Form Of Differential Equations Research Articles

Analytical and numerical solutions of squeezing nanofluid flow between parallel plates: A comparison of IRPSM and HPM

The nanofluid flow between two parallel plates has many applications across various fields of engineering. The addition of nanoparticles to conventional fluids can improve the rheological properties of nanofluids, enhance the heat transfer rate, and have other beneficial characteristics. Therefore, a squeezing flow problem between two parallel plates has been investigated in this article. The nanofluid flow is composed of alumina (Al2O3) nanoparticles and water, which is used as a base fluid. This study is featured according to the recommendations of a previous study to implement the improved residual power series method (IRPSM). The mathematical model is presented in the form of partial differential equations (PDEs), which are then transformed into ordinary differential equations (ODEs) by means of similarity variables. The considered problem is solved by two analytical methods called IRPSM, the homotopy perturbation method (HPM), and a numerical method (NDSolve). The residual errors of both applied techniques are compared to verify the validation of a new method. The comparison of both applied analytical techniques is conducted at the 17th order of approximations. From the obtained results, we confirmed that both techniques are in great agreement and have a very close relationship. The results of both methods are compared with the numerical method, and it was found that the absolute error between the IRPSM and numerical techniques is much more significant than the absolute error between the HPM and numerical techniques. Also, we have seen that the residuals of IRPSM are quite significant in terms of less residual errors than those of HPM. We confirmed that the IRPSM is a more powerful method than the HPM in terms of solving such types of mathematical problems.

Stability scrutinization of a non‐Newtonian (Williamson) ternary hybrid nanofluid past a stretching/shrinking sheet

AbstractThe thermal superiority of ternary hybrid nanofluids (THNFs) over conventional heat transfer fluid has led to growing interest in their applications. This new type of nanofluid can be customized for cooling systems, heat exchangers, and electronic cooling by carefully selecting nanoparticle types and their volume fraction. Hence, this study seeks to investigate the Heimenz flow in a Williamson THNF over a sheet that stretches or shrinks. The fundamental objective is to assess the effect of the stretching/shrinking parameter, the Weissenberg number, and the nanoparticle volume fraction on the physical quantities and flow profiles. Besides, attention is also given to the occurrences of multiple solutions in this fluid flow situation. By employing a similarity transformation, the governing equations are modified as a simpler form of ordinary differential equations (ODEs). Next, the numerical method is put to use to solve the resulting ODEs system, specifically the bvp4c solver in MATLAB. Significant changes in heat transmission occur due to variations in the Weissenberg number and volume fractions of nanoparticles, particularly when the sheet starts to shrink. The escalating Weissenberg number correlates with growing critical values of the stretching/shrinking parameter, suggesting that both parameters help to hold off the detachment of the boundary layer. These findings emphasize the capacity of THNFs to improve heat transfer performance in numerous applications. This study also reveals that while stretching sheets often have unique solutions, a shrinking sheet has multiple solutions when the shrinking parameter falls within a certain range. By scrutinizing the robustness of these solutions, it was concluded that only one of them maintains stability over an extended period. It is essential to highlight that these present discoveries apply exclusively to the mixture of copper, alumina, and titania. Various mixtures of nanoparticles can demonstrate distinct characteristics of THNFs concerning both flow dynamics and thermal transfer.

Mathematical model and its solution for water-altering-gas (WAG) injection process incorporating the effect of miscibility, gravity, viscous fingering and permeability heterogeneity

