A class of singularly perturbed convection–diffusion boundary value problems for second-order ordinary differential equations without interior turning points, involving a small perturbation parameter multiplying the highest-order derivative and subject to Dirichlet boundary conditions, is solved using a Shishkin mesh based on spline-in-compression. The proposed method effectively captures the boundary layer behavior and ensures improved numerical accuracy. To demonstrate the efficiency and robustness of the scheme, two numerical examples are presented and the results are compared with those obtained by using variable-mesh and uniform mesh approaches. The comparative analysis confirms the superior performance of the proposed method, particularly for problems exhibiting sharp boundary layers.

The flow and heat transfer characteristics of a tri-hybrid nanofluid in a porous medium are investigated under the influence of magnetohydrodynamics (MHD) and a ramped wall temperature. A Maxwell fluid is employed as the base fluid, in which three types of spherical nanoparticles, tungsten trioxide (WO3), silver (Ag), and titanium dioxide (TiO2), are suspended. The physical model is formulated using a system of partial differential equations subject to appropriate initial and boundary conditions. To enhance the novelty of the analysis, fractional derivatives are incorporated into the Maxwell fluid model along with porosity effects. Among the various definitions of fractional derivatives, the Caputo fractional derivative is preferred for its wide applicability in physical problems. The fractional-order derivatives are evaluated using the Caputo formulation, while the Crank-Nicolson numerical scheme is employed to discretize the time-dependent terms and solve the governing equations under ramped heating conditions. The proposed framework, which combines the Caputo fractional derivative with the Crank-Nicolson method to analyze tri-hybrid nanofluid flow, is a distinctive feature of this work. The Caputo derivative effectively captures memory-dependent behavior, allowing the model to account for the system's dependence on its past states. This capability is particularly important for nanofluids exhibiting nonlocal and anomalous interactions, where classical integer-order models based on simple linear stress-strain relationships fail to accurately represent the complex rheological behavior. Overall, the adopted numerical approach provides improved accuracy and flexibility in modeling complex heat transfer processes, making the present study relevant to a wide range of biomedical and industrial applications.

In this work, a scale-3 Haar wavelet collocation method is proposed for 2D time-fractional integro-differential equations with a weakly singular kernel (TFIDE). The Riemann-Liouville fractional integral operator for the scale-3 Haar wavelet (RLFIO-3H) is obtained. The proposed approach is a fully collocation-based approach wherein all temporal and spatial derivatives are discretized in terms of the scale-3 Haar wavelet and RLFIO-3H. Usually, in the available literature, the integral term in TFIDE is approximated by using Gauss quadrature, which results in additional errors. On the other hand, the proposed method directly deals with this integral term by using RLFIO-3H. The convergence analysis of the method is performed, and an exponential order of convergence is achieved in terms of wavelet resolution. Three benchmark problems are solved to confirm the accuracy of the method, and the obtained results are compared with existing methods available in the literature.

This paper proposes a Haar wavelet collocation approach to solve neutral delay differential equations on a metric star graph (NDDE-MSG) with kappa edges. The application of Haar wavelet, together with its integration on NDDE-MSG, yields a system of equations, which on solving gives unknown wavelet coefficients and subsequently the solution. The upper bound of the global error norm is established to demonstrate that the proposed method converges exponentially. We conduct some numerical experiments to test the computational convergence of our approach. In this study, the authors explore the numerical solution for NDDE on metric star graphs for the first time.

In this paper, we have studied the dynamical analysis of a new 3D chaotic system by investigating its Lyapunov exponents, time series and Hamilton energy. An optimization problem for exploring the precise ultimate bound set of the system has been solved analytically by using Lagrange multiplier method. Also, numerical simulation is applied to test the obtained results. Furthermore, an application of the obtained bound set in the synchronization of two identical chaotic systems has been discussed.

