This paper investigates the complex dynamics of Social Media Addiction (SMA) through the angle of fractional calculus. We employ a five-compartment deterministic model as Susceptible (S), Exposed (E), Addicted (A), Recovered (R), and Quit (Q) which map the transmission pathways of this behavioral addiction. The model is formulated using the Caputo fractional derivative to effectively capture the memory and hereditary properties inherent in human behavioral patterns, which integer-order models often overlook. We conduct a comprehensive qualitative analysis, establishing the existence, uniqueness, non-negativity, and boundedness of the solutions. The stability of the model’s equilibria as addiction-free and endemic states is rigorously examined through local and global stability analyses based on the basic reproduction number (R_0). To approximate the solution of the fractional system, we implement the robust Adams-Bashforth-Moulton predictor-corrector algorithm. Numerical simulations and a detailed sensitivity analysis are presented to illustrate the theoretical findings, demonstrating the influence of key parameters and the fractional order α on the system’s dynamics. The results confirm that fractional-order models offer a more nuanced and realistic framework for understanding the propagation of social media addiction. The main findings show that the fractional-order model provides slower convergence and reduced addiction peaks compared to the classical integer-order case, highlighting the significant role of memory effects. The analysis confirms that the basic reproduction number governs the persistence of addiction, while sensitivity results identify recovery and quitting rates as the most effective control parameters. These results demonstrate the importance of incorporating memory in accurately describing and controlling social media addiction dynamics.
In this paper, we focus on the stochastic Nagumo (SN) equation, perturbed in the Itô sense by advection Wiener Process. This provides a robust framework for modeling random transport phenomena in excitable media, incorporating both intrinsic dynamics and environmental variations. Comprehending these stochastic influences is essential for examining noise-induced phenomena, such as pattern formation, stability fluctuations, and signal propagation in biological and physical systems subjected to uncertain flow conditions. To achieve this, we concentrate on obtaining exact solutions to the SN equation. Utilizing suitable transformation method and Itô calculus, the SN equation is separated into a stochastic ordinary differential equation (SODE) and a deterministic Nagumo equation augmented by an additional diffusion term. The extended direct algebraic method is applied to find exact solutions to the deterministic equation. The obtained results, combined with the solution of the SODE, yield exact analytical solutions of the original stochastic Nagumo equation. In addition, these exact solutions offer deeper insight into noise-induced phenomena and provide reliable benchmarks for validating numerical simulations as well as approximate analytical methods. Numerical investigations in both two and three dimensions are further conducted to evaluate the impact of advection noise on the solution dynamics. The analysis demonstrates that multiplicative advection noise markedly influences wave behavior, inducing significant amplitude variations, phase shifts, and enhanced spatiotemporal irregularities.
Heat transfer and entropy generation in microchannels are crucial to several industrial applications, particularly in device cooling and performance enhancement. These aspects are also significant in healthcare sectors, where microchannel-based systems are employed in processes such as dialysis and diagnostics. The current study investigates these phenomena in a microchannel with a non-uniform rough surface membrane attached to both walls, under the influence of a transverse magnetic field. Unlike conventional peristaltic or smooth-wall microchannel models, the present study focuses on a non-uniform rough-membrane configuration to simultaneously examine its influence on pumping performance, thermal transport, and entropy generation in a Ree-Eyring fluid. This combined analysis provides new insights into the interactions among wall roughness, membrane actuation, magnetic effects, and irreversibility, which have been poorly investigated in the existing literature. By employing a low Reynolds number and long-wavelength approximation, the governing equations are solved analytically to obtain the velocity profile, volumetric flow rate, temperature distribution and entropy generation analysis. Furthermore, the results of the present investigation are demonstrated and analyzed using MATLAB software. The results reveal that the rate of heat transfer is enhanced with the increase in key parameters, in conjunction with the membrane propagation. An increase in entropy generation is observed with rising Brinkman number and heat source parameter; however, this effect can be mitigated by adjusting the temperature difference.
In this paper, we study a new class of coupled nonlinear fractional integral equations involving the Ψ-Katugampola fractional integral operator. By applying Dhage’s fixed point theorem in a Banach algebra, we establish sufficient conditions for the existence of solutions to the proposed coupled system and prove uniqueness via the Banach contraction principle. The obtained results extend and generalize several known existence results for fractional integral equations associated with particular choices of the kernel. An illustrative example is presented to demonstrate the applicability of the main theorem.
