AI-enhanced analytical and numerical solutions for fractional cancer tumor models in older adults using healthcare technologies

Instability of MHD Poiseuille flow of nanoparticles FeO in water cylinder flow

This study presents efficient direct numerical solutions for the nonlinear Volterra-Fredholm integro-differential equations (NVFIDEs). The main component of the numerical scheme is the collocation method, which uses shifted Chebyshev polynomials. The method converts the given NVFIDEs into a system of linear/nonlinear algebraic equations with unknown coefficients of the truncated shifted Chebyshev series. With the aid of Maple, unknown coefficients are obtained, and so, the desired approximation solution is achieved. Several numerical examples are presented to assess the accuracy by comparing them with other existing techniques in the literature and to evaluate the method's efficiency. From the tables and figures, the examples reveal that only a small number of shifted Chebyshev series terms are needed to achieve small errors.

ABSTRACT In this study, linear instability and nonlinear stability analyses are performed to investigate double‐diffusive convection driven by radiation absorption and magnetic field effects. The Rayleigh‐Bénard configuration with realistic rigid boundaries is considered, in which the fluid layer is heated and salted from below. The mathematical model comprises the Navier‐Stokes equation for an incompressible fluid, incorporating a Kelvin‐Voigt term that describes viscoelasticity, and is coupled with convection‐diffusion equations for energy and solute concentration. Linear instability thresholds are determined using the Fourier mode approach, while nonlinear stability analysis is carried out by using the energy method. The resulting eighth‐order differential eigenvalue problems from both linear and nonlinear theories are solved using the Chebyshev Collocation Method. The nonlinear analysis produces critical Rayleigh numbers that closely match those obtained from linear instability theory. However, the discrepancy between linear and nonlinear stability thresholds indicates the existence of a subcritical instability region. The Kelvin Voigt and Hartmann numbers are found to exert a stabilizing influence, whereas radiation absorption significantly destabilizes the onset of convection.

Purpose The purpose of this study is to analyse double-diffusive convection in a fluid-saturated porous layer using an extended Darcy–Brinkman model with higher-order (bi-Laplacian) thermal and solutal diffusion and to determine linear and nonlinear stability thresholds. Design/methodology/approach The governing equations are nondimensionalised and linearised about the conduction state to obtain the perturbation equations. Linear instability analysis was performed, and a nonlinear energy stability analysis was developed to determine unconditional decay thresholds for perturbations. The instability and nonlinear thresholds are computed using two high-accuracy Chebyshev collocation methods (standard and boundary-fitted). Findings Brinkman viscous diffusion and higher-order thermal/solutal diffusion act predominantly as stabilising mechanisms: they increase the critical Rayleigh numbers, reshape the neutral curves and shift the stationary–oscillatory transition in parameter space. The nonlinear (energy) threshold is consistently lower than the linear threshold, identifying a conditional-stability interval RaE < Ra < RaL in which linear stability holds but unconditional nonlinear decay is not guaranteed by the present energy estimate. In the top-heavy solutal configuration, higher-order solutal diffusion provides the strongest suppression of solutal-driven fingering. Both numerical schemes exhibit spectral convergence; however, the boundary-fitted method achieves smaller residuals at the same truncation order, indicating improved accuracy and efficiency. Originality/value To the best of the authors’ knowledge, this is the first combined linear/energy stability study of thermosolutal convection in a Darcy–Brinkman porous layer with higher-order thermal and solutal diffusion, supported by spectrally accurate Chebyshev collocation schemes for the associated high-order eigenvalue problems.

ABSTRACT This investigation explores the influence of a constant internal heat source on Darcy–Bénard–Brinkman convection in a porous medium under local thermal non‐equilibrium (LTNE) conditions. Both linear and nonlinear stability analyses are conducted for rigid‐rigid isothermal boundaries. This work presents a detailed analysis using a high‐accuracy Chebyshev collocation method under these boundary conditions. The Chebyshev method, with its high spectral resolution, effectively captures sharp boundary layer structures and higher‐order thermal interactions, providing a physically realistic representation of the system dynamics. The study reveals that the constant internal heat source can trigger subcritical instability. In addition, the inter‐phase heat transfer coefficient and the Darcy–Brinkman number exert stabilizing effects, whereas the porosity‐modified conductivity ratio and the internal heat source parameter promote instability. These findings highlight the intricate interplay between internal heating and LTNE conditions and emphasize the necessity of high‐fidelity numerical methods such as the Chebyshev approach for accurately capturing such behaviors.

