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Abstract 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.

Due to the numerous applications of the Nizhnik-Novikov-Veselov system (NNVS) in fluid mechanics, thus, the current investigation is focused on studying the fractional form of this model to reveal the ambiguity around many nonlinear phenomena that arise in different fluid medias. Accordingly, we aim to derive several families of symmetric solitons and traveling wave solutions to the (2 + 1)-dimensional fractional asymmetric NNVS (FANNVS), defined in conformable fractional derivatives’ sense. For this purpose, a groundbreaking analytical technique known as the modified extended direct algebraic method (mEDAM) is utilized to solve and analyze the FANNVS. According to this method, four cases with several families of soliton-like solutions are derived. Our research uncovers various soliton solutions, including solitary waves, periodic waves, shocks, dual shock waves (lump waves), and anti-shock waves. These solutions are graphically discussed to understand their dynamical proprieties against the fractional parameters. This broad range of soliton-like solutions supports the relevance of our findings and demonstrates the effectiveness of our methodology. These findings significantly advance the field by deepening our understanding of solitonic behavior in FANNVS and demonstrating the effectiveness of the medium approach in solving challenging nonlinear systems.

The main objective of this work is to investigate the characteristics and behavior of the ion-acoustic cnoidal waves (IACWs) in an electron–positron–ion magnetoplasma having inertial positive ions with anisotropic thermal pressure and inertialess Maxwellian positrons and electrons. We utilize the reductive perturbation technique to reduce the fluid governing equations of the present model into the Korteweg–de Vries (KdV) equation in order to achieve this objective. We calculate the periodic solution of the KdV equation, also referred to as the cnoidal wave. We investigate the impact of various related parameters, including ion pressure anisotropy, positron concentrations, and temperature ratio, on the properties of IACWs. This study, particularly in the near-Earth magnetosheath and magnetosphere, may offer an insightful analysis of space and astrophysical plasma systems displaying ion pressure anisotropy.

In this paper, the Burgers’ equations using the fractional derivative of Atangana-Baleanu sense are investigated and discussed. A Laplace variational iteration approach is used to demonstrate the fractional model's mathematical solution. The solution's existence and uniqueness are examined using fixed point theory. Several numerical simulations that enhance the efficacy of the employed derivative are presented and discussed.

To satisfy the electricity needs of a village in Tangi, northwest Pakistan, the present research can design and evaluate the environmental and economical aspects of an optimal hybrid photovoltaic-biogas-hydropower-battery energy sustainable system (PV-BG-HP-BESS). This framework integrates various renewable energy sources, delivering a modern, efficient approach to sustainable energy solutions. The HOMER Pro software is utilized to optimize the most economical and effective hybrid energy system. The results showed that the proposed hybrid system comprising 91.4 kWp PV modules, 19.6 kW hydropower, a 50 kW biogas generator (BG), 36 batteries, and a 60.6 kW converter was the most economical choice. This system, which used the cyclic charging (CC) method, had a cost of energy (COE) of 0.0728 $/kWh and a total net present cost (NPC) of $152,242. The suggested hybrid energy system for rural areas of Pakistan includes photovoltaic (PV), biogas (BG), hydro, and battery components to provide a dependable and sustainable power supply. This system minimizes the need for expensive fossil fuels while simultaneously minimizing environmental impact by lowering pollutants and greenhouse gas emissions. The system's annual electricity production is 294,782 kWh, with PV leads at 59.4%, BG at 6.02%, and hydro at 34.6%, ensuring uninterrupted power generation even in remote areas. The unmet load, extra electricity, and capacity shortage illustrate the reliability of the system and make it possible to address rural electrification challenges while supporting sustainable development and economic growth. Moreover, the outcomes of the proposed hybrid system dominate the previous studies in multiple objectives, including cost and sensitivity analysis, when compared.

Abstract The area of fractional partial differential equations has recently become prominent for its ability to accurately simulate complex physical events. The search for traveling wave solutions for fractional partial differential equations is a difficult task, which has led to the creation of numerous mathematical approaches to tackle this problem. The primary objective of this research work is to provide optical soliton solutions for the Frictional Kundu–Eckhaus equation (FKEe) by utilizing generalized coefficients. This strategy utilizes the Riccati–Bernoulli sub-ODE technique to effectively discover the most favorable traveling wave solutions for fractional partial differential equations. As a result, it facilitates the extraction of optical solitons and intricate wave solutions. The Backlund transformation is used to methodically construct a sequence of solutions for the specified equations. The study additionally showcases 3D and Density graphics that visually depict chosen solutions for certain parameter selections, hence improving the understanding of the outcomes.

Abstract This work dives into the Conformable Stochastic Kraenkel-Manna-Merle System (CSKMMS), an important mathematical model for exploring phenomena in ferromagnetic materials. A wide spectrum of stochastic soliton solutions that include hyperbolic, trigonometric and rational functions, is generated using a modified version of Extended Direct Algebraic Method (EDAM) namely r+mEDAM. These stochastic soliton solutions have practical relevance for describing magnetic field behaviour in zero-conductivity ferromagnets. By using Maple to generate 2D and 3D graphical representations, the study analyses how stochastic terms and noise impact these soliton solutions. Finally, this study adds to our knowledge of magnetic field behaviour in ferromagnetic materials by shedding light on the effect of noise on soliton processes inside the CSKMMS.

