In this study, we work on a Holling–Tanner prey-predator (HTPP) model by incorporating the impact of predator cannibalism and later introducing a delay in prey growth. We first examine its behavior without delay: we identify equilibrium points, describe their stability, and explain when the system remains stable or goes into oscillation. Then we add a time lag in the prey growth, which captures the natural lag in biological systems. We establish positivity, and by applying the Routh–Hurwitz criterion, we obtain explicit criteria for local stability and also prove global stability (GS) by applying a Lyapunov approach. Numerical simulations indicate that populations can converge in the ordinary model but oscillate indefinitely as soon as the delay passes a threshold. Aside from analysis, we also plot a detailed graph to show the findings. The results validate that delay significantly destabilizes the system, and cannibalism determines the long-term dynamics.
A numerical collocation approach employing independence polynomials of paths is formulated to approximate the solutions of high-order linear Fredholm–Volterra integro-differential problems under mixed conditions. The proposed approach transforms the given equation and conditions into a matrix form, leading to a linear system whose unknowns are the independence coefficients. To demonstrate the method’s validity and efficiency, several numerical examples are presented, with comparisons to existing results made in selected cases. The computations were performed in MATLAB.
The propagation of ultrashort optical pulses in nonlinear dissipative media is investigated through a generalized higher-order cubic–quintic complex Ginzburg–Landau equation (HG CQ-CGLE) incorporating third-order dispersion (TOD), fourth-order dispersion (FOD), self-steepening (SS), and intrapulse Raman scattering (IRS) effects. A consistent traveling-wave reduction is established, yielding a fourth-order nonlinear ordinary differential equation (NODE) subject to explicit compatibility constraints. Three complementary analytical approaches, the unified Riccati equation expansion method (UREEM), the F-expansion method, and a modified trial equation technique, are employed to construct several classes of exact nonlinear wave structures, including bright dissipative solitons, kink waves, singular solutions, rational states, and elliptic periodic patterns. The resulting solution families demonstrate how higher-order dispersive and dissipative mechanisms control wave localization, asymmetry, and nonlinear stabilization. To assess the physical relevance of the obtained structures, a modulational instability analysis of the continuous-wave (CW) background is performed. The results show that FOD can induce instability in normal-dispersion regimes, quintic nonlinearity regulates gain saturation, and the interplay between SS and IRS suppresses instability growth. Furthermore, spectral filtering significantly modifies the instability bandwidth and gain spectrum. A direct connection between modulation instability (MI) and nonlinear structure formation is established, providing a physical interpretation of the emergence and persistence of the analytical waveforms. The present study therefore offers a unified analytical, physical, and stability-oriented framework for understanding dissipative soliton formation, nonlinear wave transitions, and ultrashort pulse dynamics in generalized higher-order optical systems.
In this work, a novel numerical algorithm is presented to solve singularly perturbed parabolic problems with discontinuous convection coefficients and source terms. The problem is approximated using the Crank-Nicolson method in the temporal direction on a uniform mesh and the novel finite difference method in spatial direction. The parameter uniform convergence of the proposed scheme is investigated. The method is shown to be second-order accurate in both the temporal and spatial directions. It is also shown that the proposed method is better than some existing methods in the literature. The numerical experiments confirm the theoretical findings.MSC2020 Classification: 35B25, 65M06, 65N12
The Korteweg-de Vries (KdV) equation plays an important role in describing the propagation of water waves in shallow channels. Several analytical methods, such as tanh-function methods (TFMs), exponential function method, and Jacobi elliptic function method, provide the exact solitons to the KdV equation. Contrarily, the G ' G-expansion method, whose strength depends on the embedded Ricati equation for obtaining solitons of the KdV equation. There are a lot of Riccati equations that can be embedded in the G ' G-expansion method; however, these Riccati equations; GG '' + lambda G2 = 0 and G ' - mu G2 = 0, have not been embedded in the G ' G-expansion method to obtain the multiple solitons of the KdV equations. To bridge this gap in the literature, two different modified G ' G-expansion methods, each embedded with a Riccati equation, are therein this paper. The modified G ' G-expansion method endowed with a second-order Riccati equation GG '' + lambda G2 = 0, produced 15 solitons of the KdV equation comprising 10 singular dark solitons and 5 singular multidark solitons, each with different amplitude, kinetic energy, and potential energy. On the other hand, the new G ' G-expansion method embedded with a first-order Riccati equation G ' - mu G2 = 0 yielded only four solitons of the KdV equation. These solitons are mainly two singular bright solitons and two singular dark solitons. The fifteen solitons obtained indicate that embedding the second-order Riccati equation in the G ' G-expansion method is more effective than using the first-order Riccati equation. Surprisingly, all the solitons of the modified G ' G-expansion method embedded with a second-order Riccati equation GG '' + lambda G2 = 0 do not yield any bright soliton. On a new development, the shape and colour of each soliton of the KdV equation indicate its stability as the solitary wave propagates through a medium. This shape changes depending on the position and time it takes to propagate, as observed in the KdV equation using each of the introduced modified G ' G-expansion methods. The results demonstrate the complex dynamical behavior of the KdV equation and emphasize its wide-ranging applications in fluid dynamics, plasma physics, and optical systems.
