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In this paper, we apply the powerful sine-Gordon expansion method (SGEM), along with a computational program, to construct some new traveling wave soliton solutions for two models, including the higher-order nonlinear Boussinesq dynamical wave equation, which is a well-known nonlinear evolution model in mathematical physics, and the (1+1)-dimensional framework of the Van der Waals gas system. This study presents some new complex traveling wave solutions, as well as logarithmic and complex function properties. The 3D and 2D graphical representations of all obtained solutions, unveiling new properties of the considered model are simulated. Additionally, several simulations, including contour surfaces of the results, are performed, and we discuss their physical implications. A comprehensive conclusion is provided at the end of this paper.

This mathematical model is based on a system of two non-linear delay differential equations representing GDP (Gross Domestic Product) growth and national debt respectively. Growth of GDP is directly proportional to national debt. In the absence of national debt, there could be no or very slow GDP growth. The national debt is heavily dependent on external loans and external investments. These investments act as a catalyst for accelerating the rate of GDP. It is assumed that the external debt is never paid off fully. The availability of external investments is not immediate as per demand but takes some time for the maturation of the deal. The time delay in actual arrival of the foreign investment and its effect on GDP-National debt dynamics is the main focus of this study. This effect is studied using a delay parameter τ. The non-zero equilibrium of the system is calculated, and the stability analysis is performed on it. Hopf bifurcation is observed for a critical value of the delay parameter. Numerical simulation is performed using MATLAB code.

Zhou (2023) introduced the concept of path-factor critical avoidable graph and determined several parameter bounds for (P≥2,n) or (P≥3,n)-factor critical avoidable graphs. Certain instances are listed in Zhou (2023) to show that these toughness or isolated toughness bounds are sharp. However, the derivation processes are flawed. In this remark, we directly modify the derivation processes and obtain the revised bounds. The counterexamples are updated to satisfy κ(G)=n+2. Finally, richness bounds on binding number for a graph to be path-factor critical avoidable are determined.

The motivation of this work is to analyse the nonlinear models and their complex nature with generalized tools associated with material and history-based properties. With the help of well-known and widely used numerical scheme, we study the stimulating behaviours of the financial system in this work. The impact of parameters on price index, rate of interest, investment demand, influence changes and investment cost with respect to saving amount, and the elasticity of commercial markets demand are discussed. The consequences of generalizing the model within the arbitrary order are derived. The existence of the solution for the considered system is presented. This study helps beginner researchers to investigate complex real-world problems and predict the corresponding consequences.

One of the major causes of the intermittent nature of long-term climate changes is the interaction between the surface energy balance and the mass balance of the Cryosphere. In this paper, the pre-existing surface energy balance-mass balance model of the Cryosphere is modified by incorporating the radiative forcing of CO2 to observe the effect of global warming on the nature of the previous model. The modified model is generalized using Caputo’s non-local operator. The stability of the new model is analyzed at all the equilibrium points and also it is shown that there exists a unique and bounded solution for the modified system. Using bifurcation analysis and calculating the values of Lyapunov exponents for different fractional orders of the modified system, it is found that the system exhibits chaos for certain values of the radiating forcing of CO2 To observe and visualize the changes in the nature of the new model, the system is solved using the highly efficient 7th-order Runge–Kutta method. It is observed that with the inclusion of the radiative forcing of CO2, the nature of the system changes from asymptotically stable to chaotic as we decrease the order of the system. The Poincare map shows that the modified model even has characteristics of a strange attractor which is highly chaotic. The system becomes unstable when the value of the radiative forcing of CO2 is increased. As a result, the predictive power of the surface energy balance-mass balance model of the Cryosphere decreases. In addition to providing insights into the transition from stability to chaos in the Cryosphere model due to CO2 radiative forcing, this research offers a deeper understanding of the intricate interplay between climate dynamics and complex systems behavior.