This paper provides several new traveling wave solutions for a nonlinear partial differential equation (PDE) by applying symbolic computation and a new approach, the Riccati–Bernoulli sub-ODE method, in a computer algebra system. Herein, employing the Bäcklund transformation, we solve a nonlinear PDE associated with nanobiosciences and biophysics based on the transmission line model of microtubules for nanoionic currents. The equation introduced here in this form is suitable for critical nanoscience concerns like cell signaling and might continue to explain some of the basic cognitive functions in neurons. We employ advanced procedures to replicate the previously detected solitary waves. We offer our solutions in graphical forms, such as 3D and contour plots, using Mathematica. We can generalize the elementary method to other nonlinear equations in physics, requiring only a few steps.

In the present research work, we construct and examine the self-similarity of optical solitons by employing the Riccati Modified Extended Simple Equation Method (RMESEM) within the framework of non-integrable Coupled Nonlinear Helmholtz Equations (CNHEs). This system models the transmission of optical solitons and coupled wave packets in nonlinear optical fibers and describes transverse effects in nonlinear fiber optics. Initially, a complex transformation is used to convert the model into a single Nonlinear Ordinary Differential Equation (NODE), from which hyperbolic, exponential, rational, trigonometric, and rational hyperbolic solutions are produced. In order to better understand the physical dynamics, we offer several 3D, contour, and 2D illustrations for the independent selections of physical parameter values. These illustrations highlight the graphic behaviour of some optical solitons and demonstrate that, under certain constraint conditions, acquired optical solitons lose their stability when they approach an axis and display periodic-axial perturbations, which lead to the generation of optical fractals. As a framework, the generated optical solitons have several useful applications in the field of telecommunications. Furthermore, our suggested RMESEM demonstrates its use by broadening the spectrum of optical soliton solutions, offering important insights into the dynamics of the CNHEs, and suggesting possible applications in the management of nonlinear models.

This article utilizes the Aboodh residual power series and Aboodh transform iteration methods to address fractional nonlinear systems. Based on these techniques, a system is introduced to achieve approximate solutions of fractional nonlinear Korteweg-de Vries (KdV) equations and coupled Burger’s equations with initial conditions, which are developed by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. As a result, the Aboodh residual power series and Aboodh transform iteration methods for integer-order partial differential equations may be easily used to generate explicit and numerical solutions to fractional partial differential equations. The results are determined as convergent series with easily computable components. The results of applying this process to the analyzed examples demonstrate that the new technique is very accurate and efficient.

Damped Burger’s equation describes the characteristics of one-dimensional nonlinear shock waves in the presence of damping effects and is significant in fluid dynamics, plasma physics, and other fields. Due to the potential applications of this equation, thus the objective of this investigation is to solve and analyze the time fractional form of this equation using methods with precise efficiency, high accuracy, ease of application and calculation, and flexibility in dealing with more complicated equations, which are called the Aboodh residual power series method and the Aboodh transform iteration method (ATIM) within the Caputo operator framework. Also, this study intends to further our understanding of the dynamic characteristics of solutions to the Damped Burger’s equation and to assess the effectiveness of the proposed methods in addressing nonlinear fractional partial differential equations. The two proposed methods are highly effective mathematical techniques for studying more complicated nonlinear differential equations. They can produce precise approximate solutions for intricate evolution equations beyond the specific examined equation. In addition to the proposed methods, the fractional derivatives are processed using the Caputo operator. The Caputo operator enhances the representation of fractional derivatives by providing a more accurate portrayal of the underlying physical processes. Based on the proposed two approaches, a set of approximations to damped Burger’s equation are derived. These approximations are discussed graphically and numerically by presenting a set of two- and three-dimensional graphs. In addition, these approximations are analyzed numerically in several tables, including the absolute error for each approximate solution compared to the exact solution for the integer case. Furthermore, the effect of the fractional parameter on the behavior of the derived approximations is examined and discussed.

Abstract 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 Nanomaterials have found wide applications in many fields, leading to significant interest in the scientific world, in particular automobile thermal control, heat reservoirs, freezers, hybrid control machines, paper creation, cooling organisms, etc. The aim of the present study is to investigate the MHD non-Newtonian nanofluid and time-based stability analysis to verify the stable branch by computing the smallest eigenvalue across a slendering, extending, or shrinking sheet with thermal radiation and chemical reactions. The basic flow equations have been obtained in terms of PDEs, which are then converted to ODEs in dimensionless form via a suitable transformation. Based on the MATLAB software package bvp4c, the numerical solution has been obtained for the system of equations. A comparative study of the present and published work is impressive. The influence of evolving factors such as Prandtl number, Schmidt number, magnetic factor, heat generation/absorption, thermal, thermophoresis factor, chemical factor, second-grade fluid factor, and Brownian number on the velocities, energy, and concentration patterns is discussed through graphs. It is perceived that the temperature distribution enriches owing to the greater magnitude of the heat source. Furthermore, it is observed that a greater magnitude of radiation improves the temperature curves. It is also investigated from the present analysis that concentration and temperature profiles increase due to the growing values of the thermophoresis factor.