AbstractThe oil recovery from the water-alternating-gas (WAG) injection process is significantly impacted by gravity, viscous fingering, and the permeability heterogeneity of the reservoir. Therefore, the combined effect of these parameters cannot be neglected in the WAG injection process. This article presents the development of a mathematical model of oil recovery and its solution for the WAG injection process that takes into account the combined effects of miscibility change, viscous fingering, gravity, and permeability heterogeneity in an inclined stratified reservoir. First, the governing equations and fractional flow functions were explained in relation to the effects of gravity, permeability heterogeneity, viscous fingering, and miscibility change in an inclined stratified porous medium. Then, a mathematical model was developed using fractional flow functions and conservation equations for both injected water and solvent. The model was generated in the form of a quasi-linear first-order partial differential equation, which was solved analytically in two dimensions (2-D) utilizing vector calculus. Next, this model was solved analytically by applying wave theory to practical constant pressure boundary conditions, which generate distinct waves at different times to provide pressure and saturation at the displacement front location. The total volumetric flux and breakthrough time are calculated from the analytical solution at various times. The presented analytical solutions can be used to predict different parameters for stratified porous media in a fast and efficient way. Finally, the results of analytical solution validated with high-resolution numerical simulation for a wide range of permeability heterogeneity, which shows excellent agreement for breakthrough time, saturation, and pressure versus displacement location of different waves at different times. This analytical solution will save time and money by offering guidance to engineers for analyzing the saturation and pressure distribution at different times and predicting oil recovery. It will also improve the understanding of the physics underlying the multiphase flow WAG injection process in heterogeneous reservoirs.

An improved water strider algorithm for solving the inverse Burgers Huxley equation

In this paper, we introduce an improved water strider algorithm designed to solve the inverse form of the Burgers-Huxley equation, a nonlinear partial differential equation. Additionally, we propose a physics-informed neural network to address the same inverse problem. To demonstrate the effectiveness of the new algorithm and conduct a comparative analysis, we compare the results obtained using the improved water strider algorithm against those derived from the original water strider algorithm, a genetic algorithm, and a physics-informed neural network with three hidden layers. Solving the inverse form of nonlinear partial differential equations is crucial in many scientific and engineering applications, as it allows us to infer unknown parameters or initial conditions from observed data. This process is often challenging due to the complexity and nonlinearity of the equations involved. Meta-heuristic algorithms and neural networks have proven to be effective tools in addressing these challenges. The numerical results affirm the efficiency of our proposed method in solving the inverse form of the Burgers-Huxley equation. The best results were obtained using the improved water strider algorithm and the physics-informed neural network with 10,000 iterations. With this iteration count, the mean absolute error of these algorithms is O(10-4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(10^{-4})$$\\end{document}. Additionally, the improved water strider algorithm is nearly four times faster than the physics-informed neural network. All algorithms were executed on a computing system equipped with an Intel(R) Core(TM) i7-7500U processor and 12.00 GB of RAM, and were implemented in MATLAB.

Numerical study of an electrically conducting nanofluid flow past a vertical stretching Riga plate

The heat and mass transfer through electrically conducting and mixed convective nanofluid flow across an elongating Riga plate is studied. Riga plate is an electromagnetic sheet, in which electrodes are assembled alternatively. The arrangement of Riga plate produces electromagnetic behavior in the fluid flow. In the current analysis, the influence of Arrhenius activation energy, viscous dissipation and heat source/sink over the surface of Riga sheet is considered. The heat and mass transfer are examined by convective boundary conditions and nanoparticles (NPs) zero-mass flux. The Brownian motion and thermophoretic diffusion of fluid molecules are applied via Buongiorno's approach. Von Karman's similarity approach is implemented to reset the modeled equations into the dimensionless form of ordinary differential equations (ODEs). The system of ordinary differential equations is solved numerically via PCM (parametric continuation method). The fluid velocity, temperature field, and concentration gradients are analyzed by varying various flow parameters and graphically portrayed. The numerical results are validated by comparing them with the published studies. It can be resolved form the results that the flow rate decilnes with the effect of power index m, whereas enhances with the consequence of Hartmann number. The significance of thermophoresis term augments the temperature curve of the fluid.