The purpose of this investigation is to provide the numerical study based on the fractional order (FO) economic and environmental mathematical model (EEMM) called as FO-EEMM. The motive of this work is to find more realistic results of the EEMM of the non-integer and FO derivatives. The structure of the FO-EEMM is categorized into three dynamics, control accomplishment cost, capability of the manufacturing elements and the diagnostics cost of the technical exclusion. The solution of the FO-EEMM is numerically presented by using the scaled conjugate gradient neural networks (SCGNNs). Three cases based on the FO-EEMM have been scrutinized to indicate the numerical performances along with the selection of the statics as 76% for training, 13% for testing and 11% for certification. The correctness of the designed SCGNNs is authenticated by using the matching of the achieved and the reference solutions (Adams-Bashforth-Moulton). The validity, exactness, dependability, and competence of the SCGNNs is observed through the performances of the mean square error, regression, state transitions, error histograms and correlation.

Abstract This work explores the magneto-hydrodynamics (MHD) Jeffery–Hamel nanofluid flow between two rigid non-parallel plane walls with heat transfer by employing hybrid nanoparticles, especially Cu and Cu-Al $$_{2}$$ 2 O $$_{3}$$ 3 . Here the MHD nanofluid flow problem is extended with fuzzy volume fraction and heat transfer with diverse nanoparticles to cover the influence of thermal profiles with hybrid nanoparticles on the fuzzy velocity profiles. The nanoparticle volume fraction is described with a triangular fuzzy number ranging from 0 to $$5\%$$ 5 % . A novel double parametric form-based homotopy analysis approach is considered to study the fuzzy velocity and temperature profiles with hybrid nanoparticles in both convergent and divergent channel positions. Finally, the efficiency of the proposed method has been demonstrated by comparing it with the available results in a crisp environment for validation.

In this projected paper, we study on the Ivancevic option pricing model. We apply two important methods, namely, rational sine-Gordon expansion method which is recently developed, and secondly, modified exponential method. Via these methods, we obtain some important properties of Ivancevic option pricing model. We extract many solutions such as complex, periodic, dark bright, mixed dark-bright, singular, travelling and hyperbolic functions. We investigate the option price wave functions of dependent variable, and also, observe the modulation instability analysis in detail. Furthermore, we report the strain conditions for the valid solutions under the family conditions, as well. We simulate the 2D, 3D and counter plots by choosing the suitable values of the parameters involved. Finally, we present the top and low points of pricing in the mentioned intervals via contour simulations.

In this paper, to investigate the dynamic interplay between supply and demand, with a focus on collectability, a novel mathematical model was introduced via conformable operator. This model considers the possibility that operating expenses or a lack of raw materials causes a manufacturing delay than the supply of goods instantly matching demand. This maturation (delay) is represented by the delay factor $ \left(\tau \right) $. Stability analysis revolves around the equilibrium point other than zero. Chaotic behavior emerges through Hopf bifurcation at the critical delay parameter value. If this delay parameter is even slightly perturbed, oscillatory limit cycles can be induced in the market dynamics, leading to equilibrium with brisk market expansion, frequent recessions, and sudden collapses. We conducted sensitivity and directional analysis on a number of factors while also examining the stability and duration of the Hopf bifurcation. Numerical findings were validated using MATLAB. Additionally, the Caputo operator was used to examine the fractional of demand and supply dynamics. Importantly, we assumed a pivotal role in advancing fair labor practices and fostering economic growth on a national scale.

The essence of mathematical tools used to exemplify the practical problems that exist in daily life is as old as the world itself [...]

The current study provides the solutions of the mathematical model based on the coronavirus including the effects of vaccination and quarantine. The numerical stochastic process relying on Levenberg-Marquardt backpropagation technique (L-MB) neural networks (NN), i.e., L-MBNNs, is presented to solve the model. The entire dynamics of the proposed model depends upon the human population, which is represented by N and is further divided into multiple subgroups. The detail of these subgroups is presented in the form of susceptible population (S), exposed population (E), and infected people (I). Likewise, Q represents the quarantined and R shows the recovered or deceased individuals. Those who have been immunized are symbolized by V. All these categories make the model SEIQRV, that is based on a system of nonlinear differential equations. The statistics that is used to provide the numerical solutions of the SEIQRV model is 76% for training, 10% for testing and 14% for authorization. The correctness of the L-MBNNs is tested by using the comparison of the proposed and reference solutions (Adam method). The statistical representations are provided in order to check the reliability, competence and validity of L-MBNNs using the procedures of error histograms (EH), state transitions (ST), regression and correlation.