This study explores the application of advanced mathematical techniques to solve fractional differential equations, focusing particularly on the fractional diffusion equation. The fractional diffusion equation, used to simulate a range of physical and engineering phenomena, poses considerable difficulties when applied to fractional orders. Thus, by utilizing the mighty powers of fractional calculus, we employ the variational iteration method (VIM) with the Elzaki transform to produce highly accurate approximations for these specific differential equations. The VIM provides an iterative framework for refining solutions progressively, while the Elzaki transform simplifies the complex integral transforms involved. By integrating these methodologies, we achieve accurate and efficient solutions to the fractional diffusion equation. Our findings demonstrate the robustness and effectiveness of combining the VIM and the Elzaki transform in handling fractional differential equations, offering explicit functional expressions that are beneficial for theoretical analysis and practical applications. This research contributes to the expanding field of fractional calculus, providing valuable insights and useful tools for solving complex, nonlinear fractional differential equations across various scientific and engineering disciplines.

This paper introduces a novel numerical approach for tackling the nonlinear fractional Phi-four equation by employing the Homotopy perturbation method (HPM) and the Adomian decomposition method (ADM), augmented by the Shehu transform. These established techniques are adept at addressing nonlinear differential equations. The equation's complexity is reduced by applying the Shehu Transform, rendering it amenable to solutions via HPM and ADM. The efficacy of this approach is underscored by conclusive results, attesting to its proficiency in solving the equation. With extensive ramifications spanning physics and engineering domains like fluid dynamics, heat transfer, and mechanics, the proposed method emerges as a precise and efficient tool for resolving nonlinear fractional differential equations pervasive in scientific and engineering contexts. Its potential extends to analogous equations, warranting further investigation to unravel its complete capabilities.

This paper presents a comparative study of two popular analytical methods, namely the Homotopy Perturbation Transform Method (HPTM) and the Adomian Decomposition Transform Method (ADTM), to solve two important fractional partial differential equations, namely the fractional heat transfer and porous media equations. The HPTM uses a perturbation approach to construct an approximate solution, while the ADTM decomposes the solution into a series of functions using the Adomian polynomials. The results obtained by the HPTM and ADTM are compared with the exact solutions, and the performance of both methods is evaluated in terms of accuracy and convergence rate. The numerical results show that both methods are efficient in solving the fractional heat transfer and porous media equations, and the HPTM exhibits slightly better accuracy and convergence rate than the ADTM. Overall, the study provides a valuable insight into the application of the HPTM and ADTM in solving fractional differential equations and highlights their potential for solving complex mathematical models in physics and engineering.

This paper focuses on novel approaches to finding solitary wave (SW) solutions for the modified Degasperis-Procesi and fractionally modified Camassa-Holm equations. The study presents two innovative methodologies: the Yang transformation decomposition technique and the homotopy perturbation transformation method. These methods use the Caputo sense fractional order derivative, the Yang transformation, the adomian decomposition technique, and the homotopy perturbation method. The inquiry effectively solves the fractional Camassa-Holm and Degasperis-Procesi equations, which also provides a detailed numerical and graphical comparison of the solutions found. The results, which include accurate solutions, derived solutions, and absolute error displayed in tabular style, demonstrate the effectiveness of the suggested procedures. These procedures are iterative, which results in several answers. The estimated absolute error attests to the correctness and simplicity of these solutions. Especially in plasma physics, these approaches may be expanded to handle various linear and nonlinear physical issues, including the evolution equations controlling nonlinear waves.

This article discusses two simple, complication-free, and effective methods for solving fractional-order linear and nonlinear partial differential equations analytically: the Aboodh residual power series method (ARPSM) and the Aboodh transform iteration method (ATIM). The Caputo operator is utilized to define fractional order derivatives. In these methods, the analytical approximations are derived in series form. We calculate the first terms of the series and then estimate the absolute error resulting from leaving out the remaining terms to ensure the accuracy of the derived approximations and determine the accuracy and efficiency of the suggested methods. The derived approximations are discussed numerically using some values for the relevant parameters to the subject of the study. Useful examples are thought to illustrate the practical application of current approaches. We also examine the fractional order results that converge to the integer order solutions to ensure the accuracy of the derived approximations. Many researchers, particularly those in plasma physics, are anticipated to gain from modeling evolution equations describing nonlinear events in plasma systems.

