Nonlinear Partial Differential Equations Research Articles

Thermally radiative chemically reactive flow of Reiner-Rivlin nanofluid through porous medium with Newtonian conditions

The nanofluids have well-known implications in pharmaceutical industries, microelectronics, heat exchangers, engine cooling, hybrid-powered engines, thermal management of vehicles, refrigerator, machining, chillers, grinding, and in boiler fuel to reduce the gas temperature. Due to such dominant implications of nanofluids, the boundary layer Reiner-Rivlin nanomaterial flow over chemically reactive stretched sheet is considered. Radiation term is incorporated in the energy equation for heat transportation analysis. Newtonian thermal and solutal conditions are implemented at the boundaries of the surface. Similarity variables are utilized for the conversion of nonlinear partial differential equations into the system of one independent variable equations. The resulted system of equations is calculated analytically with the help of homotopy analysis method. Convergence of the calculated results is verified through plots and numeric benchmark. The results of various pertinent parameters on quantities of physical importance are drawn and discussed in detail. Results revealed that the incremented cross viscous constraint resulted higher velocity and lower temperature profiles. An augmentation in radiative constraint gives rise to temperature field. Concentration and temperature have reverse trends against the rising Brownian motion constraint values.

Polynomial Fuzzy Observer-Based Feedback Control for Nonlinear Hyperbolic PDEs Systems.

This article explores the observer-based feedback control problem for a nonlinear hyperbolic partial differential equations (PDEs) system. Initially, the polynomial fuzzy hyperbolic PDEs (PFHPDEs) model is established through the utilization of the fuzzy identification approach, derived from the nonlinear hyperbolic PDEs model. Various types of state estimation and controller design problems for the polynomial fuzzy PDEs system are discussed concerning the state estimation problem. To investigate the relaxed stability problem, Euler's homogeneous theorem, Lyapunov-Krasovskii functional with polynomial matrices (LKFPM), and the sum-of-squares (SOSs) approach are adopted. The exponential stabilization condition is formulated in terms of the spatial-derivative-SOSs (SD-SOSs). Additionally, a segmental algorithm is developed to find the feasible solution for the SD-SOS condition. Finally, a hyperbolic PDEs system and several numerical examples are provided to illustrate the validity and effectiveness of the proposed results.

Solitary wave type solutions of nonlinear improved mKdV equation by modified techniques

In this paper, the improved and modified version of the Sardar sub-equation method (IMSSEM) and the improved generalized Riccati equation mapping method (IGREMM) are manipulated effectively and generously to determine the exact solitary wave soliton solutions of the improved modified KdV (mKdV) equation. The purpose of this study is to provide novel exact solutions to the improved mKdV equation. Specifically, we utilized IMSSEM and IGREMM to study different solutions of the nonlinear improved mKdV equation, focusing on exponential, trigonometric, and trigonometric hyperbolic type solutions. Furthermore, the plotting of various solutions for direct viewing analysis is provided in two and three-dimensional graphs. The new strategies are straightforward, quick, and efficient and have many other advantages, whereas, they provide the most accurate and unique solution to many other types of nonlinear partial differential equations (NPDEs), which usually arise in engineering and applied sciences. It should be noted that these methodologies are novel mathematical instruments that have shown to be the most effective mathematical tools for solving higher-order nonlinear partial differential equations in mathematical physics. Symbolic computation was used to validate all of the solutions that were established. Thus, it is also hoped that these techniques will ultimately reduce the cumbersome workload involved during the process of solutions to complicated NLPDEs.

Effect of MHD and radiation on biviscous Bingham fluid flow on Marangoni boundary for heat source/sink with chemical reaction