The susceptible-infected-recovered (SIR) epidemic model of childhood disease is analyzed in the present framework with the help of q-homotopy analysis transform method (q-HATM). The considered model consists the system of three differential equations having fractional derivative, and the non-linear system exemplifies the evolution of childhood disease in a population and its influence on the community with susceptible, infected and recovered compartment. The projected method is a mixture of q-homotopy analysis method and Laplace transform. Two distinct explanatory cases are considered, and corresponding simulations have been demonstrated in terms of plots for different value of the order. The present investigation elucidates that the projected both derivative and technique play a vital role in the analysis and illustrate the behaviour of diverse mathematical models described with differential equations in human disease.

In this manuscript, we focus on the application of recently developed analytical scheme, namely, the rational sine-Gordon expansion method (SGEM). Some new exact solutions of Riemann wave system and the nonlinear Gross−Pitaevskii equation (GPE) by using this method are extracted. This method is based on the general properties of the SGEM which uses the fundamental properties of trigonometric functions. Many novel analytical solutions such as dark, bright, mixed dark–bright, hyperbolic, and periodic wave solutions are successfully extracted. Physical meanings of solutions are simulated by the various figures in 2D and 3D along with the contour graphs. Strain conditions of the existence are also reported in detail. Finally, modulation instability analysis of the nonlinear GPE is studied in detail.

The existence of factor and fractional factor in network graph in various settings has raised much attention from both mathematicians and computer scientists. It implies the availability of data transmission and network segmentation in certain special settings. In our paper, we consider P≥2-factor and P≥3-factor which are two special cases of general H-factor. Specifically, we study the existence of these two kinds of path factor when some subgraphs are forbidden, and several conclusions on the factor-deleted graph, factor critical-covered graph, and factor uniform graph are given with regards to network parameters. Furthermore, we show that these bounds are best in some sense.

In this paper, we find new travelling wave solutions to the nonlinear electrical transmission line model. Exponential function method is applied. This model is used to the voltage behaviors in electrical transmission lines. Under a suitable choice of control parameters, we plot the figures along with contour surfaces of new solutions by using computational programs such as Mathematica, Maple and Matlab. Some information about the behavior of voltage on the electrical lines is extracted. The hyperbolic, periodic or singular properties are reported.

In this paper, the modified auxiliary expansion method is used to construct some new soliton solutions of coupled Schrödinger-Boussinesq system that includes beta derivative. The new exact solution is obtained have a hyperbolic function, trigonometric function, exponential function, and rational function. These solutions might appreciate in laser and plasma sciences. It is shown that this method, provides a straightforward and powerful mathematical tool for solving the nonlinear problems. Moreover, the linear stability of this nonlinear system is analyzed.

The present study is concerned with studying the dynamical behavior of two space-dimensional nonlinear time-fractional models governing the unsteady-flow of polytropic-gas (in brief, pGas) that occurred in cosmology and astronomy. For this purpose, two efficient hybrid methods so-called optimal homotopy analysis J-transform method (OHAJTM) and J-variational iteration transform method (J-VITM) have been adopted. The OHAJTM is the hybrid method, where optimal-homotopy analysis method (OHAM) is utilized after implementing the properties of J-transform (JT), and in J-VITM is the J-transform-based variational iteration method. Banach’s fixed point approach is adopted to analyze the convergence of these methods. It is demonstrated that J-VITM is T-stable, and the evaluated dynamics of pGas are described in terms of Mittag–Leffler functions. The proposed evaluation confirms that the implemented methods perform better for the referred model equation of pGas. In addition, for a given iteration, the proposed behavior via OHAJTM performs better in producing more accurate behavior in comparison to J-VITM and the methods introduced recently.

In this paper, complex and combined dark-bright characteristic properties of nonlinear Date-Jimbo-Kashiwara-Miwa equation with conformable are extracted by using two powerful analytical approaches. Many graphical representations such as 2D, 3D and contour are also reported. Finally, general conclusions of about the novel findings are introduced at the end of this manuscript.

