Minimizing entropy generation and optimizing nanofluid performance are critical strategies for improving thermal management in advanced energy systems, including nuclear reactors, aerospace platforms, and automotive heat exchangers. This study examines the thermofluidic behavior of a Casson trihybrid nanofluid within an accelerating microchannel, focusing on entropy generation influenced by Darcy-Forchheimer drag effects and Cattaneo-Christove heat flux. The governing system of coupled nonlinear partial differential equations is transformed into a set of ordinary differential equations utilizing similarity transformations and solved numerically using the spectral local linearization method. Convergence and validation evaluate approve the accuracy and stability of the approach through comparison with established literature. Entropy generation is assessed with thermal conduction, viscous dissipation, and Joule heating as primary irreversibility mechanisms. Parametric investigations exhibit that the Brinkman number significantly improves the temperature distribution, while the Darcy-Forchheimer parameter reduces it due to increased flow resistance. The Reynolds number is shown to advance entropy generation, indicating greater thermodynamic irreversibility at higher velocities. Conversely, the Bejan number raises with both thermal radiation and Reynolds number, while declines under increased magnetic field strength and viscous dissipation effects. Furthermore, multiple linear regression and sensitivity analyses are employed to assess the influence of critical dimensionless parameters, posing statistical validation of the model's predictive capability. The findings emphasize the advanced thermophysical performance of trihybrid nanofluids and provide constructive insights into optimizing entropy generation and heat transfer characteristics in nanofluid-based thermal systems under complex flow conditions.

Fractional reaction–diffusion models provide a powerful framework for describing dynamical systems in which memory effects and spatial interactions are significant. In this study, we develop and analyze a spatially heterogeneous time-fractional reaction–diffusion model for the coupled dynamics of cocaine and heroin abuse. Memory effects associated with addiction persistence, delayed behavioral responses, and relapse are incorporated via Caputo time-fractional derivatives, whereas spatial heterogeneity is represented by space-dependent diffusion coefficients, including both smoothly varying diffusion profiles and multi-patch configurations that capture regional disparities. To efficiently approximate the resulting nonlinear fractional system, we propose a fractional extension of the Parker–Sochacki method for time-fractional reaction–diffusion equations with heterogeneous diffusion. The proposed scheme provides an explicit and computationally efficient numerical algorithm that significantly reduces the substantial memory requirements typically associated with fractional-order models. The mathematical properties of the model are investigated rigorously. The existence, uniqueness, positivity, and boundedness of solutions are established using the theory of sectorial operators, and the equilibrium states are characterized. Furthermore, global stability conditions are derived using appropriate Lyapunov functionals. Numerical simulations validate the proposed approach and demonstrate the combined effects of memory and spatial heterogeneity on cocaine–heroin dynamics. In particular, lower fractional orders are shown to promote prolonged persistence of substance abuse, while heterogeneous diffusion generates pronounced nonuniform spatial distributions. The proposed framework integrates methodological advances with epidemiological insight, offering a robust tool for studying substance-use dynamics in spatially heterogeneous environments.

This study presents a novel extension of the Parker-Sochacki method for the numerical solution of the fractional-order Brusselator model, formulated in both ordinary differential equation (ODE) and partial differential equation (PDE) settings. The Brusselator is a canonical model for oscillatory behavior in chemical and biological systems, and its fractional formulation captures memory effects in such dynamics. For the PDE case, the method is combined with the method of lines for efficient treatment of spatially extended systems. A multistage implementation is developed using the Caputo fractional derivative, with an a priori step-size selection based on convergence analysis to ensure stability and accuracy. Numerical simulations confirm the method's effectiveness across different fractional orders. Comparisons with benchmark results demonstrate its accuracy and computational efficiency. The results highlight the influence of the fractional order and show the method's potential for nonlinear fractional differential equations in epidemiology, chemical kinetics, and related fields.

This article introduces novel numerical approaches utilizing both standard and nonstandard finite difference methods to solve one-dimensional Bratu's problems. Using the quasilinearization technique, the original problem is converted into a sequence of linear problems. Chebyshev polynomials are employed to approximate the second derivative of the function y(x), after which Sumudu transform is applied to obtain a new form of trial function. The obtained trial function is then substituted into a linearized and discretized Bratu's equations. We discuss the convergence of the schemes and compare the numerical outcomes to those derived using other relevant methods. We further modify one of the new schemes and apply it to solve boundary value problem with associated Robin conditions. The results show that the proposed schemes yield accurate approximations to the solutions of the problems considered.

