The primary aim of this study is to present a novel approach that combines Fractional Shifted Vieta-Lucas Wavelets (FSV-LW) with a collocation technique to effectively address two types of delayed models involving both linear and nonlinear fractional-order differential equations, incorporating constant and proportional delays. Initially, we introduce an innovative formulation of (FSV-LW) defined over a given interval, and construct a closed-form expression for the fractional integral operator. The original models are then transformed into algebraic equations through the application of the collocation method and the fractional integral vector of these wavelets. In support of the method’s precision and convergence behavior, several mathematical theorems have been formulated and proven. Finally, we apply this method to solve specific examples from both models, highlighting its effectiveness and practical applicability.
This paper investigates the dispersive concatenation model (DCM), both in the absence and presence of local distributed delay or nonlocal delay. First, by the dynamic system method, the phase portraits of bifurcations for the DCM are categorized into five cases. Then, various bounded solitons of the DCM without a delay are derived through integration along different orbits. Finally, by using the geometric singular perturbation theory, it is proven that the DCM with local distributed delay or nonlocal delay admits the corresponding solitary wave solutions and periodic wave solutions for a certain wave speed. The analytic representations and theoretical existence of soliton solutions are helpful for deeply understanding and research on the dynamic properties of soliton solutions.
In this paper, we study the sixth-order nonlinear Schrödinger equation, appearing in the optical fiber. First, we derive this model from the AKNS system by constructing the Lax pair. Then, we obtain the first-order breather and higher-order rogue waves of the equation through the Darboux transformation. Finally, we demonstrate the higher-order dispersive term’s contribution of the breathers and rogue waves by the compression effects generated by the coefficient γ _1 to the sixth-order dispersive.
This paper investigates axial unsteady flows of nanofluid filling a vertical rectangular channel with a mobile wall with time-dependent velocity. The nanofluid is Boussinesquian and the flow is induced by time-dependent pressure gradient, the buoyancy and Lorentz forces. The magnetic field is applied uniformly and oriented normal to the fluid velocity. The channel walls are maintained at different, constant temperatures. The induced magnetic field included in the analysis as well. For the nanofluid, Buongiorno’s model is employed. The differential equations with partial derivatives that determine the fields of velocity, temperature, volume fraction of nanoparticles and the induced magnetic field are transformed into dimensionless form together with the initial and boundary conditions. A numerical scheme based on the Crank-Nicolson approximation formulas is formulated; the proposed scheme determines the values of the unknown functions at the discretization points by solving some three-diagonal systems of linear algebraic equations. The associate electric field is also determined. The solutions of concrete problems characterized by particular forms of the functions that give the velocity of the wall movement and the pressure gradient are studied and interpreted with the help of numerical values and graphic illustrations.
In this paper, we study the Gross–Pitaevskii equation modeling the dipolar Bose–Einstein condensation with a harmonic potential. We derive bright soliton-like solutions and blow-up solutions by using boundary conditions and the Jacobi elliptic function approach. In particular, the influence of the harmonic potential on the dynamics of the obtained solutions is illustrated graphically.
In this research, we investigate the uniqueness results of fractional -differential equations under certain specified integral boundary conditions. We utilize the properties of using the -concave and sub-homogeneous operators, along with a Banach fixed-point theorem, to achieve our results. Additionally, an example is provided to illustrate the main results. This work contributes to the field by examining the existence and uniqueness of positive solutions involving new integral boundary conditions with the -concave and sub-homogeneous operators, leading to illustrative numerical results.
We consider the interior transmission eigenvalue problem -u^''(x)=λρ (x)u(x) with the non-selfadjoint boundary conditions u(0)=0=u'(1)sin√(λ)/√(λ)-u(1)cos√(λ) , where the refractive index ρ∈ L^1 . We prove the existence of infinitely many transmission eigenvalues. In addition, we show the complete continuity of transmission eigenvalues.