This study examines hybrid nanofluid flow through variable porous medium between two spinning disks. Both the disks rotate with varied angular velocities. The nanoparticles of Ag and TiO2 are mixed in H2O to fabricate a hybrid nanofluid. A magnetic field of intensity B0 is used in the normal direction of motion, with effects of Joule heating and viscous dissipation. The main equation of problem has been evaluated through homotopy analysis method. An increase in TiO2 and Ag + TiO2 nanoparticle concentrations, variable porous factor, and Reynolds number enhances axial momentum transfer, increasing radial and axial skin friction in both disks. When the volumetric fraction varies from 0.01 to 0.04, the heat transfer rate increases from 0% to 11.27% at the upper disk and from 0% to 15.46% at the lower disk, indicating the maximum percentage increase at the lower disk. The work has been validated through a comparative study of the current results with published work by ensuring a strong promise among all the results. The findings of this study offer valuable insights for optimizing cooling and thermal management in rotating machinery, turbine rotors, disk brakes, and energy devices, where hybrid nanofluids and porous media improve heat transfer and overall efficiency under Joule heating conditions.

This study investigates the effects of heat generation and chemical reaction on Jeffrey nanofluid magnetohydrodynamic (MHD) flow over a linearly stretching sheet. The Buongiorno model examines Brownian motion's and thermophoresis's effects on Jeffrey's nanofluid. The joint phenomenon of heat and mass transfer is considered with prescribed convective surface heating and diffusion. Visco-elastic Jeffrey fluid model in a porous medium is developed in terms of highly nonlinear partial differential equations (NLPDEs), which upon similarity transformation is remodeled into nonlinear ordinary differential equations (NLODEs) with transformed boundary conditions. The problem is solved numerically using the shooting technique. The results are shown in several plots and discussed for embedded flow parameters. It has been found that magnetic parameters increased temperature and concentration, whereas they decreased velocity.

The study presents a novel algorithm for solving third order non-linear equations (Emden-Fowler type) with multi-singularity, which can be applied to various physical models. The algorithm uses a quintic trigonometric B-spline collocation method and a quasilinearization technique to avoid the non-linearity term in the equation. The study establishes a comprehensive error analysis for the proposed algorithm and proves that it has O(h4) convergence. The algorithm’s ability to handle singular behavior at the point ς = 0 and its faster rate of convergence exhibits a promising approach to solve such problems. The study also validates the theoretical results through numerical experiments and shows that the proposed algorithm has a faster rate of convergence in comparison to the new cubic B-spline collocation method [1] and uniform Haar wavelet resolution technique [2]. Moreover, the new method has an edge (in terms of accuracy) over other existing methods, when applied to the problems [3–6]. 2000 Mathematics Subject Classification: 65L80; 65M99; 65N35; 34B16; 65L05; 65L10; 65N55.

This study aims to investigate thin film flow with the effects of a magnetic field in association with Cu and CuO nanoparticles past an inclined plate. The second law of thermodynamics is used to optimize the production of entropy and to reduce the fluid’s friction. The leading equations have been solved through the homotopy analysis method in dimensionless form. It is observed that fluid motion is slowed down as the thickness factor, volumetric fraction of nanoparticles, and unsteadiness parameter increase, while it intensifies with higher values of mass and thermal Grashof numbers. The thermal distribution has augmented with an upsurge in volumetric fraction, magnetic factor, Eckert number, and unsteadiness factor. Bejan number as well as the rate of entropy production are opposed by Brinkman number and are supported by growth in magnetic factor. For nanoparticle volume fractions ranging from 0.01 to 0.03, the Nusselt number shows significant improvement, particularly for hybrid nanoparticles, where it increases from 4.3207 to 6.6983%, demonstrating their superiority over traditional nanoparticles. Validation of work has been ensured through comparative analysis between current results and published works.

Ecosystems are considered the basic units of nature because they witness all the interactions between living organisms and their physical environment. In this paper, the dynamical analysis of the competitive herbivore species network (ecological system) has been studied. We have investigated the relationship between the stability of ecological system and the Hamilton energy function. Also, we have analyzed the role of competitive modes in the emergence of chaotic behavior in the system. Furthermore, the ultimate bound sets for the considered ecological system have been obtained by using the Lagrange optimization and analytical technique. Finally, in order to control chaos, one chaotic ecological system is exponentially synced to another chaotic ecological system under the master-slave scheme. Numerical simulations are carried out, which validate the obtained results. Analytical and computational studies are found to be in excellent agreement.