This paper will use Laplace least square residual power series method (LLSRPSM) on nonlinear fractional partial differential equations, in particular, Korteweg–de Vries (KdV) equation, convection-reaction-diffusion (CRD) equation and the system of Burgers’ equations. The proposed method standardizes the descendants of the Laplace transform, the residue power series, and the least-square middleweight to come up with effective and accurate approximations of the solutions of the fraction-order. They compare the results of their findings with the available methods, the Yang transform homotopy perturbation method (YTHPM), the modified generalized Taylor fractional series method (MGTFSM), the Elzaki transform decomposition method (ETDM) and the Laplace residual power series method (LRPSM). It was compared to the competing methods that LLSRPSM is less erroneous and more stable and faster in convergence than the competing methods. Besides, the significance of fractional order parameter, Omega, is mentioned since a smaller value corresponds to the presence of memory and diffusion anomalies whereas larger values are expected to converge to the classical integer-order solutions. Overall, the findings support the effectiveness and relevance of LLSRPSM to nonlinear fractional PDEs that have the potential to find wide uses in physical and engineering sciences.
In this work, we investigate existence and multiplicity of nontrivial solutions for a nonlocal Neumann problem, with a particular focus on the case where the nonlocal term exhibits a decreasing behavior. Our main results are derived by utilizing the concentration-compactness principle originally introduced by Lions and extended by Bonder, Saintier and Silva. By employing trace theory and the Mountain Pass Theorem, we demonstrate that, under suitable conditions, the problem admits a nontrivial solution, and multiplicity of solutions is obtained via a truncation argument.
In this paper, we focus on the dynamical analysis and soliton solutions of the Vakhnenko–Parkes (VP) equation. Initially, an appropriate transformation is applied to the VP equation to reduce it to a nonlinear ordinary differential equation. Subsequently, a Galilean transformation is employed to obtain the corresponding dynamical system. The Hamiltonian structure and sensitivity analysis are investigated to better understand the considered model. Furthermore, the Hamiltonian structure of the system is derived, and the sensitivity of the model with respect to key parameters is examined. An external forcing term is then introduced to explore the chaotic dynamics of the system, with particular emphasis on the effects of amplitude and frequency variations. Finally, soliton solutions of the model are derived using the direct algebraic method, and the obtained results are visualized via two-dimensional, three-dimensional, and contour plots to analyze the influence of wave speed.
The present work is concerned with the following fractional diffusion problem in time and space. ∂ _t^βu + M(u_H_0^α(Ω )^2)(-Δ )^αu + (-Δ )^α∂ _t^βu = γ |u|^ρu + g(x,t), x ∈Ω , t > 0, u(x,t) = 0 x ∈ℝ^N∖Ω , t > 0, u(x,0) = u_0(x) x ∈Ω , where Ω⊂ℝ^N is a bounded domain with Lipschitz boundary, (-Δ )^α is the fractional Laplace operator (0 < α < 1), we consider the Riemann-Liouville fractional time derivative ∂ _t^β, where the fractional order satisfies 0 < β < 1. By the Galerkin method, let 0 < ρ < 4α/N-2α when γ < 0, and 0 < ρ < min{4α/N, 2α/N-2α} when γ > 0, we establish the global existence and uniqueness of weak solutions for problems of Kirchhoff type. The decay behavior is analyzed through the application of differential inequality techniques.
In this study, the Taylor wavelet collocation method (TWCM) is introduced to obtain an effective numerical solution for variable-coefficient inverse time-fractional advection–diffusion equations. The model is designed to represent the spatiotemporal evolution of pollutant concentrations in heterogeneous media, including both soil and atmospheric environments. Accordingly, the proposed approach is particularly effective for estimating unknown model quantities, specifically the state field and the unknown time-dependent boundary function, from prescribed initial and boundary data together with an overspecified interior observation, under varying environmental conditions. By applying TWCM, the inverse problem is reduced to a finite-dimensional linear algebraic system for the Taylor-wavelet coefficients. For noisy interior observations, Tikhonov regularization is incorporated to enhance the stability of the discrete reconstruction. Comparative analyses demonstrate that TWCM significantly outperforms established numerical methods, including the rationalized Haar wavelet (RHW), radial basis function (RBF) and finite-difference benchmarks. The results highlight the robustness and effectiveness of TWCM in addressing complex inverse problems governed by time-fractional advection–diffusion equations and demonstrate its potential to provide valuable insights for environmental monitoring and management.
This paper is devoted to studying the improved weak-strong uniqueness of the 2D generalized surface quasi-geostrophic equation with the singular velocity u=Λ ^1-βR^θ. Based on the continuity of the trilinear form in critical Besov spaces and energy methods, we proved that the solution θ (x, t) is unique in the class of weak solutions, if θ (x, t) lies in the following critical Besov spaces θ (x, t)∈ L^q(0,T;B^s_p,∞(ℝ^2)) with 2/p+α/q=α +β +s-2 and 2/α +β +s-2< p< ∞ . This result extends the uniqueness criterion from Lebesgue space to critical Besov spaces.