This study applies three advanced techniques based on transforms to find approximate solutions to the Lane-Emden type equation, which is often encountered in mathematical physics and astrophysics. The proposed methods utilize new trial functions derived from expressing the second-order derivative of the variable function y(x) using Bernoulli polynomials, and applying Laplace, Sumudu, and differential transforms. To assess the effectiveness of the proposed methods, the study establishes an error analysis and stability analysis, and provides numerical examples demonstrating their accuracy and efficiency. In addition, a comparison of the absolute errors is made among the three methods, namely, Laplace Transform Bernoulli Collocation Method (LTBCM), Sumudu Transform Bernoulli Collocation Method (STBCM), and Differential Transform Bernoulli Collocation Method (DTBCM), and with those obtained from prior literature. The results show that all three methods perform very well in terms of efficiency and accuracy, and can be considered as suitable techniques for solving the Lane-Emden type equation.

Abstract This paper concerns the analysis and optimal control of fractional order model of tumor cells and the body’s immunological response using Atangana-Baleanu-Caputo derivative fractional operator. The model consists of spawning cells (S), reposing cells (R) and tumor cell (T) where we have used the Hattaf-Yousfi functional response for the activation of the reposing cells in the presence of tumor cells. We investigate the model’s existence and stability and present numerical results using a modified predict-evaluate-correct-evaluate (PECE) method of Adams-Bashforth-Moulton. We also study the Fractional optimal control problem (FOCP) in order to minimize tumor cell density. The Fractional Pontryagin maximum principle (FPMP) is used to characterize the fractional optimal control problem and we implement the forward-backward PECE method to determine the extremals of the problem. We study the optimal control dynamics of tumor growth via several numerical simulations.

This paper presents effective algorithms to approximate solutions of nonlinear fin problem with temperature dependent thermal conductivity and heat transfer coefficient. We propose new approximate semi-analytical techniques, namely the Laplace homotopy variable transform method (LHVTM) and the differential variable transform method (DVTM), to obtain expressions for generalized convective straight fins problem. The linear dependence of thermal conductivity on temperature is considered, and fin efficiency and effectiveness are evaluated. We further analyzed the effects of the model parameters on surface heat loss, and the results are presented in tables and with graphical illustrations.

In this article the distribution of temperature in both smooth and stepped functions is compared. The study uses a model of one dimensional N-cross-sections domain of a vase shaped medium, where approximation of the solution in the first case uses smooth functions, and in the second one, stepped functions. The temperature distribution is described by heat equation. Analysis of temperature distribution in both cases is based on finding eigenvalues and their corresponding eigen-functions which satisfy boundary conditions at given endpoints. Mathcad software was applied to determine the eigenvalues and their corresponding eigen-functions, together with the temperature distribution in the media of concern. The temperature distribution in both cases was found to be basically the same. The problem solution for each case is presented and an example of a one-dimensional vase shaped domain of length 4 units for each case is also given.

We propose a Caputo-based fractional compartmental model for the dynamics of the novel COVID-19 pandemic. The newly proposed nonlinear fractional order model is an extension of a recently formulated integer-order COVID-19 mathematical model. Using basic concepts such as continuity and Banach fixed-point theorem, existence and uniqueness of the solution to the proposed model were shown. Furthermore, we analyze the stability of the model in the context of Ulam-Hyers and generalized Ulam-Hyers stability criteria. The concept of next-generation matrix was used to compute the basic reproduction number R0, a number that determines the spread or otherwise of the disease into the general population. We also investigated the local asymptotic stability for the derived disease-free equilibrium point. Numerical simulation of the constructed epidemic model was carried out using the fractional Adam-Bashforth-Moulton method to validate the obtained theoretical results.

This article analyzed carbon dioxide (CO2) emission from the combustion of reactive materials modeled in a cylindrical domain. Reactive materials in this case involve carbon-containing substances that react spontaneously with the oxygen of the surrounding environment under the influence of an exothermic chemical reaction. In this analysis, the reactant (oxygen) consumption was neglected. The nonlinear differential equation governing the problem was solved numerically using the Finite Difference Method embedded within the Maple software. It was found that there are kinetic parameters that enhance the emission of CO2, like the rate of reaction, and others, like the heat loss parameter, retard the CO2 emission during the exothermic chemical reaction.

Fins are commonly utilized to enhance (dissipate) heat in various engineering systems that include heat exchangers. In the present investigation, the impact of multi-boiling and thermo-geometric factors on a convective–radiative rectangular porous fin subjected to the temperature-dependent thermal conductivity of linear and non-linear variations is discussed extensively. The governing equations describing the problem were formulated with the aid of Darcy law. Similarity variables were employed to reduce the models to non-dimensional form. The solution of the governing dimensionless equation is approximated using the RK4 and spectral local linearization methods. Before parametric analysis, the agreement between the two numerical methods was established. Findings reveal that the non-linear variation of thermal conductivity shows better thermal efficiency than the linear variation. An improvement in the multi-boiling heat transfer parameter retards the temperature distribution of the fin. Furthermore, increasing the thermo-geometric parameter will result in a progressive decrease in the temperature of the fin. The results obtained in this work will aid in the design of heat exchangers and other heat transfer equipments.

