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Recognizing the flow behaviours across a Riga plate can reveal information about the aerodynamic efficiency of aircraft, heat propagation, vehicles, and other structures. These data are critical for optimizing design and lowering drag. Therefore, the purpose of the current analysis is to examine the energy and mass transfer across the mixed convective nanofluid flows over an extending Riga plate. The fluid flow is deliberated under the influences of viscous dissipation, exponential heat source/sink, activation energy, and thermal radiation. The Buongiorno’s concept is utilized for the thermophoretic effect and Brownian motion along with the convective conditions. The modelled are simplified into the lowest order by using similarity transformation. The obtained set of non-dimensional ordinary differential equations is then numerically solved through the parametric continuation method. For accuracy and validation of the outcomes, the results are compared to the existing studies. From the graphical analysis, it can be observed that the fluid velocity boosts with the rising values of the divider thickness parameter. The fluid temperature also improves with the effect of Biot number, Eckert number, and heat source factor. Furthermore, the effect of heat source sink factor drops the fluid temperature.

In this investigation, the fractional Hirota–Satsuma coupled Korteweg–de Vries (KdV) problem is solved using two modern semi-analytic techniques known as the Aboodh residual power series method (ARPSM) and Aboodh transform iteration method (ATIM). The two suggested approaches are briefly explained, along with how to use them to solve the fractional Hirota–Satsuma coupled KdV problem. Some analytical approximate solutions for the current problem are derived using the proposed techniques until the second-order approximation. To ensure high accuracy of the derived approximation, they are analyzed numerically and graphically and compared with the exact solutions of the integer cases. The offered techniques demonstrate more accuracy in their outcomes compared to other alternatives. The numerical results show that ARPSM and ATIM are highly accurate, practical, and beneficial for solving nonlinear equation systems. The current results are expected to help many physics researchers in modeling their different physical problems, especially those interested in plasma physics.

The fractional-order differential equations (FO-DEs) faithfully capture both physical and biological phenomena making them useful for describing nature. This work presents the stable and more effective closed-form traveling-wave solutions for the well-known nonlinear space–time fractional-order Burgers equation and Lonngren-wave equation with additional terms using the exp(−Φ(ξ))(-\Phi (\xi )) expansion method. The main advantage of this method over other methods is that it provides more accuracy of the FO-DEs with less computational work. The fractional-order derivative operator is the Caputo sense. The transformation is used to reduce the space–time fractional differential equations (FDEs) into a standard ordinary differential equation. By putting the suggested strategy into practice, the new closed-form traveling-wave solutions for various values of parameters were obtained. The generated 3D graphical soliton wave solutions demonstrate the superiority and simplicity of the suggested method for the nonlinear space–time FDEs.

In this research study, we focus on the generalized regularized long wave equation and the modified regularized long wave equation, which play pivotal roles in characterizing plasma waves in oceans and ion acoustic waves in shallow water, a domain deeply rooted in physical phenomena. Employing two computational techniques, namely, the optimal auxiliary function method and the Laplace iterative transform method, we approximate these equations. These formulas are used to characterize plasma waves in oceans and ion acoustic waves in shallow water. The results discovered have important ramifications for our comprehension of many physical events. Our results show that both methods are robust, easy to use, and successful. Both methods yield results that are satisfactory to each other. With the use of tables and graphs, we compared the two suggested approaches. The findings suggest that the suggested methods can be widely applied to explore other real-world problems.

This study solves the coupled fractional differential equations defining the massive Thirring model and the Kundu Eckhaus equation using the Natural transform decomposition method. The massive Thirring model is a dynamic component of quantum field theory, consisting of a coupled nonlinear complex differential equations. Initially, we study the suggested equations under the fractional derivative of Caputo-Fabrizio. The Atangana-Baleanu derivative is then used to evaluate the comparable equations. The results are significant and necessary for exploring a range of physical processes. This paper uses modern approach and the fractional operators in this situation to develop satisfactory approximations to the offered problems. The proposed approach combines the natural transform technique with the efficient Adomian decomposition scheme. Obtaining numerical findings in the form of a fast-converge series significantly improves the scheme’s accuracy. Some graphical plot distributions are presented to show that the present approach is very simple and straightforward. We performed a fractional order analysis of assumed phenomena to demonstrate and validate the effectiveness of the future technique. The behaviour of the approximate series solution for several fractional orders is shown visually. Additionally, the nature of the derived outcome has been observed for various fractional orders. The derived results demonstrate how simple and efficient the proposed method is to apply for analysing the behaviour of fractionally-order complex nonlinear differential equations that arise in related fields of engineering and science. [ABSTRACT FROM AUTHOR]

