A relaxed quasilinearized spectral method is developed for nonlinear Volterra–Fredholm integral equations of the second kind. Chebyshev spectral collocation is combined with a selective linearization strategy in which the nonlinear Volterra contribution is treated implicitly, while the Fredholm contribution is evaluated explicitly. A relaxation parameter is incorporated to control the resulting nonlinear iteration. Local error propagation is characterized by an explicit iteration matrix, whose spectral radius provides a local convergence criterion and a principled basis for selecting the relaxation parameter. In the pure nonlinear Volterra limit, the unrelaxed method exhibits at least quadratic local convergence under appropriate regularity and nonsingularity assumptions. Numerical experiments demonstrate rapid spectral accuracy for smooth solutions and close agreement between the spectral-radius prediction and the most effective relaxation range. The method also substantially reduces nonlinear iteration counts relative to standard Picard iteration as the coupling strength increases. An additional benchmark involving exponential and trigonometric nonlinearities confirms that the observed acceleration is not restricted to a particular choice of nonlinear functions. These results demonstrate that selective quasilinearization combined with relaxation provides an effective framework for accelerating the iterative solution of nonlinear Volterra–Fredholm integral equations while retaining the high-order accuracy of spectral discretization.

A residual-switched hybrid quasilinearized spectral method is developed for nonlinearVolterra–Fredholm integral equations. Chebyshev collocation reduces the integral equationto a nonlinear algebraic system. The iteration uses a selective pre-asymptotic phase, inwhich the Volterra nonlinearity is quasilinearized while the Fredholm contribution is laggedexplicitly, and a fully quasilinearized Newton phase. A normalized nonlinear residual triggersa permanent handoff between the two phases. Near a nonsingular discrete root, the residualis shown to be locally equivalent to the error and the full phase converges quadratically. Atrajectory-dependent finite-handoff theorem then justifies the residual trigger, conditional onconvergence of the selective sequence to the target root. Numerical experiments distinguishtarget-root convergence from convergence to other discrete roots and show that the residual-switched method can reach the prescribed root from initial approximations for which directfull quasilinearization fails or reaches another root. A nonlocal Euler–Bernoulli beam exampleillustrates the same Volterra–Fredholm discretization outside manufactured benchmarks. Themethod is therefore a residual-switched hybrid strategy that can provide trajectory-dependent rescue for selected initializations; it is not a globally convergent or uniformly basin-enlarging Newton method.

The development of stretchable organic solar cells (s-OSCs) demands concurrent breakthroughs in mechanical compliance and electronic properties, and the challenge is rooted in the intrinsic mechanical mismatch between organic semiconductors and metal electrodes. Here, this study proposes dual-phase interface engineering strategies to reconcile these conflicting requirements through molecularly interlocked conductive elastomers. Dynamic stress dissipation through dynamic bond plasticity is achieved by embedding a 3D interpenetrating conducting elastomer network within the electron transport layer (ETL). The strategy creates gradient modulus interfaces through Ag coordination-enabled nanocomposite bonding, suppressing crack propagation velocities and reduces the interfacial mechanical mismatch phenomenon. Eventually, the PCE of 19.58% is achieved on the small-area flexible devices, which is one of the highest PCEs for flexible organic solar cells (f-OSCs) to date. Notably, the stretchable devices retain over the PCE of 10% under 100% tensile strain, surpassing previous stretchable photovoltaic devices. To further validate the potential of this strategy for large-area module applications, 25 cm2-based flexible and stretchable modules are prepared with PCEs of 16.74% and 14.48%, respectively. The work redefines material design rules for deformable electronics by establishing a generic mechanically adaptive framework that synchronizes interfacial dynamics across molecular to macroscopic scales.

The current state of the planet has raised lot of concerns about the type of world we want the next generations to inherit. The global warming of the planet together with catastrophes (floods, bush fires, storms, hurricanes, tsunamis, earthquakes, extreme temperatures, droughts, etc.) that have hit different parts of the world these last decades, are on an increasing curve. Many voices are being heard asking people to look for solutions to reduce the consequences of global warming. The recent United Nation Climate Change Conference of the Parties (COP26) in Glasgow (Ireland) from 31 October to 13 November 2021 testifies. Rock fracture happening in our environment is one of important factors to consider in the solution seeking process, as rocks can absorb the carbon dioxide (CO2) via fracture. That is why we perform in this work a mathematical analysis of an ecosystem's rock fracture model, in which the fractal process is considered. The specific case where the fragmentation rate is dependent on the size of the rock is analyzed. The exact solution is evaluated and numerical simulation performed. The results show a dynamic partly repeating itself once, then twice, three times and so on, hereby marking the existence of self replicated zones in the convex body of rock fracture model that also happen to be chaotic. We observe that the evolution process tends to reproduce exact or partly exact pathways and this goes on over and over. Hence, there is existence in the system of chaotic self-replicating poles.

