Search

The “hard water” factor shows the management of water in the Nusa Tenggara Timur, which shows a higher ratio based on the ion’s minerals. The incessant use of hard water presents kidney dysfunction, which produces diabetic and vascular kinds of diseases. Therefore, it is essential to recognize the influences of hard water on kidney function. A novel design of a stochastic solver using the transfer radial basis function is provided by applying the Bayesian regularization neural network for solving the model. The kidney dysfunction mathematical system is divided into humans (susceptible, infected, recovered) and water components (magnesium, calcium). Twelve numbers of neurons with the radial basis transfer function have been used in the hidden layers for solving the model. The approach performance is remarked through the results comparison and further reducible absolute error found around 10−06 to 10−08 develop the scheme’s exactness. Moreover, the statistical performances including regression coefficient performances around 1 for each case of the model validate the reliability and exactness of the scheme for solving the model. [ABSTRACT FROM AUTHOR]

This paper shows the outcomes of computer virus propagation (CVP) model, represented with susceptible, exposed, infected, quarantine and recovered computers (SEIRQ), classes based mathematical model using the stochastic procedures. The systematic study of the CVP based SEIRQ model represents that the equilibrium state of virus-free is stable globally with reproduction not more than one, while the viral symmetry is attractive globally. The numerical performances of the CVP based SEIRQ model are presented by using the stochastic computational framework based on the artificial neural networks (ANNs) together with the Levenberg-Marquardt backpropagation (LBMB) called as ANNs-LBMB. The learning procedures via ANNs-LBMB for solving the CVP based SEIRQ model are implemented to indorse the statics using the testing, authorization, and training. Thirteen numbers of neurons and the data selection for training 72%, testing 12% and validation 16% are selected to solve the model. For the numerical outcomes of the CVP based SEIRQ model using the ANNs-LBMB, a dataset is considered through the Adams approach. The accuracy and reliability performances of the scheme are presented by using the values of the absolute error (AE) along with the observations of state transitions (STs), regression, mean square error (MSE) and error histograms (EHs). [ABSTRACT FROM AUTHOR]

In this work, a Liouville–Caputo fractional order (FO) derivative for the mathematical system based on the accelerating universe in the modified gravity (AUMG), i.e. FO-AUMG is proposed to get more accurate solutions. The nonlinear dynamics of the FO-AUMG is classified into five dynamics. The performances of the designed nonlinear FO-AUMG are numerically stimulated with the stochastic procedures of Levenberg–Marquardt backpropagated (LMB) scheme-based neural networks. The statics for FO-AUMS is used for the nonlinear FO-AUMG as 72%, 16% and 12% for training, authorization, and testing. Twenty neurons in hidden layers have been used to approximate the solution of the nonlinear FO-AUMS. The comparison of three different cases of the nonlinear FO-AUMS is performed with dataset generated by Adams method. To validate the uniformity, legitimacy, precision, and competence of LMB-based adaptive neural networks, the outcomes of the state transitions parameters, regression, correlation, error-histogram plots have been exploited. [ABSTRACT FROM AUTHOR]

Our main aim in this paper is to propose and investigate new models of fractional blood ethanol and fractional two‐cell cubic autocatalysis reaction models. In particular, we evaluate the numerical solutions of these models by means of the power law and the Mittag–Leffler kernel. The numerical solutions are based mma7188 the fundamental theorem of fractional calculus as well as the Lagrange polynomial interpolation. We compare the numerical solutions for the blood ethanol with exact solutions while for the two‐cell cubic autocatalysis reaction model, we compare with numerical solutions that derived by using the finite‐difference method. In the case of the fractional order of the second model, the comparison is made with the spectral collocation method. In all cases, we found an excellent agreement. The novelty of this work is to study these models with the proposed algorithm for first time. Finally, we illustrate the behaviors of the numerical solutions for different values of order of the models and their parameters. [ABSTRACT FROM AUTHOR]

