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In this work, we study the generalized shallow water wave equation to obtain novel solitary wave solutions. The application of this nonlinear model can be found in tidal waves, weather simulations, tsunami prediction, river and irrigation flows. To obtain the new exact solutions of the considered model, we have applied a novel analytical technique namely ( G ′ G ′ + G + A) -expansion method. Using the aforementioned method and computational software, we have obtained different kinds of periodic and singular solitary wave solutions of the generalized shallow water wave equation. The obtained solutions are exponential function and trigonometric function solutions. Using 2D and 3D plots of the wave solutions, the dynamic behaviors of the developed solutions are displayed. The retrieved solutions validated the effectiveness and robustness of the proposed technique. [ABSTRACT FROM AUTHOR]

In this paper, we examine a novel generalized (2 + 1) -dimensional Bogoyavlensky–Konopelchenko (gBK) model analytically via application of five mathematical methods, namely called Extended Simple Equation (ESE) method, Modified Extended Auxiliary equation mapping (MEAEM) method, (G ′ / G) -expansion method, Exp (− Ψ (ϕ)) -expansion method and modified F-expansion method, respectively, to obtain different types solitary wave solutions in the form of trigonometric, hyperbolic, rational and exponential functions as compared to solutions exist in [S. T. Chen and W. X. Ma, Math. China 13 (2018) 525; A. Sonia, A. Jamshad, S.-U. Rehman and A. Ali, Int. J. Appl. Comput. Math. 9 (2023)]. The research focuses on physical systems, physiology, mathematical applications, engineering fields, and the chemical processes of species in the porous nanoparticles, examining the impact of various parameters on the nonlinear model. The graphs of some solutions are plotted in the form of two-dimensional and three-dimensional by assigning the particular values to the paraments with the assistance of mathematica software which illustrated the physical behavior of the concern model. The derived solution has numerous applications in fluid dynamics and the interaction of Riemann waves and long waves. [ABSTRACT FROM AUTHOR]

It is known that a smooth function of exponential decay at infinity cannot be an orthonormal wavelet. Dziubański and Hernández constructed smooth orthonormal wavelets of Gevrey-type subexponential decay. We weaken the Gevrey-type decay and construct orthonormal wavelets of subexponential decay related to the so-called extended Gevrey classes. The virtue of our construction is that precise asymptotics of functions from such classes can be given in terms of the Lambert W -function. [ABSTRACT FROM AUTHOR]

Tasks of different nature and difficulty levels are a part of people's lives. In this context, there is a scientific interest in the relationship between the difficulty of the task and the persistence need to accomplish it. Despite the generality of this problem, some tasks can be simulated in the form of games. In this way, we employ data from a large online platform, called Steam, to analyze games and the performance of their players. More specifically, we investigated persistence in completing tasks based on the proportion of players who accomplished game achievements. Overall, we present five major findings. First, the probability distribution for the number of achievements is log-normal distribution. Second, the distribution of game players also follows a log-normal. Third, most games require neither a very high degree of persistence nor a very low one. Fourth, players also prefer games that demand a certain intermediate persistence. Fifth, the proportion of players as a function of the number of achievements declines approximately exponentially. As both the log-normal and the exponential functions are memoryless, they are mathematical forms that describe random effects arising from the nature of the system. Therefore our first two findings describe random processes of fragmenting achievements and players while the last three provide a quantitative measure of the human preference in the pursuit of challenging, achievable and justifiable tasks. [ABSTRACT FROM AUTHOR]

In this study, the modified exponential approach is used to investigate the fractional order longitudinal wave equation in a magneto-elastic circular rod, which represents the nonlinear interplay between dispersion and longitudinal wave velocity depending on the rod's material and geometry. The time-fractional order is used in the standpoint of conformable derivative. Comprehensive and descriptive waveform solutions, including kink-shaped, bell-shaped, and bright bell-shaped solitons, are obtained. The produced soliton solutions are advanced, unique, developed, and crucial. To demonstrate the physical appearance of the obtained solutions, 2D, 3D, and contour graphs of some of the obtained solutions are plotted. The extracted solutions demonstrate the physical significance of the model and express the pressure and electrostatic potential for the longitudinal wave equation and can be used to elucidate more complicated nonlinear models of classical and fractional order. The obtained results are helpful in finding the amplitude of tsunami waves which is of great interest to researchers in the field of ocean engineering. If a tsunami extends across a large area, it becomes a manifestation of solitons. The comparison of the obtained solutions is also given in the form of remarks 1 and 2 which shows the novelty of our work. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a spinor algebra of k-balancing numbers referred to as the k-balancing and k-Lucas-balancing spinors. First, we give k-balancing quaternions and their some algebraic properties. Then we introduce a spinor family of k-balancing numbers by defining a linear and injective correspondence between k-balancing and k-Lucas-balancing quaternions to spinors. Here, we give various algebraic properties of this spinor such as the Binet form, Catalan’s identity, d’Ocagne identity, generating and exponential generating functions. Moreover, we obtain various partial sum formulae for these sequences in closed form and also give matrix representations. [ABSTRACT FROM AUTHOR]