This study investigates the dynamics of two coupled solitary waves propagating in media characterised by spatially modulated non-linearity and variable dispersion. By employing numerical simulations of a system of coupled non-linear Schr & ouml;dinger equations (NLSEs) with varying coefficients, we analyse how inhomogeneous physical properties influence soliton stability and interaction. Our results demonstrate that spatial modulation of non-linearity can control the velocity and amplitude of wave packets, providing a mechanism for soliton trapping or acceleration. Furthermore, we examine the role of the dispersion-to-non-linearity ratio in maintaining the structural integrity of the coupled waves over long distances. These findings provide insights into optical signal processing in non-uniform fibres and matter-wave dynamics in Bose-Einstein condensates with varying trap potentials.
We obtain new computational soliton solutions characterized by topological, rational, exponential, trigonometric, and hyperbolic functions for the Fisher equation. Using a good strategy, the Kudryashov expansion method is used to find different dynamical wave structures of soliton solutions within the scope of evolutionary dynamical structures of solitary wave solutions. To facilitate understanding of the physical phenomena related to these dynamical models in mathematical physics, the physical behavior of these solutions is empirically demonstrated. In this regard, the current study offers a cohesive analytical examination of diverse soliton structures within a singular framework, enhancing the theoretical comprehension of soliton dynamics in nonlinear optical models. The results discovered may provide a valuable foundation for subsequent analytical investigations of associated nonlinear evolution equations.
This study explores the integrable coupled Kuralay equation, which is widely utilized to study the motion of induced curves. In fields such as ferromagnetic materials, nonlinear optics, and optical fibers, soliton solutions of the Kuralay equation have emerged as significant recent developments. For proposed model, diverse soliton structures are obtained analytically by using two techniques, the unified method and the sub-ordinary differential equation (ODE) approach. It is possible to extract soliton solutions for rational and polynomial functions with a unified technique. Proposed research uses the new sub-ODE method to provide accurate solutions for soliton waves, such as hyperbolic, periodic, dark, bright, trigonometric, Jacobi elliptic, and Weierstrass elliptic function solutions. Moreover, sensitivity analysis of proposed model is successfully analyzed by using different initial condition. The physical relevance of the proposed model by referring to the visual representation of the achieved soliton solutions. The derived solutions are presented in 2D, 3D, and contour plots to illustrate how system parameters influence pulse propagation behavior.
This paper is concerned with the torsional vibration of the nanobeam under a longitudinal magnetic field and surrounded by a torsional spring. The small size effect is taken into account by considering the modified couple stress theory of elasticity for continuum mechanics, and the power series method (PSM) is utilized in order to compute the eigenfrequencies of the obtained differential equation. In order to prove the validity of the PSM, the obtained results are satisfactorily compared with those already existing. The influence of several variables of the system, such as magnetic field, couple stress parameter, and torsional spring on the frequency or on the rotation is well examined. Tables with numerical values, computed using the present method demonstrate that the PSM could be a suitable numerical method for analyzing a torsional vibration of a nanobeam, making the present work a benchmark for future works.