Opportunities for magnetohydrodynamic mixed convection include modeling firefighting, combustion engineering, and cooling of electronic components. Nanotechnology has just provided a novel passive technique for improving heat transfer. Structured colloidal suspensions of magnetite nanoparticles in a base fluid are known as magneto nanofluids, and they are intended for use in thermal transfer uses, such as micro device cooling mechanisms. The current model is intended for examination of the nanofluid stagnation point flow using magnetohydrodynamics. A stretched sheet was exposed to magnetic strength and thermal radiation effects with outstanding results on heat transfer enhancements under mixed convection conditions. Furthermore, over the boundary effects of thermal slip and velocity are considered. The persuading system of partial differential equations of the governed fluid model is transformed into a scheme of coupled non-linear ordinary differential equations and expounded using a suitable numerical method. The fluid’s velocity, temperature, and concentration of nanoparticles are discussed graphically for a range of newly emerging parametric values. It has been noted that when there is a magnetic field, the fluid’s temperature increases, but its velocity decreases. After obtaining the numerical solution, parameters characterizing the flow, such as the local skin friction coefficient, local Sherwood number, and local Nusselt number, are thoroughly investigated.

Ordinary differential equations (ODEs) are fundamental tools for modeling and understanding a wide range of chemistry, physics, and biological phenomena. However, solving complex ODEs often presents significant challenges, necessitating advanced numerical approaches beyond traditional analytical techniques. Thus, a novel machine learning (ML)-based method for solving and analyzing ODEs is proposed in the current investigation. In this study, we utilize a feed-forward neural network (FNN) with five fully connected layers trained on data samples generated from the exact solutions of specific ODEs. To show the efficacy of our suggested method, we will conduct a thorough evaluation by comparing the anticipated solutions of the FNN with the exact solutions for some ODEs. Furthermore, we analyze the absolute error and present the loss functions for some ODE examples, providing valuable insights into the model’s performance and potential areas for further development.

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.

In this current study, hydrothermal method was employed to synthesize distinct nanostructures: pure nickel oxide (NiO), Zn-doped NiO, and NiO/ZnO composites. X-ray diffraction (XRD) results confirmed NiO’s cubic structure, with sizes of 17.01 nm for pure NiO and 30.87 nm for composites. Scanning electron microscope (SEM) images revealed flowerlike morphologies, energy-dispersive X-ray spectroscopy (EDX) authenticated the elemental composition without any impurities. Ultraviolet–visible (UV-Vis) spectroscopy demonstrated heightened absorption in doped and nanocomposite samples. The bandgap energy calculated using Tauc’s plot reduced from 2.95 to 2.55 eV with increase in zinc (Zn) content. In Fourier-transform infrared (FTIR) characterization, the Ni–O stretching vibration appeared at 666.7 cm−1 in doped samples, while Zn–O stretching was evident at 586.03 cm−1. Photoluminescence analysis unveiled a strong peak at 470 nm (near band edge) and another at 665 nm (indicative of defects), exhibiting intensified signals in doped and nanocomposite samples due to a higher defect density. The dielectric constant showed increased values at low frequencies attributed to real and space charge polarization. Notably, the conductivity of Zn-doped and nanocomposite samples displayed an upward trend with frequency due to a higher defect density. This study unveils insights into the intricate optical and electrical properties of these novel nanostructures, showcasing their potential in advanced energy storage applications.

Nanostructures of undoped zinc oxide and nickel-doped zinc oxide (Ni = Zn0.98Ni0.02O, Zn0.96Ni0.04O, and Zn0.94Ni0.06O) were synthesized by using the coprecipitation process, and their optical and dielectric properties were simultaneously investigated. The XRD results confirm the hexagonal structure having space group P63mc. By increasing nickel concentration, the particle size decreases, while the strain is increased. Fourier-transform infrared (FTIR) analysis was carried out in order to learn more about the phonon modes present in nickel-doped zinc oxide. UV-Vis spectroscopy further revealed that the optical band gap of nickel-doped samples varied from 3.18 eV to 2.80 eV. The SEM analysis confirms the rod shape morphology of the already synthesized samples. EDX analysis investigates the incorporation of nickel ions into the zinc oxide lattice. Using photoluminescence spectroscopy, we found that the synthesized materials had oxygen vacancies (Vo) and zinc interstitial (Zni) defects. Dielectric constant (εr) and dielectric loss (ε) are both improved in nickel-doped zinc oxide compared to undoped zinc oxide. Since more charge carriers enhanced after the nickel ions were exchanged for the Zn ions, the AC electrical conductivity (σa.c) improves by nickel doping compared to undoped zinc oxide.

Inorganic photocatalytic materials exhibiting a highly efficient response to ultraviolet-visible light spectrum have become a subject of widespread global interest. They offer a substantial prospect for generating green energy and mitigating water pollution. Zinc oxide (ZnO), among various semiconductors, proves advantageous for water-splitting applications due to its elevated reactivity, chemical stability, and nontoxic nature. However, its efficacy as a photocatalyst is hindered by limited light absorption capacity and swift charge carrier recombination. To improve charge separation and enhance responsiveness to ultraviolet-visible light photocatalysis, the formation of a heterojunction with another suitable semiconductor is beneficial. Thus, we employed hydrothermal route for the synthesis of the samples, which is a high-pressure method. The formations of ZnO/NiO heterostructures were revealed by scanning electron microscopy, X-ray diffraction analysis, energy-dispersive X-ray spectroscopy, and Fourier transform infrared spectroscopy. The nanocomposites were discovered to have a substantially higher photocatalytic activity for the generation of H2. The H2 production rates show that ZnO (i.e., 168.91 μmolg-1 h-1) exhibits good H2 production rates as compared to NiO (i.e., 135.74 μmolg-1 h-1). The best production rates were observed for ZN-30 (i.e., 247.56 μmolg-1 h-1) which is 1.46 times greater than ZnO and 1.82 times greater than NiO. This enhanced photocatalytic activity for ZN-30 is because of the good electron-hole pair separation due to the formation of depletion layer, suppression of fast charge recombination, and overcoming resistance corrosion.

