Search

Abstract In general, integer order differential equations fall short of adequately describing a range of phenomena in many disciplines of science and engineering when compared to fractional order differential equations. In the context of the Caputo fractional derivative, this study examines the nonlinear fractional order foam drainage problem as well as the fractional order modified regularized long wave equation. In this work, we offer two novel techniques by combining the Elzaki transform with the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM). We first translate the problem into its differential partner using the Elzaki transform, and then we use He’s polynomials and the Adomian polynomials that are derived from HPM and ADM, respectively. These are two very powerful supports for nonlinear problems. The efficacy and precision of the recommended techniques are illustrated through the use of numerical and graphical findings. The numerical simulations of the derived solutions make use of a number of relevant values of the order of fractional parameter λ. It is important to keep in mind that the recommended techniques can achieve higher overall performance since they need less computational effort than alternative strategies while yet retaining a high degree of numerical precision. The solutions acquired through the suggested techniques have been compared with the classical solutions and the solution obtained by ADM, new iterative method (NIM). The considerable results show that these approaches minimize the heavy calculation without restricting variables while requiring no assumptions. The present study can confirm the applicability of the new generalized Caputo type fractional operator to mathematical real-world problems. This approach shows to be one of the most efficient methods to solve fractional order nonlinear differential equations that arise in related fields of engineering and science. These methods also make clear how to use fractal calculus in real life. Furthermore, the results of this study support the value and significance of fractional operators in real-world applications.

Falkner-Skan aspects are revealed numerically for a non-homogeneous hybrid mixture of 50% ethylene glycol-50% water, silver nanomaterials Ag, and molybdenum disulfide nanoparticles MoS2 during its motion over a static wedge surface in a Darcy-Forchheimer porous medium by employing the modified Buongiorno model. The Brownian and thermophoresis mechanisms are included implicitly along with the thermophysical properties of each phase via the mixture theory and some efficient phenomenological laws. The present simulation also accounts for the impacts of nonlinear radiative heat flux, magnetic forces, and Joule heating. Technically, the generalized differential quadrature method and Newton-Raphson technique are applied successfully for solving the resulting nonlinear boundary layer equations. In a limiting case, the obtained findings are validated accurately with the existing literature outcomes. The behaviors of velocity, temperature, and nanoparticles volume fraction are discussed comprehensively against various governing parameters. As crucial results, it is revealed that the temperature is enhanced due to magnetic field, linear porosity, radiative heat flux, Brownian motion, thermophoresis, and Joule heating effects. Also, it is depicted that the hybrid nanoliquids present a higher heat flux rate than the monotype nanoliquids and liquids cases. Moreover, the surface frictional impact is minimized via the linear porosity factor. Furthermore, the surface heat transfer rate receives a prominent improvement due to the radiative heat flux inclusion.

Abstract Due to enhanced heat transfer rate, the nanofluid and hybrid nanofluids have significant industrial uses. The principal objective of this exploration is to investigate how thermal radiation influences the velocity and temperature profile. A water-based rotational nanofluid flow with constant angular speed $${\Omega }$$ Ω is considered for this comparative study. A similarity conversion is applied to change the appearing equations into ODEs. Three different nanoparticles i.e., copper, aluminum, and titanium oxide are used to prepare different nanofluids for comparison. The numerical and graphical outputs are gained by employing the bvp-4c procedure in MATLAB. The results for different constraints are represented through graphs and tables. Higher heat transmission rate and minimized skin friction are noted for triple nanoparticle nanofluid. Skin coefficients in the x-direction and y-direction have reduced by 50% in trihybrid nanofluid by keeping mixed convection levels between the range $$3

The formulation of models and solutions for various physical problems are the primary goals of scientific achievements in engineering and physics. Our paper focuses on using the Caputo fractional derivative operator to solve nonlinear fractional logistic differential equations. In order to solve general nonlinear fractional differential equations, we first introduce a novel numerical methodology termed the Homotopy perturbation transform method. The perturbation approach and the Yang transform method are combined to create the suggested strategy. Second, we introduce a new hybrid method that uses the time-fractional Caputo derivative to approximate and analytically solve nonlinear fractional logistic differential equations. This method combines the Yang transform with the decomposition method. To validate the analysis, we offer three numerical cases of nonlinear fractional logistic differential equations employing the Caputo fractional derivative operator. The resulting solutions exhibit rapid convergence and are presented in series form. In order to verify the efficacy and relevance of the suggested methodologies, the investigated issues were assessed through the implementation of different fractional orders. We examine and show that, under the specified initial conditions, the solution approaches under evaluation are accurate and effective. Graphs in two and three dimensions show the results that were obtained. Numerical simulations are presented to confirm the efficacy of the strategies. The numerical results show that an accurate, reliable, and efficient approximation can be obtained with a minimal number of terms. The results obtained demonstrate that the new analytical solution method is easy to apply and very successful in solving difficult fractional problems that occur in relevant engineering and scientific domains.