A significance of Marangoni-driven convective magnetohydrodynamics (MHD) was investigated in a planned study. The present study examines the impact of radiation and chemical reactions on the two dimensional boundary layer flow of bi-viscous Bingham fluid on a thermosolutal Marangoni boundary with magnetic field and heat source/sink. The physical flow problem is mathematically modelled into Navier–Stokes equations. These nonlinear partial differential equations are then mapped into a set of nonlinear ordinary differential equations using similarity transformation. The elegant analytical method is used to obtain analytic approximations for the resulting system of nonlinear differential equations. The analytical solution for the temperature and concentration profiles is expressed in terms of Kummer's confluent hypergeometric function. The features of flow characteristics such as velocity, temperature, and concentration profiles in response to the variations of the emerging parameters are simulated and examined with a physical explanation through graphs. Surface tension is considered to be corresponding to heat and mass. The results reveal that the velocity profile decreases with increasing the magnetic field, while the Marangoni number value increases with increasing the velocity profile. Moreover, the temperature profile rises with rising the radiation parameter and whereas the value chemical reaction parameter upsurges with decline concentration profile. The present problem has great interest due to its applications in industrial applications such as drying of silicon wafers, thin layers of paint, glues, in heat exchangers, crystal growth in space, biological fluids, blood, synovial liquids, plasma, lubricants, many pharmaceutical products, proteins, sewage sludge, china clay, and many more.

Diverse soliton wave profile assessment to the fractional order nonlinear Landau-Ginzburg-Higgs and coupled Boussinesq-Burger equations

The space–time fractional Landau-Ginzburg-Higgs equation and coupled Boussinesq-Burger equation describe the behavior of nonlinear waves in the tropical and mid-latitude troposphere, exhibiting weak scattering, extended connections, arising from the interactions between equatorial and mid-latitude Rossby waves, fluid flow in dynamic systems, and depicting wave propagation in shallow water. The improved Bernoulli sub-equation function method has been used to achieve new and wide-ranging closed-form solitary wave solutions to the mentioned nonlinear fractional partial differential equations through beta-derivative. A wave transformation is applied to renovate the fractional-order equation into an ordinary differential equation. Some standard wave shapes of multiple soliton type, single soliton, kink shape, double soliton shape type, triple soliton shape, anti-kink shape, and other types of solitons have been established. The more updated software Python is used to display the solutions by using 3D and contour plotlines to describe the physical significances of attained solutions more clearly. The findings of this study are straightforward, adaptable, and quicker to simulate. It has been notable that the improved Bernoulli sub-equation function method is practical, effective, and offers more sophisticated solutions that can help togenerate a large number of wave solutions for various models.

The analysis of traveling wave solutions and dynamical behavior for the stochastic coupled Maccari's system via Brownian motion

The stochastic coupled Maccari's system (MS) is a kind of important nonlinear partial differential equations to describe fluid flow, plasma physics, nonlinear optics and so on. In this article, the dynamical behavior and some new exact traveling wave solutions of the system are investigated. By means of complex traveling wave transformation, the system is transformed into a nonlinear ordinary differential equation. The dynamical behavior of the system as well as its perturbation case are illustrated by bifurcation theory. And then, some new stochastic traveling wave solutions of the system are extracted based on the theory of polynomial complete discrimination system. To show the effect of stochastic factor on the solutions, their structures under different Brownian motion amplitudes are compared by several sets of graphs. The results obtained in this paper have supplemented the study of the system, and the technique used to exploit the traveling wave solutions are effective.

Advancing Blood Flow in Stenotic Arteries through Magnetohydrodynamic Peristaltic Motion of Hybrid Nanoparticles

This research aims to investigate the efficacy of hybrid nanoparticles in improving magnetohydrodynamic (MHD) peristaltic blood flow in stenotic arteries. Hybrid nanoparticles combining MHD processes and peristaltic motion show promise for increasing blood flow through constricted arteries. To evaluate the impact of nanoparticles on blood flow parameters, researchers use theoretical modelling and analytical solutions. Incorporating copper and aluminium oxide nanoparticles (Cu-Al2O3) into blood, leading to the formation of an ionic solution, is a crucial step in creating a homogeneous hybrid blood suspension. The mathematical model uses approximations of the stenosis and non-dimensional parameters for a mild stenosis scenario to simplify the underlying coupled nonlinear partial differential equations. Analytical solutions for various blood flow parameters such as velocity, temperature, stream function, wall shear stress, Nusselt number, and flow resistance impedance are obtained using the perturbation technique. This study visually analyzes how different physiological variables affect arterial blood flow characteristics. The heat conductivity of blood is an essential aspect of preserving one's sustenance, which is largely determined by the concentration of hybrid nanoparticles. A streamlined pattern depicts the direction of blood flow within an artery. This research demonstrates that the adverse effects of vascular stenosis can be mitigated by using hybrid nanoparticles. In order to aid narrowed arteries, these nanoparticles lessen wall shear stress, boost axial flow speed, and enhance the velocity profile.