Partially ionised fluids subject to an external magnetic field results in a completely different dynamical behaviour. Very few studies are conducted to explore the fascinating properties of Hall current and Ion slip in fluid flows. Flow over a Riga plate with MHD Ion and Hall currents, nonlinear thermal radiation is still rare. We aim to analyse the inclined stagnation point slip flow of Couple Stress fluid over a stretched horizontal Riga plate to close this gap. The flow problem is modelled using Navier stokes theory along with energy conservation and then principal partial differential equations are converted into ordinary differential equations. The governed system is solved mathematically by numerical technique RK Fehlberg method. The behaviour of fluid is described through graphs including fluid’s motion and energy. Local surface skin friction coefficients and local heat and mass flux at surface are the concerned physical magnitudes which are described through graphs and tables and are highly recommended in practical solicitations in numerous production plants and manufacturing workshops. Thermally radiative MHD flow with Ion Hall currents are broadly used in thermal controlling systems, spectro scopy, microchip technology, Magnetohydrodynamic energy conversion campaigns, exploration and expansion for high & low efficiency release, power-driven & power generation structures, MHD accelerations & thrusters, and drift regulator in hypersonic and automobiles.

In this research, two modified forms of the Ḡ≡G′G-expansion method are employed to investigate various kinds of solitary wave solutions that include kink, lump, periodic, and rogue wave solutions within the framework of the fractional coupled Higgs system. The underlying patterns in the targeted model are revealed by using extended and generalized Ḡ-expansion methods. The first step involves converting the model into nonlinear ordinary differential equations via a fractional complex transformation. Following that, the suggested improved versions of the Ḡ-expansion approach are used to provide numerous solitary wave solutions. Some solitary wave solutions are represented by two- and three-dimensional graphs, demonstrating their typical propagating behavior. This research finishes by summarizing the vast findings and exploring their implications for high-energy physics.

In this study, we implemented the Yang residual power series (YRPS) methodology, a unique analytical treatment method, to estimate the solutions of a non-linear system of fractional partial differential equations. The RPS approach and the Yang transform are togethered in the YRPS method. The suggested approach to handle fractional systems is explained along with its application. With fewer calculations and greater accuracy, the limit idea is used to solve it in Yang space to produce the YRPS solution for the proposed systems. The benefit of the new method is that it requires less computation to get a power series form solution, whose coefficients should be established in a series of algebraic steps. Two attractive initial value problems were used to test the technique's applicability and performance. The behaviour of the approximative solutions is numerically and visually discussed, along with the effect of fraction order ς. It was observed that the proposed method's approximations and exact solutions were completely in good agreement. The YRPS approach results highlight and show that the approach may be utilized to a variety of fractional models of physical processes easily and with analytical efficiency.

The aim of this article is to present a comparison of two analytical approaches toward obtaining the solution of the time-fractional system of partial differential equations. The newly proposed approaches are the new approximate analytical approach (NAAA) and Mohand variational iteration transform approach (MVITA). The NAAA is based on the Caputo-Riemann operator and its basic properties with the decomposition procedure. The NAAA provides step wise series form solutions with fractional order, which quickly converge to the exact solution for integer order. The MVITA is based on a variational iteration procedure and uses the Mohand integral transform. The MVITA also provides a series solution without a stepwise solution. Both approaches provide a series form of solutions to the proposed problems. The analytical procedures and obtained results are compared for the proposed problems. The obtained results were also compared with exact solutions for the problems. The obtained result and plots have shown the validity and applicability of the proposed algorithms. Both approaches can be extended for the analytical solution of other physical phenomena in science and technology.

This paper investigates fractional-order partial differential equations analytically by applying a modified technique called the Laplace residual power series method. The analytical solution was utilized to test the accuracy and precision of the proposed methodologies and shown by tables and graphs. The solution is a convergent series established on Taylor's new form. When determining the series coefficients like RPSM, the fractional derivatives must be calculated every time. We only need to perform a few computations to obtain the coefficients because LRPSM only requires the concept of an infinite limit. The advantage of this method is that it does not require Adomian polynomials or he's polynomials to solve nonlinear problems. As a result, the method's reduced computation size is a strength. The outcome we got supports the idea that the suggested method is the best one for handling any non-linear models that appear in technology and science.

The two approaches to solving nonlinear Caputo time-fractional wave-like equations with variable coefficients are examined in this study. The Homotopy perturbation transform method and the Yang transform decomposition method are the names of these two techniques. Three separate numerical examples are provided to demonstrate the effectiveness and precision of the suggested methods. The results were acquired to demonstrate the effectiveness and power of the two approaches, providing estimates with better precision and closed form solutions. The solutions to these kinds of equations can be found using the suggested methods as infinite series, and when these series are in closed form, they provide the exact solution. The suggested techniques have been demonstrated to be effective and efficient in their application. Three numerical examples are used to examine the methods accuracy and effectiveness.