This paper constructs approximate solutions of the epidemic system of HIV/AIDS transmission model with the help of the conformable fractional differential transform method. The fractional derivatives considered in this paper are described in the sense of the conformable operator. This method provides the approximate solutions speedily in the form of a convergent series. The results show the efficiency of the proposed method.

In this paper, we present the sine-Gordon expansion method to prepare the mixed dark bright wave patterns to the nonlinear partial differential equations arising in mathematical physics. Then, we apply the proposed method for a credible recourse of two nonlinear physical models: the modified Vakhnenko-Parkes and modified α-equation. These exact solutions comprise the hyperbolic, trigonometric, rational and exponential function with few licentious parameter. The analytical solutions have different physical structures and they are graphically analyzed in order to show their dynamical behavior by means of 2D, 3D and contour plots.

Abstract According to the report presented by the World Health Organization, a new member of viruses, namely, coronavirus, shortly 2019-nCoV, which arised in Wuhan, China, on January 7, 2020, has been introduced to the literature. The main aim of this paper is investigating and finding the optimal values for better understanding the mathematical model of the transfer of 2019-nCoV from the reservoir to people. This model, named Bats-Hosts-Reservoir-People coronavirus (BHRPC) model, is based on bats as essential animal beings. By using a powerful numerical method we obtain simulations of its spreading under suitably chosen parameters. Whereas the obtained results show the effectiveness of the theoretical method considered for the governing system, the results also present much light on the dynamic behavior of the Bats-Hosts-Reservoir-People transmission network coronavirus model.

In this paper, we employ the sine-Gordon expansion method to shallow water wave models which are Kadomtsev-Petviashvili-Benjamin-Bona-Mahony and the Benney-Luke equations. We construct many new complex combined dark-bright soliton, anti-kink soliton solutions for the governing models. The 2D, 3D and contour plots are given under the suitable coefficients. The obtained results show that the approach proposed for these completely integrable equations can be used effectively.

This study reveals new voltage behaviors to the electrical transmission line equation in a network system by using the newly presented sine-Gordon equation function method. It is commented about these behaviors which come from different simulations of results obtained in this paper. Many illustrations are offered to validate our analytical results. Linear stability analysis is also investigated in a detailed manner.

In this article, the solution of vibration equation of fractional order is found numerically for the large membranes using a powerful technique namely q-homotopy analysis Sumudu transform technique. The parameter ħ suggests a convenient way to control convergence region. The given numerical examples depict competency and accuracy of this scheme. The results are discussed using figures taking diverse wave velocities and initial conditions. Results are also compared with other methods. The outcome divulges that q-HASTM is highly reliable, more efficient, attractive, easier to use as well as highly effective.

In this paper, we study on the conformable (2+1)-dimensional Ablowitz-KaupNewell-Segur equation in order to show the existence of complex combined dark-bright soliton solutions. To this purpose an effective method which is the sine-Gordon expansion method is used. The 2D and 3D surfaces under some suitable values of parameters are also plotted.

This article investigates the new results of three nonlinear conformable models (NLCMs). To study such varieties of new soliton structures, we perform the generalized Kudryashov (GK) method. The obtained new results are defined in the styles of the exponential and rational functions. The derived new soliton structures are stable, serviceable, and fitting to embrace the conformable dynamics, chaotic vibrations, global bifurcations, optimal control problems, fluid mechanics, plasma physics, system identification, local bifurcations, control theory and resonances, and so many. The outcomes declare that the process is hugely valuable and accessible for investigating nonlinear conformable order models treated in theoretical physics.

The pivotal aim of the present work is to find the solution for fractional Drinfeld–Sokolov–Wilson equation using q-homotopy analysis transform method (q-HATM). The proposed technique is graceful amalgamations of Laplace transform technique with q-homotopy analysis scheme, and fractional derivative defined with Atangana-Baleanu (AB) operator. The fixed point hypothesis considered in order to demonstrate the existence and uniqueness of the obtained solution for the proposed fractional order model. In order to validate and illustrate the efficiency of the future technique, we analysed the projected model in terms of fractional order. Meanwhile, the physical behaviour of the q-HATM solutions have been captured in terms of plots for diverse fractional order and the numerical simulation is also demonstrated. The achieved results illuminate that, the future algorithm is easy to implement, highly methodical as well as effective and very accurate to analyse the behaviour of coupled nonlinear differential equations of fractional order arisen in the connected areas of science and engineering. Keywords: Laplace transform, Atangana-Baleanu derivative, Drinfeld–Sokolov–Wilson equation, Homotopy analysis method, Fixed point theorem

