Higher Order Nonlinear Differential Equations Research Articles

Chebyshev spectral approximation-based physics-informed neural network for solving higher-order nonlinear differential equations

Chebyshev spectral approximation-based physics-informed neural network for solving higher-order nonlinear differential equations

Novel Approaches to Existence and Uniquenessin Nonlinear Higher-Order Differential Equations

The study of nonlinear higher-order differential equations presents significant challenges in terms of existence and uniqueness of solutions. This paper explores novel approaches to addressing these challenges, focusing on recent advancements and methodologies that offer new perspectives on these classical problems. We investigate advanced techniques including topological methods, functional analysis, and computational approaches to enhance our understanding of existence and uniqueness in nonlinear higher-order differential equations. By reviewing the latest literature and integrating new findings, this paper aims to provide a comprehensive overview of current research trends and future directions in this area.

Numerical study of hybrid nanofluid and thermal transport in sun-powered energy ship within the application of parabolic trough solar collectors

PurposeRecent advancements in technology have led to the exploration of solar-based thermal radiation and nanotechnology in the field of fluid dynamics. Solar energy is captured through sunlight absorption, acting as the primary source of heat. Various solar technologies, such as solar water heating and photovoltaic cells, rely on solar energy for heat generation. This study focuses on investigating heat transfer mechanisms by utilizing a hybrid nanofluid within a parabolic trough solar collector (PTSC) to advance research in solar ship technology. The model incorporates multiple effects that are detailed in the formulation.Design/methodology/approachThe mathematical model is transformed using suitable similarity transformations into a system of higher-order nonlinear differential equations. The model was solved by implementing a numerical procedure based on the Wavelets and Chebyshev wavelet method for simulating the outcome.FindingsThe velocity profile is reduced by Deborah's number and velocity slip parameter. The Ag-EG nanoparticles mixture demonstrates less smooth fluid flow compared to the significantly smoother fluid flow of the Ag-Fe3O4/EG hybrid nanofluids (HNFs). Additionally, the Ag-Ethylene Glycol nanofluids (NFs) exhibit higher radiative performance compared to the Ag-Fe3O4/Ethylene Glycol hybrid nanofluids (HNFs).Practical implicationsAdditionally, the Oldroyd-B hybrid nanofluid demonstrates improved thermal conductivity compared to traditional fluids, making it suitable for use in cooling systems and energy applications in the maritime industry.Originality/valueThe originality of the study lies in the exploration of the thermal transport enhancement in sun-powered energy ships through the incorporation of silver-magnetite hybrid nanoparticles within the heat transfer fluid circulating in parabolic trough solar collectors. This particular aspect has not been thoroughly researched previously. The findings have been validated and provide a highly positive comparison with the research papers.

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.

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.

Novel approach for solving higher-order differential equations with applications to the Van der Pol and Van der Pol–Duffing equations

BackgroundNumerical methods are used to solve differential equations, but few are effective for nonlinear ordinary differential equations (ODEs) of order higher than one. This paper proposes a new method for such ODEs, based on Taylor series expansion. The new method is a second-order method for second-order ODEs, and it is equivalent to the central difference method, a well-known method for solving differential equations. The new method is also simple to implement for higher-order differential equations. The proposed technique was applied to solve the Van der Pol and Van der Pol–Duffing equations. It is stable over a wide range of nonlinearity and produces accurate and reliable results. For the self-excitation Van der Pol equation, the proposed technique was applied with different values of nonlinear damping.ResultsThe results were compared with those obtained using the ODE15s solver in MATLAB. The two sets of results showed excellent agreement. For the forced Van der Pol–Duffing equation, the proposed technique was applied with different values of exciting force amplitude and frequency. It was found that for certain conditions, the solution obtained using the proposed technique differed from that obtained using ODE15s.ConclusionsThe solution obtained using the proposed technique showed good agreement with the solutions obtained using ODE45 and Runge–Kutta fourth order. The results show that the proposed approach is very simple to apply and produces acceptable error. It is a powerful and versatile tool for solving of high-order nonlinear differential equations accurately.

Dynamical Behavior and Expressions of Solutions of a Class of Higher-Order Nonlinear Difference Equations

The aim of this article is to get the forms of the solutions of the following nonlinear higher-order difference equations χs+1=χs−3k+1/±1±Πl=1kχs−3l+1,s=0,1,2,…, where the initial conditions χ−j,j=0,1,2,…,3k−1 and k∈1,2,… are arbitrary real numbers. Also, we examine stability, boundedness, oscillation, and the periodic nature of these solutions.

