The Whitham–Broer–Kaup (WBK) model describes tsunami type wave pattern due to involvedness of dispersion, dissipation as well as weedy nonlinear effects. As the wave evolutions rise with time, we are going to investigate time varying wave solutions of the nonlinear model. To do that this article applied modified simple equation (MSE) and enhanced modified simple equation (EMSE) schemes on the WBK model to extract time varying wave pattern. As a result, we found the time varying solitary wave solutions as kink and anti-kink, kinky-periodic, parabolic-periodic, multi-period and periodic wave for some free parameters with their numerical values carried some massive soliton wave which can produces tsunami. Some plentiful 3D figures, density plot and counter plot are sketch from the obtained solutions to understand the topic and their description are deliberated properly. By analyzing the achieved results, it is vibrant that the EMSE and MSE scheme is informal, reliable and effective to arise the time varying wave evolutions of such model applicable in water-wave mechanism as tsunami type massive wave.
The behavior of solitons in multimode fibers is investigated using the Connected Higher-Order Nonlinear Schrödinger Equations (CHNSE). A sophisticated algebraic technique is employed to analyze the CHNSE, yielding a variety of exact solutions, including bright and dark solitons, Weierstrass elliptic function solutions, periodic singular solutions, and hyperbolic and Jacobi elliptic solutions. These solutions are graphically represented using density and three-dimensional graphics to illustrate their physical characteristics. The CHNSE model provides deeper understanding of nonlinear optical processes in multimode fibers, essential for high-speed communication systems. The findings contribute to the development of new optical communication technologies.
The paper is devoted to the study of a direct problem with time-nonlocal conditions and inverse problems with integral overdetermination conditions for the equation of transverse vibrations of a beam. Necessary and sufficient conditions for the existence of a solution of the direct problem are obtained. The inverse problem consists of recovering an unknown time-dependent coefficient from an additional integral overdetermination condition. Using the eigenvalues and eigenfunctions of the corresponding beam vibration operator, both the direct and inverse problems are reduced to integral equations. The existence and uniqueness of the solution to the inverse problem are established by Banach’s contraction mapping principle.
This study proposes an analytical-numerical model to investigate wave resonance attenuation in hybrid coastal protection systems consisting of mangroves and trapezoidal breakwaters. The proposed model is adopted from the modified Linear Shallow Water Equations, which include the effect of vegetation-induced drag and the breakwater geometry. An analytical solution is derived to determine the natural period of the basin, whereas a staggered finite volume method is used to simulate wave propagation and resonance. The results show that mangroves can suppress resonance by themselves when sufficient vegetation-induced friction is provided, while the combination of mangroves and a rough trapezoidal breakwater can achieve resonance mitigation with a lower roughness requirement. Further sensitivity analyses demonstrate that wider mangrove belts and a shorter distance between the mangroves and the breakwater are more effective in attenuating waves. These results provide practical guidelines for the design of sustainable hybrid systems for coastal defence capable of alleviating resonance-induced wave amplification.
In this paper, we present an effective approach for constructing Lyapunov functions for a specific class of partial differential equations (PDEs) involving the fractional p-Laplacian operator, both with and without delay. We systematically examine special cases of classical and fractional spatial diffusion equations falling under this class. Additionally, the proposed method is applied to a compartmental epidemic model with general incidence rate as well as to a nonlinear delayed system describing the spread of disease in a population under an anomalous diffusion.