This work investigates the nonlinear Schrödinger equation (NLSE) with group velocity dispersion and second order spatiotemporal dispersion coefficients. The governing model is reduced into the classical nonlinear ordinary differential equation. Extended direct algebraic method (EDAM) is implemented to construct many novel mixed dark, and complex optical solutions. As a result, some important analytical solutions such as travelling mixed dark, and complex travelling wave solutions for the model are extracted.

Modern science is one of the most-used commodities globally, and it is especially important in determining the sources of various threats faced by the world [...]

The aim of this study is to provide the numerical outcomes of a nonlinear HIV infection system of latently infected CD4+ T cells exists in bioinformatics using Morlet wavelet (MW) artificial neural networks (ANNs) optimized initially with global search of genetic algorithms (GAs) hybridized for speedy local search of sequential quadratic programming (SQP), i.e., MW-ANN-GA-SQP. The design of an error function is presented by designing the MW-ANN models for the differential equations along with the initial conditions that represent the HIV infection system involving latently infected CD4+ T cells. The precision and persistence of the presented approach MW-ANN-GA-SQP are recognized through comparative studies from the results of the Runge-Kutta numerical scheme for solving the HIV infection spread system in case of single and multiple trails of the MW-ANN-GA-SQP. Statistical estimates with ‘Theil’s inequality coefficient’ and ‘root mean square error’ based indices further validate the sustainability and applicability of proposed MW-ANN-GA-SQP solver.

This projected work applies the generalized exponential rational function method to extract the complex, trigonometric, hyperbolic, dark bright soliton solutions of the cubic nonlinear Schrödinger’s equation. Moreover, trigonometric, complex, strain conditions and dark-bright soliton wave distributions are also reported. Furthermore, the modulation instability analysis is also studied in detail. To better understand the dynamic behavior of some of the obtained solutions, several numerical simulations are presented in the paper. According to the obtained results, it is clear that the method has less limitations than other methods in determining the exact solutions of the equations. Despite the simplicity and ease of use of this method, it has a very powerful performance and is able to introduce a wide range of different types of solutions to such equations. The idea used in this paper is readily applicable to solving other partial differential equations in mathematical physics.

This manuscript aims to construct a family of travelling wave solutions to the high nonlinearity differential equation for obtaining new hyperbolic function solutions by using an analytical method, which is based on the exponential function. A number of new hyperbolic solutions for the model have been newly derived by using the method. We take advantage of some computer programming for all numerical calculations, and surfaces of solutions obtained in this paper. Keywords: Benjamin-Bona-Mahony-Peregrine-Burgers equation, The modified exponential function method, Hyperbolic function solution

In this paper, we study the extraction of some analytical solutions to the nonlinear perturbed sine-Gordon equation with the long Josephson junction properties. The model studied was formed to observe the long Josephson junction properties separated by two superconductors. Moreover, it is also used to explain the Josephson effect arising in the highly nonlinear nature of the Josephson junctions. This provides the shunt inductances to realize a Josephson left-handed transmission line. A powerful scheme is used to extract the complex function solutions. These complex results are used to explain deeper properties of Josephson effects in the frame of impedance. Various simulations of solutions obtained in this paper are also reported.

This paper anatomizes the exact solutions of the resonant non-linear Schrödinger’s equation (R-NLSE) with the Kerr law non-linearity with the assistance of the new extended direct algebraic technique. The secured soliton erections are newfangled and unreservedly invigorating for investigators. The graphically comprehensive report of some specific solutions is embellished with the well-judged values of parameters to illustrate their propagation. Then a planer dynamical system is introduced and the bifurcation analysis has been executed to figure out the bifurcation structures of the non-linear and super non-linear traveling wave solutions of the heeded model. All possible phase portraits are exhibited with specific values of parameters. Furthermore, a precise class of non-trivial and first-order conserved quantities is enumerated by the intervention of the multiplier approach.