The Impact of Thermal Radiation on Mixed Convective Unsteady Nanofluid Flow in a Revolving Vertical Cone

This study investigates the effects of an unsteady mixed convection nanofluid flow in a rotating vertical cone submerged in spinning nanofluid. Our analysis considered the impacts of heat flux, chemical reactions, and thermal radiation, with the thermal and concentration Biot numbers serving as constraints at the boundary. The governing unsteady and coupled partial differential equations were solved through appropriate similarity transformations, addressing the nonlinear terms inherent in these equations. The spectral quasi-linearisation method (SQLM) was employed to solve the higher-order nonlinear differential equations. This study elucidates and assesses the impact of diverse physical constraints and parameters through the use of graphical representations. Notably, the temperature distribution of the liquefied substance was intensified as the thermal and solutal Biot numbers increased.

Heat and mass transfer phenomena for fractal interfacial layer model on metallic-nanomaterials nanofluids flow with chemical reaction effect

This study presents a comprehensive investigation into the intricacies of single-phase simulation concerning the behavior of fractal interfacial magnetized metallic chemical-reactive nanoparticles within fluid systems. The research focuses on unraveling the multifaceted phenomena associated with heat and mass transfer in complex nanoscale environments. To achieve this, we employ numerical techniques to address the intricate fluid flow problems that arise due to the presence of these nanoparticles. Fractal complexities are linked to critical phenomena, such as mixing and mass transport, at the turbulent/non-turbulent interface of shear layers and boundary layers. This study primarily concentrates on the phenomena of heat and mass transfer, with a focus on estimating thermal conductivity models that have significant impacts on the flow of metallic nanofluids under convective conditions. The combination of a high thermal conductivity model with a novel interfacial fractal theory reveals a substantial effect on heat and mass transfer enhancement. Furthermore, this research explores hybrid types of porous channel geometries under the combined influence of magnetohydrodynamics and chemical reaction mechanisms in the presence of metallic nanoparticles. A comprehensive numerical analysis of higher-order nonlinear differential equations is conducted, which exposes the flow field’s behavior concerning momentum, energy, and mass transfer. These insights are envisioned through single-phase simulations of nanofluids. The influence of the nanoparticles’ radius and diameter is found to play a significant role in thermal performance, with nanoparticles ranging from 1% to 8% exhibiting a notable impact on thermal conductivity. Highlights A new mathematical relation of mass, as well as heat transfer for metallic-nanomaterial being affected by thermal conductivity (T.C.), has been established. The important role of the physical number is observed under the effect of T.C. and heat and mass transfer through orthogonal porous surfaces. An injection/suction, as well as an expansion/contraction phenomenon related to the porous surface is observed.

Global attractivity of a higher order nonlinear difference equation with decreasing terms

In the present paper, we further study the asymptotical behavior of the following higher order nonlinear difference equation x ( n + 1 ) = a x ( n ) + b f ( x ( n ) ) + c f ( x ( n − k ) ) , n = 0 , 1 , … where a , b and c are constants with 0 < a < 1 , 0 ≤ b < 1 , 0 ≤ c < 1 and a + b + c = 1 , f ∈ C [ [ 0 , ∞ ) , [ 0 , ∞ ) ] with f ( x ) > 0 for x > 0 , and k is a positive integer, which has been recently studied in: On global attractivity of a higher order difference equation and its applications [Electron. J. Qual. Theory Diff. Equ. 2022, No. 2, 1–14]. We obtain some new sufficient conditions for the global attractivity of positive solutions of the equation, and show the applications of these results to some population models.