In this article application of neural network using Levenberg–Marquardt Back-propagation is implemented on differential model to study and analyze ringworm infectious disease. The formulated system of differential equations is consisting of following parts, namely; S(t) population which is at verge of being infected by ringworm, E(t) shows the environment effected by dermetophytosis fungus, I(t) whereas represent the infected individuals, R(t) and shows the population which has recovered from the infection. The solutions of different categories are represented by considering distinct datasets modeled and designed using LMB neural network. The numerical scheme Adam has been employed to establish a reference data set of the designed LMB neural network. The approximate outcomes of the SEIR based on dispersing and curing are discussed using the authentication, testing and training procedures to truncate the mean square error in function with help of LMB. The mean square error, regression, error histograms are generated to produce efficiency, effectiveness and correctness of proposed LMB neural network scheme.

A theoretical novel model in the context of the photo-thermoelasticity theory is examined. The novel mode is studied under the impact of the external magnetic field with strong intensity. The semiconductor medium is excited by a beam of light and the governing equations are taken during a one-dimensional (1D) deformation. The main parameters (thermo-elastic, electrical conductivity and optical constants) of the medium depending on the depth of the medium which is defined as functionally graded (FG). The hyperbolic two-temperature theory is taken into account to describe the acceleration of two distinct temperature. During the excitation transport processes and magnetic field, the main parameters can be chosen as a function of the axial distance and Hall current appears, respectively. The main equations are solved analytically using the Laplace transform technique. The complete solutions have been obtained in space–time domain when the convers of Laplace transform is used numerically under some mechanical, thermal and plasma conditions. The evaluation results are compared with the corresponding computational results obtained under the impacts of some parameters.

Abstract The current paper discusses the numerical simulation results of the NePCM melting process inside an annulus thermal storage system. The TES system consists of a wavy shell wall and a cylindrical tube equipped with three fins. The enthalpy-porosity method was utilized to address the transient behavior of the melting process, while the Galerkin FE technique was used to solve the system governing equations. The results were displayed for different inner tube positions (right-left–up and down), inner cylinder rotation angle (0 ≤ α ≤ 3π/2), and the nano-additives concentration (0 ≤ ϕ ≤ 0.04). The findings indicated that high values of nano-additives concentration (0.4), bigger values of tube rotation angle (3π/2), and location of the tube at the lower position accelerated the NePCM melting process.

An anisotropic horizontal porous layer saturated with viscoelastic liquids of the Oldroyd-B type is explored to determine how the internal heat source affects thermal convection. As a momentum equation, a modified Darcy–Oldroyd model is used that takes into account the anisotropy of the porous layer. The energy equation is formulated in such a way that the influence of internal heat sources and anisotropy in thermal diffusivity on the stability criterion may be easily identified. The effects of anisotropy, viscoelasticity, and internal heat generation on the onset of thermal convection are investigated using linear stability analysis. It is understood that convection begins via an oscillatory mode instead of a stationary mode because viscous relaxation, thermal diffusions, and internal heat generation mechanisms compete with one another. Both steady and unsteady finite-amplitude convections are studied using nonlinear stability analysis with the truncated Fourier series method. The effect of different governing parameters on the system’s stability and on convective heat transfer is studied. The present investigation has been significantly validated by the recovery of several prior results as special situations. The findings presented in this work are anticipated to have significant implications for a number of real-world applications, including modeling of oil reservoirs, crude oil extraction, crystal growth, the pharmaceutical and medical industries, and the use of geothermal energy, among others.

