Nonlinear Coupled Equations Research Articles

Formation of Optical Fractals by Chaotic Solitons in Coupled Nonlinear Helmholtz Equations

Formation of Optical Fractals by Chaotic Solitons in Coupled Nonlinear Helmholtz Equations

Non-similar solution of Casson fluid flow over a curved stretching surface with viscous dissipation; Artificial neural network analysis

PurposeThe main focus is to provide a non-similar solution for the magnetohydrodynamic (MHD) flow of Casson fluid over a curved stretching surface through the novel technique of the artificial intelligence (AI)-based Lavenberg–Marquardt scheme of an artificial neural network (ANN). The effects of joule heating, viscous dissipation and non-linear thermal radiation are discussed in relation to the thermal behavior of Casson fluid.Design/methodology/approachThe non-linear coupled boundary layer equations are transformed into a non-linear dimensionless Partial Differential Equation (PDE) by using a non-similar transformation. The local non-similar technique is utilized to truncate the non-similar dimensionless system up to 2nd order, which is treated as coupled ordinary differential equations (ODEs). The coupled system of ODEs is solved numerically via bvp4c. The data sets are constructed numerically and then implemented by the ANN.FindingsThe results indicate that the non-linear radiation parameter increases the fluid temperature. The Casson parameter reduces the fluid velocity as well as the temperature. The mean squared error (MSE), regression plot, error histogram, error analysis of skin friction, and local Nusselt number are presented. Furthermore, the regression values of skin friction and local Nusselt number are obtained as 0.99993 and 0.99997, respectively. The ANN predicted values of skin friction and the local Nusselt number show stability and convergence with high accuracy.Originality/valueAI-based ANNs have not been applied to non-similar solutions of curved stretching surfaces with Casson fluid model, with viscous dissipation. Moreover, the authors of this study employed Levenberg–Marquardt supervised learning to investigate the non-similar solution of the MHD Casson fluid model over a curved stretching surface with non-linear thermal radiation and joule heating. The governing boundary layer equations are transformed into a non-linear, dimensionless PDE by using a non-similar transformation. The local non-similar technique is utilized to truncate the non-similar dimensionless system up to 2nd order, which is treated as coupled ODEs. The coupled system of ODEs is solved numerically via bvp4c. The data sets are constructed numerically and then implemented by the ANN.

Quantitative analysis of heat and mass transfer in MoS2-Al2O3/EG hybrid flow between parallel surfaces with suction/injection by numerical modeling of HPM method

This description focuses on how the magnetic field affects mass and heat transfer in a hybrid nanofluid (Hnf) between two parallel, rotating plates. By dispersing aluminum oxide (Al2O3) and molybdenum disulfide (MoS2) nanoparticles (NPs) in ethylene glycol (EG), a hybrid nanofluid (Hnf) is created. This research aims to analyze the heat and mass transfer characteristics in the flow of a hybrid nanofluid (MoS2-Al2O3/EG) between two rotating parallel plates under the influence of a magnetic field. Furthermore, the statistical technique of response surface methodology (RSM) has been employed to optimize the parameters of velocity, temperature, and concentration of the nanofluid within the flow region bounded by the rotating plates. Dimensionless differential equations have been calculated and checked using the Homotopy perturbation method. This study introduces a novel approach by utilizing the RSM method to identify optimal points for velocity and temperature parameters of nanofluids between two stretching plates for the first time. Additionally, the article innovatively applies the exact HPM method to validate dimensionless linear and non-linear coupling equations. As the Reynolds number and the suction/injection coefficient of nanofluids flowing between two plates under tension increase, the results indicate a decrease in the velocity field. This decrease in velocity field can be attributed to the reduction in fluid diffusion as viscous forces diminish with varying Reynolds numbers. The ideal temperature distribution for nanofluids flowing between two parallel plates occurs when they are uniformly dispersed at the midpoint between them. As the distance from the initial point of nanofluid entry to the end of the plates increases, along with the vertical distance from the bottom plate, the temperature gradient diminishes, reducing the thickness of the thermal boundary layer. The velocity gradient and the rate of heat flux transfer between the nanofluid and plate rise by 34 % when the volume percentage is raised from 1 % to 5 %.

An investigation of heat transfer and optimization of entropy in bio-convective flow of Eyring-Powell nanomaterial with gyrotactic microorganisms

Entropy generation in nanofluid flows is a critical parameter that influences the performance, efficiency, and sustainability of thermal systems. Understanding and optimizing entropy creation can lead to noteworthy advancements in numerous engineering applications, from renewable energy systems to industrial processes and biomedical equipments. The purpose of this article is to examine the entropy produced in the bioconvection flow of Eyring-Powell nanomaterial with gyrotactic microorganisms. The mathematical equations representing the flow are modeled considering the flow towards a porous surface of a cylinder. Together with the effects of the governing parameters, the mass and heat transfer components of the problem are discussed. The thermal effects of different natures are used in a new way. Overall, entropy creation is designed with regard to the second law of thermodynamics. Activation energy, chemical reaction, thermal radiation, and internal friction force effects are accounted for the development of the mathematical model. Coupled non-linear dimensional equations have been altered into ordinary differential equations (ODEs) and then treated using the MATLAB Finite Difference Method (FDM). Consequence of diverse sundry parameters on entropy production, thermal field, mass concentration profile, Bejan quantity, and motile density of microorganisms is deliberated via plots. Through tables, engineering quantities are analyzed. According to the findings, the velocity profile rises as the curvature and Eyring-Powell fluid parameters rise, while it falls when the magnetic parameter is enhanced. Additionally, it is noted that as the curvature variable enhances, the rate of heat transfer and skin friction coefficient decays.