This study presents a novel discrete fractional-order mathematical framework for modeling and forecasting the coupled progression of diabetes and cardiovascular complications, with emphasis on population dynamics in Saudi Arabia. Using the discrete Caputo fractional operator, the model captures memory effects and long-term disease dependence not represented by classical integer-order systems. The population is divided into five interacting compartments: susceptible, exposed, diabetic without major complications, diabetic with severe complications, and cardiovascular-affected individuals. A rigorous qualitative analysis establishes equilibrium points and examines their local stability under fractional discrete dynamics. Stability regions are identified in parameter space, clarifying the mechanisms governing disease persistence and progression. A feedback-based control strategy is proposed to restore stability of the disease-free equilibrium when destabilization occurs. To enhance predictive performance, a Reservoir Computing framework is integrated and compared with Long Short-Term Memory networks, demonstrating improved accuracy and lower computational cost. Numerical simulations validate theoretical findings and highlight key epidemiological influences. Future work will extend the model by incorporating optimal control strategies aimed at minimizing infections through targeted interventions such as vaccination, improved hygiene, and environmental sanitation. The integration of fractional-order dynamics with control theory offers valuable insights for public health decision-making and epidemic mitigation.
In this work, we investigate a transmission problem for the Timoshenko system with distributed delay terms acting on the rotation-angle equations of a beam. The model describes a structure composed of two different materials connected at an interface, which introduces transmission conditions that complicate the stability analysis. Distributed delays are incorporated into the internal feedback laws associated with the variables v_i , which may lead to destabilizing effects if they are not properly balanced with non-delayed damping mechanisms. Under suitable assumptions on the relative weights of the delayed and non-delayed feedback terms, as well as on the wave propagation speeds, we first establish the well-posedness of the system by using semigroup theory. In the case of equal propagation speeds, we construct a suitable Lyapunov functional and apply the energy method to derive an exponential decay estimate for the total energy of the system. This result shows that the combined action of distributed delay and non-delayed feedback is capable of guaranteeing uniform stability, provided that the delay contribution remains sufficiently small. Finally, we highlight the importance of the equal wave speed condition, which plays a crucial role in ensuring exponential convergence to equilibrium.
This paper systematically investigates the integrability, extended structures, and exact solutions of a class of coupled Korteweg–de Vries (KdV) models derived from supersymmetric evolution equations. First, the coupled system consisting of bosonic and fermionic field components is obtained by superfield expansion. Within the framework of exterior differential forms, a differentially closed ideal is constructed by employing the method of extended structures. Furthermore, an explicit representation of the Lax pair is derived by using Lie algebra representation theory and the 𝔰𝔩(4,ℂ) algebra, thus establishing the integrability of the system. On this basis, the first-order and higher-order Darboux transformations are systematically derived, which generate explicit expressions for multi-soliton solutions. Finally, exact single- and double-soliton solutions with supersymmetric locking are obtained by taking the zero background as the seed solution. It is demonstrated that the bosonic and fermionic components propagate cooperatively in the form of supermultiplets. This work provides a systematic framework for the algebraic analysis and exact solution construction of supersymmetric integrable systems.
Electromagnetically controlled transport in microstructured fluids is of considerable interest in advanced thermofluidic and microfluidic technologies, particularly when heat and mass transfer are strongly coupled. This study investigates steady micropolar fluid flow and coupled thermo-solutal transport in a transpiration-controlled channel driven by an upper Riga plate. The mathematical model accounts for non-uniform Riga-induced Lorentz forcing, thermal radiation, Joule dissipation, and Soret–Dufour cross-diffusion effects, leading to a highly nonlinear system of coupled boundary-value equations. The novelty of the work lies in examining the combined influence of these mechanisms within a confined micropolar channel configuration, which has received limited attention in the existing literature. The governing equations are transformed into a dimensionless form and solved using a fourth-order Hybrid Block Runge–Kutta shooting scheme with analytical initialization. The results demonstrate that micropolar material parameters suppress the velocity, microrotation, and concentration fields while enhancing the temperature distribution. Stronger electromagnetic forcing generated by the Riga plate and larger modified Hartmann effects significantly attenuate the flow and microrotation characteristics. Moreover, Dufour and Soret cross-diffusion mechanisms substantially modify thermal and concentration transport, whereas increasing thermal and mass Peclet numbers enhance local heat and mass transfer rates. These findings provide new insight into the control of coupled momentum, thermal, and species transport in electromagnetically actuated micropolar channel flows and offer useful guidance for the design of advanced thermal-management and microfluidic systems.