A reduced ideal MHD system for nonlinear magnetic field turbulence in plasmas with approximate flux surfaces

This paper studies the nonlinear evolution of magnetic field turbulence in proximity of steady ideal Magnetohydrodynamics (MHD) configurations characterized by a small electric current, a small plasma flow, and approximate flux surfaces, a physical setting that is relevant for plasma confinement in stellarators. The aim is to gather insight on magnetic field dynamics, to elucidate accessibility and stability of three-dimensional MHD equilibria, as well as to formulate practical methods to compute them. Starting from the ideal MHD equations, a reduced dynamical system of two coupled nonlinear partial differential equations for the flux function and the angle variable associated with the Clebsch representation of the magnetic field is obtained. It is shown that under suitable boundary and gauge conditions such reduced system preserves magnetic energy, magnetic helicity, and total magnetic flux. The noncanonical Hamiltonian structure of the reduced system is identified, and used to show the nonlinear stability of steady solutions against perturbations involving only one Clebsch potential. The Hamiltonian structure is also applied to construct a dissipative dynamical system through the method of double brackets. This dissipative system enables the computation of MHD equilibria by minimizing energy until a critical point of the Hamiltonian is reached. Finally, an iterative scheme based on the alternate solution of the two steady equations in the reduced system is proposed as a further method to compute MHD equilibria. A theorem is proven which states that the iterative scheme converges to a nontrivial MHD equilbrium as long as solutions exist at each step of the iteration.

Comparative numerical study featuring magnetized nanofluids configured by elongating sheet with thermophoresis and Brownian motion

This study investigates the heat and mass transfer characteristics of MHD rotating Fe₃O₄-Al₂O₃/H₂O hybrid nanofluid flow over a three-dimensional stretching surface. Comparative analysis was conducted among the base fluid H₂O, Al₂O₃-H₂O nanofluid, and Fe₃O₄-Al₂O₃/H₂O hybrid nanofluid, focusing on velocity, temperature, and concentration distributions influenced by thermophoresis, Brownian motion, Hall current, temperature ratio, magnetic parameter, rotation, and thermal radiation. The governing nonlinear partial differential equations were transformed into ordinary differential equations using similarity adaptations and solved numerically with the BVP-5C method in MATLAB. Results indicate that increasing the thermophoresis and Brownian motion parameters elevates the temperature profile while affecting the concentration distribution differently. The hybrid nanofluid exhibited a higher temperature distribution compared to the nanofluid and base fluid, whereas the base fluid showed a greater concentration gradient.

Impacts of fossil fuel thermophoretic convective heat transfer on climate change with variable viscosity and thermal conductivity

The main aim of the current study is to analyze the impacts of fossil fuel thermophoretic convective heat transfer on climate change with variable viscosity and thermal conductivity. Furthermore, the purpose of the proposed problem is to develop a mathematical model based on three regions: source region (in terms of rectangular coordinates), plume region (in terms of cylindrical coordinates), and atmospheric region (in terms of spherical coordinates). The fossil fuels release thermophoretic particles, such as carbon dioxide, methane, black carbon, and many others, during burning process in the source region, and then release through the plume region. These particles are then distributed into the atmosphere, where the impact of thermophoretic particles on climate change is analyzed. The modeled nonlinear partial differential equations are transformed into a dimensionless form using suitable non-dimensional scaling variables. The proposed model is solved using finite difference approach in order to analyze the impacts of fossil fuel thermophoretic particles in the atmosphere in terms of climate change. In this regard, the effect of dimensionless parameters, viscosity variation parameter γ, Schimdt number Sc, thermal conductivity variation parameter ε, coefficient of thermophoretic process k, and thermophoresis parameter Nt on the velocity, temperature, and thermophoretic concentration fields are discussed. The main novelty of current work is that three models in three regions are coupled via trans-boundaries in term of temperature differences. It is very interesting to note that the concentration of thermophoretic particles, along with temperature profile, is maximum at α=π rad and minimum at α=1.5 rad in the atmospheric region.