Wavelet transforms or wavelet analysis represent a recently created mathematical tool for assistance in resolving various issues. Wavelets can also be used in numerical analysis. In this study, we solve pantograph delay differential equations using the Modified Laguerre Wavelet method (MLWM), an effective numerical technique. Fractional derivatives are defined using the Caputo operator. The convergence of the suggested strategy is carefully examined. The suggested strategy is straightforward, effective, and simple in comparison with previous approaches. Specific examples are provided to demonstrate the current scenario's reliability and accuracy. Compared with other methodologies, our results show a higher accuracy level. With the aid of tables and graphs, we demonstrate the effectiveness of the proposed approach by comparing results of the actual and suggested methods and demonstrating their strong agreement. For better understanding of the proposed method, we show the pointwise solution in the tables provided which confirm the accuracy at each point of the proposed method. Additionally, the results of employing the suggested method to various fractional-orders are compared, which demonstrates that when a value shifts from fractional-order to integer-order, the result approaches the exact solution. Owing to its novelty and scientific significance, the suggested technique can also be used to solve additional nonlinear delay differential equations of fractional-order. [ABSTRACT FROM AUTHOR]

This article investigates different nonlinear systems of fractional partial differential equations analytically using an attractive modified method known as the Laplace residual power series technique. Based on a combination of the Laplace transformation and the residual power series technique, we achieve analytic and approximation results in rapid convergent series form by employing the notion of the limit, with less time and effort than the residual power series method. Three challenges are evaluated and simulated to validate the suggested method's practicability, efficiency, and simplicity. The analysis of the acquired findings demonstrates that the method mentioned above is simple, accurate, and appropriate for investigating the solutions to nonlinear applied sciences models. [ABSTRACT FROM AUTHOR]

This study examines approximate long wave and the modified Boussinesq equations, as well as their complexities with the Atangana–Baleanu fractional derivative operator in the Caputo sense. The analytical solution of the aforementioned model is discussed using the Elzaki transform and the Adomian decomposition method. These problems are indispensable for defining the characteristics of surface water waves by applying a particular relationship of dispersion. We used Elzaki transformation on time-fractional approximate long wave and modified Boussinesq equations, followed by inverse Elzaki transformation, to achieve the results of the equations. To validate the methodology, we concentrated on two systems and compared them to the actual solutions. The numerical and graphical results demonstrate that the proposed method is computationally precise and straightforward for investigating and resolving fractionally coupled nonlinear phenomena that occur in scientific and technological. [ABSTRACT FROM AUTHOR]

Theoretically, this work describes the exact solutions of fractional Casson fluid through a channel under the effect of MHD and porous medium. The unsteady fluid motion of the bottom plate, which is confined by parallel but perpendicular sidewalls, supports the flow. By introducing the dimensionless parameters and variables, the momentum equation, as well as the initial and boundary conditions, has been transformed to a dimensionless form. A mix of Laplace and Fourier transformations is used to get the exact solution for the momentum equation. The constitutive equations for Caputo-Fabrizio's time-fractional derivative are also incorporated for recovering the exact solutions of the flow problem under consideration. After recovering the exact solutions for flow characteristics, three different cases at the surface of the bottom plate are discussed, by addressing the limiting cases under the influence of the side walls. Moreover, these solutions are captured graphically, and the effects of the Reynolds number Re , fractional parameter α , effective permeability K eff , and dimensionless parameter for Casson fluid β on the fluid's motion are observed. [ABSTRACT FROM AUTHOR]

Applying the electric field to a fluid flowing on an infinite vertical plate is the most recent technique used for studying fluid movement. This technique is known as electro-osmotic flow (EOF). Therefore, the core aim of the present research work is to examine the time-dependent electro-osmotic flow of viscoelastic fluid on a vertical flat plate together with the effects of heat generation and chemical reaction. The classical system of governing equations has been fractionalised by means of Caputo–Fabrizio’s time-fractional derivative. Governing equations have been non-dimensionalised by using relative dimensionless quantities. The exact solutions for the momentum, temperature and concentration equations have been developed by implementing the Laplace transform technique. For graphical analysis, the solutions have been plotted against the inserted parameters using the computational software Mathematica. It is interesting to mention that the time-fractional model provides more than one fluid layer for the analysis of the fluid motion, heat distribution and mass concentration, which is not possible by assuming the classical mathematical model. It is also very important to mention that the velocity profile shows declination for greater values of electro-osmotic parameter Es. [ABSTRACT FROM AUTHOR]