The hydrodynamic instability of a thin liquid film possessing odd viscosity and flowing down a slippery inclined substrate is investigated under the combined influence of an imposed tangential shear stress and a normal electric field. Employing long-wave approximation techniques, a nonlinear evolution equation is derived that incorporates the effects of odd viscosity, slip length, normal electric field, and bidirectional tangential shear. Linear stability analysis reveals that odd viscosity and reverse-direction tangential shear exert stabilizing influences, whereas slip length, the normal electric field, and forward-direction tangential shear promote destabilization. The critical Reynolds number is determined analytically and its dependence on the governing system parameters is examined systematically. Through multi-scale analysis, the weakly nonlinear instability is investigated by deriving a complex Ginzburg–Landau equation that delineates four distinct dynamical regimes: subcritical instability, unconditional stability, explosive instability, and supercritical stability. These regimes are characterized by the signs of the coefficients appearing in the amplitude equation. Parametric studies demonstrate that, within the supercritical stable regime, the nonlinear wave amplitude and phase velocity increase with the electrical parameter in the flow direction but diminish with increasing odd viscosity coefficient. The present results show excellent agreement with established theoretical and experimental investigations reported in the literature.
In the present work, a numerical study of the incompressible laminar flow of a Newtonian fluid circulating through a rectangular microchannel of parallel plates with permeable walls is carried out considering slip conditions. For this purpose, the corresponding governing equations, i.e., the mass and momentum conservation equations, are solved using the finite element technique with the free software FreeFEM++ to analyze the hydrodynamics of the flow under consideration, obtaining the velocity and pressure profiles. The main results show that by increasing the dimensionless filtration parameter β , the transverse velocity increases, which causes the longitudinal velocity to decrease due to mass conservation. This behavior is maintained even considering the influence of the dimensionless slip parameter δ , which is also reflected in the volumetric flow rate, since as this parameter increases, the volumetric flow rate in the longitudinal direction is slightly enhanced.
This paper studies the first-passage reliability of a two-degree-of-freedom (2-DOF) nonlinear stochastic rotational vibration energy harvester under random excitation. Firstly, a nonlinear dynamical model of the system is developed and reduced to a one-dimensional energy diffusion process via the stochastic averaging method (SAM) for quasi-non-integrable Hamiltonian systems. Secondly, the backward Kolmogorov (BK) equation and the generalized Pontryagin (GP) equation are derived to characterize the first-passage behavior. By imposing appropriate boundary conditions, the conditional reliability function, the conditional probability density function (CPDF), and statistical moments of the first-passage time are obtained by solving the associated BK and GP equations. Finally, the SAM results were compared with Monte Carlo (MC) simulations to verify the effectiveness of the proposed approach. Based on this foundation, a systematic analysis was conducted to evaluate the effects of initial energy, noise intensity, and key dynamic parameters on the conditional reliability function, the CPDF of the first-passage time, and mean first-passage time (MFPT). Results indicate that increasing the generator natural frequency and electromagnetic damping ratio markedly enhances system reliability, whereas mechanical damping and Coulomb friction provide weaker but beneficial effects within the investigated ranges. The initial energy and noise intensity are identified as the dominant factors in the sensitivity analysis, while the cubic-stiffness-related characteristic frequency, mechanical damping, and Coulomb friction play relatively minor roles. The present study provides a reduced diffusion modeling and first-passage analysis framework for nonlinear stochastic dynamical systems, while also offering reliability-oriented insights for rotational vibration energy harvesters.
This study presents novel exact solutions for the (2+1)-dimensional Kundu–Mukherjee–Naskar (KMN) equation, a model governing optical soliton propagation in nonlinear media. By incorporating two arbitrary functions G(y) and H(ξ ) into the solution ansatz. Four types of solutions are constructed: (i) basic solitons exhibiting cross-shaped breather dynamics; (ii) periodic breathers combining rational spatial localization with temporal oscillations; (iii) localized breathers featuring static periodic backgrounds coupled to moving bright solitons; and (iv) fractal rogue waves emerging from rational fractional functions. These solutions exhibit rich dynamics including spatiotemporal coupling dynamics, breathing behaviors, directional propagation ( v_x = α k ), and extreme localization of wave energy. Parameter analysis demonstrates control over amplitude (C), localization(K, L, P), and energy distribution ( A_0, B_0 ) , with numerical simulations visually confirming the predicted structures. These results significantly expand the KMN solution space, enhancing the modeling of complex wave phenomena in optical materials like birefringent fibers and photonic crystals.