Heat transfer effect on peristaltic Newtonian flow in an annular region of eccentric cylinders is considered. An approximate solution for temperature and velocity is obtained using perturbation technique. For constant flow rate, the variation of pressure gradient and velocity with respect to several non-dimensional parameters is studied using graphical representation. It is seen in these graphs that velocity, pressure gradient and temperature enhance with raise in eccentricity and amplitude ratio. Heat diffuses with faster rate with depletion in area of cross section eccentric annulus through parameter inner cylinder radius. Also, velocity rises with increase in heat transfer parameters such as Grashof number and heat transfer co-efficient. Thus velocity of peristaltic flow in eccentric annulus enhances rapidly in presence of heat transfer compared to absence of heat transfer. A study of peristaltic flow in concentric annulus in presence of heat transfer is obtained analytically and compared with present study at eccentricity equal to zero, to validate the approximate solution. It is seen that flow rate increases more in case of eccentric annulus compared to concentric in presence of heat transfer. Present study has application in endoscope, which is important to diagnose problems in internal organs. Also, the variation of pressure gradient helps to maintain flow rate which is essential during insertion of catheter into artery.

This communication elaborates the irreversibility analysis of the flow of Prandtl nanofluid along with thermal radiation past a permeable stretched surface embedded in a Darcy-Forchheimer medium. The activation and chemical impressions along with effects of thermophoretic and Brownian motion are as well examined. The flow symmetry of the problem is modeled mathematically and leading equations are rehabilitated into nonlinear ordinary differential equations (ODEs) through the assistance of suitable similarity variables. The Keller-box technique in MATLAB is employed to draw the impacts of the contributing elements on the velocity field, temperature distribution, and concentration. The impact of the Prandtl fluid parameter has mounting performance for the velocity whereas conflicting behavior is examined in the temperature profile. The achieved numerical results are matched correspondingly with the present symmetrical solutions in restrictive cases and fantastic agreement is scrutinized. In addition, the entropy generation uplifts for the growing values of the Prandtl fluid parameter, thermal radiation, and Brinkman number and decreases for growing numbers of the inertia coefficient parameter. It is also discovered that the coefficient of friction decreases for all parameters involved in the momentum equation. Features of nanofluids can be found in a variety of real-world fields, including microfluidics, industry, transportation, the military, and medicine.

The importance of Phase change material (PCM) energy storage systems is no longer new in the industry. However, the influence of using branched fins inside the energy storage system on the melting process of alumina nanoparticles and n-octadecane has not been reported in the literature. Consequently, the outcome of a study on the numerical simulation for optimizing the melting performance of a PCM in various tubes, including those with branching fins is presented in this report. Four examples were assessed in relation to a suspension of alumina nanoparticles and n-octadecane paraffin that contains heated fins. A numerical technique based on the Galerkin finite element method (GFEM) was used to solve the dimensionless governing system. The average liquid percentage over the flow zone in question was computed. The primary results indicated that altering the number of heated fins might affect the flow structures, the system’s irreversibility, and the melting process. Case four, with eight heated fins, likewise produces the greatest average liquid fraction values and completes the melting process in 850s. Additionally, when 6% nano-enhanced PCM was used instead of pure PCM, the melting process is accelerated by 28.57 percent.

This research uses a novel analytical method known as the modified Extended Direct Algebraic Method (mEDAM) to explore families of soliton solutions for the complex structured Coupled Fractional Biswas–Arshed Model (CFBAM) in Birefringent Fibers. The Direct Algebraic Method (DAM) is extended by the mEDAM’s methodology to compute more analytical solutions that would otherwise be difficult to acquire. We use this method to derive several families of soliton solutions and examine their characteristics. We also look at how different model parameters, such as amplitude, width, and propagation speed, affect the dynamics of soliton. Our use of 2D and 3D graphics to illustrate the soliton solutions also makes it possible to see the soliton dynamics more clearly. The outcomes also demonstrate that the method suggested has proven successful in producing soliton solutions for intricate structures such as the CFBAM.

In this investigation, we utilize advanced versions of the Extended Direct Algebraic Method (EDAM), namely the modified EDAM (mEDAM) and r+mEDAM, to explore families of optical soliton solutions in the Fractional Perturbed Radhakrishnan–Kundu–Lakshmanan Model (FPRKLM). Our study stands out due to its in-depth investigation and the identification of multiple localized and stable soliton families, illuminating their complex behavior. We offer visual validation via carefully designed 3D graphics that capture the complex behaviors of these solitons. The implications of our research extend to fiber optics, communication systems, and nonlinear optics, with the potential for driving developments in optical devices and information processing technologies. This study conveys an important contribution to the field of nonlinear optics, paving the way for future advancements and a greater comprehension of optical solitons and their applications.