The Legendre wavelet based method has been employed in this paper to investigate neutral delay differential equations. The highest order derivative is approximated by Legendre wavelet using the integral operator technique. Then integrations of Legendre wavelet are used to approximate the lower order derivatives and unknown function. To get an algebraic system of linear or nonlinear equations, approximated values of unknown function and its derivatives are substituted in neutral delay differential equations. On solving the developed system, we get unknown wavelet coefficients and subsequently the approximate solution. To analyze the theoretical usability of the approach, the upper bound of error norm is established. Moreover, the theoretical results are confirmed through few numerical experiments. A comparison of the results of presented method with method available in literature is given to conclude the superiority of the proposed method.

The ecology of marine life and other biotic processes, such as septicity, are influenced by the drive of microorganism cells in the fluid. Considering the transference mechanism in nanofluids including a microbial suspension is imperative for several chemical and medical applications. This study examines the bioconvection of an Al2O3-Graphene-CNT/water ternary hybrid nanofluid between two infinitely parallel spinning disks in a porous media under the influence of heat source/sink and radiation. The Cattaneo-Christov model has been employed to inspect the transference mechanism of mass and heat. The MATLAB function "bvp4c" is used to numerically tackle the governing equations. The relationship between the most relevant variables and the motile microorganism's density, velocity, temperature, concentration of nanoparticles and other characteristics is graphically represented. Last, figures are provided to illustrate how several important elements relate to the Nusselt and Sherwood number, and local motile microbe density number. The heat transfer rate is seen to be higher at the upper disk than at the lower disk. A rise in the volume fraction of nanoparticles causes the heat transfer rate at both disks to increase. The outcomes of this study will be useful for a wide range of architectural designs, transportation systems, oil recovery systems with microbes, medical sectors and other industries that utilize nanofluids.

This study is devoted to the numerical investigation of linear and nonlinear hyperbolic telegraph equation. We have proposed a wavelet collocation method based on Legendre polynomials for approximating the solution. Both the spatial and temporal variables, along with their derivatives, are approximated using the Legendre wavelet and its integration. The present approach is simple, consistent and straightforward. To assure the theoretical consistency of the method, an estimate for the upper bound of the error norm is provided. We have proved an exponential order of convergence which is better than the methods available in the literature. Some numerical experiments are carried out to justify the theoretical results and the outcomes confirm the computational efficiency of the proposed method.

A wavelet collocation method based on Hermite polynomials is proposed for the study of a class of 2D quasi-linear elliptic partial differential equations (PDEs) that arise frequently in applied mathematics, electromagnetic theory, nonlinear optics, weather forecasting, etc. The application of Hermite wavelets and their integration to 2D quasi-linear elliptic PDEs yielded a system of equations. For the theoretical aspect, the upper bound of the error norm is established to guarantee the convergence of the method. The proposed approach has an exponential rate of convergence, so it converges very rapidly. The proposed method can be uniformly adapted to investigate the solution of 2D singularly perturbed elliptic PDEs without modifying the current scheme. Some numerical simulations have been done to validate the theoretical findings. The maximum absolute errors are calculated for different numbers of collocation grids. The comparison of numerical findings with the existing methods concludes the superiority of the proposed method.

In this paper, we design and analyse a high-order numerical algorithm based on the improvised quintic B-spline collocation method for solving the fourth-order fractional diffusion equation. The time-fractional derivative is approximated by Caputo’s time derivative. The space derivative is approximated by the collocation method based on improvised quintic B-spline functions. It is shown that the proposed algorithm is unconditionally stable. Through rigorous convergence analysis, the method is shown (2-β ) order convergent in time and almost sixth-order convergent in space direction. It is also shown that the theoretical rate of convergence is the same as that acquired experimentally. To confirm the theoretical results and to test the efficiency and robustness, the method is tested on three problems. The main contribution of the developed algorithm is that the order of convergence and numerical results obtained are better than the existing methods, like the sextic B-spline collocation method (Roul and Goura in Appl Math Comput 366:124727, 2020), the quintic B-spline method (Siddiqi and Arshed in Int J Comput Math 92(7):1496–1518, 2015), and the quintic spline method (Tariq and Akram in Numer Methods Part Differ Equ 33(2):445–466, 2017). It has been proved that the order of convergence of the proposed method is six, which is two orders of magnitude higher than the other spline collocation methods.