In this paper, novel soliton solutions to the conformable time fractional Benjamin–Bona–Mahony and seventh-order time fractional Sawada–Kotera–Ito equations are investigated via application of two mathematical methods. These models are reformulated into a fractional order derivative by applying the conformable derivative operator. The derived solutions have been obtained including periodic and solitons solutions in the form of trigonometric, hyperbolic and exponential function solutions. For physical phenomena of concern models, some solutions are plotted in two-dimensional and three-dimensional by assigning the particular values to the parameters under the constrain conditions with the assistance of Mathematica software. Hence, all results obtained by our proposed methods are novel and have not been yet reported in any literature. Moreover, the proposed schemes are enormously superbly mathematical tools to review wave solutions of several fractional models in nonlinear science. [ABSTRACT FROM AUTHOR]

In this paper, we primarily focus on the nonlinear paraxial wave equation in Kerr media, a generalization of the nonlinear Schrödinger equation utilized to characterize the dynamics of optical beam propagation. By using three potent analytical methods, namely, the Sine-Gordon expansion method, the functional variable method, and the Bernoulli (G′/G)-expansion method, numerous novel soliton solutions are derived. These solutions, comprising hyperbolic, trigonometric and exponential functions, represent significant additions to the field of optics. Furthermore, we elucidate the physical characteristics of these novel optical soliton solutions by presenting a series of three-dimensional (3D) and two-dimensional (2D) graphs with appropriate parameter values. [ABSTRACT FROM AUTHOR]

A model condition for the propagation of ultrashort pulses in optical fiber frameworks could be established through the extensive generalization of the nonlinear Schrödinger equation introduced by Triki and Biswas. This study uses two significant and effective trustworthy methods to examine the Triki–Biswas (TB) equation. Sardar Subequation Methods (SSM) and the generalized Kudryashov (KUD) method generate solutions for trigonometric, hyperbolic, exponential, and rational functions. These approaches are specifically designed for handling solitary wave patterns, dark-singular-mixed solitons, combined dark-bright solitons, singletons, bright solitons, exponential and rational functions, as well as periodic solitons. All of these solution classes contribute to the physical dynamics of outcomes. These findings represent an innovative extension of the soliton domain within the TB model and are being reported for the first time in our investigation. In addition, Mathematica simulations are used to display the 3D and 2D graphs to explain the identified solutions' physical dynamics. [ABSTRACT FROM AUTHOR]

A type of general fractional calculus with respect to another function is introduced in this paper. The kernel function within the fractional derivative and integral is specified as the Exponential function in a typical way. Based on the excellent property of the newly defined general fractional calculus, we proposed two viscoelastic models in the fractional manner compared with the classical calculus. [ABSTRACT FROM AUTHOR]

The exponential decay of lattice Green functions is one of the main technical ingredients of the Bałaban's approach to renormalization. We give here a self-contained proof, whose various ingredients were scattered in the literature. The main sources of exponential decay are the Combes–Thomas method and the analyticity of the Fourier transforms. They are combined using a renormalization group equation and the method of images. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new convexity notion for inter-valued functions, known as Geometrically–Arithmetically Cr-h-convex functions (abbreviated as GA-Cr-h-CFs) and explore its properties. The family of GA-Cr-h-CFs simultaneously covers the family of GA-CFs, GA-h-CFs and GA-Cr-CFs. On the other hand, we demonstrate its equivalence with convex functions (CFs), h-CFs, Cr-CFs and Cr-h-CFs via exponential function. We also investigate necessary and sufficient conditions for GA-Cr-h-CFs via two corresponding real-valued GA-h-CFs. Due to significance and applications of mathematical inequalities, we apply our findings and propose several inequalities such as Jensen–Mercer inequality (JMI), Hermite–Hadamard inequality (HHI) and weighted Hermite–Hadamard-type inequalities (wHHIs) for GA-Cr-h-CFs. Through our results, we re-capture many known results and inequalities for subclasses of GA-Cr-h-CFs studied in the recent literature. [ABSTRACT FROM AUTHOR]