The current research work studies an inverse boundary value problem (InBVP) for the two-dimensional sixth-order Boussinesq model equation under additional constraints. First, we introduce an auxiliary InBVP and establish its equivalence (in a specified sense) to the original problem. To analyze the auxiliary problem, we employ the method of separation of variables. Using this approach, the solution of the direct boundary value problem (for a given unknown function) is reduced to solving a problem with undetermined coefficients. This leads to a countable system of integrodifferential equations (IDEs) governing these coefficients. The system is then reformulated as a single IDE for the desired solution. Next, by incorporating the additional conditions of the auxiliary inverse problem, we derive a system of two nonlinear integral equations to determine the unknown functions. Consequently, resolving the auxiliary InBVP essentially reduces to examining a system of three coupled nonlinear IDEs for the unknown functions. To address this system, we construct a specialized Banach space. Within a ball in this space, we apply contraction mapping principles to demonstrate both the existence and uniqueness of solutions to the nonlinear integrodifferential system, thereby obtaining the solution to the auxiliary inverse problem. Finally, leveraging the equivalence between the auxiliary and original problems, we showcase that the original InBVP possesses exactly one classical solution in the specified function space.MSC2020 Classification : 35R30, 35D30, 35A01
Tuberculosis (TB), a fatal infectious disease caused by Mycobacterium tuberculosis, spreads through aerosol droplets from active cases. The Global TB Report 2025 estimates similar to 10.7 million incident cases and 1.23 million deaths worldwide. This study aimed to investigate seasonal patterns in TB incidence in India and to develop univariate time series models using monthly nationwide active TB cases from January 2017 to June 2024 obtained from NIKSHAY portal. The time series forecasting of TB was approached by both traditional and machine learning-based models, including seasonal auto-regressive (AR) integrated moving average (MA) (SARIMA), AR neural networks (ARNNs), hybrid model (SARIMA-ARNN) and Bayesian structural time series (BSTS) model. The performance of the models was evaluated using accuracy measures like root mean square error (RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE). Bayesian information criterion (BIC) and Akaike information criterion (AIC) were estimated. The BSTS model achieved the highest forecasting performance, yielding the lowest RMSE (27,634.42), MAE (23,300.73) and MAPE (10.55). These findings highlight the potential of integrating Bayesian frameworks to enhance the accuracy of time series forecasting for TB cases, particularly in limited time series.
This research work provides a comprehensive investigation of the M-fractional paraxial wave equation (M-fPWE) in describing complex optical phenomena in telecommunication systems and nonlinear media, focusing on the dynamical analysis of optical soliton solutions, the impact of M-fractional parameters, stability, multistability, and the chaotic nature of the proposed model. To examine optical soliton solutions for the time M-fPWE model, we employ three advanced analytical methods, such as the Expa-function, improved Kudryashov, and unified solver techniques. These methods yield diverse soliton structures, such as the Expa-function technique, which produces kinky periodic waves, kink, and anti-kink waves, and double periodic waves; the improved Kudryashov method reveals solitary periodic waves, kink, and periodic waves, various periodic breather waves, and interactions such as kink-periodic lump and anti-kink-periodic lump waves; the unified solver technique uncovers double periodic waves, periodic breather waves, and kink-bell shape interactions. Moreover, by employing the Galilean transformation, we formulated the dynamical system of the equation, facilitating a comprehensive chaotic analysis, 2D and 3D phase portraits, Poincar & eacute; plots, and multistability analysis that uncovered essential data transmission systems. Finally, we compare our results and outcomes with a published work. The obtained results are significant in understanding key physical phenomena in optical fiber communication.