In the past, the existence and uniqueness of the solutions of fractional differential equations have been investigated by many researchers theoretically in various approaches in the literature. In this paper, there is no discussion of the existence of solutions for the nonlinear differential equations with fractal fractional operators. The objective of this work is to present novel contraction approaches, notably the $ \varpropto $-$ \psi $-contraction $ \varpropto $-type of the $ \tilde{\texttt{F}} $-contraction, within the context of $ \hat{F} $-metric and orbital metric spaces. The aim of this study is to illustrate certain fixed point theorems that offer a new and direct approach to establish the existence and uniqueness of the solution to the general partial differential equations by employing the fractal fractional operators.

The customization of hybrid nanofluids to achieve a particular and controlled growth rate of thermal transport is done to meet the needs of applications in heating and cooling systems, aerospace and automotive industries, etc. Due to the extensive applications, the aim of the current paper is to derive a numerical solution to a wall jet flow problem through a stretching surface. To study the flow problem, authors have considered a non-Newtonian Eyring-Powell hybrid nanofluid with water and CoFe2O4 and TiO2 nanoparticles. Furthermore, the impact of a magnetic field and irregular heat sink/source are studied. To comply with the applications of the wall jet flow, the authors have presented the numerical solution for two cases; with and without a magnetic field. The numerical solution is derived with a similarity transformation and MATLAB-based bvp4c solver. The value of skin friction for wall jet flow at the surface decreases by more than 50% when the magnetic field MA=0.2 is present. The stream function value is higher for the wall jet flow without the magnetic field. The temperature of the flow rises with the dominant strength of the heat source parameters. The results of this investigation will be beneficial to various applications that utilize the applications of a wall jet, such as in car defrosters, spray paint drying for vehicles or houses, cooling structures for the CPU of high-processor laptops, sluice gate flows, and cooling jets over turbo-machinery components, etc.

In this paper, we study a high-order unconditionally stable numerical method to approximate the class of multi-term time-fractional diffusion equations. This type of problem appears in the modelling of transport of certain quantities such as heat, mass, energy, solutes in ground water and soils. The multi-term time-fractional derivative is approximated by using the Crank-Nicolson method for the Caputo's time derivative. The space derivative is approximated by using the collocation method based on quintic B-spline basis functions. We have established the stability and convergence analysis of the proposed numerical scheme thoroughly, and it is shown that the order of convergence in space variable is almost four and in the time variable is O (Delta t(2-max{gamma,gamma i})). To prove the accuracy and efficiency of the developed method, we consider four numerical examples and perform the numerical simulation. The developed algorithm works well andvalidate the theoretical results. The developed method is fourth-order convergent in the space variable, which is almost two orders of magnitude higher than the other spline collocation methods.

Abstract This paper investigates numerical solution of generalized space-time fractional Klein–Gordon equations (GSTFKGE) by using Gegenbauer wavelet method (GWM). The developed method makes use of fractional order integral operator (FOIO) for Gegenbauer wavelet, which is constructed by employing the definition of Riemann–Liouville fractional integral (RLFI) operator and Laplace transformation. The present algorithm is based on Gegenbauer wavelet jointly with FOIO to convert a GSTFKGE into a system of equations which is solved by using Newton’s technique. Additionally, the upper bound of error norm of the proposed method is calculated to validate the theoretical authenticity of the developed method. The comparison of numerical outcomes with the existing results in the literature and graphical illustrations show the accuracy and reliability of our method.

In this manuscript, the Korteweg-de Vries-Burgers (KdV-Burgers) partial differential equation (PDE) is investigated under nonlocal operators with the Mittag-Leffler kernel and the exponential decay kernel. For both fractional operators, the existence of the solution of the KdV-Burgers PDE is demonstrated through fixed point theorems of $ \alpha $-type $ \digamma $ contraction. The modified double Laplace transform is utilized to compute a series solution that leads to the exact values when fractional order equals unity. The effectiveness and reliability of the suggested approach are verified and confirmed by comparing the series outcomes to the exact values. Moreover, the series solution is demonstrated through graphs for a few fractional orders. Lastly, a comparison between the results of the two fractional operators is studied through numerical data and diagrams. The results show how consistently accurate the method is and how broadly applicable it is to fractional nonlinear evolution equations.