This paper aims to identify the most effective fractional derivative for image denoising within an anisotropic regularized Perona–Malik framework, and to clarify whether non-singular-kernel or singular-kernel operators perform better in this setting. To address this, we present a comparative analysis of five fractional derivatives. We systematically evaluate both singular-kernel operators (Caputo and Caputo–Hadamard) and non-singular kernel operators (Caputo–Fabrizio, Atangana–Baleanu, and Caputo–Odibat). For each model we develop specialized numerical schemes incorporating an implicit–explicit (IMEX) parameter to enhance stability and flexibility, along with tailored finite-difference solvers that respect each operator’s mathematical characteristics. The stability of all proposed numerical schemes is rigorously analyzed using the von Neumann method. The denoising performance is rigorously quantified using three complementary metrics: Peak Signal-to-Noise Ratio (PSNR) for reconstruction accuracy, Structural Similarity Index (SSIM) for perceptual quality, and a novel adaptation of Chamfer Distance (CD) that treats images as 3D point clouds to measure structural and intensity discrepancies simultaneously. We further benchmark our best-performing model against the classical Perona–Malik, Catté, and Weickert models as well as existing fractional diffusion-based denoising models that generalize the Perona–Malik equation, specifically the LF, EHAB and Kassimi models. Quantitative evaluations on grayscale, medical, and color test images under Gaussian noise reveal the superior performance of specific fractional operators in achieving an optimal balance between noise suppression and feature preservation, and demonstrate that our model is competitive with, and in several cases outperforms, these established approaches.
This paper investigates the stability properties of a variable-order fractional (VOF) reaction-diffusion (RD) epidemic model based on the SIR framework. Unlike existing models that assume constant fractional order (FO) or ignore spatial effects, our model incorporates a time-varying Caputo fractional derivative to capture evolving memory effects (e.g., waning immunity or behavioral changes) together with spatial diffusion to represent population mobility. We conduct a detailed analysis of both local and global asymptotic stability (LAS and GAS) for the disease-free and endemic equilibria (DFE and EE). Using linearization and novel Lyapunov functionals (LFs) adapted to the VOF-RD setting, we derive sufficient stability conditions with and without diffusion. Numerical simulations validate the theoretical results, demonstrating convergence under homogeneous Neumann boundary conditions. The study extends classical integer-order models to the VOF domain and provides new insights into the interplay between time-dependent memory, spatial spread, and epidemic outcomes.
In this paper, we investigate the Cauchy problem for modified Korteweg de Vries-sine-Gordon equation with a Riesz fractional-order derivative containing time-dependent variable coefficients(vcRfmKdV-sG). We show that the inverse scattering transform(IST) method can also be used to obtain a soliton solutions of the vcRfmKdV-sG equation. We have determined the time evolution of the scattering data for the Ablowitz-Kaup-Newell-Segur system associated with the solution of the vcRfmKdV-sG equation. Then, using the solution of the inverse scattering problem with respect to the time-dependent scattering data, we recover the desired solution of the vcRfmKdV-sG equation. For the one- and two-soliton cases, explicit formulas for the solutions of the problem under consideration are derived as examples to simulate their spatial structures and analyze their structural properties by selecting different variable coefficients and fractional orders. The results indicate that both the variable coefficients and the fractional order significantly influence the velocity of soliton propagation, while no energy dissipation occurs throughout the entire motion process. This study contributes valuable insights into nonlinear dynamics and soliton behavior, improving the understanding of multifaceted wave interactions in nonlinear fractional equations.
This study presents a numerical investigation of free convection-driven melting of nano-encapsulated phase change material (NEPCM) within a semi-bottle-neck-shaped porous enclosure. The cavity is differentially heated, with the left vertical wall kept hot, the right wall kept cold, and the remaining walls adiabatic. The enthalpy–porosity method is employed to model the melting process, while the Darcy–Brinkman model governs flow within the porous matrix. Parametric analyses are performed across a range of Rayleigh numbers (Ra = 10³–10⁵), fusion temperatures (θf = 0.1–0.9), Stefan numbers (Ste = 0.1–0.9), nanoparticle volume fractions (ϕ = 0–0.05), Darcy numbers (Da = 10⁻4–10⁻1), and Frank-Kamenetskii numbers (Fk = 0–1). Results reveal that increasing the nanoparticle volume fraction enhances thermal conductivity, leading to an improvement of up to 12.5% rise in the mean Nusselt number (Nua). A rise in fusion temperature θf from 0.1 to 0.9 leads to a 5.7% reduction in Nua. Stefan number increases from 0.2 to 0.7, producing a 9% decline in Nua, highlighting the latent heat damping effect. The Frank-Kamenetskii number, which characterizes internal heat generation due to exothermic reactions, significantly influences thermal transport: an increase in Fk from 0 to 1 decrease Nuaby approximately 7.7%, indicating thermal thickening and slower melting fronts. Additionally, the semi-bottle-neck geometry intensifies localized circulation, while the application of a magnetic field attenuates convection, offering a controllable method to regulate thermal performance. These insights are vital in developing NEPCM-based phase change thermal energy storage systems in porous environments.