Fractional stochastic equations are gaining more and more importance for the modelling of real systems where memory and randomness are involved. In this paper, we study a class of Itô-Doob fractional stochastic systems with proportional delays and fractional derivative of Caputo-Katugampola type. The main difference between our work and the mean-square analysis is that we use the more general pth moment framework, with p≥ 2. By choosing this way, it is not only more robust but also more reliable. First, we prove that the system is well-posed by showing existence, uniqueness, and continuous dependence of the solution on the initial data under reasonable assumptions. The derivation employs a combination of fixed-point arguments and stochastic calculus. In the following step, we establish the Ulam–Hyers stability in the pth moment sense, demonstrating that the solution remains close to the exact solution in instances where the system experiences a small perturbation. We also prove the averaging principle. This principle allows for the substitution of a complex and rapidly oscillating system with a more straightforward, averaged version. This is particularly useful for systems with multiple time-scale dynamics. We give a rigorous proof of the closeness of the solutions of the averaged and the original systems. We demonstrate our theoretical results with some numerical examples using a suitable adaptation of the Euler–Maruyama scheme to the fractional case.
Efficient thermal management in modern engineering systems often relies on manipulating flow behavior under magnetic, rotational, and porous boundary conditions. This study investigates the transient magnetohydrodynamic (MHD) behavior of a non-Newtonian Casson nanofluid embedded with uniformly dispersed dust particles in an inclined rotating porous channel. Wall suction and injection are introduced as active boundary control mechanisms to regulate flow and heat transfer. The governing equations for momentum and energy of both the fluid and particulate phases are formulated, non-dimensionalized, and analytically solved using the Poincaré–Lighthill perturbation method. The results reveal that the Casson parameter suppresses velocity in both the fluid and dust phases, reflecting shear-thinning behavior, while higher thermal Grashof numbers enhance buoyancy-driven convection. Increasing magnetic field strength or rotation rate reduces the axial velocity due to the Lorentz and Coriolis forces, respectively. Suction decreases the momentum boundary layer thickness, whereas injection enhances it. Nanoparticle loading elevates the temperature distribution but lowers the velocity because of increased viscosity. Thermal radiation significantly alters heat transfer rates, enhancing the Nusselt number at the wall. Key engineering indicators such as the skin-friction coefficient and Nusselt number are computed to quantify the system’s hydrodynamic and thermal performance. The findings suggest that coupled magnetic, rotational, and suction/injection mechanisms provide a controllable means for optimizing convective heat transport in rotating porous configurations relevant to cooling technologies, microfluidic devices, and thermal regulation systems.
This paper is concerned with the existence of ground state solution for a class of subcritical Schrödinger-Possion systems with irregular potentials. The irregular potentials cause difficulty in recovering the compactness of Palais-Smale sequences by classical method. To conquer this obstacle, with the help of weak∗-compactness method, we obtain a new weak∗-type splitting lemma, by which we utilize interaction effects among irregular potentials to generate the compactness of (PS) sequences. Some new compactness conditions are introduced to obtain ground state solutions.
In Almomani and Almomani [6], the authors established a priori estimation for the solution of the quasi-inverse problem. More recently, Almeflah et al. [3], the authors get some estimations of the classical solution of the same problem from the series representation of the solution [8]. Also, they establish an a priori estimation of a higher order than that in [6]. In this paper, we get the sufficient conditions for the convergence of the modified quasi-inverse method for the generalized solution. Our results play an important role in optimal control theory. Many works have been devoted to this problem, including the control problem of heat conduction with the inverse direction of time and integral boundary conditions [1, 4, 5, 7–14, 17, 18].
The random time-fractional coupled Drinfeld-Sokolov-Wilson system that show up in shallow water flow models when the shallow water equations are simplified under some assumptions, is solved by q-homotopy analysis transform method (q-HATM). Caputo sense is used to define the derivatives of fractional order. The parameters and the initial conditions of this system are analyzed by Normal and Uniform distributions. The characteristics of the solutions are obtained and the 2D and 3D plots of these are simulated in Maple software. The error tables for the obtained solutions of this system are constructed. It is observed that this method is high effectiveness, too fast and extremely powerful.