The present research investigates symmetric soliton solutions for the Fractional Coupled Konno–Onno System (FCKOS) by using two improved versions of an Extended Direct Algebraic Method (EDAM) i.e., modified EDAM (mEDAM) and r+mEDAM. By obtaining precise analytical solutions, this research explores the characteristics and behaviours of symmetric solitons in FCKOS. Further, the amplitude, shape and propagation behaviour of some solitons are visualized by means of a 3D graph. This investigation fosters a more thorough comprehension of non-linear wave phenomena in considered systems and offers helpful insights towards soliton behavior in it. The outcomes reveal that the recommended techniques are successful in constructing symmetric soliton solutions for complex models like the FCKOS.

The q-homotopy analysis transform method (q-HATM) is a powerful tool for solving differential equations. In this study, we apply the q-HATM to compute the numerical solution of the fractional-order Kolmogorov and Rosenau–Hyman models. Fractional-order models are widely used in physics, engineering, and other fields. However, their numerical solutions are difficult to obtain due to the non-linearity and non-locality of the equations. The q-HATM overcomes these challenges by transforming the equations into a series of linear equations that can be solved numerically. The results show that the q-HATM is an effective and accurate method for solving fractional-order models, and it can be used to study a wide range of phenomena in various fields.

The phase change heat transfer of nano-enhanced phase change materials (NePCMs) was addressed in a heatsink filled with copper metal foam fins. The NePCM was made of 1-Tetradecanol graphite nanoplatelets. The heatsink was an annulus contained where its outer surface was subject to a convective cooling of an external flow while its inner surface was exposed to a constant heat flux. The governing equations, including the momentum and heat transfer with phase change, were explained in a partial differential equation form and integrated using the finite element method. An artificial neural network was employed to map the relationship between the anisotropic angle and nanoparticles fractions with the melting volume fraction. The computational model data were used to successfully train the ANN. The trained ANN showed an R-value close to unity, indicating the high prediction accuracy of the neural network. Then, ANN was used to produce maps of melting fractions as a function of design parameters. The impact of the geometrical placement of metal foam fins and concentrations of the nanoparticles on the surface heat transfer was addressed. It was found that spreading the fins (large angles between the fins) could improve the cooling performance of the heatsink without increasing its weight. Moreover, the nanoparticles could reduce the thermal energy storage capacity of the heatsink since they do not contribute to heat transfer. In addition, since the nanoparticles generally increase the surface heat transfer, they could be beneficial only with 1.0% wt in the middle stages of the melting heat transfer.

In this work, we have performed an investigation to increase our understanding of the motion of a hybrid nanofluid trapped inside a three-dimensional container. The room also includes a three-dimensional heated obstacle of an elliptic cross-section. The top wall of space is horizontally movable and adiabatic, while the lower part is zigzagged and thermally insulated as well. The lateral walls are cold. The container’s space is completely replete with Al2O3-Cu/water; the concentration of nanoparticles is 4%. The space is also characterized by the permeability, which is given by the value of the Darcy number (limited between 10−5 and 10−2). This studied system is immersed in a magnetic field with an intensity is defined in terms of Hartmann number (limited between 0 and 100). The thermal buoyancy has a constant impact (Gr = 1000). This study investigates the influences of these parameters and the inclination angle of the obstacle on the heat transfer coefficient and entropy generation. The Galerkin finite element method (GFEM) was the principal technique for obtaining the solution of the main partial equations. Findings from our work may be exploited to depict the conditions for which the system is effective in thermal cooling and the case in which the system is effective in thermal insulation.

Thermophoresis represents one of the most common methods of directing micromachines. Enhancement of heat transfer rates are of economic interest for micromachine operation. This study aims to examine the heat transfer enhancement within the shell and tube latent heat thermal storage system (LHTSS) using PCMs (Phase Change Materials). The enthalpy–porosity approach is applied to formulate the melting situation and various shapes of inner heated fins are considered. The solution methodology is based on the Galerkin finite element analyses and wide ranges of the nanoparticle volume fraction are assumed, i.e., (0% ≤ φ ≤ 6%). The system entropy and the optimization of irreversibility are analyzed using the second law of the thermodynamics. The key outcomes revealed that the flow features, hexagonal entropy, and melting rate might be adjusted by varying the number of heated fins. Additionally, in case 4 where eight heated fins are considered, the highest results for the average liquid percentage are obtained.