Chaotic systems play a crucial role in science and engineering due to their complex and unpredictable behavior. In this study, we investigated a nonlinear chaotic system known as the multi-bond orbital chaotic attractor (MBOCA), which we modeled using the Caputo Fabrizio fractional operators. We first established the existence and uniqueness of solutions for the system after applying these operators to the MBOCA. We then presented a numerical scheme and analyzed its stability and convergence. To validate the proposed numerical scheme, we performed numerical simulations to visualize the system's behavior for both integer and fractional order cases. The results confirmed that the generated bond-orbital attractors exhibit chaotic behavior, highlighting the influence of fractional order operators on the system's dynamic complexity.

This paper analyzes a generalized chaotic system of differential equations characterized by attractors with bondorbital structures. Both classical and fractional-order cases are examined analytically and numerically, with convergence and stability analyses provided. The numerical findings confirm the presence of bondorbital attractors in the classical system. In contrast, bondorbital attractors also emerge in the fractional model employing the Caputo-Fabrizio operator, albeit with significant perturbations for specific fractional orders. To validate these results, an electric circuit implementation of the fractional-order system using an field-programmable gate array board was conducted, yielding consistent outcomes. This study highlights the potential of fractional calculus, particularly the Caputo-Fabrizio operator, in capturing the memory effects and complex dynamics of chaotic systems. The work bridges theoretical modeling and practical hardware applications, offering valuable insights for modeling complex systems.

In the modern world of technology, thermal performance of the working fluid can be greatly enhanced through adding one or more nanoparticles into a base fluid. Therefore, the intention of this work is to scrutinize heat transfer in magnetized mixed convective flow of Williamson ternary (tri)-hybrid nanofluid (THNF) across a horizontal circular cylinder in a porous medium with suction/injection, partial slip and convective boundary conditions. The novelty of the study is enhanced by utilizing features of varying fluid properties, nonlinear radiation and heat source/sink for analysis of heat transfer characteristics of the working THNF flow. The formation of THNF is achieved through the sequential suspension of copper, alumina and titania onto water as base fluid. The dimensionless conservation equations are handled via the domain-decomposition bivariate spectral local linearization method. The deportment of particular parameters on velocity field and thermal dissemination along with quantities of engineering interest are disclosed. We found that temperature is enhanced by considering non-linear thermal radiation, variant thermal conductivity and convective boundary conditions. The use of tri-hybrid nanoparticles contributes towards thermal augmentation of the Williamson working fluid. The heat transfer coefficients have higher magnitude for THNF of assisting flow, but lower magnitude in the opposing flow. Reported findings can be used as a reference for inspecting the capability of THNF in minimizing the production cost than when metal nanofluid is used.

This work inspects entropy generation and heat transfer induced by a bioconvection slip flow of nonlinear radiative Carreau-Yasuda hybrid nanofluid (NF) over a convectively heated sphere. Activation energy for microbes is contemplated in order to comprehend its contribution toward flow features. The original partial differential equations (PDEs) are rendered non-dimensional through appropriate conversions, and the resulting PDEs are solved using the overlapping grid spectral collocation algorithm. The impact of diverse flow factors on flow profiles, entropy generation, skin friction, heat, mass, and motile microbes transport rates is analyzed. Key outcomes reveal that hybrid NF flow demonstrates a supplementary indispensable feature in the operation of heat transport compared to mono NF flow. More entropy is generated in the system by using the hybrid NF model along with magnetic field, heat source, convective heating, nonlinear radiation, and viscous dissipation. Thermal fields and rate of heat transport are improved by including nonlinear radiative heat flux in the system. Mass and motile microbes transport rates are respectively enhanced by chemical and microbial reactions, but both quantities are reduced by activation energy. The effects of microbial reactions on flow quantities substantiate the significance of these features for the dynamics of microorganisms. The findings of this study can be useful in the upsurge of thermal performance of the working fluid and contribute toward the improvement of microbial fuel cell performance.