Temperature-dependent density effects on oscillatory mixed convective flow across a non-conducting horizontal circular cylinder embedded in a porous medium

Temperature-dependent density and aligned magnetic field influence on mixed thermal convection heat and current density aspects over a non-conducting porous circular horizontal cylinder is the novelty of this investigation. The magnetic effects over the surface are minimal but maximum away from the surface. The density of the fluid is assumed as an exponential variable of temperature to record the heat and current density rates. The pertinent form of partial differential equations is designed using suitable dimensionless quantities. The computational outputs of fluid velocity, magnetic fields and temperature variations are drafted using fluctuating-stokes quantities. The primitive quantities are utilised to make similar and asymptotic programs in FORTRAN for geometrical and numerical outcomes. The transient and oscillating behaviour of gradient values is sketched by using steady results in the oscillating formula. The finite difference methodology is applied to examine the physical properties with the Gaussian elimination matrix scheme. Effects of buoyant force ( λ ), Prandtl (Pr), magnetic force ( ξ ), density factor ( m ) and MHD Prandtl factor ( γ ) on the thermal behaviour of magneto-hydrodynamic analysis are applied. Results are explored around three angles π / 6 , π / 3 , π of the cylinder. Increasing magnitude in fluid velocity is deduced with high amplitude as porosity, buoyant and density parameters are enhanced.

Rate dependency and fragmentation response of phase field models with micro inertia and micro viscosity terms

We present elliptic, parabolic, and hyperbolic phase field (PF) equations, referred to as EPF, PPF, and HPF, for cohesive zone model (CZM) and Linear Elastic Fracture Mechanics (LEFM) PF models. The micro viscosity and micro inertia terms result in PPF and HPF, that can be solved explicitly in time. The additional advantage of the HPF is implying a finite damage propagation speed and a more favorable time advance limit. The desired micro-viscosity for the HPF is shown to correspond to a damping factor of one. The appropriate length and time scales, nondimensional equations, and (asymptotic) strain-stress solutions are provided for each differential equation. Two sources of rate sensitivity are discussed. First, when the fields are spatially uniform (0D solution) the rate effect arises from the form of differential equation. Asymptotic solutions show that at high loading rates, the energy dissipation has the rates of 2/3 and 1 for the PPF and HPF, respectively, with the former matching the Grady’s rate-sensitivity. The dynamic strength and failure strain show half of this rate. The analysis is extended to damage models with stress-based and bounded driving forces. Second, the rate effect arises from the deviation of 1D from 0D solutions as the damage exceeds a critical value and fragments form. Since this critical value tends to zero for quasi-static loading, this rate-effect is mainly present for low loading rates. This results in lower- and upper-shelf energy limits for the EPF at low and high loading rates, resembling some experimentally observed sigmoidal rate models for energy dissipation.

Statistical analysis of periodic MHD chemically radiative casson flow over an inclined cylinder with entropy optimization

Statistical analysis (SA) stands as an indispensable tool in materials science and engineering, aiding in the understanding and optimization of heat and mass transport processes. This study delves into the interplay between periodic magneto-hydrodynamics (MHD) and chemical reactions concerning entropy generation over an inclined porous cylinder (IPC). We employ a statistical approach, amalgamating the robust 6th order Runge-Kutta (R-K) integration algorithm with the Nachtsheim-Swigert (N-S) shooting iteration method, subsequent to converting the governing equations into Ordinary Differential Equation (ODE) form. The primary objective of this study is to analyze the behavior of entropy generation, unveiling the intricate relationship between entropy dynamics, radiative periodic MHD on IPC, and chemical processes. The numerical findings encapsulate the dynamics of temperature, velocity, and entropy generation, visually portrayed across various dimensionless parameters. The primary findings of this investigation indicate that Casson fluid flow generates 18.99 % more entropy in comparison to Newtonian fluid flow. Furthermore, entropy is 9.87 % larger in an inclined cylinder than in a level cylinder. In contrast, entropy production falls by 3.34 % in the presence of periodic MHD as opposed to non-periodic MHD. Regression study reveals a genuine and significant interaction between variables inside the entropy-generating systems, as well, with a correlation value (R2) of 99.82 % at a 95 % confidence level. The correlation study shows that there is a significant 98.13 % relationship between entropy generation and chemical processes. Understanding the heightened entropy associated with Casson fluid flow benefits tissue engineering, drug delivery systems, and renewable energy. Increased entropy in inclined cylinders aids in optimizing drainage systems, pipeline design, and aerodynamic efficiency in aerospace engineering.

