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In this study, we introduce a novel iterative method combined with the Elzaki transformation to address a system of partial differential equations involving the Caputo derivative. The Elzaki transformation, known for its effectiveness in solving differential equations, is incorporated into the proposed iterative approach to enhance its efficiency. The system of partial differential equations under consideration is characterized by the presence of Caputo derivatives, which capture fractional order dynamics. The developed method aims to provide accurate and efficient solutions to this complex mathematical system, contributing to the broader understanding of fractional calculus applications in the context of partial differential equations. Through numerical experiments and comparisons, we demonstrate the efficacy of the proposed Elzaki-transform-based iterative method in handling the intricate dynamics inherent in the given system. The study not only showcases the versatility of the Elzaki transformation but also highlights the potential of the developed iterative technique for addressing similar problems in various scientific and engineering domains. [ABSTRACT FROM AUTHOR]

The steady MHD boundary-layer axis-symmetric flow of a third-grade fluid passing through an exponentially expanded cylinder in the vicinity of a magnetic field is investigated in this study. The problem is mathematically modeled. Suitable similarity transformations are carried out to convert the partial differential equations into nonlinear ordinary differential equations. The Runge–Kutta fourth-order shooting technique is used to solve the transmuted system of nonlinear ODEs. Graphical representations of numerical findings are used to examine the effects of various physical factors on the velocity and temperature profiles. The influence of fluid variables on the velocity curve, such as third-grade parameters, second-grade parameters, and Reynolds number, is illustrated and explored. The skin-friction coefficient expression is computed and given. The widths of the velocity and momentum boundary layers are revealed to be increasing functions of the curvature parameter. It is found that the third-grade fluid has a higher velocity profile than Newtonian and second-grade fluids. Also, the stretched cylinders cause a more progressive shift in heat and mass pattern for flow than flat plates do. [ABSTRACT FROM AUTHOR]

Proper wastewater disposal is crucial in various manufacturing and ecological systems. This study aims to prevent and regulate pollution in the water supply. It examines how the pollutant discharge concentration affects the flow of non-Newtonian nanofluids (NNNFs) over a porous Riga surface. Two different types of NNNFs, namely, Walter's B and second-grade fluids, have been examined. The fluid flow is conveyed in the form of a system of partial differential equations (PDEs), which are first reduced to a non-dimensional set of ordinary differential equations (ODEs) and then to first-order differential equations. The numerical approach parametric continuation method is employed to solve these ODEs. It has been noticed that the energy curve declines with increasing numbers of TiO2-nanoparticles (NPs). The effect of the external pollutant source variation factor enriches the concentration of pollutants in both fluid cases. Furthermore, the viscoelastic parameter K1 plays a notable role in determining the behavior of the fluids. Particularly in NNNFs, the variation of K1 enhances the fluid flow, whereas the rise of second-grade fluid factor decreases the velocity of the fluid. Our findings indicate a substantial impact of the parameters under consideration on the concentration of pollutant discharge. Significantly, it was observed that an increase in the amount of NPs and the thermal radiation parameter led to an improvement in the thermal conductivity of the nanofluid, consequently decreasing the concentration of pollutants in the discharge. The nanofluid has greater efficiency in boosting the energy transfer rate of the base fluid. In the case of the second-grade fluid, the energy propagation rate increases up to 6.25%, whereas, in the case of Walter's fluid B, it increases up to 7.85%. [ABSTRACT FROM AUTHOR]

This research explores the solution of the time-fractional Newell–Whitehead–Segel equation using two separate methods: the optimal auxiliary function method and a new iterative method. The Newell–Whitehead–Segel equation holds significance in modeling nonlinear systems, particularly in delineating stripe patterns within two-dimensional systems. Employing the Caputo fractional derivative operator, we address two case study problems pertaining to this equation through our proposed methods. Comparative analysis between the numerical results obtained from our techniques and an exact solution reveals a strong alignment. Graphs and tables illustrate this alignment, showcasing the effectiveness of our methods. Notably, as the fractional orders vary, the results achieved at different fractional orders are compared, highlighting their convergence toward the exact solution as the fractional order approaches an integer. Demonstrating both interest and simplicity, our proposed methods exhibit high accuracy in resolving diverse nonlinear fractional order partial differential equations. [ABSTRACT FROM AUTHOR]