Thermal and Computational Analysis of MHD Dissipative Flow of Eyring-Powell Fluid: Non-Similar Approach via Overlapping Grid-Based Spectral Collocation Scheme

The aim of the present study is to numerically investigate the non-similar flow and heat transfer in a dissipative Eyring-Powell fluid (EPF) over a stretching surface. A constant magnetic field is applied perpendicular to the stretched surface to explore the impact of the Lorentz force. Both viscous and magnetic dissipation are considered to comprehensively examine their effects on heat transfer. The problem in hand does not admit self-similar solutions as the non-Newtonian fluid parameter varies with the spatial variable along the stream-wise direction. Consequently, the set of nonlinear partial differential equations, modeling the flow problem is nondimensionalized primarily by employing a pseudo-similarity variable and stream-wise coordinate. The non-dimensional set of nonlinear partial differential equations is solved by a newly developed and efficient “overlapping multi-domain bivariate spectral local linearization method (OMD-BSLLM)”. The current study includes residual error analysis and convergence tests to demonstrate the accuracy of the numerical method applied to the current mathematical model. Graphs show fluid flow and heat transfer results for different flow parameters, while tables display skin friction and Nusselt number values. The results indicate that the non-Newtonian fluid parameter enhances both the velocity profile and temperature distribution. The fluid decelerates with increasing the dimensionless stream wise coordinate and Hartmann number. Viscous dissipation and dimensionless stream-wise coordinate enhances the temperature profile.

Significances of melting heat transfer and bioconvection phenomena in nanofluid flow over a three different geometries

This study addresses the numerical investigation of the steady and two-dimensional flow of magnetohydrodynamic nanofluid containing motile microorganisms over three distinct configurations: a wedge, a plate, and the stagnation point of a flat plate. The impacts of activation energy, melting phenomena, thermophoresis, and Brownian motion are taken into account. The dimensionless form of the governing coupled nonlinear partial differential equations is obtained by using similarity transformations. The system of ordinary differential equations is numerically solved using the “bvp4c” solver in MATLAB, and obtained results are also validated with an analytical approach Optimal Auxiliary Function Method (OAFM). Skin friction, heat, mass and motile microorganisms transfer rates are discussed through graphs and tables. Findings indicate that profile for Nusselt number enhances for higher inputs of melting parameter while Sherwood number lowers with increasing inputs of activation energy parameter. An improving pattern of velocity profiles is magnificent for stagnation point flow [f(η = 1) = 0.943711] compared to wedge [f(η = 1) = 0.915698]and horizontal plate [f(η = 1) = 0.868617]with respect to wedge angle parameter keeping velocity ratio parameter A < 1. For the wedge surface, the temperature profiles decrease [θ(η = 1.5) = 0.933075, 0.895245, 0.858931] for increasing inputs of radiation parameter (2.5 ≤ Nr ≤ 4.5) while motile microorganism profiles enhance [χ(η = 1.3) = 0.532505, 0.621569, 0.690635] with bioconvection Schmidtt number (0.3 ≤ Sb ≤ 0.7) respectively. Furthermore, it was found that, as velocity slip parameter increases then velocity of fluid also improves over a plate, wedge, and stagnation point of a flat plate considering velocity ratio parameter A < 1 .

Thermal conductivity impact on MHD convective heat transfer over moving wedge with surface heat flux and high magnetic Prandtl number

The present evaluation focused on heat transfer and magnetic flux along the moving non-conducting wedge shape with thermal conductivity and surface heat flux effects. A suitable and well-known similarity transformation for integration is used for the coupled non-linear partial differential equations with stream formulations. The Keller Box technique is utilized to integrate the final non-similar equations numerically. The discretized algebraic equations are displayed graphically and numerically using the MATLAB software. The physical characteristics like temperature, magnetic field, velocity profiles, skin friction, heat transfer and magnetic intensity are investigated with various factors. The influence of relevant characteristics like thermal conductivity ξ, Prandtl number Pr, wedge parameter β, variable viscosity ε, and Biot number Bi is interpreted graphically and numerically. It is noted that velocity graph effects are declined at lower value of Biot number and enhanced value is obtained at the higher value of Biot number. The fluid temperature with highest thermal slip effects is displayed with minimum value of thermal conductivity but minimum value is obtained at large value of thermal conductivity. The maximum heat flux value is obtained at large value of moving parameter.