The Klein Kramer equation (KKE) stands for the probability distribution function (PDF) that describes the diffusion of particles subjected an external force in the presence of friction. It is applicable in statistical and stochastic treatments of chemical dynamics, in particular the diffusive description of chemical reactions. Here, we are concerned with finding the exact solutions of the conformable fractional derivative (CFD) KKE. An approach is presented to transform linear partial differential equations (PDEs) to a set of first order PDEs. On the other hand the CFD is shown to be reduced to the classical one's by using similarity transformations. Here, the objective of this work is to find the exact solutions of CFD-KKE. To this issue, the approach presented is applied. The solutions are found by implementing extended unified method. It is found that, the integrability condition is that the external force is constant. The numerical results of the solutions are calculated and the are shown graphically. Calculations are carried by MATHEMATICA 12. [ABSTRACT FROM AUTHOR]

This paper explores the class C G , consisting of functions g that satisfy a specific subordination relationship with Gregory coefficients in the open unit disk E. By applying certain conditions to related coefficient functionals, we establish sharp estimates for the first five coefficients of these functions. Additionally, we derive bounds for the second and third Hankel determinants of functions in C G , providing further insight into the class's properties. Our study also investigates the logarithmic coefficients of log (g (t) t) and the inverse coefficients of the inverse functions (g − 1) within the same class. [ABSTRACT FROM AUTHOR]

In this work, a comparative study of seven well-known mathematical techniques for the coupled Burgers' equations is reported. The techniques involve in this comparison are as follows: Laplace transform Adomian decomposition method, Laplace transform homotopy perturbation method, Variational iteration method, Variational iteration decomposition method, Variational iteration homotopy perturbation method, the optimal homotopy asymptotic method, and OHAM with Daftardar-Jafari polynomial. Here we considered a practical example which consists of coupled Burgers' equations with the kinematic viscosity ε = 1. Convergence and stability analysis is a major part of this analysis. After a careful observation, it is found that the variational iteration method has faster convergence than all the remaining methods. Adomian decomposition method and Homotopy perturbation method show weaker stability in comparison with other involved techniques. [ABSTRACT FROM AUTHOR]

This article is dealing with the analytical solution of Fornberg–Whitham equations in fractional view of Caputo operator. The effective method among the analytical techniques, natural transform decomposition method, is implemented to handle the solutions of the proposed problems. The approximate analytical solutions of nonlinear numerical problems are determined to confirm the validity of the suggested technique. The solution of the fractional-order problems are investigated for the suggested mathematical models. The solutions-graphs are then plotted to understand the effectiveness of fractional-order mathematical modeling over integer-order modeling. It is observed that the derived solutions have a closed resemblance with the actual solutions. Moreover, using fractional-order modeling various dynamics can be analyzed which can provide sophisticated information about physical phenomena. The simple and straight-forward procedure of the suggested technique is the preferable point and thus can be used to solve other nonlinear fractional problems. [ABSTRACT FROM AUTHOR]

In this paper, we study an accurate numerical method for solving the multi space-fractional Gardner equation (MSFGE) with the Caputo–Fabrizio (CF) and Atangana–Baleanu (AB) fractional derivatives where the space-fractional terms are under the sense of Caputo. To the best knowledge of the reader, we are first to use the spectral collocation method based on the third Chebyshev approximations to reduced the multi space-fractional Gardner equation to a system of ordinary differential equations by using the properties of Chebyshev polynomials and then solved them via the finite difference method (FDM). By computing the absolute errors we present the effectiveness and accuracy of the proposed methods. The present paper investigates the dynamics of Gardner Equation by considering two fractional operators that is the Caputo–Fabrizio and Atangana–Baleanu, this is entirely new idea by using the two operators on a Gardner equation. Our results prove that the given procedure is an easy and efficient tool to investigate the solution of nonlinear equations with local and non-local singular kernels. • Investigation of certain new models of the multi space-fractional gardner equation. • We evaluate the approximation solutions for this model via Caputo–Fabrizio and Atangana–Baleanu (AB) fractional derivatives where the space-fractional terms is under the sense of Caputo fractional derivative. • We use the spectral collocation method based on the third Chebyshev approximations. [ABSTRACT FROM AUTHOR]

