for this model. Granting appropriate values to parameters, we obtain various types of soliton solutions such as periodic singular solitons, bright solitons, dark solitons, singular soliton, combined dark-bright solitons, and some other wave solutions. These novel solitons and other wave results have significant applications in engineering and applied sciences. The graphical sketchings of the results are illustrated to purify the impact of the SSM. Furthermore, the executed technique can be utilized for further studies to discuss the realistic phenomena developing in physical and engineering problems.

Nonlinear science is a fundamental science frontier that include studies and the common properties of nonlinear phenomena. This article is devoted to the study of sub-pico second optical pluses in birefringent fibers for Kaup-Newell equation (KNE) without four-wave mixing. Three prominent integrations techniques are successfully implemented on KNE in coupled vector form. Variety of soliton solutions namely dark, bright, periodic singular, singular and bright-singular combo solitons are constructed for the KNE in birefringent fibers. The obtained solutions are reckoned with their respective existence criterion. In addition, two-dimensional and three-dimensional graphs are drawn to exhibit the physical behavior of the obtained solutions.

In this study, the monochromatic beam propagation interprets non-scattering and non-dissipation spatiotemporally localized wave parcels proliferated in the announcement of optical Kerr. By using the mathematical techniques some elliptic, rational, and soliton solutions of dimension-less (time-dependent) paraxial wave structure are established. The Sardar subequation method (SSM) and mathematica 11.0 are used to find the exact solution of the paraxial wave equation. The solutions which we will be obtained explain some key implementations in engineering and physics. Some graphical representation has been explained in modulus, real and imaginary graph by considering relevant values of the framework. The solidity of this work explains that the soliton solutions are secure and perfect.

In this paper, the extended simple equation method (ESEM) and the generalized Riccati equation mapping (GREM) method are applied to the nonlinear third-order Gilson–Pickering (GP) model to obtain a variety of new exact wave solutions. With the suitable selection of parameters involved in the model, some familiar physical governing models such as the Camassa–Holm (CH) equation, the Fornberg–Whitham (FW) equation, and the Rosenau–Hyman (RH) equation are obtained. The graphical representation of solutions under different constraints shows the dark, bright, combined dark–bright, periodic, singular, and kink soliton. For the graphical representation, 3D plots, contour plots, and 2D plots of some acquired solutions are illustrated. The obtained wave solutions motivate researchers to enhance their theories to the best of their capacities and to utilize the outcomes in other nonlinear cases. The executed methods are shown to be practical and straightforward for approaching the considered equation and may be utilized to study abundant types of NLEEs arising in physics, engineering, and applied sciences.

The purpose of this article is to achieve new soliton solutions of the Gilson–Pickering equation (GPE) with the assistance of Sardar’s subequation method (SSM) and Jacobi elliptic function method (JEFM). The applications of the GPE is wider because we study some valuable and vital equations such as Fornberg–Whitham equation (FWE), Rosenau–Hyman equation (RHE) and Fuchssteiner–Fokas–Camassa–Holm equation (FFCHE) obtained by particular choices of parameters involved in the GPE. Many techniques are available to convert PDEs into ODEs for extracting wave solutions. Most of these techniques are a case of symmetry reduction, known as nonclassical symmetry. In our work, this approach is used to convert a PDE to an ODE and obtain the exact solutions of the NLPDE. The solutions obtained are unique, remarkable, and significant for readers. Mathematica 11 software is used to derive the solutions of the presented model. Moreover, the diagrams of the acquired solutions for distinct values of parameters were demonstrated in two and three dimensions along with contour plots.

In this paper, the geodesic motion of neutral and test particles around the time conformal (TC) Dilaton black hole (BH) is investigated using the eϵg(t) as the time conformal factor in which g(t) is an arbitrary function of time and ϵ is a perturbation parameter. The function g(t) leads to (ta) by utilizing the well-known approximate Noether symmetry (ANS). Furthermore, we discuss the effect of magnetic fields and find the location of stable and unstable orbits w. r. t time, graphically. After that, in the presence and absence of a magnetic field, we interrogate the crucial physical parameters such as effective potential (Ueff), effective force (Feff) and escape velocity (ν⊥). We find the unstable and stable regions of particles for different values of angular momentum (Lz) and magnetic field (B) near the TC Dilaton BH. Moreover, the effects of the Dilaton parameter (μ) on neutral and charged particles are also discussed, which provide some new features. The important results in this study could estimate the powerful relativistic jets originating from the BH.