Comparison of Classical and Inverse Calibration Equations in Chemical Analysis.

Chemical analysis adopts a calibration curve to establish the relationship between the measuring technique's response and the target analyte's standard concentration. The calibration equation is established using regression analysis to verify the response of a chemical instrument to the known properties of materials that served as standard values. An adequate calibration equation ensures the performance of these instruments. There are two kinds of calibration equations: classical equations and inverse equations. For the classical equation, the standard values are independent, and the instrument's response is dependent. The inverse equation is the opposite: the instrument's response is the independent value. For the new response value, the calculation of the new measurement by the classical equation must be transformed into a complex form to calculate the measurement values. However, the measurement values of the inverse equation could be computed directly. Different forms of calibration equations besides the linear equation could be used for the inverse calibration equation. This study used measurement data sets from two kinds of humidity sensors and nine data sets from the literature to evaluate the predictive performance of two calibration equations. Four criteria were proposed to evaluate the predictive ability of two calibration equations. The study found that the inverse calibration equation could be an effective tool for complex calibration equations in chemical analysis. The precision of the instrument's response is essential to ensure predictive performance. The inverse calibration equation could be embedded into the measurement device, and then intelligent instruments could be enhanced.

Well-posedness and stability of a stochastic neural field in the form of a partial differential equation

A system of partial differential equations representing stochastic neural fields was recently proposed with the aim of modelling the activity of noisy grid cells when a mammal travels through physical space. The system was rigorously derived from a stochastic particle system and its noise-driven pattern-forming bifurcations have been characterised. However, due to its nonlinear and non-local nature, standard well-posedness theory for smooth time-dependent solutions of parabolic equations does not apply. In this article, we transform the problem through a suitable change of variables into a nonlinear Stefan-like free boundary problem and use its representation formulae to construct local-in-time smooth solutions under mild hypotheses. Then, we prove that fast-decaying initial conditions and globally Lipschitz modulation functions imply that solutions are global-in-time. Last, we improve previous linear stability results by showing nonlinear asymptotic stability of stationary solutions under suitable assumptions.

A new convex integration approach for the compressible Euler equations and failure of the local maximal dissipation criterion

Abstract In this paper we establish a new convex integration approach for the barotropic compressible Euler equations in two space dimensions. In contrast to existing literature, our new method generates not only the momentum for given density, but also the energy and the energy flux. This allows for a simple way to construct admissible solutions, i.e. solutions which satisfy the energy inequality. Moreover using the convex integration method developed in this paper, we show that the local maximal dissipation criterion fails in the following sense: There exist wild solutions which beat the self-similar solution of the one-dimensional Riemann problem extended to two dimensions. Hence the local maximal dissipation criterion rules out the self-similar solution. The convex integration machinery itself is carried out in a very general way. Hence this paper provides a universal framework for convex integration which not even specifies the form of the partial differential equations under consideration. Therefore this general framework is applicable in many different situations.

Thermal analysis of mathematical model of heat and mass transfer through bioconvective Carreau nanofluid flow over an inclined stretchable cylinder

In this proceeding the main focus to scrutinize the heat and mass transfer rate increment due to the involvement of motile microorganisms in the Carreau nanofluid flow through an inclined stretchable cylinder under the consequences of activation energy and thermal radiation. This phenomenon has many physical applications in the field of civil, mechanical and seismographic engineering. The research focuses on creating and analyzing new nanocomposites with applications in energy storage, showcasing the transformative impact of nanotechnology on the advancement of high-performance materials. First of all, for the numerical treatment of the above physical model mathematical model is developed in the form of partial differential equations (PDE's) by considering all involving quantities. The resultant model is highly nonlinear and of higher order. Appropriate similarity variables are defined to transform this system of PDE's into first order ordinary differential equations (ODE's). For numerical results we develop a numerical algorithm from nonlinear ODE's and use ‘bvp4c’ built in package of MATLAB. Numerical results are acquired from software in the form of graphical visuals and tabular data as narrated in the results and discussion section. Using this tabular data streamlines are constructed for the better understanding of heat and mass transfer through this phenomenon under the defined constraints. From results, velocity profile increased by the augmentation of values of curvature parameter. Thermal profile is stronger when values of thermophoresis parameter is boosted, Concentration of nanoparticle profile get smoother when temperature coefficient increased, and motile microorganism's density profile is enlarged upon inclining the values of bioconvection lewis number.