Exponential space‐based heat source with shear‐thinning/thickening behavior offers innovative insights into the optimization of thermal transport process in complex systems, such as in the cooling of microelectronics, energy storage technologies, and biomedical devices, where accurate thermal control is critical. This study explores the investigation of an exponential space‐based heat source with effects of shear thinning/thickening behavior of a ternary infinite shear rate viscosity‐dependent Carreau water‐based nanofluid [Cu, Fe3O4, SiO2] over a three‐dimensional wedge in a magnetized environment. The interaction between heat transport and magnetic fields is explored through the behavior of the ternary nanofluid by classifying the both shear‐thinning and shear‐thickening effects. The incorporation of physical assumptions in the model gives the set of partial differential equations (PDEs) and similarity transformations are launched to convert them into ordinary differential equations (ODEs). Furthermore, bvp4c scheme is used to fetch the numerical solution. Temperature profile is increasing with exponential‐based source heat parameter due to a reduction in the rate of heat absorption by the fluid. Ternary nanofluids exhibit the highest rate of temperature increment due to the effects of multiple nanoparticles as compared to hybrid or single‐nanoparticle nanofluids. Rate of velocity decrement in ternary nanofluids compared to bi‐hybrid nanofluids and single‐particle nanofluids. [ABSTRACT FROM AUTHOR]

In this paper, the Homotopy perturbation Laplace method is implemented to investigate the solution of fractional-order Whitham–Broer–Kaup equations. The derivative of fractional-order is described in Caputo's sense. To show the reliability of the suggested method, the solution of certain illustrative examples are presented. The results of the suggested method are shown and explained with the help of its graphical representation. The solutions of fractional-order problems as well as integer-order problems are determined by using the present technique. It has been observed that the obtained solutions are in significant agreement with the actual solutions to the targeted problems. Computationally, it has been analyzed that the solutions at different fractional-orders have a higher rate of convergence to the solution at integer-order of the derivative. Due to the analytical analysis of the problems, this study can further modify the solution of other fractional-order problems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and implement the optimal auxiliary function method to solve a system of fractional-order Whitham–Broer–Kaup equations, a class of nonlinear partial differential equations with broad applications in mathematical physics. This method provides a systematic and efficient approach to finding accurate solutions for complex systems of fractional-order equations. We give a full analysis using tables and figures to demonstrate the reliability and accuracy of our approach. We confirm the effectiveness of our suggested method in solving the considered equations using numerical simulations and comparisons, emphasizing its potential for applications in a variety of scientific and engineering areas. [ABSTRACT FROM AUTHOR]

The variational homotopy perturbation method VHPM is used for solving n-dimensional Burgers’ system. Some examples are examined to validate that the method reduced the calculation size, treating the difficulty of nonlinear term and the accuracy. [ABSTRACT FROM AUTHOR]

This research deals with a comparative numerical analysis of chaos in two systems with non-integer derivatives. The one-scroll system and circle equilibrium system with different hidden attractors are simulated considering the fractal derivative, Khalil and Atangana conformable derivatives, and the truncated M-derivative considering a constant and variable-order. Phase portraits are shown, as well as bifurcation diagrams, and Lyapunov exponents are obtained. Later, 0–1 test, dynamic death analysis, and sensitivity to initial conditions are considered to choose which derivative produces richer chaotic behaviors. According to those mentioned above, we could observe that the M-derivative not only generalizes Khalil's type conformable derivative but also its two non-integer orders produce interesting dynamic behaviors compared to the remaining derivatives. In the numerical results, we observe that the variable order makes the system more sensitive to the change in the initial conditions. The new chaotic behaviors with constant and variable order are used to develop a synchronization scheme of two identical one-scroll systems via the backstepping technique with the truncated M-derivative involved. [ABSTRACT FROM AUTHOR]