The Klein–Fock–Gordon equation (KFGE), defined as the equation of relativistic wave related to NLEEs, has numerous implications for energy particle physics and is useful as a model for several types of matter, with deviation in the basic stuffs of particles and in crystals. In this work, the Sardar subequation method (SSM) is used for finding the solution of this KFGE. The advantage of SSM is that it provides many different kinds of solitons, such as dark, bright, singular, periodic singular, combined dark–singular and combined dark–bright solitons. The results show that the SSM is very reliable, simple and can be functionalized to other nonlinear equations. It is verified that all the attained solutions are stable by modulation instability process. To enhance the physical description of solutions, some 3D, contour and 2D graphs are plotted by taking precise values of parameters using Maple 18.

Graph theory has provided a very useful tool, called topological index, which is a number from the graph M with the property that every graph N isomorphic to M value of a topological index must be same for both M and N. Topological index is a descriptor in graph theory which is used to quantify the physio-chemical properties of the chemical graph. In this paper, we computed closed forms of M-polynomials for line graphs of H-naphtalenic nanotubes and chain silicate network. From M-polynomial, we obtained some topological indices based on degrees.

Internet of things grew swiftly and many services, software, sensors-embedded electronic devices and related protocols were developed and still in progress with full swing. Internet of things enabling physically existing things to see, hear, think and perform a notable task by allowing them to talk to each other and share useful information while making decision and caring-on/out their important tasks. Internet of things is greatly promoted by wireless sensor network as it becomes a perpetual layer for it. Wireless sensor network works as a base-stone for most of the Internet of things applications. There are severe general and specific threats and technical challenges to Internet of things–based sensor networks which must overcome to ensure adaptation and diffusion of it. Most of the limitations of wireless sensor networks are due to its resource constraint objects nature. The specified open research challenges in Internet of things–based sensor network are power consumption, network lifespan, network throughput, routing and network security. To overcome aforementioned problems, this work aimed to prolong network lifetime, improve throughput, decrease packet latency/packet loss and further improvise in encountering malicious nodes. To further tune the network lifetime in terms of energy, wireless harvesting energy is suggested in proposed three-layer cluster-based wireless sensor network routing protocol. The proposed mechanism is a three-tier clustering technique with implanted security mechanism to encounter malicious activities of sensor nodes and to slant them into blacklist. It is a centred-based clustering protocol, where selection of cluster head and grid head is carried out by sink node based on the value of its cost function. Moreover, hardware-based link quality estimators are used to check link effectiveness and to further improve routing efficiency. At the end, excessive experiments have been carried out to check efficacy of the proposed protocol. It outperforms most of its counterpart protocols such as fuzzy logic–based unequal clustering and ant colony optimization–based routing hybrid, Artificial Bee Colony-SD, enhanced three-layer hybrid clustering mechanism and energy aware multi-hop routing in terms of network lifetime, network throughput, average energy consumption and packet latency.

In this paper, the Kudryashov’s method and generalized tanh method (GTM) are employed to retrieve the new distinct optical solitons for highly dispersive NLSE. These two schemes effectively acquire the exact solution of the model in the form of dark, singular, singular periodic and bright-singular combo optical solitons. These solutions assist the mathematicians and physicians to realize the physical phenomena of present model. The solutions in presents work indicate that these schemes offer us an influential mathematical tools for solving NLSEs in various areas of applied sciences. In addition, the graphical illustrations of some solutions are also illustrated.

The purpose of this paper is to introduce the notion of strongly h,s-nonconvex functions and to present some basic properties of this class of functions. We present Schur inequality, Jensen inequality, Hermite–Hadamard inequality, and weighted version of the Hermite–Hadamard inequality.

Abstract In this work we develop the square integral estimates for the functions of n variables which are subharmonic with respect to some variables and for other remaining variables are superharmonic. It is in a sense a generalization of reverse Poincaré type inequalities for the difference of superharmonic functions developed in (J. Inequal. Appl. 2015: doi:10.1186/s13660-015-0916-9 , 2015).

The approximate solution to a class of sixth order boundary value problems is obtained using the reproducing kernel space method. The numerical procedure is applied on linear and nonlinear boundary value problems. The approach provides the solution in terms of a convergent series with easily computable components. The present method is simple from the computational point of view, resulting in speed and accuracy significant improvements in scientific and engineering applications.It was observed that the errors in absolute values are better than compared (Che Hussin and Kiliçman (2011) and, Noor and Mahyud-Din (2008), Wazwaz (2001), Pandey (2012)).Furthermore, the nonlinear boundary value problem for the integrodifferential equation has been investigated arising in chemical engineering, underground water flow and population dynamics, and other fields of physics and mathematical chemistry. The performance of reproducing kernel functions is shown to be very encouraging by experimental results.