In this paper, we investigate the semi-analytical solution of Kaup-Kupershmidt equations with the help of a modified method known as the new iteration transformation technique. This method combines the Yang transform and the new iteration technique. The nonlinear terms can be calculated straightforwardly by a new iteration method. The numerical simulation results have been presented to demonstrate the reliability and validity of the proposed approach. The result confirms that the suggested technique is the best tool for dealing with any nonlinear problems arising in technology and science. In addition, in terms of figures for varying fractional order, the physical behavior of new iteration transformation technique solutions has been shown and the numerical simulation is also exhibited. The solutions of the new iteration transformation technique reveal that the projected technique is reliable, competitive and powerful for studying complex nonlinear fractional type models.

Incorporating self-diffusion and super-cross diffusion factors into the modeling approach enhances efficiency and realism by having a substantial impact on the scenario of pattern formation. Accordingly, this work analyzes self and super-cross diffusion for a predator-prey model. First, the stability of equilibrium points is explored. Utilizing stability analysis of local equilibrium points, we stabilize the properties that guarantee the emergence of the Turing instability. Weakly nonlinear analysis is used to get the amplitude equations at the Turing bifurcation point (WNA). The stability analysis of the amplitude equations establishes the conditions for the formation of small spots, hexagons, huge spots, squares, labyrinthine, and stripe patterns. Analytical findings have been validated using numerical simulations. Extensive data that may be used analytically and numerically to assess the effect of self-super-cross diffusion on a variety of predator-prey systems.

In this study, the ψ-Haar wavelets operational matrix of integration is derived and used to solve linear ψ-fractional partial differential equations (ψ-FPDEs) with the fractional derivative defined in terms of the ψ-Caputo operator. We approximate the highest order fractional partial derivative of the solution of linear ψ-FPDE using Haar wavelets. By combining the operational matrix and ψ-fractional integration, we approximate the solution and its other ψ-fractional partial derivatives. Then substituting these approximations in the given ψ-FPDEs, we obtained a system of linear algebraic equations. Finally, the approximate solution is obtained by solving this system. The simplicity and effectiveness of the proposed method as a mathematical tool for solving ψ-Fractional partial differential equations is one of its main advantages. The sparse nature of the operational matrices improves the ability of the proposed method to execute with less computation complexity. Numerical examples are provided to show the efficiency and effectiveness of the method.

This paper solves a fractional system of non-linear Whitham-Broer-Kaup equations using a natural decomposition technique with two fractional derivatives. Caputo-Fabrizio and Atangana-Baleanu fractional derivatives were applied in a Caputo-manner. In addition, the results of the suggested method are compared to those of well-known analytical techniques such as the Adomian decomposition technique, the Variation iteration method, and the optimal homotopy asymptotic method. Two non-linear problems are utilized to demonstrate the validity and accuracy of the proposed methods. The analytical solution is then utilized to test the accuracy and precision of the proposed methodologies. The acquired findings suggest that the method used is very precise, easy to implement, and effective for analyzing the nature of complex non-linear applied sciences.

In this article, a new fractional mathematical model is presented to investigate the contagion of the human immunodeficiency virus (HIV). This model is constructed via recent improved fractional differential and integral operators. Other operators like Caputo, Riemann-Liouville, Katugampola, Jarad and Hadamard are being extended and generalized by these improved fractional differential and integral operators. Banach's and Leray-Schauder nonlinear alternative fixed point theorems are utilized to examine the existence and uniqueness results of the proposed fractional HIV model. Moreover, different kinds of Ulam stability for the fractional HIV model are established. It is simple to recognize that the extracted results can be reduced to some results acquired in multiple works of literature.

This paper proposes an analytical solution for a fractional fuzzy acoustic wave equation. Under the fractional Caputo-Fabrizio operator, we use the Laplace transformation and the iterative technique. In the present study, the achieved series type result was determined, and we approximated the estimated values of the suggested models. All three problems used two various fractional-order simulations between 0 and 1 to obtain the upper and lower portions of the fuzzy results. Since the exponential function is present, the fractional operator is non-singular and global. Due to its dynamic behaviors, it provides all fuzzy form solutions that happen between 0 and 1 at any level of fractional order. Because the fuzzy numbers return the solution in a fuzzy shape with upper and lower branches, the unknown quantity likewise incorporates fuzziness.

This paper studies the fractional system of the third-order Korteweg-De Vries equation under the fractional derivatives of Caputo, Atangana-Baleanu and Yang-Srivastava-Machado. The Homotopy perturbation transform technique defined with the Caputo, Atangana-Baleanu and Yang-Srivastava-Machado operators is applied to achieve the non-linear model’s actual results. This non-linear scheme generally arises as a description of waves in traffic flow, electric circuits, elastic media and electrodynamics, multi-component plasmas, shallow water waves etc. This research aims to develop a new class of methods that do not require small parameters to find approximate solutions of fractional coupled systems and eliminate unrealistically and linearization factors. Analytical simulation demonstrates that the suggested method is reliable, efficient, and straightforward to various physical systems. This study demonstrates that Homotopy perturbation transform method numerical solutions are very accurate and effective for analyzing the non-linear behavior of a system. According to the research, the Homotopy perturbation transform method is much easier, more convenient, and more efficient than other analytical techniques.