The goal of the current study is to analyze several nonlinear two-dimensional time-fractional Rosenau–Hyman equations. The two-dimensional fractional Rosenau–Hyman equation has extensive use in engineering and applied sciences. The fractional view analysis of two-dimensional time-fractional Rosenau–Hyman equations is discussed using the homotopy perturbation approach, Adomian decomposition method, and Yang transformation. Some examples involving two-dimensional time-fractional Rosenau–Hyman equations are provided to better understand the suggested approaches. The solutions appear as infinite series. We offer a comparison between the accurate solutions and those that are generated employing the proposed approaches to demonstrate the effectiveness and applicability of the proposed techniques. The results are graphically illustrated using two-dimensional and three-dimensional graphs. It has been noted that the obtained results and the targeted problems real solutions are quite similar. Calculated solutions at various fractional levels describe some of the problems useful dynamics. A comparison between the numerical solutions of the models under study and the exact solutions in cases when a solution is known serves as a clear demonstration of the viability and dependability of the suggested approaches. Other fractional problems that arise in other fields of science and engineering can be solved using a modified version of the current techniques. [ABSTRACT FROM AUTHOR]

This research paper aims to investigate the peristaltic transport of a nanofluid (NF) in a tapered asymmetric channel. Initially, the governing equations for the balance of mass, momentum, temperature, and volume fraction for the NF using Reiner–Philippoff (RP) based NF are formulated. Subsequently, these equations are employed to analyze long wavelength and small Reynolds number scenarios. The numerical results for various flow features are thoroughly examined and discussed. Dual solutions have been examined for some factors. So, stability assessment is implemented to find stable solution. Novelty of the existing is to investigate the peristaltic motion of Buongiorno's NF model and its stability which has not investigated in the previous literatures. It has been demonstrated that modifying the RPF parameter leads to a transition in the fluid's velocity, changing it from a dilatant liquid to a Newtonian fluid and from Newtonian to pseudoplastic. The findings indicate that the temperature curves rise as Brownian motion and thermophoretic factors increase, while they decrease as the Prandtl number increases. Furthermore, a concise mathematical and graphical analysis is carried out to examine the impact of each key parameter on the flow characteristics. [ABSTRACT FROM AUTHOR]

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 symmetry features of fractional differential equations allow effective explanation of physical and biological phenomena in nature. The generalized form of the fractional differential equations is the variable-order fractional differential equations that describe the physical and biological applications. This paper discusses the closed-form traveling wave solutions for the nonlinear space-time variable-order fractional modified Kawahara and (2 + 1)-dimensional Burger hierarchy equations. The variable-order fractional differential equation has a derivative operator in the Caputo sense that is converted into the integer-order ordinary differential equation (ODE) by fractional transformation. The obtained ODE is solved by the exponential rational function method, and as a result, new exact solutions are constructed. Two problems are proposed to confirm the solutions of the space-time variable-order fractional differential equations. [ABSTRACT FROM AUTHOR]

In this work, we use two unique methodologies, the homotopy perturbation transform method and Yang transform decomposition method, to solve the fractional nonlinear seventh-order Kaup–Kupershmidt (KK) problem. The physical phenomena that arise in chemistry, physics, and engineering are mathematically explained in this equation, in particular, nonlinear optics, quantum mechanics, plasma physics, fluid dynamics, and so on. The provided methods are used to solve the fractional nonlinear seventh-order KK problem along with the Yang transform and fractional Caputo derivative. The results are significant and necessary for exploring a range of physical processes. This paper uses modern approaches and the fractional operator to develop satisfactory approximations to the offered problem. To solve the fractional KK equation, we first use the Yang transform and fractional Caputo derivative. He's and Adomian polynomials are useful to manage nonlinear terms. It is shown that the suggested approximate solution converges to the exact one. In these approaches, the results are calculated as convergent series. The key advantage of the recommended approaches is that they provide highly precise results with little computational work. The suggested approach results are compared to the precise solution. By comparing the outcomes with the precise solution using graphs and tables we can verify the efficacy of the offered strategies. Also, the outcomes of the suggested methods at various fractional orders are examined, demonstrating that the findings get more accurate as the value moves from fractional order to integer order. Moreover, the offered methods are innovative, simple, and quite accurate, demonstrating that they are effective for resolving differential equations. [ABSTRACT FROM AUTHOR]

This study delved into the dynamics of wave solutions within the Estevez-Mansfield-Clarkson equation in fractional nonlinear space-time. Utilizing conformable fractional derivatives, the equation governing shallow water phenomena and fluid dynamics was transformed into a nonlinear ordinary differential equation. Applying the Riccati Bernoulli sub-ODE approach yielded a finite series representation. Notably, our findings revealed novel solitary wave solutions characterized by kink, anti-kink, periodic, and shock functions. Visualized through 3D and contour graphs, kink and periodic waves emerged as distinct observable manifestations. Intriguingly, the diversity of results surpassed previous results, contributing to a deeper understanding of the intricate dynamics inherent in the system. [ABSTRACT FROM AUTHOR]