To date, when considering the dynamics of water conveying multi-walled carbon nanoparticles (MWCNT) through a vertical Cleveland Z-staggered cavity where entropy generation plays a significant role, nothing is known about the increasing Reynold number, Hartmann number, and Darcy number when constant conduction occurs at both sides, but at different temperatures. The system-governing equations were solved using suitable models and the Galerkin Finite Element Method (GFEM). Based on the outcome of the simulation, it is worth noting that increasing the Reynold number causes the inertial force to be enhanced. The velocity of incompressible Darcy-Forchheimer flow at the middle vertical Cleveland Z-staggered cavity declines with a higher Reynold number. Enhancement in the Hartman number causes the velocity at the center of the vertical Cleveland Z-staggered cavity to be reduced due to the associated Lorentz force, which is absent when Ha = 0 and highly significant when Ha = 30. As the Reynold number grows, the Bejan number declines at various levels of the Hartmann number, but increases at multiple levels of the Darcy number.

This paper presents a numerical simulation of a magneto-convection flow in a 3D chamber. The room has a very specific permeability and a zigzag bottom wall. The fluid used in this study is Al2O3-Cu/water with 4% nanoparticles. The Galerkin finite element technique (GFEM) was developed to solve the main partial equations. The hybrid nanofluid inside the container is subjected to the horizontal motion of the upper wall, an external magnetic field, and a thermal buoyancy force. The present numerical methodology is validated by previous data. The goal of this investigation was to understand and determine the percentage of heat energy transferred between the nanofluid and the bottom wall of the container under the influence of a set of criteria, namely: the movement speed of the upper wall of the cavity (Re = 1 to 500), the amount of permeability (Da = 10−5 to 10−2), the intensity of the external magnetic field (Ha = 0 to 100), the number of zigzags of the lower wall (N = 1 to 4), and the value of thermal buoyancy when the force is constant (Gr = 1000). The contours of the total entropy generation, isotherm, and streamline are represented in order to explain the fluid motion and thermal pattern. It was found that the heat transfer is significant when (N = 4), where the natural convection is dominant and (N = 2), and the forced convection is predominant.

Nanofluids have become important working fluids for many engineering applications as they have better thermal properties than traditional liquids. Thus, this paper addresses heat transfer rates and entropy generation for a Fe3O4/MWCNT-water hybrid nanoliquid inside a three-dimensional triangular porous cavity with a rotating cylinder. The studied cavity is heated by a hot wavy wall at the bottom and subjected to a magnetic field. This problem is solved numerically using the Galerkin finite element method (GFEM). The influential parameters considered are the rotating cylinder speed, Hartmann number (Ha), Darcy number (Da), and undulation number of the wavy wall. The results showed that higher Da and lower Ha values improved the heat transfer rates in the cavity, which was demonstrated by a higher Nusselt number and flow fluidity. The entropy generation due to heat losses was also minimized for the enhanced heat transfer rates. The decrease in Ha from 100 and 0 improved the heat transfer by about 8%, whereas a high rotational speed and high Da values yield optimal results. For example, for Ω = 1000 rad/s and Da = 10−2, the enhancement in the average Nusselt number is about 38% and the drop in the Bejan number is 65% compared to the case of Ω = 0 rad/s and Da = 10−5. Based on the applied conditions, it is recommended to have a high Da, low Ha, one undulation for the wavy wall, and high rotational speed for the cylinder in the flow direction.

The construction of a specific splitting-type preconditioner in block formulation applied to a class of group relaxation iterative methods derived from the centred and rotated (skewed) finite difference approximations has been shown to improve the convergence rates of these methods. In this paper, we present some theoretical convergence analysis on this preconditioner specifically applied to the linear systems resulted from these group iterative schemes in solving an elliptic boundary value problem. We will theoretically show the relationship between the spectral radiuses of the iteration matrices of the preconditioned methods which affects the rate of convergence of these methods. We will also show that the spectral radius of the preconditioned matrices is smaller than that of their unpreconditioned counterparts if the relaxation parameter is in a certain optimum range. Numerical experiments will also be presented to confirm the agreement between the theoretical and the experimental results.