This article evaluates the Soret-Dufour diffusion, Joule heating and heat-dissipation effects on two-dimensional unsteady gravity-driven electromagnetic flow of Casson nanofluids over a vertically slanted cone entrenched in a porous medium with variable thermal source/sink and higher-order chemical reactions. Specifically, two distinct nanofluids are synthesized by mixing two types Ag and Cu nanoparticles with the blend of water and ethylene-glycol as the base fluid. The explicit DuFort-Frankel finite difference scheme has been executed to discretize and solve the ultimate partial differential system. The research findings uncovered that the temperature field noticeably increased for both nanofluids by the Eckert parameter, radiation, Soret and Dufour impacts but it decays with nanoparticles capacity fraction. The Soret parameter and chemical reaction order elicited to expand concentration field for both nanofluids but chemical reaction and Schmidt number revealed opposite effect. Likewise, the velocity field lowered for both nanofluids by the cone’s inclination angle, magnetic intensity and Casson parameter whereas the porosity, radiation and Eckert number caused to inflate the velocity field for both nanofluids. The surface-friction improved for both nanofluids by the magnetic field, Casson and Dufour parameters but it was decayed by the nanoparticle’s concentration. An upsurge in magnetic field, thermal source and sink caused to downgrade Nusselt number for both nanofluids but the nanoparticle’s volume fraction exhibited opposite trend. The Sherwood number for both nanofluids enhanced with Schmidt number but it was decreased with chemical reaction and nanoparticles volume-fraction.
The present study theoretically investigates the mixed convection flow and heat transfer of a Ree-Eyring non-Newtonian fluid in a vertical channel driven by rhythmic membrane contraction. The main objective of this work is to understand how variable fluid properties, thermal effects, and non-Newtonian behaviour influence fluid motion, pumping performance, and heat transfer in membrane-based microfluidic systems. The pressure gradient induced by a moving membrane, combined with buoyancy forces, drives the fluid through a vertical microchannel. To capture realistic thermal behaviour, temperature-dependent viscosity and thermal conductivity are incorporated into the model. Variable viscosity and variable thermal conductivity are considered to accommodate variable fluid properties. The basic nonlinear equations are then derived using the assumptions of a small Reynolds number, a long wavelength, and the lubrication approximation. Because of the nonlinearity of the governing equations, a regular perturbation technique is developed to obtain semi-analytical expressions for the velocity and temperature distributions. From these obtained velocity and temperature fields, the pressure gradient, skin friction, Nusselt number, flow rate, and stream function were then evaluated. Extensive parametric analysis has been performed in MATLAB R2024b to investigate the influence of variable viscosity, variable thermal conductivity, the heat-source parameter, the Grashof number, and the Ree-Eyring fluid parameter on the flow and thermal characteristics. Results indicate that variable viscosity improves flow behaviour, whereas heat generation and buoyancy forces significantly affect pumping and heat characteristics. The present findings provide useful guidance for the design and optimisation of thermally controlled micro-pumps, MEMS cooling devices, lab-on-chip technologies, and biomedical microchannels that require accurate regulation of fluid flow and heat transfer.