The homotopy perturbation transform method was examined in the present research to address the nonlinear time-fractional Korteweg-de Vries equations using a nonsingular kernel fractional derivative that Caputo-Fabrizio recently developed. We devoted our research to the nonlinear time-fractional Korteweg-de Vries equation and certain associated phenomena because of some physical applications of this equation. The results are significant and necessary for illuminating a range of physical processes. This paper considered an innovative method and fractional operator in this context to obtain satisfactory approximations to the provided issues. To solve nonlinear timefractional Korteweg-de Vries equations, we first considered the Yang transform of the Caputo-Fabrizio fractional derivative. In order to confirm the applicability and efficacy of the provided method, we took into consideration two cases of the nonlinear time-fractional Korteweg-de Vries equation. He's polynomials were useful in order to manage nonlinear terms. In this method, the outcome was calculated as a convergent series, and it was demonstrated that the homotopy perturbation transform method solutions converge to the exact solutions. The main benefit of the suggested method was that it offered solutions with a high degree of precision while requiring minimal computation. Graphs were also used to illustrate the series solution for a certain non-integer orders. Finally, a comparison of both examples outcomes were examined using diagrams and numerical data. These graphs showed how the approximated solution's graph and the precise solution's graph eventually converged as the non-integer order gets closer to integer order. When ζ = 1, several numerical comparisons were conducted with the exact solutions. The numerical simulation was offered to illustrate the efficiency and reliability of the proposed approach. In addition, the behavior of the provided solutions was explained using a number of fractional orders. The theoretical analysis matched with the findings obtained using the current technique, and the suggested technique can be extended to tackle many higher-order nonlinear dynamics problems. [ABSTRACT FROM AUTHOR]

The complex Hirota-dynamical model (CHDM) has applications in the study of plasma physics, the investigation of fusion energy, astrophysical research, and space studies. The CHDM may also be used to investigate turbulent flows to study shocks and other nonlinear phenomena, and light waves venturing through the fibers. Nowadays, plasma physics, fusion energy, astrophysical research, and space studies are very interesting topics in the modern research. So, we need to shed light on this model as a good application in these fields. For deep investigation of these physical problems, we need to find their analytical solutions. In this study, we explore a variety of soliton solutions with different geometrical structures for the CHDM via the double variable expansion method. By means of this method, we have obtained three types of soliton solutions, namely, hyperbolic, trigonometric, and rational function solutions. The graphical interpretation of these solutions gives us some popular shapes such as singular-periodic, kink, bell, and singular shapes. The performed method is an efficient technique to execute and provides reliable analytical soliton solutions which are very important to further advanced investigation of the mention equation. [ABSTRACT FROM AUTHOR]

In this study, we employ the effective iterative method to address the fractional Wu-Zhang Equation within the framework of the Caputo Derivative. The effective iterative method offers a practical approach to obtaining approximate solutions for fractional differential equations. We seek to provide insights into its solution and behavior by applying this method to the Wu-Zhang Equation. Through numerical analysis and the presentation of relevant tables and Figures, we demonstrate the accuracy and efficiency of this method in solving the fractional Wu-Zhang Equation. This research contributes to the understanding and solution of fractional-order differential equations and their applications in various scientific and engineering domains. [ABSTRACT FROM AUTHOR]

Carbon nanotubes (CNTs) influenced nanofluid is gaining popularity in the industry for solar energy and scratch heat exchanger applications. Consequently, this research focuses on evaluating the impact of nonlinear thermal radiation from a CNT-based nanofluid on an unsteady three-dimensional nonasymmetric Homann stagnant flux as a function of length and radius. CNTs have remarkable thermal physical properties that appear to be critical for nonlinear thermal transport. As a result, the nonlinear heat transfer properties of H2O composed of single or multiple wall CNTs are studied. The nanomaterial has a length and radius of approximately 3 nm ≤ L ≤ 7 0 nm and 1 0 nm ≤ R ≤ 4 0 nm. Partially differential equations with appropriate similarity transformations serve as a mathematical model for the process. The numerical solution of the simplified system of equations is achieved via the use of the well-known Runge–Kutta (RK) method in conjunction with the shooting approach. An effective way to show how a component affects velocity and temperature, skin frictions in both direction and Nusselt number are utilized in graphical representations. Increasing the unsteadiness parameter causes a reduction in the temperature profile and the velocity profile in both directions. As ε grows larger with ϕ , the skin friction in both directions decreases, while the Nusselt number profile grows larger. In addition, the variation in the Nusselt number is included in the tables, along with a comparison of the model without radiation to the model with radiation. [ABSTRACT FROM AUTHOR]