Chebyshev Collocation Method for Time Fractional Generalized Equal Width Model Equation in Shallow Water Channels

The normal mode model is widely used to solve underwater acoustic propagation problems in horizontally layered waveguides. However, ocean elastic bottoms are often modeled as fluids or replaced by an equivalent reflection coefficient in most current models; for this reason, the solution of elastic modes and seismic wavefields cannot be solved. To overcome this constraint, a solving method utilizing the Chebyshev collocation method (CCM) for the seismo-acoustic normal mode model within arbitrarily layered inhomogeneous fluid-solid media is proposed. The CCM is applied to discretize the modal equations of compressional wave (P-wave) and shear wave (S-wave), along with the boundary and interface conditions. A complex matrix eigenvalue system is established by restructuring the global discrete modal equation matrix, on which proper interface conditions (fluid-solid and solid-solid) as well as boundary conditions have been enforced. Solving this eigenvalue problem yields the horizontal wavenumbers and the full-depth modal functions; the full-wavefield is then constructed by summing up contributions of each of the modes. Numerical experiments demonstrate that the proposed method accurately computes both the acoustic and seismic wavefields. Moreover, detailed analyses reveal that its accuracy is comparable to that of rpress and significantly surpasses that of kraken and krakenc.

The Chebyshev collocation method with trigonometric transformation is a spectral numerical solution technique for initial and/or boundary value problems in computational mechanics. This approach leverages the trigonometric representation of Chebyshev polynomials to enable analytic differentiation, eliminating the need for numerical differentiation matrices. This technique has demonstrated superior accuracy and computational efficiency across diverse applications in engineering and physical sciences. This paper presents a comprehensive analysis of the trigonometric Chebyshev collocation method (TCM) and its application to four diverse problems: heat transfer in a triangular fin, torsion of a rectangular shaft, nonlinear heat conduction with temperature-dependent conductivity, and an integro-differential equation. Our results demonstrate that the TCM achieves spectral accuracy with significantly fewer computational nodes compared to conventional finite difference, and finite element methods, making it a viable alternative for problems requiring high-precision solutions.

ABSTRACT We consider heat transfer by natural convection with radiation effects on a rough, wet, semispherical fin. The wetness on the fin surface with varying roughness, compounded with the internal heat generation and their influence on the fin's thermal profile and heat transfer rate, is investigated. The governing equation of the model is nondimensionalized, and the resulting nonlinear differential equation is numerically solved by the Keller‐box method and validated using the Chebyshev collocation method subject to quasilinearization. The robustness of these numerical methods is tabulated, and the impact of the critical dimensionless parameters on the thermal profile of the extended surface and the heat transfer rate along it is analyzed graphically and quantitatively. Fin efficiency is also computed, and the influence of the pertinent parameters on it is inferred. The study reveals that the fin's thermal profile is enhanced with a surge in the generation number, temperature‐dependent heat generation parameter, surface roughness, ambient temperature, and exponent of the convective heat transfer, while there is a significant drop in the heat transfer rate with a surge in these parameters. The wetness, convection, and radiation parameters assist the heat transfer rate throughout the fin and degrade its thermal profile.

Abstract This paper presents and establishes the Chebyshev collocation method, which generates numerical solutions for nonlinear fractional partial differential equations such as the fractional diffusion, wave, and Korteweg–De Vries equations. To obtain the novel fractional derivative operational matrix of the shifted Chebyshev polynomials, new theorems and lemmas have been proved. Caputo’s fractional-order derivative definition is used to represent the fractional-order terms. This approach transforms the problem under discussion into a nonlinear algebraic system of equations that Newton’s method is applied to solve numerically. The error analysis is determined to justify the proposed technique. Several numerical examples are provided to illustrate the accuracy and applicability of the suggested method. The estimated, residual, and absolute errors are computed for each numerical example, and comparisons with other approaches are shown to strengthen the reliability and effectiveness of the suggested method.