This paper presents an a-priori and a-posteriori error analysis for the time-dependent case of a coupled continuum pipe-flow in three dimensions on anisotropic meshes (i.e. elements with very large aspect ratio). Our investigation includes nonconforming discretizations as well as different elements and different versions of Crouzeix-Raviart finite element methods. Under suitable regularity assumptions on the exact solution, we derive optimal a priori error estimates in the H1-like norm. Leveraging residual elements, local error indicators and a global estimator are generated, demonstrating reliability and efficiency. The efficiency of error estimators holds unconditional, though its reliability involves the alignment measure notion. Mesh requirements are established to ensure the optimality of the error estimator.
The present article aims to investigates the characteristics of flow of blood through a porous bifurcated artery by assuming blood as a micropolar fluid. The effects of porous wall and branching of artery are assimilated to account for the physiological conditions includes permeability and resistance to flow. Artery under consideration is assumed to be symmetric about it’s own axis and circular cylinders of restricted length. Partial differential equations governing the flow are formulated with reference to micropolar fluid framework in cylindrical polar co-ordinate system. These equations are made non-dimensional and coordinate transformation is used to convert the irregular geometry to a regular. The transformed equations along with concern boundary conditions are initially solved by finite difference scheme. A feedforward neural network is trained utilizing the Levenberg - Marquardt backpropagation method to model responses for various parameter alterations on a reference dataset produced by the numerical solver. Present study concentrates on the influence of Darcy number on physical parameters. These results shows enhanced permeability and lower resistance, resulting in higher velocity, increased flow rate, more stable shear stress distribution, and decreased impedance, thereby resembling healthy arterial flow behavior. Findings of this work provide potential implications for understanding pathological conditions such as atherosclerosis and stenosis in porous arterial segments.
This current paper examines the magnetohydrodynamic (MHD) Williamson nanofluid flow over a stretching superficial in the influence of thermal radiation, viscous dissipation, heat generation, and chemical retort force. Boundary layer estimates of the momentum, energy and the concentration equalities are used to derive the mathematical archetypal of the fluid flow. After suitable similarity transformations, the regulatory non-linear partial differential equalities are transmuted into a classification of ordinary differential equations. The MATLAB inbuilt solver used bvp4c is a Lobatto IIIA based solver to solve these equations numerically. The stimulus of numerous corporal factors of magnetic constraint (M), Williamson parameter (λ), radiation stricture (Rd), Prandtl numeral (Pr), Eckert numeral (Ec), heat generation parameter (S), thermophoresis constraint (Nc), Schmidt number (Sc), Brownian motion parameter (Nb), and chemical retort constraint (Cr) on the velocity, temperature, and concentration distributions are premeditated. The change in skin friction coefficient, Nusselt number and Sherwood numeral are likewise discussed using tabular and graphical formats. The findings indicate that magnetic actions decrease the velocity field, thermal radiation and dissipation due to viscous actions improve the temperature distribution and the nature of the mass transmission is greatly affected by the Schmidt numeral, Brownian gesture and chemical reactions constraint. These results demonstrate the significance of the governing constraints in regulating the process of heat and mass transfer in Williamson nanofluid flows in reference to engineering use.
This paper presents a new approach for solving a class of boundary value problems for the 1+1-dimensional Kardar-Parisi-Zhang equation on generic bounded intervals. The Hopf-Cole transformation is used to solve the given problem, based on a closed-form solution for the linear reaction-diffusion equation on finite intervals, published recently. The exact solution to the deterministic equation with fixed Neumann boundary conditions is derived in the time domain using inverse Laplace transforms. Even if analytic inverses cannot be found in transform tables, highly efficient algorithms are available. Numerical inverses in the time domain are always feasible, regardless of the complexity of the Laplace domain expressions. In comparison with solutions derived using series expressions or numerical methods, closed-form solutions, even in the Laplace domain, offer novel insights. Furthermore, the utilization of the numerical inverse Laplace transforms is shown to be a more efficient computational approach, thereby establishing a reference point for numerical and semi-analytical methods.