Buongiorno’s radiative Prandtl-Eyring nanofluid flow through porous medium over a stretching surface with cross-diffusion effects

The main focus of the current analysis is to describe the thermo-magnetic flow, heat and mass transport features of two-dimensional dissipative Prandtl-Eyring nanofluid (PE-NF) flow over a stretching sheet under the influence of porous medium and magnetic Ohmic dissipation numerically. An innovative Buongiorno’s nanofluid model in terms of Brownian motion and thermophoresis is deployed to accurately simulate the nano behavior in Prandtl-Eyring within the boundary layer regime. The thermal radiation and heat source/sink effects are also included to describe the thermal transport process. In addition to this, the thermo-diffusion and diffusion-thermo effect are included to demonstrate the temperature and concentration diffusion mechanism under the presence of chemical reaction process. The emerged coupled nonlinear two-dimensional time-independent partial differential equations are rendered to their dimensionless form through appropriate similarity transformations and solved by deploying Matlab-based BVP4C technique. The graphical visualization showed that, the increasing magnetic number diminished the flow field and enhanced the temperature and concentration profiles in the flow regime. Rising porous parameter decayed the Prandtl-Eyring nanofluid velocity. Increasing radiation and Eckert numbers enhance the temperature field in the flow regime. Magnifying Prandtl-Eyring parameter suppressed the velocity diffusion. Increasing Brownian motion and thermophoresis parameters increases the temperature profile. Amplifying Soret parameter upsurges the concentration profile. Skin-friction coefficient amplified with enhanced magnetic and porous numbers. Heat transfer rate acts as increasing function of thermophoresis and Brownian motion parameters. The main objective of this study is the addition of magnetic Ohmic dissipation, radiation, porous medium, thermal source/sink, Soret and Dufour effects and Buongiorno nanofluid model and this consideration generalizes former studies and gives a new re-defined mathematical formulation of heat and mass transport features of Prandtl-Eyring nanofluid flow over a stretching sheet. Finally, the numerical accuracy of the current similarity results is validated with available solutions and noticed a good agreement.

On Laplace invariants of two-dimensional nonlinear equations of the second order with homogeneous polynomial

We study two-dimensional nonlinear partial differential equations of the second order with variable coefficients. The left-hand side of these equations is a homogeneous polynomial of the second degree in unknown function and its derivatives. We consider a set of linear multiplicative transformations of the unknown function which keep the form of the initial equation. By analogy with linear equations, the Laplace invariants are determined as the invariants of this transformation. The expressions for the Laplace invariants in terms of the coefficients of the equation and their first derivatives are obtained. For the considered equations, we found the equivalent systems of the first order equations containing the Laplace invariants. It is shown that if one of the Laplace invariants equals zero, the corresponding system is reduced to one equation of the first order. Also in this case, the solution of the initial equation can be obtained in quadratures if some additional conditions on the coefficients are met. The investigations are executed for a hyperbolic equation with a mixed derivative and for a nonlinear second order equation of the general form with a homogeneous polynomial of the second degree in unknown function and its derivatives. We obtained for these cases the Laplace invariants and equivalent systems of the first order equations.

Dufour and Soret effects on MHD squeezed nanoliquid flow between parallel plates by differential transform method

Purpose This paper aims to examine the Dufour and Soret combined effects on the study of two-dimensional squeezed flow of copper water nanofluid between parallel plates along with applied (external) magnetic field. Impact of higher order chemical reaction is also considered. Design/methodology/approach The nonlinear partial differential equations (PDEs) are changed into system of ordinary differential equations (ODEs) by employing suitable similarity transformations. These transformed ODEs are then solved by means of a semianalytical method called differential transform method (DTM). Effects of several changing physical parameters on fluid flow, temperature and concentration have been deliberated through graphs. Findings It is observed that Dufour and Soret numbers are directly related to temperature profile and a reverse trend was observed in the concentration profile. Temperature enhancement is perceived for the enhanced Dufour number. Enhancement in Dufour number shows a direct association with Sh and Nu for all values of squeezing parameter. Practical implications The combined Dufour and Soret effects are used in separation of isotopes in mixture of gases, oil reservoirs and binary alloys solidification. The squeeze nanoliquid flow can be used in the field of composite material joining, rheological testing and welding engineering. Social implications This study is mainly useful for geosciences and chemical engineering. Originality/value The uniqueness in this research is the study of the impact of cross diffusion on chemically reacting squeezed nanoliquid flow with the chemical reaction order more than one in the presence of applied magnetic force using a semianalytical procedure, named DTM.