Analysis of multi-term arbitrary order implicit differential equations with variable type delay

Due to their capacity to simulate intricate dynamic systems containing memory effects and non-local interactions, fractional differential equations have attracted a great deal of attention lately. This study examines multi-term fractional differential equations with variable type delay with the goal of illuminating their complex dynamics and analytical characteristics. The introduction to fractional calculus and the justification for its use in many scientific and technical domains sets the stage for the remainder of the essay. It describes the importance of including variable type delay in differential equations and then applying it to model more sophisticated and realistic behaviours of real-world phenomena. The research study then presents the mathematical formulation of variable type delay and multi-term fractional differential equations. The system’s novelty stems from its unique combination of variable delay, generalized multi terms fractional differential operators (n and m), and integral implicit parameters, and studying the stability of the the newly formulated system as compared to the work in the existing literature. While the variable type delay is introduced as a function of time to describe instances where the delay is not constant, the fractional order derivatives are generated using the Caputo approach. The existence, uniqueness, and stability of solutions are the main topics of the theoretical analysis of the suggested differential equations. In order to establish important mathematical features, the inquiry makes use of spectral techniques, and fixed-point theorems. The study finishes by summarizing the major discoveries and outlining potential future research avenues in this developing field. It highlights the potential contribution of multi-term fractional differential equations with variable type delay to improving the control and design of complex systems. Overall, this study adds to the growing body of knowledge in the field of fractional calculus and provides insightful information about the investigation of multi-term fractional differential equations with variable type delay, making it pertinent for academics and practitioners from a variety of fields.

Modeling of adsorptive treatment of lead (II) ions contaminated effluents using cashew nut shell alkali activated carbon: Batch kinetic, isothermal and thermodynamic studies

The numerical study of the decontamination of lead(II) ions-rich effluent employing activated carbon derived from cashew nut shell was the central focus of this work. Chemical activation with zinc chloride (ZnCl2) was used to produce powdered activated carbon at optimized activation conditions of impregnation ratio (0.531), activation temperature (1083 K) and activation time (63min). The physiochemical properties of raw and activated cashew nut shell were determined using Fourier transform infrared spectroscopy, scanning, X-ray diffractor (XRD), scanning electron microscopy (SEM), and Brunauer-Emmet-Teller (BET) analyses. Batch adsorption experiments were performed at varied process conditions of temperature (303–323) K, contact time (5–80min), and initial lead(II) ion concentrations (100–400 mg.L−1). The adsorption behavior of the batch–type operation was mathematically described in the form of non-linear parabolic partial differential equations (PDE’S) incorporating film, pore and surface diffusion mechanisms. The batch mathematical model was solved by implementing a custom MATLAB program-OkiyNwabannebatch and calibrated to predict the desired response (percentage removal) using measured adsorption data. The equilibrium adsorption, kinetics and thermodynamics of lead(II) ions adsorption onto activated cashew nut shell were also assessed using nonlinear isotherm, kinetic and thermodynamic models. Sensitivity analysis unveiled that temperature with sensitivity value of 37.31 % had a predominant effect on the model performance. The simulated equilibrium data was well explicated by the Freundlich model for lead(II) ions adsorption onto activated cashew nut shell. The pseudo-second order kinetic model best reproduced the simulated adsorption data for lead(II) ions uptake by cashew nut shell adsorbent. Likewise, the kinetic data followed Boyd model, indicating that the adsorption process is controlled by external (film) diffusion for the uptake of lead(II) ions by alkali activated cashew nut shell. Thermodynamic analysis using the Gibbs and Van’t Hoff’s models indicated that lead(II) ions adsorption by modified cashew nut shell is spontaneous and exothermic. The physical nature of the adsorption process was elucidated from the low values of the isosteric heats of adsorption (ΔHX) evaluated (<80 kJ mol−1). This study showed that zinc chloride activated cashew nut shell has good potential to be used as a cost-effective and efficient adsorbent for the uptake of lead(II) ions from aqueous solution.