The present study investigates the axisymmetric stagnation point radiative flow of a Cu-Al2O3/water hybrid nanofluid over a radially stretched/shrunk disk. In this paper, a new mathematical model has been developed by taking into consideration the concept of different nanoparticles concentration in a hybrid nanofluid, which are Brownian motion and thermophoresis of nanoparticles. A new model for entropy generation has also been provided in the present study. The non-dimensional governing equations of the developed mathematical model are solved using newly developed and efficient overlapping grid spectral collocation method. Numerical stability and residual error test are provided here to show the accuracy of the numerical method in this mathematical model. The outcomes of fluid flow, temperature, and two different types of concentration profiles are depicted, and described in graphical and tabular forms. For the limiting instances, comparison shows excellent agreement among current and results established in the literature. Increasing the strength of magnetic field is seen to increase the radial component of fluid velocity as well as the entropy generated within the system. Two different nanofluid concentration profiles are increasing and decreasing with rising thermophoresis and Brownian motion parameters, respectively, from a particular height above the disk because of the revised nanofluid boundary condition. Temperature profile increases here with increasing Biot number, and increasing Brinkman number causes higher entropy generation number for both stretching and shrinking disks. The enhanced thermal characteristics of the hybrid nanofluid over the single particle nanofluid has been observed.

This article inspects entropy generation and mixed convection boundary layer flow of carbon nanotubes (CNTs) Casson nanoliquid via semi-infinite vertical cylinder which moves with nonlinear velocity in Darcy-Forchheimer porous medium. Appropriate similarity variables have been employed to convert the original partial differential equations into ordinary differential equations that have been solved using overlapping grid spectral quasilinearisation method (SQLM). Comparison of accuracy, convergence and stability between the two methods is made. Rate of entropy generated, flow fields and engineering quantities are discussed for different embedding parameters. It is revealed that single-wall CNTs are more efficacious to improve heat transport features than multi-wall CNTs. The fluid flow and thermal dispersion processes improve with consideration of curved surface, CNTs, non-Newtonian fluid and injection of the fluid. The curvature surface and suction of the fluid contribute towards growth of skin friction factor and rate of thermal transference. Entropy generated expands by accounting for viscous nature of the fluid, strong nonlinearity, suction, non-Newtonian fluid, convective boundary condition and CNTs. The study finds applications in various processes, which are massively impacted by heat transport enhancement and high porosity. Boundary layer flow and heat transfer analysis through cylinders are relevant to various metallurgical and engineering solicitations.KEYWORDS: CNTs Casson nanofluidmixed convectionDarcy-Forchheimer relationvertical moving cylinderentropy productionoverlapping grid spectral collocation algorithm Disclosure statementNo potential conflict of interest was reported by the author(s).

The improvement in thermal performance of fluid and the control of energy loss are equitably significant. Therefore, the purpose of this study is to analyze entropy generation, stagnation point flow, and thermal characteristics of non-Newtonian third-grade modified hybrid nanofluid generated by a stretchable/shrinkable Riga plate in a porous medium with varying flow viscosity. In this analysis, a modification of hybrid nanofluid is considered by using pure water as a base fluid and three various nanomaterials (aluminium oxide, copper, and nickel) as nanoparticles in the characterization of heat transfer. Furthermore, the contribution of heat source/sink and viscous dissipation are accounted for in the model. The suited transformations are enforced to remodel the governing mathematical equations to produce ordinary differential equations that are conveniently tackled via spectral quasilinearization method (SQLM) along with the overlapping grid idea to yield numerical solutions. The preference of this approach over others has been justified through discussion of error bound theorems, residual and solution errors, computational time, and conditioning of matrices. The physical significance of disparate governing parameters on flow variables, velocity gradient, thermal rate, and entropy generation are scrutinized through graphs and tables. Crucial findings of the study include that temperature of the modified hybrid nanofluid enhances quickly (better thermal conductor) than temperature of single nanofluid, hybrid nanofluid, and conventional third-grade fluid for higher Biot number, variable viscosity, and heat source parameters. Mass suction enhances fluid flow and physical quantities of interest, but suppresses the fluid temperature. An increase in variable fluid viscosity, modified Hartmann number, and third-grade parameters enhances the wall drag coefficient while lowering the rate of heat transfer, and the opposite is true for porous media. More entropy is generated in the system by high variable fluid viscosity, suction, viscous dissipation, modified Hartman number, and non-Newtonian parameters. Owing to high velocity and temperature associated with modified hybrid nanoparticles, modified hybrid technology is recommended in enhancing the physical attributes of the fluid with minimal cost effects. In engineering and industrial point of view, this study can contribute significantly in thermal improvement of the working fluid.