In this work, we studied the dynamics of a chaotic neural network using the fractional-conformable operator and non-integer order derivatives. We first systematically show the dynamics behavior of the network using the conformable operator. Then, we show the dynamics of the network using fractional-order derivatives. We compared the different effects and behaviors when using conformable operators and fractional-order derivatives. Next, we studied the dynamics when applying both operators to the network simultaneously, and we graphically show the effects of each operator on the chaotic neural network. Following the dynamical analysis, we used a fractional-variable order in the derivative operator to compare the system behavior when we used a fractional constant order in the derivative operator. We used a smooth function in all our simulations in the interval (0, 1]. Our results show that either the conformable operator or the fractional derivative, both of them, affect the dynamics of the network showing interesting behaviors. We show that when the order in the conformable operator decreases an attractor is generated which takes the system into a stable state, and when the order in the fractional derivative decreases the system loses its dynamics. Contrary, when we used the fractional-variable order in the derivative operator, the system does not lose its dynamics; instead, it gains more chaotic behavior, which means that the system is susceptible to the initial conditions fractional-variable order function and its initial point. Finally, after depicting the different system behavior under both operators, we proposed a full master-slave synchronization using the fractional-conformable approach. We designed a fractional order controller to reach synchronization using optimal gains means Cuckoo search algorithm. Our results show the different behaviors when using the fractional-conformable approach in the master and slave systems. As we expected, the gains in a combination of the non-integer order allow achieving the synchronization between the master and the slave system even when the dynamics in the slave system was regular. [ABSTRACT FROM AUTHOR]

This paper is about Newtonian nanofluid flow manifested with thermal radiation and heat generation/absorption. Further, the study of interest are mixed convection, stagnation point, magnetic field, Joule heating, temperature stratification, concentration stratification and chemical reaction. The flow field is caused by the inclined stretching cylinder. The mathematical model is developed in the form of coupled partial differential framework and is later on, descend to coupled ordinary differential framework by means of admissible similarity transformation. The numerical findings are presented by Runge–Kutta Fehlberg method along with shooting scheme. The temperature trend towards higher values of heat generation enhances whereas the reverse trend is observed for heat absorption. The fluid temperature depicts the uplift for thermal radiation. The approximations for skin friction coefficient and local Nusselt number are also examined. • A Newtonian nanofluid with thermal radiation and heat generation. • The mixed convection, Joule heating are considered. • The equations are solved by Runge–Kutta Fehlberg method with shooting scheme. • The temperature increases for heat generation and decreases for heat absorption. • The values of skin friction coefficient are found numerically. [ABSTRACT FROM AUTHOR]

Finite element simulations for the dynamics of Casson fluid flow over time-dependent two-dimensional stretching sheet subjected to magnetic field and variable time and space-dependent temperature are studied numerically through Galerkin finite element method implementation. For this, weak form of the governing boundary value problems is derived through their residuals. Domain is discretized using two nodes per element, and assembly process is performed. The system of algebraic nonlinear equations is linearized through Picard's linearization algorithm. Linear system of algebraic equations is solved iteratively with computational tolerance 10 - 8 . The independent variable is searched through several computational experiments, and code is tested by comparing the results for special case with already published benchmarks. After the validation of code, simulations are performed in order to capture the dynamics of the physical situation against the variation of the pertinent parameters. Behavior of stresses and heat flux for different values of the physical parameters is studied. The temperature decreases when the intensity of radiation in the form of electromagnetic waves is increased. Boundary layer thickness for the Casson fluid is less than the boundary layer thickness of Newtonian fluid. However, opposite trend of thermal boundary layer thickness is noted. The magnetic is responsible for producing a hindrance to flow. Consequently, wall shear stress increases. Heat flux at the surface of stretching sheet increases when the values of unsteadiness parameter are increased, whereas there is a decreasing trend in the rate of heat transfer when the value of Eckert number is increased. Shear stresses are increasing function of the temperature. However, there is an increasing trend in the rate of heat transfer. [ABSTRACT FROM AUTHOR]