Novel analytical superposed nonlinear wave structures for the eighth-order (3+1)-dimensional Kac-Wakimoto equation using improved modified extended tanh function method

<p>Higher-order nonlinear partial differential equations, such as the eighth-order Kac-Wakimoto model, are useful for studying wave turbulence in fluids, where energy transfers across a range of wave numbers. This phenomenon is observed in oceanographic research involving sea surface and internal waves, where intricate multi-dimensional interactions play a crucial role. In this work, we use the improved modified extended tanh function method for the first time to extract the exact solutions of the eighth-order (3+1)-dimensional Kac-Wakimoto equation, which describes the dynamics of fields and the structure of solutions in various physical and mathematical contexts. The proposed method is simple and quick to execute, and it offers more innovative solutions than other methods. As a consequence, through the donation of suitable assumptions for the parameters, some new solutions for dark and singular soliton, as well as Jacobi elliptic, exponential, hyperbolic, and singular periodic forms, are developed. Furthermore, to enhance understanding, graphical representations of certain solutions are included.</p>

MHD flow of a rotating micropolar fluid between two plates by using DTM

The primary purpose of this work is to investigate the influence of a magnetic field on a micropolar fluid, which is rotating between two horizontally arranged parallel plates. A uniform magnetic field acts perpendicularly to the porous plate, which consumes fluid with a steady suction velocity. The obtained higher-order nonlinear partial differential equations (PDEs) are converted into nonlinear ordinary differential equations (ODEs) by using similarity transformation. A semi-analytical methodology has been applied to obtain the result to the problem. The influence of various flow parameters of the flow field like coupling parameter, viscosity parameter, Reynolds number, porosity parameter, rotation parameter and magnetic field parameter has been examined and explained by graphs. A resemblance between the results obtained by solving by the differential transform method (DTM) and the outcomes of the former research works has also been displayed which shows the correctness and efficiency of the used analytical method. The numerical values of the coefficient of skin friction at the lower plate are also displayed in tabular form and compared with the literature.

Computational analysis of melting radiative heat transfer for solar Riga trough collectors of Jeffrey hybrid-nanofluid flow: A new stochastic approach

Current investigation deals with the melting heat transfer for the Jeffrey hybrid-nanofluid flow in parabolic trough solar collectors through Darcy Forchheimer porous media over a variable thick vertical elongation Riga surface under the effect of solar radiation. The impacts of viscous dissipation and Joule heating are also investigated. Equations governing the Jeffrey’s hybrid nanofluid flow are higher-order non-linear partial differential equations (PDEs). These governing PDEs are transformed into the non-linear ordinary differential equations (ODEs) by introducing appropriate similarity transformations and dimensionless parameters. Runga Kutta’s fourth-order numerical scheme is implemented with the shooting technique to solve coupled higher-order ODEs. Results for velocity profile, temperature profile, drag coefficient and Nusselt number are discussed for various influential parameters. Artificial neural networking is also performed to predict Nusselt numbers in different cases and scenarios. The Artificial Neural Network gives the desired outputs with the highest possible accuracy. It is observed that the temperature profile rises with increase in Hartmann number, porosity parameter, and Forchheimer number. However thermal profile worsens with escalates in the exponential index. Radiant energy of the Sun is a renewable energy source available in considerable amounts in our living environment. A parabolic trough solar collector is an efficient solar collector that stores the concentrated incoming radiant energy from the Sun to fulfill the requirement of high temperatures in thermal energy storage systems. It has various applications in solar-powered appliances for cooking and air conditioning, solar-powered systems for treating wastewater, photovoltaic lighting, solar-powered cars, aircraft, etc.

Non-Newtonian Slippery Nanofluid Flow Due to a Stretching Sheet Through a Porous Medium with Heat Generation and Thermal Slip

The domains of engineering, electrical, and medicine all have a significant demand for nanofluids. Applications for nanofluid flow include electronic device storage, industrial cooling and heating frameworks, and associated medicinal management information systems. Nanofluids are utilized generally as coolants in heat exchangers such as thermostats, electronic cooling systems, and radiators due to their enhanced thermal characteristics. This study aims to explain the mixed convection phenomenon’s applications on the thermal impact of Maxwell nanofluid. The mass diffusivity is supposed to be a function of concentration, whereas the thermal conductivity and viscosity of Maxwell nanofluid are assumed to be functions of temperature. It is recommended to consider the additional thermal effects of thermal slip, magnetic fields, and heat generation phenomena. The fluid flow motion was caused by the vertically stretched sheet. The dimensionless formulation of the suggested physical model is shown by the suitable variables interacting. The shooting approach is used in the numerical simulations, and it is based on lowering higher-order nonlinear differential equations to first-order. The slip velocity and the magnetic parameters have a direct impact on the local skin friction coefficient and velocity, as indicated by the research findings. Also, the increase in values of the Maxwell parameter, porous parameter, and viscosity parameter leads to the enhancement of temperature distribution, while the decline in velocity distribution can be attributed to the same factors. A comparison is also made with the results described in the literature that is currently available, and a superb agreement is discovered.