Host (base) fluids are unable to deliver efficient heating and cooling processes in industrial applications due to their limited heat transfer rates. Nanofluids, owing to their distinctive and adaptable thermo-physical characteristics, find a widespread range of practical applications in various disciplines of nanotechnology and heat transfer equipment. The novel effect of this study is to determine the effects of mixed convection, and activation energy on 3D Sutterby nanofluid across a bi-directional extended surface under the impact of thermophoresis diffusion and convective heat dissipation. The flow equations are simplified in terms of partial differential equations (PDEs) and altered to non-dimensional ODEs by implementing classical scaling invariants. Numerical results have been obtained via the bvp4c approach. The physical insights of crucial and relevant parameters on flow and energy profiles are analysed through plotted visuals. Some factors have multiple solutions due to shrinking sheets. So stability analysis has been adapted to analyses stable solutions. Graphical representations demonstrate the reliability and accuracy of the numerical algorithm across a variety of pertinent parameters and conditions. A comparison between existing results and previously published data shows a high degree of compatibility between the two datasets. The present study extensively explored a multitude of practical applications across a diverse spectrum of fields, including but not limited to gas turbine technology, power generation, glass manufacturing, polymer production, wire coating, chemical production, heat exchangers, geothermal engineering, and food processing. [ABSTRACT FROM AUTHOR]

This work investigates the complex dynamics of the stochastic fractional Kuramoto–Sivashinsky equation (SFKSE) with conformable fractional derivatives. The research begins with the creation of singular stochastic soliton solutions utilizing the modified extended direct algebraic method (mEDAM). Comprehensive contour, 3D, and 2D visual representations clearly depict the categorization of these stochastic soliton solutions as kink waves or shock waves, offering a clear description of these soliton behaviors within the context of the SFKSE framework. The paper also illustrates the flexibility of the transformation-based approach mEDAM for investigating soliton occurrence not only in SFKSE but also in a wide range of nonlinear fractional partial differential equations (FPDEs). Furthermore, the analysis considers the effect of noise, specifically Brownian motion, on soliton solutions and wave dynamics, revealing the significant influence of randomness on the propagation, generation, and stability of soliton in complex stochastic systems and advancing our understanding of extreme behaviors in scientific and engineering domains. [ABSTRACT FROM AUTHOR]

In this paper, we examined the approximations to the time-fractional Kawahara equation and modified Kawahara equation, which model the creation of nonlinear water waves in the long wavelength area and the transmission of signals. We implemented two novel techniques, namely the homotopy perturbation transform method and the Elzaki transform decomposition method. The derivative having fractional-order is taken in Caputo sense. The Adomian and He's polynomials make it simple to handle the nonlinear terms. To illustrate the adaptability and effectiveness of derivatives with fractional order to represent the water waves in long wavelength regions, numerical data have been given graphically. A key component of the Kawahara equation is the symmetry pattern, and the symmetrical nature of the solution may be observed in the graphs. The importance of our suggested methods is illustrated by the convergence of analytical solutions to the precise solutions. The techniques currently in use are straightforward and effective for solving fractional-order issues. The offered methods reduced computational time is their main advantage. It will be possible to solve fractional partial differential equations using the study's findings as a tool. [ABSTRACT FROM AUTHOR]

In this study, we used two unique approaches, namely the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM), to derive approximate analytical solutions for nonlinear time-fractional Zakharov–Kuznetsov equations (ZKEs). This framework demonstrated the behavior of weakly nonlinear ion-acoustic waves in plasma containing cold ions and hot isothermal electrons in the presence of a uniform magnetic flux. The density fraction and obliqueness of two compressive and rarefactive potentials are depicted. In the Liouville–Caputo sense, the fractional derivative is described. In these procedures, we first used the Yang transform to simplify the problems and then applied the decomposition and perturbation methods to obtain comprehensive results for the problems. The results of these methods also made clear the connections between the precise solutions to the issues under study. Illustrations of the reliability of the proposed techniques are provided. The results are clarified through graphs and tables. The reliability of the proposed procedures is demonstrated by illustrative examples. The proposed approaches are attractive, though these easy approaches may be time-consuming for solving diverse nonlinear fractional-order partial differential equations. [ABSTRACT FROM AUTHOR]

In comparison to fractional-order differential equations, integer-order differential equations generally fail to properly explain a variety of phenomena in numerous branches of science and engineering. This article implements efficient analytical techniques within the Caputo operator to investigate the solutions of some fractional partial differential equations. The Adomian decomposition method, homotopy perturbation method, and Elzaki transformation are used to calculate the results for the specified issues. In the current procedures, we first used the Elzaki transform to simplify the problems and then applied the decomposition and perturbation methods to obtain comprehensive results for the problems. For each targeted problem, the generalized schemes of the suggested methods are derived under the influence of each fractional derivative operator. The current approaches give a series-form solution with easily computable components and a higher rate of convergence to the precise solution of the targeted problems. It is observed that the derived solutions have a strong connection to the actual solutions of each problem as the number of terms in the series solution of the problems increases. Graphs in two and three dimensions are used to plot the solution of the proposed fractional models. The methods used currently are simple and efficient for dealing with fractional-order problems. The primary benefit of the suggested methods is less computational time. The results of the current study will be regarded as a helpful tool for dealing with the solution of fractional partial differential equations. [ABSTRACT FROM AUTHOR]