This article uses non-Fourier theory to investigate the transport phenomenon in Carreau liquid of temperature dependent diffusion coefficients. The governing problems are solved by finite element method (FEM) and simulated results for different values of Prandtl and Schmidt numbers, viscosity parameter, local Wiessenberg number, chemical reaction parameter, heat generation/absorption parameter and thermal and concentration relaxation parameters are displayed graphically. The relaxation parameters have a tendency to decrease the temperature and concentration field for both cases of variable and constant dynamic viscosity. Thermal relaxation time has significant decreasing behavior on the temperature. Thermal boundary layer relaxation time has significant decreasing behavior on the temperature. Thermal boundary layer thickness in the Carreau liquid obeying classical Fourier's law is greater than that in the Carreau liquid obeying Cattaneo–Christov heat flux model further, the thermal boundary layer thickness associated with Carreau liquid of variable viscosity is greater or smaller than the thermal boundary layer thickness associated with Carreau liquid of constant viscosity. More heat transfer in liquids obeying classical Fourier's laws of conduction is possible rather than transfer of heat in liquids obeying Cattaneo–Christov flux model. Thermo-elasticity in fluid flow has shown a decreasing trend in the transport of heat and mass. • Computational simulation for transport of mass and heat in Carreau liquid of variable viscosity and thermal conductivity is studied. • The diffusion of heat and mass takes place according to Cattaneo–Christov flux laws. • FEM is utilized to simulate heat generation and chemical reaction effects. • The momentum boundary layer thickness in the Newtanion liquid is greater than that in the Carreau liquid. [ABSTRACT FROM AUTHOR]

This study examines the unsteady 3D non-axisymmetric Homann flow of an electrically conducting nanofluid in the presence of buoyancy forces. We consider the uniform external magnetic field B 0 , by neglecting induced magnetic field and examine the three possible directions of magnetic field which coincides with the direction of axes. A similarity solution is derived which involves the important physical dimensionless parameters like the nanoparticles volume fraction φ , the unsteadiness parameter ω , the buoyancy parameter λ , Hartmann number M and shear to strain ratio γ. We have treated the case for forced convection when λ = 0 which arise from the singularity γ = ∓ 1. We found that, for large γ and λ , the leading terms of the solutions are independent of M and ω , and the effects of φ in that solutions are negligible. Numerical results are found for illustrative values of all the flow parameters by using bvp4c scheme in MATLAB. The critical values λ c of λ are seen in opposing flow for small rate of deceleration parameter ω while it changes to assisting flow for large value of ω. • Unsteady 3D non-axisymmetric Homann flow of an electrically conducting nanofluid in the presence of buoyancy forces is examined. • Magnetic field is considered in three directions. • Dual solution exists for the opposing flow. [ABSTRACT FROM AUTHOR]

In this work, we propose a new optimal perturbation iteration method for solving the generalized Fitzhugh–Nagumo equation with time-dependent coefficients. This research reveals that the new proposed technique, with the aid of symbolic computations, provides a straightforward and impressive mathematical tool for solving nonlinear partial differential equations. Implementing this method to Fitzhugh–Nagumo equation illustrates its potency. Convergence analysis also shows that OPIM, unlike many other methods in literature, converges fast to exact analytical solutions of the nonlinear problems at lower order of approximations. • We proposed a new efficient technique, optimal perturbation iteration technique, as an alternative method to some-well known methods in solving NPDEs. • The presented method has been favorably implemented to obtain the solution of the nonlinear generalized Fitzhugh–Nagumo equations. • Two illustration displays that the OPIM is forceful and prepotent mathematical tool to handle with these types of nonlinear equations. In this technique, it is basically crucial to get the parameters P0; P1; ::: and this makes OPIM time consuming, particularly for large n. • After first three iterations, we encounter a problem of high cost of calculations that takes up computer's memory. However, we have also witnessed that this technique converges fast at lower order of approximations for the nonlinear problems in this study. • Additionally, since the method is frequently toilsome to struggle by hand, one has to use a symbolic computer program to find approximate solutions. In this paper, we have used Mathematica 9.0 to carry out the complex computations in examples. It should be also said that one can get better solutions with more powerful processors or computers. [ABSTRACT FROM AUTHOR]