This study examines the diffusive-bioconvective magnetohydrodynamic (MHD) flow of a hybrid nanofluid past an exponentially stretching sheet embedded in a porous medium, incorporating thermal radiation, viscous dissipation, Soret-Dufour effects, and gyrotactic microorganisms. The hybrid nanofluid, consisting of nanoparticles (Ag + TiO2) suspended in water, is employed to enhance thermal conductivity and overall heat transfer performance. The governing nonlinear partial differential equations for the incompressible, steady, and two-dimensional flow are transformed into a system of ordinary differential equations via appropriate similarity variables and solved numerically using the finite element method (FEM). Comprehensive parametric analyses reveal that the fluid velocity decreases with increasing magnetic field strength and suction parameter, while it rises under the influence of viscous dissipation, thermal radiation, and the Dufour effect. The temperature field is significantly augmented by thermal radiation, viscous dissipation, and coupled Soret-Dufour mechanisms. The concentration field diminishes with stronger chemical reactions and higher Lewis numbers but grows with the Soret effect. Furthermore, skin friction coefficient increases notably with magnetic field intensity, viscous dissipation, and Soret-Dufour parameters. Both the local Nusselt number and Sherwood number exhibit declining trends with rising heat generation and chemical reaction rate, respectively. The computed results demonstrate excellent agreement with established benchmark solutions.
This study investigates the radiative magnetohydrodynamic (MHD) flow of Williamson and hybrid Williamson nanofluids (Cu-water and Cu + Ag-water) over a porous, linearly stretching sheet with suction and internal heat generation. Although nanofluids have been extensively studied, limited research addresses the combined influence of non-Fourier heat flux, viscous dissipation, thermal radiation, and entropy generation in Cu + Ag/water Williamson hybrid nanofluids. The novelty of this work lies in its comprehensive analytical modeling of these coupled transport phenomena and in providing a comparative assessment between Cu-water and Cu + Ag-water Williamson nanofluids under realistic boundary conditions. The governing nonlinear equations are reduced using similarity transformations and solved analytically through the Homotopy Analysis Method (HAM), which offers excellent convergence control for highly nonlinear systems. Results show that the Cu-water nanofluid exhibits higher velocity due to lower viscosity, while the Cu + Ag-water hybrid nanofluid demonstrates superior thermal conductivity and enhanced heat transfer. The wall shear (skin friction) increases with We but decreases with M, K, and S, reflecting the competing roles of elastic effects versus magnetic braking, porous drag, and suction-induced momentum reduction. Meanwhile, the Nusselt number decreases with M, K, and Q, but increases with Nr, S, and gamma, indicating that radiative transport, boundary-layer thinning, and thermal relaxation steepen -theta '(0), thereby offering a concise guideline for balancing heat-transfer enhancement against frictional penalties. An increase in the thermal relaxation parameter (gamma) delays the heat-flux response in the Cattaneo-Christov model, which weakens thermal diffusion, lowers the boundary-layer temperature, and improves thermal stability. As a result, the wall temperature gradient steepens, increasing -theta '(0) and enhancing the Nusselt number. Entropy generation is intensified by the magnetic parameter (M), porous resistance (permeability) parameter (K), and Brinkman number (Br), but it is suppressed by the Weissenberg number (We) and temperature-difference effects. Overall, the findings provide valuable insight for optimizing hybrid nanofluid-based systems in cooling, micro-electro-mechanical systems (MEMS), and sustainable thermal management applications.
This study investigates characteristics of the mass transport for Maxwell fluids driven by a time-periodic oscillatory electroosmotic flow (EOF) in annular microchannels under high zeta potential conditions. A finite difference method is employed to solve the nonlinear Poisson-Boltzmann equation, the Maxwell fluid momentum equation, the convection-diffusion equation, and the time- and space-averaged mass transport rates are obtained by using the composite trapezoidal rules, and the reliability of the present numerical results for the low zeta potential is verified by comparing the analytical approximate results obtained by using the Debye-H & uuml;ckel (D-H) linear approximation. The effects of key dimensionless parameters-including electrokinetic width, angular Reynolds number, wall potential ratio, relaxation time, and the inner-to-outer radius ratio-on flow velocity, solute concentration distribution, and average mass transport performance are systematically analyzed. The results show that: (1) a high zeta potential enhances velocity and concentration, while suppressing the spatio-temporal average mass transport rate; (2) the elastic effects of Maxwell fluids, characterized by relaxation time, significantly modulate the structure of velocity and concentration fields under periodic electric forcing, thereby enhancing local mixing or enabling spatially selective separation; (3) at low Reynolds number, the flow remains relatively uniform, facilitating species separation, whereas at higher Reynolds number, amplified elastic responses near the walls lead to increased nonuniformity in velocity and concentration distributions, promoting mixing; (4) asymmetric zeta potentials induce elastic responses in the core region, further intensifying concentration nonuniformity and enabling species separation under specific parameter combinations; and (5) geometric factors such inner-to-outer radius ratio and electrokinetic width significantly affect radial gradient intensity, resulting in switching phenomena between fast- and slow-diffusing species.