Fractional calculus is at this time an area where many models are still being developed, explored, and used in real-world applications in many branches of science and engineering where non-locality plays a key role. Although many wonderful discoveries have already been reported by researchers in important monographs and review articles, there is still a great deal of non-local phenomena that have not been studied and are only waiting to be explored. As a result, we can continually learn about new applications and aspects of fractional modelling. In this study, a precise and analytical method with non-singular kernel derivatives is used to solve the Caudrey–Dodd–Gibbon (CDG) model, a modification of the fifth-order KdV equation (fKdV). The fractional derivative is taken into account by the Caputo–Fabrizio (CF) derivative and the Atangana–Baleanu derivative in the Caputo sense (ABC). This model illustrates the propagation of magneto-acoustic, shallow-water, and gravity–capillary waves in a plasma medium. The dynamic behaviour of the acquired solutions has been represented in a number of two- and three-dimensional figures. A number of simulations are also performed to demonstrate how the resulting solutions physically behave with respect to fractional order. The significance of the current research is that new solutions are obtained by using a strong analytical approach. Utilizing a fractional derivative operator to solve equivalent models is another benefit of this approach. The results of the present work have similar aspects to the symmetry of partial differential equations. [ABSTRACT FROM AUTHOR]

This paper examines two methods for solving the nonlinear fractional Phi-four problem with variable coefficients. One of the distinct states of the Klein–Gordon model yields the Phi-four equation. It is also used to simulate the kink and anti-kink solitary wave connections that have recently emerged in biological systems and nuclear particle physics. The approaches that are being suggested consist of the Yang transform, the homotopy perturbation approach, the decomposition approach, and the fractional derivative as stated by Caputo. The advantages of the proposed techniques are their capability of combining two dominant approaches for attaining precise and approximate solutions of nonlinear equations. It is important to keep in mind that the suggested methods can perform better in general as they need less computational effort than the alternative methods, while keeping a high level of numerical precision. The actual and estimated outcomes are demonstrated in graphs and tables to be quite similar, demonstrating the usefulness of the proposed approaches. Additionally, several simulations are used to show the physical behaviors of the found solutions with regard to fractional order. The article's results possess complimentary properties that relate to the symmetry of partial differential equations. [ABSTRACT FROM AUTHOR]

Little is known about the rising impacts of Coriolis force and volume fraction of nanoparticles in industrial, mechanical, and biological domains, with an emphasis on water conveying 47 nm nanoparticles of alumina nanoparticles. We explored the impact of the volume fraction and rotation parameter on water conveying 47 nm of alumina nanoparticles across a uniform surface in this study. The Levenberg–Marquardt backpropagated neural network (LMB-NN) architecture was used to examine the transport phenomena of 47 nm conveying nanoparticles. The partial differential equations (PDEs) are converted into a system of Ordinary Differential Equations (ODEs). To assess our soft-computing process, we used the RK4 method to acquire reference solutions. The problem is investigated using two situations, each with three sub-cases for the change of the rotation parameter K and the volume fraction ϕ. Our simulation results are compared to the reference solutions. It has been proven that our technique is superior to the current state-of-the-art. For further explanation, error histograms, regression graphs, and fitness values are graphically displayed. [ABSTRACT FROM AUTHOR]

AbstractThe current study employs the Cattaneo–Christov diffusion law to analyze the thermal tendency of Maxwell nanomaterials flowing over an unsteady stretching sheet. Thermophoretic and Brownian diffusion aspects of nanosolids are incorporated to investigate thermal and nanoparticle’s concentration distribution. Through flow similarities, the governing PDEs (Partial Differential Equations) are articulated into a set of nonlinear ODEs (Ordinary Differential Equations). The homotopy analysis approach, a recognized semi-analytical technique, is implemented to solve these differential equations. Results are presented graphically, revealing that higher magnetic field strengths diminish the flowing field. Temperature and nanosolids concentration distributions escalate with increasing thermal and singular relaxation time phenomena. Furthermore, when nanoparticle concentration decreases and Brownian motion intensifies, the temperature field rises. It is also observed that heightened resistive heating enhances the fluid’s capability to carry thermal energy. Overall, this investigation sheds light on the intricate interplay between several factors influencing the thermal behavior of Maxwell nanomaterials over a stretching board, offering insights into their practical applications. [ABSTRACT FROM AUTHOR]