Scale-dependent vibration and stability of three-dimensional functionally graded (FG) porous Rayleigh nanobeams with axial motion in varying environmental conditions are investigated by considering scale effects in the thickness orientation. It is supposed that the effective material characteristics of the nanobeam in the longitudinal, width, and thickness orientations are graded according to exponential and power-law functions. Different patterns involving uniform, non-uniform, and logarithmic distributions are considered to model the porosity impacts. Based on the nonlocal strain gradient Rayleigh beam model and incorporating thickness effects, the dynamical equation of the system is derived by considering linear, parabolic, and sinusoidal elastic foundations. The Galerkin discretization technique and Laplace transform are adopted to solve the eigenvalue problem and accomplish the stability analysis. Several comparative studies are performed with the available data in the open literature to validate the solution approach and results. Also, detailed parametric investigations are accomplished to interpret the importance of scale factor, rotary inertia factor, porosity characteristics, magneto-hygro-thermal loads, foundation parameters, gradient indices, and boundary conditions on the dynamical behavior, critical divergence, and flutter axial speeds. The results revealed that, contrary to the effect of the material gradient in thickness orientation, the longitudinal and width gradation of materials can lead to the increment of vibrational frequencies. The outcomes declared that the instability threshold improves significantly by considering the scale effects along the thickness orientation. Also, it is demonstrated that when the middle surface of the cross-section has the highest porosity, the stability regions expand by increasing the porosity coefficient. The present research outcomes can offer valuable insights into the optimum design of high-speed moving multi-directional graded porous nanosystems. [ABSTRACT FROM AUTHOR]

This study uses the modified Sardar sub-equation method to find novel soliton solutions to the nonlinear three-component coupled nonlinear Schrödinger equation (NLSE), which is used for pulse propagation in nonlinear optical fibers. Multi-component NLSE equations are widely used because they can represent a wide range of complex observable systems and more dynamic patterns of localized wave solutions. The optical solutions proposed in this study are novel and can be described using hyperbolic, trigonometric, and exponential functions. These solutions are categorized as bright, dark, singular, combo bright-singular, and periodic solutions. Some solutions' dynamic behaviors are demonstrated by selecting appropriate physical parameter values. The results and computational analysis indicate that the techniques provided are simple, effective, and adaptable. They can be applied to a variety of nonlinear evolution equations, whether stable or unstable, and can be used in fields such as mathematics, mathematical physics, and applied sciences. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear space–time fractional form of Cahn–Allen equation (FCAE) with beta and M-truncated derivatives. Cahn–Allen equation (CAE) is commonly used in many problems of physics and engineering, such as, solidification problems, phase separation in iron alloys and others. We apply the improved tan (Ψ (η) 2) -expansion method (ITEM). We obtain four types of traveling wave solutions, including, trigonometric, hyperbolic, rational and exponential function solutions. We demonstrate some of the extracted solutions using definitions of the beta (BD) and M-truncated derivatives (MTD) to understand their dynamical behavior. We observe the fractional effects of the aforementioned derivatives on the related physical phenomena up to possible extent. [ABSTRACT FROM AUTHOR]

This paper proposes a novel seismic demand hazard model conditional on the separation distance between two adjacent buildings to fully explore the possibility of seismic pounding occurrence between buildings during their design life. The model uses an exponential function with two model parameters estimated through a two-point approach to yield an explicit expression for deriving a Seismic Demand Hazard Curve (SDHC) for adjacent buildings. The resulting SDHC provides a basis for seismic pounding risk assessment of adjacent buildings with a specific separation distance and a design tool for the risk-targeted separation distance between buildings. To verify the model, this paper compares the SDHCs derived for six pairs of adjacent buildings having different heights and natural period combinations with corresponding numerical simulation results. The results show that the SDHCs derived from the model, within the engineering interest range, closely agree with numerical simulation results. [ABSTRACT FROM AUTHOR]

The univalent function ϕ SG (z) : = 2 / (1 + e − z) maps the open unit disk | z | < 1 onto a sigmoid domain | log (w / (2 − w)) | < 1. The sharp radii constants have been computed for the classes involving ratios f / g and g / (z p) of analytic functions f , g and p defined in | z | < 1 such that p is subordinate to ϕ SG , the other two ratios are subordinate to 1 + z , e z or z + 1 + z 2 and the quantity z f ′ / f lies in a certain region of the right half plane. [ABSTRACT FROM AUTHOR]

This research is concerned with some modernistic optical solitons of beta-fractional density-dependent diffusion-reaction model. The collected optical solutions can be executed in exposing this model in a prominent form. The obtained results include the trigonometric, hyperbolic trigonometric, rational and exponential functions. Verification of achieved results is also done by using Mathematica tools. Three distinct techniques named as exp a function, extended sinh-Gordon equation expansion (EShGEE) and modified extended tanh function expansion techniques are employed to protect the results. The achieved results are also illustrated by 2-D, 3-D and contour plots. The gained results can also be fruitful for the development of model in future. [ABSTRACT FROM AUTHOR]