An innovative subdivision collocation algorithm for heat conduction equation with non-uniform thermal diffusivity.

This paper presents a subdivision collocation algorithm for numerically solving the heat conduction equation with non-uniform thermal diffusivity, considering both initial and boundary conditions. The algorithm involves transforming the differential form of the heat conduction equation into a system of equations and discretizing the time variable using the finite difference formula. The numerical solution of the system of heat conduction equations is then obtained. The feasibility of the algorithm is verified through theoretical and numerical analyses. Additionally, numerical and graphical representations of the obtained numerical solutions are provided, along with a comparison to existing methods. The results demonstrate that our proposed method outperforms the existing methods in terms of accuracy.

Passive control and damage assessment with a pendulum tuned mass damper using mean square equations and second-order eigen-perturbation techniques

This paper proposes a novel framework for damage detection and retuning of a Pendulum Tuned Mass Damper (PTMD) addressing effects of stochasticity via mean-square equations and establishing rapid change detection. The framework was made possible via a Stochastic Differential Equation (SDE) formulation of the equations of motion considering a two degree of freedom base and structure system equipped with a PTMD. The system was assumed to be excited at the base with Gaussian white noise and the solutions were computed using a stochastic integration scheme. A Second Order Eigen-Perturbation (SOPS) estimate is used to carry out detection of sudden stiffness reduction in the base floor of the structure. Once damage is detected, mean square equations on monitored displacements were used to retune the PTMD by adapting the length of the pendulum. It was found that damage could be detected rapidly and successive retunings were able to minimize the mean square displacement of the damaged structure and stabilise it over time.

A magnetohydrodynamic flow of a water-based hybrid nanofluid past a convectively heated rotating disk surface: A passive control of nanoparticles

Abstract One of the basic fluid mechanics problems of fluid flows over a revolving disk has both theoretical and real-world applications. The flow over a rotating disk has been the subject of numerous theoretical studies because it has many real-world applications in areas like rotating machinery, medical equipment, electronic devices, and computer storage. It is also crucial for engineering processes. Therefore, this article deals with a time-independent water-based hybrid nanofluid flow containing copper oxide and silver nanoparticles past a spinning disk. The Newtonian flow is taken into consideration in this analysis. The influence of magnetic field, thermophoresis, nonlinear thermal radiation, Brownian motion, and activation energy has been considered. The present analysis is modeled in a partial differential equation form and is then converted to ordinary differential equations using appropriate variables. A numerical solution using the bvp4c technique is accomplished using MATLAB software. The current results are matched with the previous literature and established a close relationship with previous studies. The purpose of this investigation is to numerically investigate the time-independent hybrid nanofluid flow comprising copper oxide and silver nanoparticles over a rotating disk surface. The results show that the increased magnetic parameters increase the friction force at the surface, which decreases the radial and azimuthal velocity distribution. At the sheet surface, the radial velocity of the hybrid nanofluid shows dominant performance compared to the nanofluid. On the other hand, the magnetic factor has dominant behavior on the azimuthal velocity component of the nanofluid flow compared to the hybrid nanofluid flow. The higher volume fraction and magnetic factor enhance the skin friction at the disk surface. Furthermore, greater surface drag is found for the hybrid nanofluid flow. The higher solid volume fraction, temperature ratio, and Biot number enhance the rate of heat transmission. Also, a higher rate of heat transmission is observed for the hybrid nanofluid flow.