In this work, a novel technique is considered for analyzing the fractional-order Jaulent-Miodek system. The suggested approach is based on the use of the residual power series technique in conjunction with the Laplace transform and Caputo operator to solve the system of equations. The Caputo derivative is applied to express the fractional operator, which is more suitable for modeling real-world phenomena with memory effects. As a real example, the proposed technique is implemented for analyzing the Jaulent-Miodek equation under suitable initial conditions. Additionally, the proposed technique’s validity (accuracy and effectiveness) is examined by studying some numerical examples. The obtained solutions show that the suggested technique can provide a reliable solution for the fractional-order Jaulent-Miodek system, making it a helpful tool for researchers in different areas, including engineering, physics, and mathematics. We also analyze the absolute error between the derived approximations and the analytical solutions to check the validation and accuracy of the obtained approximations. Many researchers can benefit from both the obtained approximations and the suggested method in analyzing many complicated nonlinear systems in plasma physics and nonlinear optics, and many others.

This article focuses on the investigation and computation of solutions to fuzzy fractional-order Cahn–Hilliard and Gardner equations. The study hybridizes the fuzzy Gardner and Cahn–Hilliard equation into two equations using hybrid techniques and the concept of a parametric fuzzy number. To explore these equations, a combination of a novel iterative approach and the Shehu transformation is employed. The article presents detailed procedures for computing a series of solutions to the fractional-order Cahn–Hilliard and Gardner problem. The applied techniques not only offer precision, simplicity, and efficacy but also outperform other existing technologies. Additionally, several examples are solved to validate the proposed theoretical solution.

In this study, we present a comprehensive comparison of two powerful analytical techniques, Aboodh Adomian decomposition method (AADM) and homotopy perturbation transform method (HPTM), for obtaining series solutions of nonlinear partial differential equations, specifically focusing on Camassa–Holm (CH) and Degasperis–Procesi (DP) equations. These equations are widely used to describe various nonlinear wave phenomena in fluid mechanics, optical fibers, and other applications. By applying both AADM and HPTM to CH and DP equations, we demonstrate the effectiveness and efficiency of each method in terms of accuracy, convergence, and computational complexity. Furthermore, we provide a detailed analysis of the series solutions obtained by each method and discuss their respective advantages and limitations. The results reveal that both methods are capable of providing accurate and convergent series solutions for the considered equations. However, AADM shows a slightly better performance in terms of convergence rate and ease of implementation, making it a preferable choice for solving CH and DP equations. This comparative study serves as a useful reference for researchers and practitioners working in the field of nonlinear partial differential equations and their applications.

The time-fractional coupled Schrödinger-KdV equation is an interesting mathematical model because of its wide and significant application in mathematics and applied sciences. A fractional coupled Schrödinger-KdV equation in the sense of Caputo derivative is investigated in this article. Namely, we provide a comparative study of the considered model using the Adomian decomposition method and the homotopy perturbation method with Shehu transform. Approximate solutions obtained using the Adomian decomposition and homotopy perturbation methods were numerically evaluated and presented in graphs and tables. Then, these solutions were compared to the exact solutions, demonstrating the simplicity, effectiveness, and good accuracy of the applied method. To demonstrate the accuracy and efficiency of the suggested techniques, numerical problem are provided.

In this article, we solved pantograph delay differential equations by utilizing an efficient numerical technique known as Chebyshev pseudospectral method. In Caputo manner fractional derivatives are taken. These types of problems are reduced to linear or nonlinear algebraic equations using the suggested approach. The proposed method's convergence is being studied with particular care. The suggested technique is effective, simple, and easy to implement as compared to other numerical approaches. To prove the validity and accuracy of the presented approach, we take two examples. The solutions we obtained show greater accuracy as compared to other methods. Furthermore, the current approach can be implemented for solving other linear and nonlinear fractional delay differential equations, owing to its innovation and scientific significance.

The major goal of this research is to use a new integral transform approach to obtain the exact solution to the time-fractional convection-reaction-diffusion equations (CRDEs). The proposed method is a combination of the Elzaki transform and the homotopy perturbation method. He's polynomial is used to tackle the nonlinearity which arise in our considered problems.Three test examples are considered to show the accuracy of the proposed scheme. In order to find satisfactory approximations to the offered problems, this work takes into account a sophisticated methodology and fractional operators in this context. In order to achieve better approximations after a limited number of iterations, we first construct the Elzaki transforms of the Caputo fractional derivative (CFD) and Atangana-Baleanu fractional derivative (ABFD) and implement them for CRDEs. It has been found that the proposed method's solution converges at the desired rate towards the accurate solution. We give some graphical representations of the accurate and analytical results, which are in excellent agreement with one another, to demonstrate the validity of the suggested methodology. For validity of the present technique, the convergence of the fractional solutions towards integer order solution is investigated. The proposed method is found to be very efficient, simple, and suitable to other nonlinear problem raised in science and engineering.