We consider a class of nonhomogeneous elliptic equations with fractional Laplacian and nonlinear gradient terms, namely (− Δ) α 2 u = V (x) u + g (u , ∇ u) + f in ℝ n , where 0 < α < n , g is the nonlinearity, V the potential and f is a forcing term. Some examples of nonlinearities dealt with are u | u | ρ − 1 , | ∇ u | ρ and | u | ρ 1 | ∇ u | ρ 2 , covering large values of ρ , ρ 1 , ρ 2 , and particularly variational supercritical powers for u and super- α ones for | ∇ u | (superquadratic if α = 2). Moreover, we are able to consider some exponential growths, g belonging to certain classes of power series, or g satisfying some conditions in the Lipschitz spirit. We obtain results on existence, uniqueness, symmetry, and other qualitative properties in a new framework, namely modulation-type spaces based on Lorentz spaces. For that, we need to develop properties and estimates in those spaces such as complex interpolation, Hölder-type inequality, estimates for product, convolution and Riesz potential operators, among others. In order to handle the nonlinearity, other ingredients are estimates for composition operators in our setting. [ABSTRACT FROM AUTHOR]

The work aims to explore the exponential rational function technique and the generalized Kudryashov procedure to investigate one of the most important equations which is the equation of cold bosonic atoms in a zig-zag optical lattice. The considered equation is reduced to a supreme equation by using the continuum approximation which describes the soliton's dynamics with the implication pulse variables. The adopted techniques are accurate, simple, straightforward, and succinct to compute. [ABSTRACT FROM AUTHOR]

Global warming is becoming a big concern for the environment since it is causing serious and often unexpected impacts on species, affecting their abundance, genetic composition, behavior and survival. So, the modeling study is necessary to investigate the effects of global warming in predator–prey dynamics. This research paper analyzed the memory effect evaluated by Caputo fractional derivative on predator–prey interaction using an exponential fear function with a Holling-type II function in the presence of global warming effect on prey and predator species. It is assumed that the densities of prey and predator species decrease due to the increase of global warming. It is considered that both prey and predator species are contributing to the increase of global warming. Also, it is considered that global warming is increasing constantly and decreasing due to the natural decay rate. All possible equilibria of the system are determined, and the stability of the system around all equilibria points is investigated. Around the interior equilibrium point, the Hopf bifurcation is also theoretically and numerically studied. A number of numerical simulation results are presented to demonstrate the impacts of fear, fractional order and global warming on the behavior of the model. It is observed that the global warming effect on predator species may destabilize the system but ultimately the system may become stable. Again, it is obtained that the natural decay rate of global warming can stabilize the system initially but a higher decay rate may destabilized the system. It is also found that the fractional-order model is determined to be more stable than the integer-order model. [ABSTRACT FROM AUTHOR]

Let f be a Hecke–Maass cusp form of weight zero for S L 2 (ℤ) and λ f (n) be the nth Fourier coefficient. For almost all , we have ∑ n ≤ x λ f 2 (n) e (n) ≪ x 0. 8 1 2 5 + , which improves the result of Acharya [Exponential sums of squares of Fourier coefficients of cusp forms, Proc. Indian Acad. Sci. Math. Sci. 130 (2020) 24], who showed an upper bound larger than x 0. 8 2 9 7. For all α > 1 of type τ < ∞ , we also show that ∑ 1 ≤ n ≤ x λ f 2 ([ α n + β ]) − α − 1 ∑ 1 ≤ n ≤ [ α x + β ] λ f 2 (n) ≪ x 1 3 / 1 6 + + x 1 − 3 / 4 (τ + 1) + , where ℬ α , β : = { [ α n + β ] } n = 1 ∞. This result relies heavily on a generalized double sum (see Theorem 1.3). [ABSTRACT FROM AUTHOR]

The action of a right-angled Artin group on its extension graph is known to be acylindrical because the cardinality of the so-called r-quasi-stabilizer of a pair of distant points is bounded above by a function of r. The known upper bound of the cardinality is an exponential function of r. In this paper we show that the r-quasi-stabilizer is a subset of a cyclic group and its cardinality is bounded above by a linear function of r. This is done by exploring lattice theoretic properties of group elements, studying prefixes of powers and extending the uniqueness of quasi-roots from word length to star length. We also improve the known lower bound for the minimal asymptotic translation length of a right-angled Artin group on its extension graph. [ABSTRACT FROM AUTHOR]

The reconstruction of a bandlimited function from its finite sample data is fundamental in signal analysis. It is well known that oversampling of a bandlimited function leads to exponential convergence in its reconstruction. A simple and efficient Gaussian regularized Shannon sampling formula has been proposed in G. W. Wei [Quasi wavelets and quasi interpolating wavelets, Chem. Phys. Lett. 296 (1998) 215–222] with such an exponential convergence ability. We show that all hyper-Gaussian regularized formulas share this desired property. The analysis is built on estimates on the Fourier transform of the hyper-Gaussian functions. We also establish error bounds for the reconstruction of derivatives of a univariate bandlimited function, and for multivariate bandlimited functions. [ABSTRACT FROM AUTHOR]