Numerical investigation of the magnetohydrodynamic hybrid nanofluid flow over a stretching surface with mixed convection: A case of strong suction

The present work explores the physical aspects of the alumina and silver nanoparticles on the magnetohydrodynamic (MHD) flow of mixed convection micropolar hybrid nanofluid with ethylene glycol + water [Formula: see text] base fluid via stretching surface embedded in a porous medium. A strong magnetic field is employed normally in the flow direction. The behavior of the suction on the presented flow analysis is discussed strongly. Heat transport phenomena are analyzed. The current model’s mathematical modeling is based on higher-order nonlinear partial differential equations, which are then translated into higher-order nonlinear ordinary differential equations using appropriate similarity transformations. The modeled higher-order nonlinear ordinary differential equations are solved using NDSolve technique. The physical significance of the different flow parameters on the velocity, microrotation, and temperature profiles of the hybrid nanofluid are described in a graphical form. In a tabular form, the skin friction coefficients for nanofluid and hybrid nanofluid against various flow parameters are calculated. Some important results from this investigation are demonstrated that the velocity of the hybrid nanofluid is higher for the stretching ratio parameter and it is detected that the suction parameter enhanced the microrotation profile of the hybrid nanofluid. From the comparison, it is noted that the velocity, microrotation, and temperature of the hybrid nanofluid are higher as compared to the velocity, microrotation, and temperature of the alumina nanofluid and silver-nanofluid.

Mechanical characteristics of MHD of the non-Newtonian magnetohydrodynamic Maxwell fluid flow past a bi-directional convectively heated surface with mass flux conditions

In engineering and manufacturing industries, stretching flow phenomena have numerous real-world implementations. Real-world applications related to stretched flow models are metalworking, crystal growth processes, cooling of fibers, and plastics sheets. Therefore, in this work, the mechanical characteristics of the magnetohydrodynamics of the non-Newtonian Maxwell nanofluid flow through a bi-directional linearly stretching surface are explored. Brownian motion, thermophoresis, and chemical reaction impacts are considered in this analysis. Additionally, thermal convective and mass flux conditions are taken into consideration. The mathematical framework of the existing problem is constructed on highly non-linear partial differential equations (PDEs). Suitable similarity transformations are used for the conversion of partial differential equations into ordinary differential equations (ODEs). The flow problem is tackled with the homotopy analysis method, which is capable of solving higher-order non-linear differential equations. Different flow profiles against various flow parameters are discussed physically. Heat and mass transference mechanisms for distinct flow factors are analyzed in a tabular form. The outcomes showed that both primary and secondary velocities are the declining functions of magnetic and Maxwell fluid parameters. The heat transfer rate rises with the cumulative values of the Brownian motion and thermal Biot number. In addition, the mass transfer rate decreases with the rising Schmidt number, Brownian motion parameter, and chemical reaction parameter, while it increases with the augmenting thermophoresis parameter. It has been highlighted that streamlines in the current work for Maxwell and Newtonian models are in fact different from one another.

Computation of Cattaneo-Christov heat and mass flux model in Williamson nanofluid flow with bioconvection and thermal radiation through a vertical slender cylinder

Nanofluids are in high demand in the medical, electrical, and engineering fields. Nanofluid flow is used in a variety of applications including industrial cooling and heating frameworks, hyperthermia, electronic device batteries, and related pharmaceutical administration systems. Because of their improved thermal properties, nanofluids are commonly used as coolants in heat exchangers also including heating systems, electronic cooling systems, and radiators. Many scientists have studied heat transfer via a cylinder. The current study uses a vertical thin cylinder to scrutinize the effects of activation energy, motile microorganisms, bioconvection, and WU's slip on the Williamson nanofluid. Brownian motion, thermal radiation, Cattaneo-Christov heat, mass flow, and thermophoresis are all considered. Using appropriate similarity variables, the structures of PDEs are translated into the structure of ordinary differential equations. Using the well-known shooting strategy (Bvp4c) in the mathematical program MATLAB, higher-order nonlinear differential equations (DEs) are reduced to first-order. The effects of important flow parameters are shown graphically and numerically. It is noted that the momentum distribution field is upsurged for mixed convection parameters and decreased for the fluid parameter. The thermal profile is declining for the Prandtl number while increasing for the thermal radiation parameter and heat source-sink parameter. It is also observed that the concentration distribution field decreased for the Lewis number and concentration relaxation parameter while boosted up for larger magnitude of activation energy and thermophoresis parameter. It is analyzed that the microorganisms profile is upsurged for bioconvection Rayleigh number while declined for Peclet number. A comparison with the reported results from available literature is also done, and an excellent agreement is found. The discovery has far-reaching ramifications in the fields of medicine, modern aircraft technology, and power generation. In this study, we combine metallic nanoparticles with the base fluid water. Metallic nanoparticles offer a wide range of uses in everyday life, including catalysts, biosensors, medicine, and medication delivery, soil pollutant removal, electronics, coatings, packaging, semiconductors, bioengineering, paints, telecommunications, cosmetics, automobiles, and water treatment.