The significance of hybrid nanofluids in controlling heat transmission cannot be overemphasized. Therefore, this article scrutinizes the electromagnetized flow of Carreau hybrid nanofluid towards a stretching surface in a Darcy–Forchheimer porous medium with the occurrence of slip conditions. To form the hybrid nanofluid, the amalgamation of silver and alumina nanoparticles (NPs) embedded in water as conventional fluid is assumed. For accurate interception of the rate of heat and mass transport, thermal conductivity and mass diffusion conductance are presumed to be temperature variants. The modeling system of partial differential equations has been translated into a nondimensional form by means of suitable similarity conversions. Then, the subsequent system of ordinary differential equations is handled using overlapping domain decomposition spectral local linearization method to acquire numerical solutions. The choice of the method has been justified through the provision of errors, condition numbers, and computation time. The behavior of distinct fluid parameters on the flow features, quantities of engineering curiosity, and entropy is analyzed. Findings of paramount importance constitute that the superior thermal conductivity, heat transfer efficiency, and low production cost can be achieved through the hybridization of silver and alumina NPs. The role of thermal radiation and temperature‐variant thermal conductivity is to enhance the thermal transport performance of Carreau hybrid nanofluids. The velocity, energy, and mass profiles grow with the utilization of injection effects. The principal aspiration of the second law of thermodynamics (minimizing the rate of entropy generation) can be achieved by considering shear‐thinning Carreau fluid while reducing the porosity parameter and Brinkman number in the existence of velocity slip conditions in the flow system. Outcomes of the current flow model can play a significant role in biomedical, technological, and various manufacturing processes. The approximation of entropy contributes towards power engineering and aeronautical propulsion to anticipate the smartness of the overall system.

This paper scrutinizes Hall current, Soret and Dufour impacts on natural convective flow of nanofluid attributable to variation in sinusoidal surface temperature over a vertical plate through a porous medium with strong transverse magnetic field applied normally to the flow. In this regard, the silver metal is considered as nanoparticles with water as base fluid. The overlapping multi-domain bivariate spectral local linearization method (OMD-BSLLM) has been utilized to solve the dimensionless governing equations which are attained by means of appropriate transformations. The obtained results are portrayed via graphical and tabular formations to inspect flow fields, shear stresses, heat and mass transmission characteristics for varying thermo-physical parameters. We found that there is an enhancement in the flow fields, shear stresses and heat transportation with the use of silver nanoparticles. Thermal and concentration boundary layer thickness enhance by using porous material, whereas diminish with improvement in Hall effect and ratio of buoyancy forces. The flow characteristics decline with the inclusion of porous medium while elevates with increment in Hall parameter. Moreover, an upgrade in thermal-diffusion correspond to a substantial growth in concentration field along with mass transfer rate. Current analysis can be useful in MHD energy generators, and industrial applications such as heating and cooling procedures owing to the involvement of nanoparticles having superior thermal conductivity features.

We generate a fractal using a finite collection of generalized cyclic contraction mappings, belonging to a particular category of mappings defined on a partial metric space. As a consequence, different results are attained for iterated function system that satisfy a different set of generalized cyclic contraction conditions. To substantiate the proven results, an example together with some applications are presented. With these results, we extend, unify and generalize some common results in contemporary literature.

This article is concerned with the numerical solution of three-dimensional elliptic partial differential equations (PDEs) using the trivariate spectral collocation approach based on the Kronecker tensor product. By using the quasilinearization method, the nonlinear elliptic PDEs are simplified to a linear system of algebraic equations that can be discretized using the spectral collocation method. The method is based on approximating the solutions using the triple Lagrange interpolating polynomials, which interpolate the unknown functions at selected Chebyshev–Gauss–Lobatto (CGL) grid points. The CGL points are preferred to ensure simplicity in the conversion of continuous derivatives to discrete derivatives at the collocation points. The collocation process is carried out at the interior points to reduce the size of differentiation matrices. This work is aimed at verifying that the algorithm based on the method is simple and easily implemented in any scientific software to produce more accurate and stable results. The effectiveness and spectral accuracy of the numerical algorithm is checked through the determination and analysis of errors, condition numbers and computational time for various classes of single or system of elliptic PDEs including those with singular behavior. The communicated results indicate that the proposed method is more accurate, stable and effective for solving elliptic PDEs. This good accuracy becomes possible with the usage of few grid points and less memory requirements for numerical computation.