Using Symmetries to Investigate the Complete Integrability, Solitary Wave Solutions and Solitons of the Gardner Equation

In this paper, using a scaling symmetry, it is shown how to compute polynomial conservation laws, generalized symmetries, recursion operators, Lax pairs, and bilinear forms of polynomial nonlinear partial differential equations, thereby establishing their complete integrability. The Gardner equation is chosen as the key example, as it comprises both the Korteweg–de Vries and modified Korteweg–de Vries equations. The Gardner and Miura transformations, which connect these equations, are also computed using the concept of scaling homogeneity. Exact solitary wave solutions and solitons of the Gardner equation are derived using Hirota’s method and other direct methods. The nature of these solutions depends on the sign of the cubic term in the Gardner equation and the underlying mKdV equation. It is shown that flat (table-top) waves of large amplitude only occur when the sign of the cubic nonlinearity is negative (defocusing case), whereas the focusing Gardner equation has standard elastically colliding solitons. This paper’s aim is to provide a review of the integrability properties and solutions of the Gardner equation and to illustrate the applicability of the scaling symmetry approach. The methods and algorithms used in this paper have been implemented in Mathematica, but can be adapted for major computer algebra systems.

Investigating thermal conduction dynamics in ternary Carreau–Yasuda nanofluids under convective boundary conditions and exponential heat source/sink effects

The analysis of the Carreau–Yasuda nanofluid (CYNF) across a stretching surface has practical applications in several fields, such as heat exchange, engineering, and material science. Scholars may discover novel perspectives for increasing heat transfer performance, establishing advanced materials, and enhancing the efficiency of multiple engineering systems by studying the behavior of CYNF in this particular instance. The energy transfer through trihybrid CYNF flow with the effect of magnetic dipole across a stretching sheet is examined in the present study. The ternary hybrid nanofluid (THNF) has been prepared by the addition of ternary nanoparticles (NPs) in the water (50%) and ethylene glycol (50%). Titanium dioxide (TiO2), silicon dioxide (SiO2), and aluminum oxide (Al2O3) are used in base fluid. The fluid velocity and heat transfer are examined under the impact of Darcy–Forchheimer, chemical reaction, convective condition, activation energy, and exponential heat source. The fluid flow has been stated in form of velocity, energy, and concentration equations. The set of modeled equations is simplified to non-dimensional form of ordinary differential equations (ODEs) by using similarity substitutions. Numerically, the system of lowest order ODEs is calculated through the parametric continuation method. It has been noticed that the temperature field augments against the variation of viscous dissipation and heat source term. Furthermore, the magnetic dipole has significant impact to enhance the thermal curve of THNF whereas falloffs the velocity profile. It can be noticed that the energy propagation rate enhances with the rising numbers of ternary NPs, from 1.22% to 5.98% (nanofluid), 2.30% to 9.61% (hybrid nanofluid), and 3.23% to 11.12% (trihybrid nanofluid).

An efficient computational method for solving the fractional form of the European option price PDE with transaction cost under the fractional Heston model

This paper introduces a new method for precise option pricing analysis by utilizing the fractional form of the option price partial differential equation (PDE) and implementing a collocation technique based on the first Fermat polynomial sequence. The key innovation of this study is the exploration of the fractional formulation of European option pricing within the context of the fractional Heston model, which employs a collocation scheme utilizing Fermat polynomials to generate operational matrices characterized by numerous zeros, thereby enhancing computational efficiency. To achieve this, we first derive the option price PDE and convert it to its fractional form in the Caputo sense. We then solve the fractional PDE using fractional Fermat functions, expressing the solution as a series of multivariate Fermat functions with unknown coefficients. Following this, we compute the operational matrices for the Caputo fractional derivative and related partial derivatives, demonstrating how this computational framework transforms the primary problem into a nonlinear system of equations. Additionally, we conduct a convergence analysis of the collocation method. We conclude by presenting several numerical examples that illustrate the applicability and effectiveness of the proposed method, with its robust theoretical foundations and successful numerical tests indicating significant potential for practical applications in finance.