This study examines an innovative mathematical framework for magnetohydrodynamic (MHD) nanofluid passages over an angled cone dipped in a porous substance, highlighting the combined effects of radiation, heat dissipation, chemical reactions, and nanoparticle transport. The research novelty lies in capturing the flow incited by Brownian motion and thermophoresis, integrated with cross–diffusion effects, offering a unified outline for the analysis of complex conical MHD nanofluid systems. The governing steady, nonlinear partial differential equations are transformed via similarity techniques into dimensionless form and dealt with using the high–precision finite element method. Detailed parametric analysis reveals the effects of the key dimensionless parameters on the velocity, temperature, and concentration fields, as well as quantitative valuations of the wall shear stress, Nusselt and Sherwood numbers. The results indicate that greater thermal radiation, cross–diffusion, heat dissipation, and porosity impact fluid flow, whereas an increased Prandtl number and angle inclination decrease it. The presence of a magnetic field diminishes fluid velocity, whereas its absence accelerates fluid velocity. The dominance of Brownian motion, thermophoresis, viscous dissipation, Dufour, and radiation effects in the flow led to enhanced nanofluid temperature. Likewise, thermophoresis and Soret effects caused the fluid concentration to evolve, whereas Brownian motion and the Lewis number condensed it. The wall friction increases with thermophoresis, Brownian motion, and radiation, but decreases with increasing viscous dissipation and porosity. Furthermore, the model is validated against reference solutions, demonstrating its accuracy and its ability to inform the design and optimization of thermal systems, reactive flow dynamics, and porous media engineering processes.
This work investigates how Hall currents influence the convection flow of a hydromagnetic nanofluid past a porous stretching cylinder through hollow tubes in an artificial kidney. Dialysis removes more metabolic excretory products and naturally occurring nitrogenous waste from a patient’s blood by using a man made device, an artificial kidney. It is used in patients whose both natural kidneys are damaged and can no longer filter blood. In this study, blood has been considered an incompressible biofluid that flows during hemodialysis. When a magnetic flux is applied perpendicularly in the direction of the porous cylinder that functions as the dialyzer for the prosthetic kidney, the flow becomes irregular. The study’s novelties include the use of Hall currents and nanoparticles to the blood taken from the patient during dialysis in an artificial kidney to enhance the nanofluid’s thermal transfer rate, its dynamic viscosity, heat conduction coefficient, and thermal penetration rate. This study aims to improve hemodialysis sessions, which usually last two to six hours. In an artificial kidney, magnetically influence flow of convection fluid with suspension of sized nanoparticles via hollow membranes across a porous, stretched cylinder influenced by Hall currents is controlled by coupled and nonlinear equations. The Crank-Nicolson method is implemented to tackle the set of coupled and non-linear system equations. The findings obtained after MATLAB simulation are then shown graphically and discussed. It has been noted when a patient’s blood is filtered by an artificial kidney, dialysis time, temperature, and nanofluid velocity can all be enhanced by increasing Hall currents, adjusting the stretching cylinder parameter as a heat source, and adding nanoparticles to the base fluid.
The article constructs an explicit solution to the Cauchy problem for a high-even-order equation involving a fractional derivative in the sense of Jrbashyan-Nersesyan.
This paper presents a nonlinear, non-Markovian model of subdiffusive transport influenced by a chemotactic gradient affecting cellular mobility. In the model, both the stochastic waiting time and the escape rate are modulated by the chemotactic gradient. We derive the subdiffusive fractional master equation, examine its diffusive limit, and implement Monte Carlo simulations to analyse particle transport under varying chemotactic conditions. Simulation results show that, in the absence of chemotaxis, particles exhibit symmetric subdiffusion consistent with the standard continuous time random walk behaviour. A constant chemotactic gradient has little effect on particle distribution, whereas spatially varying gradients—linear or quadratic—produce pronounced effects. Specifically, particles drift away from regions with high chemotactic intensity and tend to aggregate in areas of minimal chemotactic influence, with the strongest aggregation observed under quadratic gradients. These findings highlight the significant role of spatially dependent chemotaxis in shaping anomalous subdiffusive transport dynamics.