The approximate solution of the Kersten-Krasil'shchik coupled Korteweg-de Vries-modified Korteweg-de Vries system is obtained in this study by employing a natural decomposition method in association with the newly established Atangana-Baleanu derivative and Caputo-Fabrizio derivative of fractional order. The Korteweg-de Vries equation is considered a classical super-extension in this system. This nonlinear model scheme is commonly used to describe waves in traffic flow, electromagnetism, electrodynamics, elastic media, multi-component plasmas, shallow water waves and other phenomena. The acquired results are compared to exact solutions to demonstrate the suggested method's effectiveness and reliability. Graphs and tables are used to display the numerical results. The results show that the natural decomposition technique is a very user-friendly and reliable method for dealing with fractional order nonlinear problems.

This article shows how to solve the time-fractional Fisher's equation through the use of two well-known analytical methods. The techniques we propose are a modified form of the Adomian decomposition method and homotopy perturbation method with a Yang transform. To show the accuracy of the suggested techniques, illustrative examples are considered. It is confirmed that the solution we get by implementing the suggested techniques has the desired rate of convergence towards the accurate solution. The main benefit of the proposed techniques is the small number of calculations. To show the reliability of the suggested techniques, we present some graphical behaviors of the accurate and analytical results, absolute error graphs and tables that strongly agree with each other. Furthermore, it can be used for solving fractional-order physical problems in various fields of applied sciences.

In this study, we propose a method to study fractional-order shock wave equations and wave equations arising from the motion of gases. The fractional derivative is taken in Caputo manner. The approaches we used are the combined form of the Yang transform (YT) together with the homotopy perturbation method (HPM) called homotopy perturbation Yang transform method (HPYTM) and also Yang transform (YT) with the Adomian decomposition method called Yang transform decomposition method (YTDM). The HPYTM is a combination of the Yang transform, the homotopy perturbation method and He's polynomials, whereas the YTDM is a combination of the Yang transform, the decomposition method and the Adomian polynomials. Adomian and He's polynomials are excellent tools for handling nonlinear terms. The manipulation of the recurrence relation, which generates the series solutions in a limited number of iterations, is the essential innovation we describe in this study. We give several graphical behaviors of the exact and analytical results, absolute error graphs, and tables that highly agree with one another to demonstrate the reliability of the suggested methodologies. The results we obtained by implementing the proposed approaches indicate that it is easy to implement and computationally very attractive.

In this article we consider the homotopy perturbation transform method to investigate the fractional-order equal-width equations. The homotopy perturbation transform method is a mixture of the homotopy perturbation method and the Yang transform. The fractional-order derivative are defined in the sense of Caputo-Fabrizio operator. Several fractions of solutions are calculated which define some valuable evolution of the given problems. The homotopy perturbation transform method results are compared with actual results and good agreement is found. The suggested method can be used to investigate the fractional perspective analysis of problems in a variety of applied sciences.

In this paper, we used the Natural decomposition approach with nonsingular kernel derivatives to explore the modified Boussinesq and approximate long wave equations. These equations are crucial in defining the features of shallow water waves using a specific dispersion relationship. In this research, the convergence analysis and error analysis have been provided. The fractional derivatives Atangana-Baleanu and Caputo-Fabrizio are utilised throughout the paper. To obtain the equations results, we used Natural transform on fractional-order modified Boussinesq and approximate long wave equations, followed by inverse Natural transform. To verify the approach, we focused on two systems and compared them to the exact solutions. We compare exact and analytical results with the use of graphs and tables, which are in strong agreement with each other, to demonstrate the effectiveness of the suggested approaches. Also compared are the results achieved by implementing the suggested approaches at various fractional orders, confirming that the result comes closer to the exact solution as the value moves from fractional to integer order. The numerical and graphical results show that the suggested scheme is computationally very accurate and simple to investigate and solve fractional coupled nonlinear complicated phenomena that exist in science and technology.

The primary goal of this study is to create and characterise solitary wave solutions for the conformable Fractional Coupled Boussinesq-Whitham-Broer-Kaup Equations (FCBWBKEs), a model that governs shallow water waves. Through wave transformations and the chain rule, the authors used the modified Extended Direct Algebraic Method (mEDAM) for transforming FCBWBKEs into a more manageable Nonlinear Ordinary Differential Equation (NODE). This accomplishment is particularly noteworthy because it surpasses the drawbacks linked to both the Caputo and Riemann–Liouville definitions in complying to the chain rule. The study uses visual representations such as 3D, 2D, and contour graphs to demonstrate the dynamic nature of solitary wave solutions. Furthermore, the investigation of diverse wave phenomena such as kinks, shock waves, periodic waves, and bell-shaped kink waves highlights the range of knowledge obtained in the study of shallow water wave behavior. Overall, this study introduces novel methodologies that produce valuable and consistent results for the problem under consideration.