In this paper, we aim to obtain some new common attractors with the assistance of finite families of generalized contractive mappings, that belong to the special class of mappings defined on a partial metric space. Consequently, a variety of results for iterated function systems satisfying a different set of generalized contractive conditions are acquired. We present some examples to reinforce the results proved herein. These results generalize, unify and extend a variety of results that exist in current literature.

The application of the recently proposed integral and differential operators known as the fractal-fractional derivatives and integrals has opened doors to ongoing research in different fields of science, engineering, and technology. These operators are a convolution of the fractal derivative with the generalized Mittag-Leffler function with Delta-Dirac property, the power law, and the exponential decay law with Delta-Dirac property. In this paper, we aim to extend the work in the literature by applying these operators to a modified stretch–twist–fold (STF) flow based on the STF flow related to the motion of particles in fluids that naturally occur in the dynamo theorem. We want to capture the dynamical behavior of the modified STF flow under these operators. We will present the numerical schemes that can be used to solve these nonlinear systems of differential equations. We will also consider numerical simulations for different values of fractional order and fractal dimension.

The novelty of this work rests upon the use of the domain partitioning technique in time variable when discretizing the domain of solution in spectral collocation algorithm. The single domain multivariate spectral collocation based methods have been proven to be effective in solving time-dependent partial differential equations (PDEs) defined over small time domains. However, there is a significant loss of accuracy as time computational domain proliferates and also when the number of grid nodes approaches a definite particular number. Therefore, the establishment of the new innovative multi-domain multivariate spectral quasilinearisation method (MDMV-SQLM) is described for the purpose of solving (2+1) dimensional nonlinear PDEs defined on large time intervals. The main output of this study is confirmation that minimizing the size of time computational domain at each subinterval assures sufficiently accurate results that are attained using minimal number of nodal points and less computational time. The solution algorithm involves partitioning the time domain into multiple non-overlapping sub-domains, simplification of the nonlinear PDEs using the quasilinearisation method and assumption of approximate solutions using triple Lagrange interpolating polynomials with Chebyshev–Gauss–Lobatto (CGL) points. The multi-domain spectral collocation procedure is executed on the linear systems of algebraic equations, where the subsequent matrix systems are solved separately in every time sub-interval with the continuity equation essentially used in obtaining initial conditions in the next subintervals. MATLAB software is used to implement the solution algorithm and numerical results are demonstrated graphically and in tabular form. To highlight the efficaciousness and accuracy of the MDMV-SQLM, error estimates, condition numbers and computational time are presented for well known (2+1) dimensional nonlinear initial-Dirichlet boundary value problems. The adoption of the domain decomposition technique is efficacious in suppressing the numerical challenges linked to large matrices and ill-conditioned nature of the resulting coefficient matrix. Also, the communicated results confirm that the numerical scheme is computationally cheap, fast and yield extremely accurate and stable results with the aid of fewer number of grid points for large time domains.

The widespread application of chaotic dynamical systems in different fields of science and engineering has attracted the attention of many researchers. Hence, understanding and capturing the complexities and the dynamical behavior of these chaotic systems is essential. The newly proposed fractal-fractional derivative and integral operators have been used in literature to predict the chaotic behavior of some of the attractors. It is argued that putting together the concept of fractional and fractal derivatives can help us understand the existing complexities better since fractional derivatives capture a limited number of problems and on the other side fractal derivatives also capture different kinds of complexities. In this study, we use the newly proposed Caputo-Fabrizio fractal-fractional derivatives and integral operators to capture and predict the behavior of the Lorenz chaotic system for different values of the fractional dimension $ q $ and the fractal dimension $ k $. We will look at the well-posedness of the solution. For the effect of the Caputo-Fabrizio fractal-fractional derivatives operator on the behavior, we present the numerical scheme to study the graphical numerical solution for different values of $ q $ and $ k $.

In this paper, we present a numerical scheme and mathematical analysis for the famous three-dimensional quadratic autonomous self-govern system that happens to be chaotic with the coexistence of multi-scroll attractors. The scheme is based on the Atangana-Baleanu fractional derivative in the Caputo sense. The formulation of these schemes introduces the non-local and non-singular kernel to the fractional derivatives. The fractional derivative is then approximated using the family of the Adams–Bashforth schemes. The results are presented in both numerical and graphical as the fractional order β varies between 0<β⩽1. We study the proposed model in the both generalized case that is 0<β<1 and the case where β=1, which is the integer standard case. Due to the impact of the generalized case, the proposed model is able to maintain the coexistence of multi-scroll attractors.