This paper presents a comprehensive investigation of the free vibration characteristics of beams with variable cross-sections and elastically restrained boundaries using a high-precision Chebyshev collocation method. The methodology employs Chebyshev–Gauss–Lobatto collocation points with cosine transformation and direct analytical differentiation formulas to reduce the governing differential equations to algebraic eigenvalue problems. Three distinct structural configurations are analyzed: (i) linearly tapered beams with flexible ends representing non-ideal structural connections, (ii) beams with exponentially varying properties and damaged boundaries for structural health monitoring applications, and (iii) functionally graded beams with general elastic constraints relevant to advanced material systems. The implementation utilizes multiprecision arithmetic to ensure numerical stability and reliable eigenvalue separation. Extensive validation against three independent analytical and numerical solutions demonstrates exceptional agreement. The method requires only 20–25 collocation points compared to 40–60 nodes typically needed in finite element or differential quadrature methods, while maintaining spectral accuracy. Natural frequencies accurate to six decimal places are presented for various boundary conditions, different taper, varying damage parameters, and different material gradation indices .The results provide valuable benchmark data for structural design optimization, damage detection algorithms, and validation of commercial finite element software.

Singularity-removing Chebyshev collocation methods for nonlinear fractional differential equations with blow-up

Perforated microbeams are widely employed in micro- and nano-electromechanical systems due to their lightweight structures and tunable mechanical properties. At small scales, however, classical elasticity fails to capture size effects and porosity-induced stiffness variations. This study develops a comprehensive framework for the static, dynamic, and electrostatic behavior of perforated Timoshenko microbeams with graded porosity, based on the Chebyshev collocation method. The model integrates nonlocal elasticity to account for size effects, graded porosity distributions to represent realistic microstructural variations, and electrostatic actuation to analyze pull-in instability. Validation against published results confirms the accuracy and efficiency of the proposed approach. Parametric investigations reveal that porosity distribution strongly affects stiffness and deflection responses, while size-dependent effects enhance rigidity and increase natural frequencies. Electrostatic pull-in voltage is found to increase with both porosity ratio and length scale parameter, indicating improved stability for micro- and nano-electromechanical systems devices. The findings demonstrate that the Chebyshev collocation method framework provides a robust and efficient tool for the design and optimization of next-generation perforated microstructures.

This study presents a rigorous and in-depth comparative analysis of the Chebyshev Collocation and Spectral Galerkin methods for solving nonlinear singular initial value problems, which are fundamental in modeling astrophysical phenomena, with a focus on the Lane-Emden equation as a model test case, by applying both methodologies, performance was evaluated based on precise metrics including accuracy, rate of convergence, computational efficiency, the results unequivocally demonstrate the superiority of the Spectral Galerkin method across all critical performance aspects, the method achieved true exponential convergence for smooth solutions, reaching a precision approaching machine accuracy using significantly lower approximation degrees than the collocation method. Furthermore, the Galerkin method exhibited superior robustness and a higher algebraic rate of convergence in cases where the solution is less regular, while maintaining excellent numerical stability and better-conditioned algebraic systems, we conclude that the integral nature of the orthogonal projection in the Galerkin method endows it with a structural advantage

A micropolar Maxwell fluid's unsteady two-dimensional flow over a stretching surface embedded in a porous medium, subject to thermal radiation, heat generation/absorption, suction/injection, and microstructural parameters, is studied in this work. The governing nonlinear partial differential equations (PDEs) are subsequently converted into a system of coupled ordinary differential equations (ODEs) by means of appropriate similarity variables after being derived from considerations of momentum, energy, and micro-rotation. The highly efficient and robust Chebyshev-collocation method is used to solve these equations numerically, and a comprehensive parametric analysis is performed to examine the detailed effects of various physical parameters, such as the unsteadiness parameter, suction/injection, elasticity, material parameter, resistance of the porous medium, and radiation on the flow and thermal fields. Not only do the findings indicate how the temperature, velocity, and angular velocity profiles are controlled, but they also show how these factors influence the modified skin friction and the Nusselt number.