In this manuscript, new dark and trigonometric function traveling wave soliton solutions to the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation by using the modified exponential function method are successfully obtained. Along with novel dark structures, trigonometric solutions are also extracted. For deeper investigating of waves propagation on the surface, 2D and 3D graphs along with contour simulations via computational programs such as Wolfram Mathematica, Matlap softwares and so on are presented.

This research paper focuses on the application of the tanh function method to find the soliton solutions of the (2+1)-dimensional nonlinear electrical transmission line model. Materials used to form a transmitting line are very important to transmit electric charge. In this sense, we find some new voltage behaviors such as dark, trigonometric, and complex function solutions. Choosing some suitable values of parameters, we present some various surfaces of results obtained in this paper. These results play an important role in telecommunications lines used to stand for wave propagations.

The security of a network is closely related to the structure of the network graph. The denser the network graph structure is, the better it can resist attacks. Toughness and isolated toughness are used to characterize the vulnerable programs of the network which have been paid attention from mathematics and computer scholars. On this basis, considering the particularity of the sun component structures, sun toughness was introduced in mathematics and applied to computer networks. From the perspective of modern graph theory, this paper presents the sun toughness conditions of the path factor uniform graph and the path factor critical avoidable graph in P≥2-factor and P≥3-factor settings. Furthermore, examples show that the given boundaries are sharp.

Ontology is one of the oldest terminologies in physics and is used to describe the origin and most essential attributes of all things in the world. With the development of contemporary science, ontology was given a specific definition and then introduced into the computer science as a conceptual model to describe the relationship between objects. In the past decade, the algorithms and applications in the ontology-related field have attracted the attention of many scholars. In this work, a support vector machines based multi-dividing ontology learning algorithm is proposed. We pay attention to the similarity of topological indices in chemical graph theory, and apply SVM-based multi-dividing ontology learning algorithms to give some calculation results of similarity between topological indices.

The stomach is usually considered as a hollow muscular sac, which initiates the second segment of digestion. It is the most sophisticated endocrine structure having unique biochemistry, physiology, microbiology, and immunology. The pivotal aim of the present study is to propose the nonlinear mathematical model of the nervous stomach system based on three compartments namely, tension (T), food (F), and medicine (M). The detailed description of each compartment is provided along with the mathematical form and different rates/factors, such as sleep factor, food rate, tension rate, medicine term, and death rate. The solution of the designed model is presented numerically by using the well-known differential transformation technique. The behavior of the obtained solution has been captured with respect to time as well as presentations of the numerical simulations.

In this paper, we analyzed and found the solution for a suitable nonlinear fractional dynamical system that describes coronavirus (2019-nCoV) using a novel computational method. A compartmental model with four compartments, namely, susceptible, infected, reported and unreported, was adopted and modified to a new model incorporating fractional operators. In particular, by using a modified predictor–corrector method, we captured the nature of the obtained solution for different arbitrary orders. We investigated the influence of the fractional operator to present and discuss some interesting properties of the novel coronavirus infection.

In this article, a fractional model of the Capsicum annuum (C. annuum) affected by the yellow virus through whiteflies (Bemisia tabaci) is examined. We analyzed the model by equilibrium points, reproductive number, and local and global stability. The optimal control methods are discussed to decrease the infectious B. tabaci and C. annuum by applying the Verticillium lecanii (V. lecanii) with the Atangana–Baleanu derivative. Numerical results described the population of plants and comparison values of using V. lecanni. The results show that using 60% of V. lecanni will control the spread of the yellow virus in infected B. tabaci and C. annuum in 10 days, which helps farmers to afford the costs of cultivating chili plants.