In this work, we execute the generalized exponential rational function (GERF) method to construct numerous and a large number of exact analytical solitary wave solutions of the nonlinear extended Zakharov–Kuznetsov (EZK) dynamical equation in (3+1)-dimensions. The implemented method is one of the best, most reliable, and efficient techniques in the present time for determining numerous closed-form wave analytic solutions to NPDEs. We have accomplished a variety of solitary wave solutions related to some arbitrary parameters under various family cases. These solutions take the following forms based on the free parameters chosen: exponential functions form, trigonometric functions form, and hyperbolic functions form. The obtained solutions are dissimilar and entirely new from the previous findings available in the literature. The dynamics of obtained solutions, namely, soltion, singular soliton wave, a periodic wave, bell-shape, anti-bell-shape wave, breather wave, and multisoliton wave solutions by the special-choice of parameters, are shown graphically in 3D, 2D, and corresponding density profiles. The results demonstrate that the employed computational strategy is efficient, direct, concise, and can be executed in various complex phenomena with symbolic computations. Furthermore, it is revealed that the generalized exponential rational function technique can be effectively utilized for several other NPDEs in engineering, sciences, and mathematical physics. [ABSTRACT FROM AUTHOR]

By consideration of an Einstein-dilaton nonlinear charged gravitating system, it has been shown that this theory is confronted with the problem of indeterminacy. It means that the number of independent differential equations is one less than the number of unknowns. To overcome this problem, the power-law and exponential ansatz functions have been used, separately. Through solving the field equations, in the presence of a Coulomb-like electric field, it has been found that this theory includes two novel classes of charged black holes (BHs) with unusual asymptotic behavior, for each ansatz. It has been found that, under some circumstances, both of the ansatz functions lead to the same results. The novel exact solutions show BHs with one horizon, two horizons and without horizon. Using a Smarr-type mass formula, the validity of the first law of BH thermodynamics (FLT) has been proved, after calculating the thermodynamic and conserved quantities. Making use of thermodynamical and geometrical approaches, thermal stability of the BHs has been analyzed. Results of the aforementioned methods have been compared by use of the plots. [ABSTRACT FROM AUTHOR]

This paper investigates the cross-kink and solitary wave solutions to the nonlinear vibration and dispersive wave systems. The solutions include periodic, cross-kink and solitary wave solutions. The bilinear form is considered in terms of Hirota derivatives. Accordingly, we utilize the Cole–Hopf algorithm to get the exact solutions of the (2 + 1) -dimensional modified dispersive water-wave system. The analytical treatment of cross-kink, solitary wave solutions is studied and plotted in three forms of two, density and three-D plots. A nonlinear vibration system will be investigated. Employing appropriate mathematical assumptions, the novel kinds of the cross-kink and solitary wave solutions are derived and constructed in view of the combination of kink, periodic and soliton for cross-kink and also a combination of two kinks in terms of exponential functions for solitary of the governing equation. To achieve this, the illustrative example of the (2 + 1) -D modified dispersive water-wave system is furnished to exhibit the feasibility and reliability of the procedure utilized in this research. The trajectory solutions of the traveling waves are presented explicitly and graphically. The effect of the free parameters on the behavior of designed figures of a few obtained solutions for two nonlinear rational exact cases was also considered. By comparing the suggested technique with the other existing schemes, the results present that the execution of this technique is concise, simple and straightforward. [ABSTRACT FROM AUTHOR]

This study investigates the Hamiltonian amplitude equation by using the generalized exponential rational function method. As a result, a variety of soliton solutions have been established with different wave structures such as optical exponential, dark, singular, periodic, periodic-singular, and hyperbolic solutions. 2D and 3D graphics of certain acquired solutions are drawn with the help of computer program. By picking appropriate values for the free parameters in the obtained solutions, the physical phenomena for these obtained solutions are graphically studied and depicted. [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, a (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation is studied from the governing equations of dusty plasma by means of multiscale analysis and reduced perturbation method. We utilize the generalized exponential rational function method to give various forms of analytical solutions with free parameters of the corresponding equation by using Mathematica software calculation. Then, appropriate parameter values are selected to draw the three-dimensional (3D) diagram and contour diagram of the analytic solutions. We also give the influence of these physical parameters on wave amplitude, waves profile, and stability of waves. The results show the generalized method is a reliable mathematical method to solve similar nonlinear equations in mathematical physics and these solutions can enrich the physical behavior of the equation in dusty plasma. [ABSTRACT FROM AUTHOR]

In this paper, we consider (2 + 1) and (3 + 1) -dimensional Boiti–Leon–Manna–Pempinelli equations. We obtain the soliton solutions by using the new extended direct algebraic method. The resulting solutions carry a variety of new families including the singular solution of type 1 and 2, dark, dark-bright, and dark-singular soliton solution. These solutions contain trigonometric, hyperbolic, and exponential type functions. Moreover, to highlight the important features of the obtained solutions, some three-dimensional and two-dimensional diagrams of these solutions are plotted with the suitable choice of the values of the involving parameters. [ABSTRACT FROM AUTHOR]