Micro-polar nanofluid in the presence of thermophoresis, hall currents, and Brownian motion in a rotating system

This study examined a rotating system by a micro-polar nanofluid between two parallel plates in the presence of magnetic and electric fields. The flow study has been performed in a steady-state. The governing equations of the present problem are transformed into nonlinear and coupled equations with appropriate similarity variables. The impacts of the Nusselt number, skin friction, and Sherwood number on temperature, velocity, and concentration distribution have been discussed. This research has mainly investigated the effects of the rotation, Brownian motion, thermophoresis analysis, and Hall current of micro-polar nanofluid. Results demonstrate for weak concentration [Formula: see text] and strong concentration [Formula: see text], that Nusselt number (Nu) increased with higher value of [Formula: see text], [Formula: see text] and decreased when [Formula: see text], Sc and [Formula: see text] increased. Also, by increasing the [Formula: see text] and Sc numbers, the temperature profile is decreased and increased by increasing [Formula: see text], [Formula: see text], and [Formula: see text]. In addition, at a higher value of the [Formula: see text], the velocity profile in the [Formula: see text]-direction increased because of increasing the fluid motion and by fast ionization, increasing the [Formula: see text] parameter raised the velocity profile in the [Formula: see text]-direction. There was a decrease in the velocity profile in the [Formula: see text]-direction and the micro-rotation velocity profile. Our results show that the method used is very efficient and practical for solving this category of coupled equations, and that the solution of higher-order nonlinear differential equations in engineering is very consistent. Also, by comparing the obtained results with the previous results, the obtained values differ by about 6%.

Global attractivity of a higher order nonlinear difference equation with unimodal terms

In the present paper, we study the asymptotic behavior of the following higher order nonlinear difference equation with unimodal terms \[x(n+1)= ax(n)+ bx(n)g(x(n)) + cx(n-k)g(x(n-k)), \quad n=0, 1, \ldots,\] where \(a\), \(b\) and \(c\) are constants with \(0\lt a\lt 1\), \(0\leq b\lt 1\), \(0\leq c \lt 1\) and \(a+b+c=1\), \(g\in C[[0, \infty), [0, \infty)]\) is decreasing, and \(k\) is a positive integer. We obtain some new sufficient conditions for the global attractivity of positive solutions of the equation. Applications to some population models are also given.

Analysis of a Squeezing Flow of a Casson Nanofluid between Two Parallel Disks in the Presence of a Variable Magnetic Field

The present article deals with the MHD flow of a Casson nanofluid between two disks. The lower disk was fixed as well as permeable. The upper disk was not permeable, but it could move perpendicularly up and down toward the lower disk. Titanium dioxide was selected as nanoparticles and water as a base fluid. The governing higher-order nonlinear partial differential equations were transformed into a set of nonlinear ordinary differential equations by using similarity transformation. The differential transform method (DTM) was applied to solve the nonlinear ODEs. The nature of the velocity profiles for the different values of the suction injection parameter, the squeeze number, the Casson fluid parameter, and the volume fraction parameter of the nanofluid are pictorially discussed in this paper. The coefficient of skin friction was tabulated for the novelty of the research. The comparison of the results was determined by the DTM and the numerical methods. The profile values were also compared with the literature work and found to agree. This comparative study proves the accuracy and efficiency of the method. It is concluded from this research that the flow properties behave oppositely for all parameters during suction and injection.