Fractional calculus (FC) has become more popular during the past four decades due to its extensive applications in mathematics, physics, engineering, and statistics. B-spline functions offer flexible and incredibly precise approximations because of their piecewise polynomial structure and smoothness at knots. For a class of third-order conformable boundary value problems (BVPs), we develop approximation solutions using the quartic B-spline method. The conformable fractional derivative (CFD) is utilized to formulate fractional problems. More specifically, singularities are used to modify the category of conformable Lane-Emden models. Three numerical examples are shown and examined to demonstrate the approach's effectiveness. The numerical results are highly accurate and require less computational work, and they closely match the exact solutions.MSC2020 Classification: 39A12, 39B62, 33B10, 26A48, 26A51.
This paper presents the generalized synchronization of neuronal networks in a unidirectionally coupled drive-response system with interconnected nodes exhibiting chaotic behavior. The study focuses on the synchronization of the Hindmarsh-Rose (HR) neuronal mathematical model through connections among the output, input, and hidden layers. Based on the classical Lyapunov method, sufficient conditions are proposed to synchronize HR neuronal networks using appropriate coupling parameters. The results are derived from classical Lyapunov theory, which employs a continuous time chaotic response system to monitor synchronized dynamics. It is shown that the proposed approach is capable of establishing a response network that achieves generalized synchronization with the drive system via controller design, without relying on dynamic cancelation or feedback. Finally, several simulation examples are presented to demonstrate the feasibility of generalized synchronization in HR neuronal networks.
We present new relations derived from Noether’s identity that reveal the compatibility between the components of the Hessian matrix of the Lagrangian, the infinitesimal symmetry transformation of the configuration variables and time, and a constant of motion. Using these relations, we develop two new methods to incorporate symmetry requirements directly into the inverse problem of mechanics, thereby restricting the set of acceptable Lagrangians. We accomplish this by combining these relations with Helmholtz’s conditions, which allow us to construct Lagrangians whose actions exhibit specific symmetries from the outset. The theory is illustrated with one- and two-dimensional examples.
Nonlocal information, such as material deformation, genetic genes, or the history of the disease, are essential as they provide us with additional details that increase the numerical solution's accuracy. With the help of the phase delay, we may also predict the future of the phenomena we are researching. This study involves consideration of both the nonlocal conditions and the phase delay effect. The phase-lag integro-partial differential equation (I-PDE) with nonlocal conditions is investigated in order to achieve this, transforming it into a two-dimensional mixed integral equation (2-D MIE). The I-PDE, thus, has a unique solution, as shown by the Banach fixed point theorem. Furthermore, the solution's convergence has been demonstrated using Picard's approach. Since the obtained MIE equation requires a specific approach to find its solution. In this investigation, MIE is numerically addressed using the product Nystr & ouml;m method (PNM). Ultimately, numerical results were obtained by solving various types of applications. Therefore, many interesting conclusions were derived.
In this article, a highly generalized way of studying nonlinear evolution equations (NLEEs) with time-dependent variable coefficients is provided. The innovative exact solutions of the Kadomtsev-Petviashvili (KP) equation and the modified Korteweg-de Vries (mKdV) equation with temporal variable coefficients are evaluated by using the extended generalized G ' G2-expansion method. Ion-acoustic waves, ferromagnets, and traffic flow models are all described by the mKdV equation, while shallow-water waves, plasma physics, and nonlinear optics in two dimensions are explained by the KP equation. The governing equations must incorporate variable coefficients that account for the nonuniform properties of the medium in space and time. The obtained exact solutions of NLEEs are in both hyperbolic and trigonometric forms. The achieved closed-form solutions are illustrated in the form of 3-D and contour plots with the aid of Mathematica-13.3. Dynamical structures, solitary waves, and periodic solitary waves resemble these results.