This paper investigates the strict stability of multi-term non-impulsive pseudo-fractional differential equations defined with respect to different kernel functions. By employing Caputo-type fractional derivatives, we extend stability concepts from impulsive systems to the non-impulsive framework. Lyapunov-like functions are constructed to derive sufficient conditions that ensure strict stability. Moreover, comparison principles are established to relate the associated fractional operators to explicit stability bounds. An illustrative example is presented to illustrate the effectiveness of the theoretical results and to highlight the influence of the multi-term structure on the stability behavior of the system.
This paper presents a higher-order compact finite difference method for solving third-order boundary value problems subject to Robin boundary conditions. The proposed scheme achieves sixth-order accuracy in the interior domain and incorporates a new boundary closure technique for Robin conditions to maintain this accuracy at the boundaries. A detailed error analysis is performed to establish the theoretical convergence properties of the method. Numerical experiments on several benchmark problems are conducted to validate the accuracy and efficiency of the scheme. The computed results confirm that the method attains the expected sixth-order rate of convergence and demonstrate its superiority over existing lower-order methods in terms of accuracy and computational efficiency. The proposed approach provides a reliable and effective tool for solving third-order boundary value problems subject to Robin boundary conditions.
We develop a K–R defect framework for classical inequalities centred on the normalised defect functional Φ _f(x,y;K,R) = D_f/(KR(x-y)^2), where D_f(x,y;K,R) = Kf(x)+Rf(y)-f(Kx+Ry) for an admissible pair satisfying K≥ 0, R≥ 0, K+R=1. The framework establishes a complete forward-inverse curvature theory. The Curvature Recovery Theorem proves that Φ _f equals 12f”(ξ ) for some interior point ξ, making Φ _f a derivative-free curvature estimator. The Local Limit Theorem establishes exact pointwise curvature recovery in the limit, with an explicit error bound. The Uniqueness Principle proves that two C^2 functions with identical normalised defect fields differ by at most an affine function. The Constant Defect Characterisation identifies quadratic polynomials as precisely the C^2 functions with constant normalised defect. The Reconstruction Formula recovers f from its defect measurements by double integration. The K–R Stability Theorem proves that a C^2 function with small defect magnitude satisfying homogeneous boundary conditions is uniformly close to zero, with an explicit constant. A K–R deficit theory for the Hölder inequality is established, including a new K–R Isoperimetric Deficit Ratio, with equality conditions strictly stronger than the classical case. Applications to the nonlinear boundary value problem -u”=f(u) include derivative-free source term curvature bounds, a polynomial reconstruction algorithm with a proved positive-definite Hessian guarantee, and a derivative-free convexity diagnostic for solution profiles. A multivariable extension connects Φ _f to the Rayleigh quotient of the Hessian, motivating a programme for full Hessian recovery from defect measurements.
Motivated by lower bound estimates of Payne and Escobar for the classical Steklov problem, we study sharp lower bounds for the first eigenvalue of Steklov-type problems on compact surfaces with smooth boundary. We prove that, if the Gaussian curvature satisfies K ≥ -α and the geodesic curvature of the boundary satisfies k_g≥ c > 0, then the first eigenvalue of the Steklov-type eigenvalue problem satisfies an optimal inequality and equality holds only for the Euclidean disk. In particular, we obtain a sharp lower bound for the first eigenvalue of a Schrödinger-Steklov eigenvalue problem. If the Gaussian curvature satisfies K ≥ -α and the geodesic curvature satisfies k_g≥ c > 0, then the first eigenvalue σ _1 satisfies σ _1 + α/σ _1≥ c, where equality holds if and only if the surface is a Euclidean disk of radius 1/c and α = 0. We also prove that the first eigenvalue of a fourth-order Steklov-type problem is bounded below by 2c under nonnegative Gaussian curvature, with equality again characterizing the Euclidean disk.
In this work, a modified numerical scheme is developed based on the scale-3 Haar wavelet for solving elliptic partial differential equations (PDEs) that describe the Helmholtz equation and 3D Poisson equation. The spatial derivatives are discretized scale-3 Haar wavelet expansions, which are then integrated and extended to a two and three dimensional solution via Kronecker tensor product, incorporating boundary conditions through integration constants. Furthermore, theoretical convergence analysis of the proposed method is discussed and supported with numerical evaluation of maximum absolute errors (L_∞), mean squared errors (L_2), and the computational convergence rate across resolution levels, validating the method’s convergence. Computational simulations are executed using MATLAB programming. The wavelet method is compared with the existing finite difference method and scale-2 Haar wavelet method, and the results demonstrate that while all three approaches effectively solve elliptic PDEs, the scale-3 Haar wavelet method outperforms the others by delivering more accurate approximate solutions with greater efficiency. The results of this investigation establish the potential and reliability of Haar wavelet methods for solving intricate PDEs in various engineering domains.