Some sufficient conditions on certain constants which are involved in some first and second-order differential subordinations associated with certain functions with positive real part like modified Sigmoid function, exponential function and Janowski function are obtained so that the analytic function p normalized by the condition p (0) = 1 , is subordinate to Janowski function. The admissibility conditions for Janowski function are used as a tool in the proof of the results. As application, several sufficient conditions are also computed for Janowski starlikeness. [ABSTRACT FROM AUTHOR]

In this paper, the generalized rational function procedure was used to produce new solitary solutions to the fractional Jaulent–Miodek hierarchy equation. The equation is shortly called the Jaulent–Miodek equation, which was first derived by Jaulent and Miodek and associated with energy-dependent Schrödinger potentials. The equation is converted into a fourth-order partial differential equation using a transformation. In order, we find some solitary solutions such as soliton, rational and hyperbolic function solutions of Jaulent–Miodek hierarchy equation by the help of generalized exponential rational function procedure. Then, for some parameters, we draw three-dimensional graphics of real values of some exact solutions that we acquired by using this procedure. [ABSTRACT FROM AUTHOR]

In this paper, we pay attention to the nonlinear dynamical behavior of ultra-short pulses in optical fiber. The new Hamiltonian amplitude equation is used as a governing model to analyze the pulses. We secure the ultra-short pulses in the forms of bright, dark, singular, combo and complex soliton solutions by the utilization of three of sound computational integration techniques that are protracted (or extended) Fan-sub equation method (PFSEM), the generalized exponential rational function method (GERFM), extended Sinh-Gordon equation expansion method (ShGEEM). Moreover, Jacobi's elliptic, trigonometric, and hyperbolic functions solutions are also discussed as well as the constraint conditions of the achieved solutions are also presented. In addition, we discuss the different wave structures by the assistance of logarithmic transformation. The findings demonstrate that the examined equation theoretically contains a large number of soliton solution structures. By selecting appropriate criteria, the actual portrayal of certain obtained results is sorted out graphically in 3D and 2D profiles. The results suggest that the procedures used are concise, direct, and efficient, and that they can be applied to more complex phenomena. The resulting solutions are novel, intriguing, and potentially useful in understanding energy transit and diffusion processes in mathematical models of several disciplines of interest, including nonlinear optics. Our new results have been compared to these in the literature. [ABSTRACT FROM AUTHOR]

The generalized exponential rational function method, which is one of the strong methods for solving nonlinear evolution equations, is applied to the conformable resonant nonlinear Schrödinger's equation in this study. This equation plays a significant role in nonlinear fiber optics. It also has many important applications in photonic crystal fibers. The procedure implemented in this paper can be recommended in solving other equations in the field. All calculations and graphing are performed using powerful symbolic computational packages in Mathematica software. All calculations and graphing are performed using powerful symbolic computational packages in Mathematica software. [ABSTRACT FROM AUTHOR]

In this study, the strain wave equation in micro-structured solids which take an important place in solid physics is presented for consideration. The generalized exponential rational function method is used for this purpose which is one of the most powerful methods of constructing abundantly distinct, exact solutions of nonlinear partial differential equations. In micro-structured solids, wave propagation is based on the structure of the strain wave equation. As a consequence, we successfully received many different exact solutions, including non-topological solutions, periodic singular solutions, topological solutions, singular solutions, like periodic lump solutions. Furthermore, in order to better understand their physical interpretation, 2D, 3D, and counter plots are graphed for each of the solutions acquired. [ABSTRACT FROM AUTHOR]

In this study, the Lonngren-wave equation, which is physically semiconductor, is taken into consideration. Traveling wave solutions of this equation are presented with generalized exponential rational function method, which is one of the mathematically powerful analytical methods. These solutions are produced in bright (non-topological) soliton and complex trigonometric-type traveling wave solutions. Three-dimensional (3D), 2D and contour graphics are presented with the help of a ready-made package program with special values given to constants in these solutions. The effect of the change in wave velocity on the traveling wave solution showing energy transport is presented with the help of simulation. It is argued that velocity is one of the important factors in wave diffraction. In the results and discussion section, the advantages and disadvantages of the method are discussed. [ABSTRACT FROM AUTHOR]

A user-friendly approach depending on nonlocal kernel has been constituted in this study to model nonlocal behaviors of fractional differential and difference equations, which is known as a generalized proportional fractional operator in the Hilfer sense. It is deemed, for differentiable functions, by a fractional integral operator applied to the derivative of a function having an exponential function in the kernel. This operator generalizes a novel version of Čebyšev-type inequality in two and three variables sense and furthers the result of existing literature as a particular case of the Čebyšev inequality is discussed. Some novel special cases are also apprehended and compared with existing results. The outcome obtained by this study is very broad in nature and fits in terms of yielding an enormous number of relating results simply by practicing the proportionality indices included therein. Furthermore, the outcome of our study demonstrates that the proposed plans are of significant importance and computationally appealing to deal with comparable sorts of differential equations. Taken together, the results can serve as efficient and robust means for the purpose of investigating specific classes of integrodifferential equations. [ABSTRACT FROM AUTHOR]