This paper introduces an innovative optimization model employing fuzzy logic to address the challenge of minimizing variance while considering costs in the context of a two-stage stratified random sampling model with a randomized response approach. The proposed model leverages the alpha-cut technique to establish an optimal allocation strategy, making it possible to effectively manage the trade-off between cost constraints and variance reduction objectives. To illustrate the practical application of the model, we provide numerical examples, demonstrating its efficacy in real-world scenarios. This research contributes to the field by offering a comprehensive framework for decisionmakers to enhance the quality of data collection processes, particularly in situations where preserving respondent privacy is crucial. The integration of fuzzy logic and randomized response techniques presents a novel approach to addressing the inherent challenges of collecting sensitive information while maintaining data integrity and cost-efficiency. [ABSTRACT FROM AUTHOR]

The main features of scientific effort in physics and engineering are the development of models for various physical issues and the development of solutions. In this paper, we investigate the numerical solution of time-fractional non-linear dispersive K(m, n, 1) type equations using two innovative approaches: the homotopy perturbation transform method and Yang transform decomposition method. Our suggested approaches elegantly combine Yang transform, homotopy perturbation method (HPM) and adomian decomposition method (ADM). With the help of the Yang transform, we first convert the problem into its differential partner before using HPM to get the He's polynomials and ADM to get the Adomian polynomials, both of which are extremely effective supports for non-linear issues. In this case, Caputo sense is used for defining the fractional derivative. The derived solutions are shown in series form and converge quickly. To ensure the effectiveness and applicability of the proposed approaches, the examined problems were analyzed using various fractional orders. We analyze and demonstrate the validity and applicability of the solution approaches under consideration with given initial conditions. Two and three dimensional graphs reflect the outcomes that were attained. To verify the effectiveness of the strategies, numerical simulations are presented. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. The results of this study demonstrate that the studied methods are effective and strong in solving nonlinear differential equations that appear in science and technology. [ABSTRACT FROM AUTHOR]

It is essential for mathematicians, physicists, and engineers to construct fractional mathematical models for specific phenomena and develop numerical or analytical solutions for these models. In this work, we implement the natural decomposition approach with nonsingular kernel derivatives to investigate the solution of nonlinear fractional forced Korteweg–de Vries (FF-KdV) equation. We first investigate the FF-KdV equation under the Caputo–Fabrizio fractional derivative. The similar equations are then examined using the Atangana–Baleanu derivative. This approach combines the decomposition method with the Natural transform method. The series solution of the suggested equations is thus obtained using the natural transform. The key benefit of this novel approximate-analytical approach is that it may provide an analytical solution for the FF-KdV problem in the form of convergent series with simple computations. For each equation, three unique situations are chosen to demonstrate and test the viability of the proposed method. To guarantee the competence and dependability of the proposed method, the nature for various values of the Froude number F r have been provided. The present approach is also used to calculate solutions at various fractional orders. The approximate series solution's behavior for various fractional orders has been graphically displayed. The outcomes demonstrate that the methodology is simple to use and reliable when applied to numerous fractional differential equations. [ABSTRACT FROM AUTHOR]

This document is a correction notice for an article titled "Analysis of some dynamical systems by combination of two different methods" published in Scientific Reports. The correction states that there was an error in the original article regarding the affiliation of one of the authors, A. M. Zidan. The correct affiliation is the Department of Mathematics, College of Science, King Khalid University, Saudi Arabia. Additionally, there was an error in the Acknowledgements section, which has been corrected. The authors express their appreciation to the Deanship of Scientific Research at King Khalid University for funding the work through a large group Research Project under grant number RGP.2/16/45. [Extracted from the article]

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]

AbstractA hyperbolic tangent fluid model characterized by shear rate dependent viscosity together with the effects of hybrid nanoparticles, Brownian motion, thermo-diffusion, and Nield’s conditions, is utilized for thermal enhancement purpose. Five percent copper nanoparticles and twenty percent alumina nanoparticles are submerged into ethylene glycol with the parametric influence of tangent hyperbolic fluid to obtain the tangent hyperbolic hybrid nano suspension. Numerical solutions of the nondimensionalized conservation laws are found by the Keller-box method. The influence of various parametric factors on velocity, temperature, concentration, Nusselt number, and Sherwood number has been discussed. A constant Nusselt number is noted with the influence of Nield’s constraints. A 90% increase in thermal conductivity is achieved by dispersing 5% copper and 20% alumina nanoparticles into the ethylene glycol with the influence of hyperbolic tangent fluid. The error and convergence analysis of the numerical solution have been discussed, along with the validation of the obtained numerical results. [ABSTRACT FROM AUTHOR]