We propose a Caputo-based fractional compartmental model for the dynamics of the novel COVID-19 epidemic taking into consideration social distancing and the influence of the environment. Using basic concepts such as continuity and Banach fixed-point theorem, the existence and uniqueness of the solution to the proposed model were shown. Furthermore, we analyze the stability of the model in the context of Ulam-Hyers and generalized Ulam-Hyers stability criteria. The concept of next-generation matrices was used to compute the basic reproduction number $R_0,$ a number that determines the spread or otherwise of the disease into the general population. Numerical simulation of the disease dynamics was carried out using the fractional Adam-Bashforth-Moulton method to validate the obtained theoretical results.

This paper presents new efficient numerical methods for solving Volterra integro-differential equations and a system of nonlinear delay integro-differential equations which arises in biology. The principal idea of these approaches is based on a careful blend of the Petrov-Galerkin technique and the Sumudu transform method. In the proposed methods, using Lagrange polynomials and zeros of Jacobi polynomials, the considered system of linear and nonlinear integro-differential equations, with their associated initial conditions are reduced to linear and nonlinear systems of algebraic equations in the unknown expansion coefficients. Solving the resulting algebraic systems by Gaussian elimination and Newton's methods respectively, approximate solutions of the integro-differential problems are constructed. Detailed error analysis of the proposed methods is carried out to establish and ascertain the reliability and effectiveness of the methods. The methods are then tested on several examples, and the results are compared with those obtained via existing methods in the literature. The numerical results showed that the proposed methods are accurate, efficient, and reliable for solving all kinds of integro-differential equations.

In this paper, based on Parker-Sochacki method for solving a system of differential equations, a multistage technique is developed for solving the nonlinear boundary layer equations of powerlaw fluid on infinite domain. The problem domain is split into subintervals over which the boundary value problem is replaced with a sequence of subproblems. In a shooting-like approach, the boundary condition at infinity is converted to an equivalent initial condition. By recasting the problem as a polynomial system of first-order autonomous equations, the sub-problems are solved with ParkerSochacki method with very high accuracy. The interval of convergence of the solution is derived a-priorly in terms of the parameters of the polynomial system, which guides optimal choice of the discretization parameter. The technique yielded a convergent piecewise continuous solution over the problem domain. The results obtained, demonstrated graphically and in tables, compared well with existing ones in the literature.

In this paper, a nonstandard finite difference scheme is employed to approximate the solution of a nonlinear second order differential equation obtained from the model of heat transfer in extended surfaces. The proposed method for the system enjoys the nonlocal approximation of nonlinear terms and requires neither any iterative procedure nor the computation of Jacobian unlike the standard finite difference method. Accuracy of the method and conditions for generating expected positive solution are also discussed in detail.

The article introduces a new multistage technique for solving a polynomial system of nonlinear initial and boundary value problems of differential equations. The radius of convergence R of the series solution to the problem is derived a-priorly in terms of the parameters of the polynomial system. Then guided by the convergence-control parameter h

In this article, a hybrid collocation method for solving highly nonlinear boundary value problems is presented. This hybrid method combines Chebyshev collocation method with Laplace and differential transform methods to obtain approximate solutions of some highly nonlinear two-point boundary value problems of ordinary differential equations. The efficiency of the method is demonstrated by applying it to ordinary differential equations modelling Darcy-Brinkman-Forchheimer momentum problem, laminar viscous flow problem in a semi-porous channel subject to transverse magnetic field, fin problem with a temperature-dependent thermal conductivity, transformed equations modelling two-dimensional viscous flow problem in a rectangular domain bounded by two moving porous walls and two-dimensional constant speed squeezing flow of a viscous fluid between two approaching parallel plates. The results obtained are compared with the existing methods and the results show that the new method is quite reasonable, accurate and efficient.

In this article, we introduce a new method to obtain an approximate analytical solution of the highly unstable Troesch’s problem. In the proposed method, without recourse to any hyperbolic tangent transformation or finite term approximation of the hyperbolic sine function, the problem is recast as a system of projectively polynomials which allows straightforward computation of the series solution of the problem. The radius of convergence of the series solution to the problem is derived a-priorly in terms of the parameters of the polynomial system. Using a step length ; the problem domain is divided into subintervals, where corresponding subproblems are defined and solved with Parker-Sochacki method with very high accuracy. Highly accurate piecewise continuous approximate solution is thus obtained on the entire integration interval. The obtained solution, which is valid for every choice of the Troesch parameter , showed comparable accuracy to known numerical solutions in the literature. In particular, new results are presented for large values of in the range [20;500].

In this article, analytical expressions for fin efficiency and effectiveness of convective straight fins with temperature-dependent thermal conductivity are derived using the Parker-Sochacki method. The effect of various parameters of the fin problem on fin efficiency and effectiveness were investigated. The obtained results, presented both graphically and in tables, showed excellent agreement with those obtained from existing notable techniques in the literature.