Uncertainty is an unavoidable components of decision making process. The ranking of fuzzy numbers which deals with such uncertainties play again a significant role in the process. Fuzzy numbers must be ranked in order to take the appropriate action by a decision maker in any real life situation. A few numbers of ranking techniques have been encountered in last few decades. However, existing techniques are situation-dependent which have drawbacks/shortcomings. In this regard, this paper presents a sophisticated ranking method based on the concept of the exponential area of the fuzzy numbers. The outputs obtained from this approach are obtained to be more efficient in comparison to the other ranking methods and outperform in all situations. The novelty and validity have been established through comparison with existing works. Furthermore, the ranking approach has been applied in medical decision making problem and the results obtained by the approach absolutely conform with analytical results and human intuitions as well. [ABSTRACT FROM AUTHOR]

This paper presents analytical solutions for bending and buckling of nonlocal functionally graded (FG) Euler–Bernoulli (EB) nanobeams. Material gradation along the thickness direction could be defined by a power function (P-FG), a sigmoidal function (S-FG), and an exponential function (E-FG). Laplace transform is applied to the differential form of the equation of motion of the nonlocal elasticity theory. Closed-form expressions for bending deflection and critical buckling load of FG nanobeams are derived. Effects of material gradations as well as the nonlocal parameter are examined. It is found that bending displacements and critical buckling loads could be controlled by an appropriate choice of material distribution parameter for P-FG nanobeams. The presented results also demonstrate the influences of factors such as the choice of material gradation, power-law index, and nonlocal parameter on bending and buckling behavior. [ABSTRACT FROM AUTHOR]

In this paper, we use the improved modified extended tanh-function method to obtain exact solutions for the nonlinear longitudinal wave equation in magneto-electro-elastic circular rod. With the aid of this method, we get many exact solutions like bright and singular solitons, rational, singular periodic, hyperbolic, Jacobi elliptic function and exponential solutions. Moreover, the two-dimensional and the three-dimensional graphs of some solutions are plotted for knowing the physical interpretation. [ABSTRACT FROM AUTHOR]

In this paper, the Heisenberg ferromagnetic spin chain equation, which is a model with different magnetic interactions in the classical and semiclassical limits, is investigated using the generalized exponential rational function method. The reduction of the governing equation to a simpler ordinary differential equation by wave transformation is the first step of the procedure. A plurality of the exact solution is obtained by using the relevant method. Physical interpretations of some obtained solutions are also included. Particularly, upon choosing appropriate parameters, various plots are depicted. We achieve also a numerical solution corresponding to the initial value problem by the Elzaki–Adomian decomposition method and give comparative results in a table. Moreover, by using Maple, some graphical simulations are done to see the behavior of these solutions with choosing the suitable parameters. [ABSTRACT FROM AUTHOR]

In this study, analytical solutions and physical interpretations are presented for the Riemann wave equation (RWE), which has an important physical property in fluid dynamics. The solutions of the RWE, which models the formation, interaction and breaking of the waves that occur as a result of any external effect on the ocean surface, are examined using the generalized exponential rational function method (GERFM). Bright (nontopological) soliton, singular soliton and solitary wave solutions are produced with advantages of GERFM over other traditional exponential methods. The factors in which solitary wave solutions cause breaking of wave are examined. The effects of parameters on wavefunctions and the physical interpretations of these effects are discussed and supported by graphics and simulations. [ABSTRACT FROM AUTHOR]

In this paper, at first, we introduce fractional differential equations with Z -valuation. Then, we propose a numerical method to approximate the solution. The proposed method is a hybrid method based on the corrected fractional Euler's method and the probability distribution function. Moreover, the corrected fractional Euler's method based on the generalized Taylor formula and the modified trapezoidal rule is proposed that this method can be used in the problems' limitation section of the Z -fractional Initial value problem of order α ∈ (0 , 1) with the fuzzy Caputo fractional differential (fractional derivatives are defined on the basis of the Hukuhara differences and the generalized fuzzy derivatives). The probability function is based on exponential distribution function and used to represent the reliability of the problem limitation part. Finally, by two examples, we show that the proposed method can arbitrarily approximate the fractional differential equations with Z -valuation. [ABSTRACT FROM AUTHOR]

Based on the distributions space on 𝒮 ℂ ′ (denoted by ℱ 𝜃 ∗ ( 𝒮 ′ ℂ)) which is the topological dual space of the space of entire functions with exponential growth of order 𝜃 and of minimal type, we introduce a new type of differential equations using the Wick derivation operator and the Wick product of elements in ℱ 𝜃 ∗ ( 𝒮 ′ ℂ). These equations are called generalized Bernoulli Wick differential equations which are the analogue of the classical Bernoulli differential equations. We solve these generalized Wick differential equations. The present method is exemplified by several examples. [ABSTRACT FROM AUTHOR]