Parameter Inference Based on Gaussian Processes Informed by Nonlinear Partial Differential Equations

Parameter Inference Based on Gaussian Processes Informed by Nonlinear Partial Differential Equations

Qualitative analysis and optical soliton solutions galore: scrutinizing the (2+1)-dimensional complex modified Korteweg–de Vries system

This investigation discusses the (2+1)-dimensional complex modified Korteweg–de Vries (cmKdV) system. The cmKdV system describes the nontrivial dynamics of water particles from the surface to the bottom of a water layer, providing a more comprehensive understanding of wave behavior. The cmKdV system finds applications in various fields of physics and engineering, including fluid dynamics, nonlinear optics, plasma physics, and condensed matter physics. Understanding the behavior predicted by the cmKdV system can lead to insights into the underlying physical processes in these systems and potentially inform the design of novel technologies. A new version of the generalized exponential rational function method (nGERFM) is utilized to discover diverse soliton solutions. This method uncovers analytical solutions, including exponential function, singular periodic wave, combo trigonometric, shock wave, singular soliton, and hyperbolic solutions in mixed form. Moreover, the planar dynamical system of the concerned equation is created, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore, after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. To gain a deeper understanding of the dynamic behavior of the solutions, analytical results are supplemented with numerical simulations. These obtained outcomes provide a foundation for further investigation, making the solutions useful, manageable, and trustworthy for the future development of intricate nonlinear issues. This study’s methodology is reliable, strong, effective, and applicable to various nonlinear partial differential equations (NLPDEs). As far as we know, this type of research has never been conducted to such an extent for this equation before. The Maple software application is used to verify the correctness of all obtained solutions.

Analysis of multi-wave solitary solutions of (2+1)-dimensional coupled system of Boiti–Leon–Pempinelli

This work examines the (2+1)-dimensional Boiti–Leon–Pempinelli model, which finds its use in hydrodynamics. This model explains how water waves vary over time in hydrodynamics. We provide new explicit solutions to the generalized (2+1)-dimensional Boiti–Leon–Pempinelli equation by applying the Sardar sub-equation technique. This method is shown to be a reliable and practical tool for solving nonlinear wave equations. Furthermore, different types of solitary wave solutions are constructed: w-shaped, breather waved, chirped, dark, bright, kink, unique, periodic, and more. The results obtained with the variable coefficient Boiti–Leon–Pempinelli equation are stable and different from previous methods. As compared to their constant-coefficient counterparts, the variable-coefficient models are more general here. In the current work, the problem is solved using the Sardar Sub-problem Technique to produce distinct soliton solutions with parameters. Plotting these graphs of the solutions will help you better comprehend the model. The outcomes demonstrate how well the method works to solve nonlinear partial differential equations, which are common in mathematical physics.With the help of this method, we may examine a variety of solutions from significant physical perspectives.

Exploring solutions to the fractional Boiti–Leon–Manna–Pempinelli equation characterizing wave dynamics in incompressible fluids

In this study, the (3 + 1)-dimensional space-time fractional Boiti–Leon–Manna–Pempinelli (BLMP) equation is investigated utilizing the Kudryashov method (KM) and the modified Kudryashov method (MKM). These two efficient methods are implemented to acquire exact closed-form solutions to the considered fractional BLMP equation, and novel solitary wave solutions are constructed involving hyperbolic functions, and exponential rational functions with arbitrary constant parameters. With the aid of a unique wave transformation, the nonlinear fractional differential equation is transformed into an ordinary differential equation. In this unique wave transformation, the conformable fractional derivative is considered to which the chain rule is applied. The obtained solutions are indeed beneficial for analyzing the dynamic behavior of the fractional BLMP equation in describing fluid propagation. A few three-dimensional representations of the attained exact analytic solutions are offered by accounting for the proper selection of the appropriate parameters with the help of mathematical software. As a result, novel soliton solutions, such as the anti-kink type and singular kink type waves, are acquired from the attained solutions. This approach is relatively straightforward and efficient, making it possible to utilize it to obtain exact solutions for higher-order nonlinear partial differential equations.