The integration of photovoltaic-thermal (PVT) systems with advanced cooling techniques is crucial for improving energy efficiency and overcoming performance limitations in solar energy conversion. This study investigates the combined effects of solar thermal radiation, Cattaneo– Christov non-Fourier heat transport and magnetohydrodynamic (MHD) rotating flow on PVT cooling performance. The purpose is to provide a deeper understanding of how thermal and electromagnetic interactions influence heat transfer and system efficiency, particularly under conditions where conventional Fourier-based models fall short. A carboxymethylcellulose (CMC)–water-based base fluid enhanced with nanoparticles zirconium oxide (ZrO 2 ), copper (Cu) and aluminum oxide (Al 2 O 3 ) is employed as the working medium to further boost thermal conductivity and system efficiency. A mathematical model is developed to capture the dynamics of radiative heat flux, magnetic field, rotation and nonlinear thermal relaxation effects. The governing equations are transformed into dimensionless form and solved numerically using a Chebyshev collocation method. The results demonstrate that the inclusion of ZrO 2 , Cu and Al 2 O 3 nanoparticles in the CMC–water base fluid significantly improves thermal conductivity, thereby enhancing the system’s cooling capacity. Under solar thermal radiation, ternary hybrid nanofluids demonstrate approximately 20% superior thermal regulation. The novelty of this work lies in the synergistic analysis of MHD rotating flow, Cattaneo–Christov heat transport and nanoparticle-engineered CMC-based fluids for PVT cooling, providing new physical insights and practical guidelines for the design of high-performance renewable energy systems.

Abstract Cancer remains a critical global health challenge, causing severe pain and high mortality rates. Conventional treatment approaches—such as surgery, chemotherapy, and radiotherapy—often damage healthy cells, posing significant limitations. This study presents a mathematical investigation of a targeted drug delivery system (TDDS) employing magnetohydrodynamic (MHD) tri‐hybrid nanoparticles to enhance precision in cancer therapy. The model incorporates the Cross non‐Newtonian fluid framework and accounts for key physical effects, including viscous dissipation, Joule heating, porous medium resistance, and heat generation, alongside thermal radiation and MHD influences. The governing partial differential equations (PDEs) are transformed into a system of ordinary differential equations (ODEs) through suitable similarity transformations. The resulting ODE system is solved using the Chebyshev collocation method (CCM). Findings indicate that thermal radiation, enhanced thermal conductivity, and heat generation significantly improve heat dispersion, thereby increasing the effectiveness of the targeted delivery mechanism within porous tumor tissues. The Nusselt number increases with the radiation parameter (Nd) for all nanofluid types, showing enhanced heat transfer with stronger radiative effects. For the nanofluid, Nu rises from 0.17138 to 0.69760 with higher impacts of Nd. For the di‐hybrid nanofluid, it increases from 0.24767 to 0.75657 over the same range. The tri‐hybrid nanofluid records the largest gain, from 1.24988 to 1.96246. This trend confirms that tri‐hybrid nanofluids consistently deliver superior thermal performance compared to mono and di‐hybrid types. This work offers new insights into optimizing nanoparticle‐assisted TDDS for improved cancer treatment outcomes.

Chebyshev Collocation Method for Fractional Temporary Heat Transfer Model in Composite Cylinders with Hollow Interiors

ABSTRACTThis paper introduces a highly accurate method for solving nonlinear differential equations. The proposed approach is applied to the analysis of magnetohydrodynamics (MHD) Jeffery–Hamel nanofluid flow within a porous medium. The effects of the Hartmann number, the Reynolds number, nanofluid volume fraction, and porosity on the velocity profiles in both divergent and convergent channels are thoroughly examined. Comparative results with the previous studies are presented to validate the accuracy and applicability of the method. The findings demonstrate that the proposed technique yields precise results that are consistent with those reported in the literature.