This work addresses the four main inducements: Lump, rogue wave, Homoclinic breather and multi-wave solutions for (2+1)-Modified Veronese Web (MVW) equation via Hirota bilinear approach and the ansatz technique. This model is a linearly degenerate integrable nonlinear partial differential equation (NLPDE) and can also be used to admit a differential covering with nonremoval physical parameters. By assuming the function f in the Hirota bilinear form of the presented model as the general quadratic function, trigonometric function and exponential function form, also with appropriate set of parameters, we have prevented the lump, rogue wave, breather and multi-wave solutions successfully. A precise compatible wave transformation is utilized to obtain multi-wave solutions of governing model. Also, the motion track of the lump, Rogue wave and multi-waves is also explained both physically and theoretically. These new results contain some special arbitrary constants that can be useful to spell out diversity in qualitative features of wave phenomena. [ABSTRACT FROM AUTHOR]

This paper discusses preventive replacement policies for an independent damage process, in which successive shocks occur at random times, and the independent damages caused by shocks are random variables. It is assumed that a unit fails when the damage has exceeded a prespecified level K. We consider three models: Model 1: the unit is replaced at the M th shock for damage Z , at damage K , or at time T , whichever occurs first. Model 2: the unit is replaced at the M th shock for damage Z , at damage K or at shock N , whichever occurs first. Model 3: the unit is replaced at the M th shock for damage Z , at damage K , at shock N or at time T , whichever occurs first. We obtain the expected cost rates for each model and discuss analytical optimal T ∗ and N ∗ to minimize their expected cost rates. Numerical examples are given when the damage has an exponential distribution and shocks occur at a gamma distribution. [ABSTRACT FROM AUTHOR]

The prime objective of this paper is to obtain the exact soliton solutions by applying the two mathematical techniques, namely, Lie symmetry analysis and generalized exponential rational function (GERF) method to the (2+1)-dimensional generalized Camassa–Holm–Kadomtsev–Petviashvili (g-CHKP) equation. First, we obtain Lie infinitesimals, possible vector fields, and commutative product of vectors for the g-CHKP equation. By the means of symmetry reductions, the g-CHKP equation reduced to various nonlinear ODEs. Subsequently, we implement the GERF method to the reduced ODEs with the help of computerized symbolic computation in Mathematica. Some abundant exact soliton solutions are obtained in the shapes of different dynamical structures of multiple-solitons like one-soliton, two-soliton, three-soliton, four-soliton, bell-shaped solitons, lump-type soliton, kink-type soliton, periodic solitary wave solutions, trigonometric function, hyperbolic trigonometric function, exponential function, and rational function solutions. Consequently, the dynamical structures of attained exact analytical solutions are discussed through 3D-plots via numerical simulation. A comparison with other results is also presented. [ABSTRACT FROM AUTHOR]

In this paper, we study the spectral property of the self-affine measure μ R , 𝒟 generated by an expanding real matrix R = diag (b , b) and the four-element digit set 𝒟 = { 0 0 , 1 0 , 0 1 , − 1 − 1 }. We show that μ R , 𝒟 is a spectral measure, i.e. there exists a discrete set Λ ⊆ ℝ 2 such that the collection of exponential functions { e − 2 π i 〈 λ , x 〉 : λ ∈ Λ } forms an orthonormal basis for L 2 (μ) , if and only if b = 2 k for some k ∈ ℕ. A similar characterization for Bernoulli convolution is provided by Dai [X.-R. Dai, When does a Bernoulli convolution admit a spectrum? Adv. Math.231(3) (2012) 1681–1693], over which b = 2 k. Furthermore, we provide an equivalent characterization for the maximal bi-zero set of μ R , 𝒟 by extending the concept of tree-mapping in [X.-R. Dai, X.-G. He and C. K. Lai, Spectral property of Cantor measures with consecutive digits, Adv. Math.242 (2013) 187–208]. We also extend these results to the more general self-affine measures. [ABSTRACT FROM AUTHOR]

We study a competition model of two competing species in population biology having exponential and rational growth functions described by Alexander et al. [1992]. They observed that, for some choice of parameters, the competition model has a chaotic attractor A for which the basin of attraction is riddled. Here, we give a detailed analysis to illustrate what happens when the normal parameter in this model changes. In fact, by varying the normal parameter, we discuss how the geometry of the basin of attraction of A , the region of coexistence or extinction, changes and illustrate the transitions between the set A being an asymptotically stable attractor (extinction of rational species), a locally riddled basin attractor and a normally repelling chaotic saddle (extinction of exponential species). Additionally, we show that the riddling and the blowout bifurcation occur. Numerical simulations are presented graphically to confirm the validity of our results. In particular, we verify the occurrence of synchronization for some values of parameters. Finally, we apply the uncertainty exponent and the stability index to quantify the degree of riddling basin. Our observation indicates that the stability index is positive for Lebesgue for almost all points whenever the riddling occurs. [ABSTRACT FROM AUTHOR]