This paper introduces a novel method to find first integrals for two-dimensional transcendental function systems. We propose a self-fission similarity learning method, which enables a symbolic mapping model to capture underlying mathematical relationships and thereby find first integrals. Experiments demonstrate that the model successfully discovers new and different first integrals for some transcendental function systems, with high accuracy and wonderful predictive generalization capabilities. In addition, famous mathematical software Maple’s DEtools fails for the above examples. It makes a new way for finding first integrals of transcendental function systems.
This paper applies a new tool—q-Phragmén-Lindelöf Indicator, which resolves the problem of determining the growth of solutions to q-difference equations in the absence of a dominant coefficient. To the best of our knowledge, this paper presents the first time that the q-Phragmén-Lindelöf indicator is linked with complex equations and utilized to characterize the growth properties of their solutions. Our results encompass and generalize some of the earlier research findings in this area.
This paper introduces a high-accuracy numerical framework for solving a generalized form of the fractional regularized long-wave Burgers (FRLWB) equation, incorporating the Caputo fractional derivative. The proposed method is based on a space-time pseudo-spectral collocation (PSC) approach, which leverages global interpolation via Lagrange polynomials and shifted Legendre-Gauss-Lobatto (LGL) nodes to achieve exponential convergence for smooth solutions. Unlike traditional schemes such as finite difference or finite element methods, the PSC technique offers superior spectral accuracy and computational efficiency, particularly in handling fractional operators. A key novelty of this work lies in the simultaneous spectral treatment of both spatial and temporal domains, enabling the construction of operational matrices that directly approximate fractional derivatives without resorting to discretization or auxiliary transformations. By collocating the governing equation along with its initial and boundary conditions, the problem is transformed into a system of nonlinear algebraic equations. These systems are solved efficiently using iterative solvers, ensuring stability and scalability across a range of fractional orders. The method also introduces a direct formulation for the mixed derivative term Q_ηηκ , which is often neglected or approximated in existing literature. This enhances the model’s fidelity and allows for more accurate simulation of wave propagation phenomena in complex media. A rigorous convergence analysis is provided to establish the reliability of the approach, and several benchmark problems are examined to demonstrate its precision and robustness. Overall, the proposed PSC framework represents a significant advancement in the numerical treatment of fractional PDEs. Its flexibility, accuracy, and ease of implementation make it a promising tool for future applications in fluid dynamics, nonlinear wave modeling, and fractional-order systems.
We present the first comprehensive benchmark revealing that discretization strategy selection in Physics-Informed Neural Networks (PINNs) profoundly impacts performance for Fredholm integral equations of the second kind. Our kernel-adaptive framework systematically compares discrete coordinate, endpoint, and midpoint methods across smooth, singular, and regularized kernels. Key findings: discrete coordinate methods achieve 35
This paper develops a mathematically rigorous and computationally efficient decoupled reduced-order framework for the dynamic analysis of asymmetric piecewise linear oscillators under harmonic excitation. Such systems appear widely in mechanical and structural applications involving unilateral constraints, gaps, or asymmetric stiffness distributions, and they are governed by non-smooth second-order differential equations with switching restoring forces. The non-smoothness induces switching dynamics and nonzero mean offsets that challenge classical averaging and harmonic balance techniques. To address these difficulties, the system response is approximated using a decoupled reduced-order representation that explicitly includes a constant basis function to capture asymmetry-induced mean-offset dynamics. A Galerkin projection yields a low-dimensional nonlinear system of ordinary differential equations that preserves the mechanical structure and the piecewise linear character of the original model. Existence and uniqueness of reduced-order solutions are established under standard Lipschitz-type conditions, while nonlinear stability is studied using energy-based Lyapunov arguments, invariance principles, and linearized spectral analysis. We show that the reduced-order model inherits Lyapunov and asymptotic stability properties of the full non-smooth system, and that the stability conditions depend explicitly on the asymmetry parameters. For numerical realization, an implicit Newmark time-integration scheme combined with Newton-type iterations is employed to maintain stability across switching regimes. Numerical investigations, reported in tabular form, demonstrate excellent agreement with direct numerical integration and improved accuracy compared with classical harmonic balance approximations, and show that the proposed approach achieves high accuracy with substantially reduced computational cost, making it well suited for parametric studies and design-oriented analyses.