In this work a powerful method is presented to solve the Klein–Gordon equation (KGE). Firstly, a discretization is implemented on the original equation. Then a geometric technique is applied to obtain the approximate solutions. Some examples containing one-dimensional and two-dimensional are solved to prove the power and the capability of the offered scheme. Indeed, graphs of the exact solution, numerical solution, absolute error and the contour plot of error are provided successfully. Keywords: Method of line, Nonlinear Klein–Gordon equation, Group preserving scheme

This article developed a functional integration matrix via the Hermite wavelets and proposed a novel technique called the Hermite wavelet collocation method (HWM). Here, we studied two models: the coupled system of an ordinary differential equation (ODE) is modeled on the digestive system by considering different parameters such as sleep factor, tension, food rate, death rate, and medicine. Here, we discussed how these parameters influence the digestive system and showed them through figures and tables. Another fractional model is used on the COVID-19 pandemic. This model is defined by a system of fractional-ODEs including five variables, called S (susceptible), E (exposed), I (infected), Q (quarantined), and R (recovered). The proposed wavelet technique investigates these two models. Here, we express the modeled equation in terms of the Hermite wavelets along with the collocation scheme. Then, using the properties of wavelets, we convert the modeled equation into a system of algebraic equations. We use the Newton–Raphson method to solve these nonlinear algebraic equations. The obtained results are compared with numerical solutions and the Runge–Kutta method (R–K method), which is expressed through tables and graphs. The HWM computational time (consumes less time) is better than that of the R–K method.

This paper presents a novel and general analytical approach: the rational sine-Gordon expansion method and its applications to the nonlinear Gardner and (3+1)-dimensional mKdV-ZK equations including a conformable operator. Some trigonometric, periodic, hyperbolic and rational function solutions are extracted. Physical meanings of these solutions are also presented. After choosing suitable values of the parameters in the results, some simulations are plotted. Strain conditions for valid solutions are also reported in detail.

The focus of the present study is to present a stochastic numerical computing framework based on Gudermannian neural networks (GNNs) together with the global and local search genetic algorithm (GA) and active-set approach (ASA), i.e., GNNs-GA-ASA. The designed computing framework GNNs-GA-ASA is tested for the higher order nonlinear singular differential model (HO-NSDM). Three different nonlinear singular variants based on the (HO-NSDM) have been solved by using the GNNs-GA-ASA and numerical solutions have been compared with the exact solutions to check the exactness of the designed scheme. The absolute errors have been performed to check the precision of the designed GNNs-GA-ASA scheme. Moreover, the aptitude of GNNs-GA-ASA is verified on precision, stability and convergence analysis, which are enhanced through efficiency, implication and dependability procedures with statistical data to solve the HO-NSDM.

The main objective of the present study is to analyze the nature and capture the corresponding consequences of the solution obtained for the Gardner–Ostrovsky equation with the help of the q-homotopy analysis transform technique (q-HATT). In the rotating ocean, the considered equations exemplify strong interacting internal waves. The fractional operator employed in the present study is used in order to illustrate its importance in generalizing the models associated with kernel singular. The fixed-point theorem and the Banach space are considered to present the existence and uniqueness within the frame of the Caputo–Fabrizio (CF) fractional operator. Furthermore, for different fractional orders, the nature has been captured in plots. The realized consequences confirm that the considered procedure is reliable and highly methodical for investigating the consequences related to the nonlinear models of both integer and fractional order.

In this paper, we use the modified exponential function method in terms of Kf(x) instead of ef(x)and the extended sinh-Gordon method to find some new family solution of the M-fractional paraxial non-linear Schrödinger equation. The novel complex and real optical soliton solutions are plotted in 2-D, 3-D with a contour plot. Moreover, the dark exact solutions, singular soliton solutions, kink-type soliton solution, and periodic dark-singular soliton solutions for M-fractional paraxial non-linear Schrödinger equation are constructed. We guarantee that all solutions are new and verified the main equation of the M-fractional paraxial wave equation. For existence, the constraint condition is also added.

The analytical solution of the longitudinal wave equation in the MEE circular rod is analyzed by the powerful sine-Gordon expansion method. Sine - Gordon expansion is based on the well-known wave transformation and sine - Gordon equation. In the longitudinal wave equation in mathematical physics, the transverse Poisson MEE circular rod is caused by the dispersion. Some solutions with complex, hyperbolic and trigonometric functions have been successfully implemented. Numerical simulations of all solutions are given by selecting the appropriate parameter values. The physical meaning of the analytical solution explaining some practical physical problems is given.

Mathematical models have been frequently studied in recent decades in order to obtain the deeper properties of real-world problems [...]