Solitary Wave Solutions to the General Class of Nonlocal Nonlinear Coupled Wave Equations

In this paper, we study a general class of nonlocal nonlinear coupled wave equations that includes the convolution operation with kernel functions. For appropriate selections of the kernel functions, the system becomes well-known nonlinear coupled wave equations, for instance Toda lattice system, coupled improved Boussinesq equations. A numerical scheme is proposed for the solitary wave solutions of the system using the Pethiashvili method. Using the different kernels, the validity of the numerical method has been tested.

Ferromagnetic and ohmic effects on nanofluid flow via permeability rotative disk: significant interparticle radial and nanoparticle radius

The significance of interparticle spacing and nanoparticle radius for the case of single-phase nanofluid flow has often been neglected. Tremendous applications of this phenomenon can be witnessed in different fields, especially in electron microscopes, heat exchange processes, and many others. This research highlights this vital aspect of Ohmic heating in nanofluid flow over a spinning disk. To ensure the novelty, a ferromagnetic nanoparticle (Manganese ferrite) has been incorporated to examine interparticle spacing and particle radius to explore the features of heat transfer. The ferromagnetic nanofluids are vital in carriers for drug delivery systems, in cancer treatment, design of systems for hyperthermia therapy, in microfluidic devices used for chemical synthesis, etc. The quantiles of dimensional equations are converted into dimensionless ones by adopting similarity transformations and to solve highly coupled nonlinear equations numerically, built-in bvp5c MATLAB tool is utilized. The effect of a few revealed factors, the velocity and temperature distributions, are examined via visualization. Furthermore, streamlined plots are also visualized. The outcomes produced showed excellent agreement with those made in the literature in the same direction by assuming some exceptional cases on different gradients. Further, the outstanding results are reported as; the permeability of the surface produces the suction velocity, and the enhanced suction velocity attenuates the fluid velocity in either of the case of pure and nanofluid. The increase in thermal radiation boosts up the heat transfer rate whereas the augmentation in the Eckert number retards it significantly.

The large strain snap-through effect in free torsion of highly elastic soft thin-walled tubes with exact closed-form solutions

It is found for the first time that, as the applied torque attains a critical value, a freely twisted highly elastic tube may jump from one to another state over large strain range. Unlike usual approximate buckling analyses of thin shell structures with Hooke’s law for small strain, such large strain snap-through effect needs to be studied with a large strain model that can accurately simulate nonlinear elastic behaviors of soft materials. As such, nonlinear coupling issues have to be treated both in large strain kinematics and in hyper-elastic constitutive formulation.It appears that exact results would be rare even for buckling analyses with small strain but large deflections. Here, the above coupling issues are treated with a novel and decoupled approach. To this end, explicit and decoupled forms of the dual potentials for hyper-elastic stress–strain and strain–stress relationships are constructed with well-designed invariants of the Hencky strain and the deviatoric Cauchy stress and, hence, multiple data sets from benchmark tests are accurately matched with these forms in a decoupled sense. It is then shown that the nonlinear coupling equations for large torsion of elastic thin tubes can be worked out to produce exact closed-form solutions for the average radius ratio, the axial length ratio and the thickness ratio as well as the applied torque. With these solutions, the large strain snap-through effect is disclosed with an explicit criterion. Furthermore, numerical examples are provided with comparison to test data and, in particular, the large torsion response features with the snap-through effect are discussed with reference to possible correlations to certain symptoms in vascular abnormalities of blood vessels, etc.

Simplified expression for transverse mode instability threshold in high power fiber lasers.

In this work, we propose an analytical expression for calculating the transverse mode instability (TMI) threshold power, which clearly shows the role of various fiber parameters and system parameters. The TMI threshold expression is obtained by solving the heat conduction equation and the nonlinear coupling equation using the fundamental mode fitted by Gaussian functions. The calculation results of the proposed TMI threshold expression are consistent with the experimental phenomena and simulation results from the well-recognized theoretical model. The influence of some special parameters on the TMI threshold and the power scaling is also investigated. This work will be helpful for fiber design and TMI mitigation of high-power fiber lasers.

Theoretical and experimental research on a Quasi-Zero-Stiffness-Enabled nonlinear piezoelectric energy harvester

Piezoelectric material provides a convenient and efficient avenue to convert ambient vibration energy into electrical energy to power wireless sensor network nodes of Internet of Things. However, the design of effective ultralow-frequency and low-amplitude harvesters using piezoelectric materials remains a challenge. To address this challenging problem, this paper devises a quasi-zero-stiffness-enabled (QZSE) piezoelectric energy harvester based on a geometrical nonlinear structure. The basic configuration and operation principles of the QZSE piezoelectric energy harvester ware elaborated firstly, then the nonlinear electromechanical coupling equation is derived, whose solution informs on both the dynamic and electrical responses of the harvester. The electromechanical-coupled equations are solved with the aid of the fourth-order Runge-Kutta algorithm. The energy harvesting performance is then investigated with respect to the parameter effects of excitation, damping and proof mass. For validating numerical results and demonstrating the advantages of the proposed design, a prototype is fabricated and experimentally tested. The results confirm the ability of the QZSE piezoelectric energy harvester in producing effective vibration-to-electricity energy conversion in ultralow frequency range. This study provides new impetus to energy harvesting from ultralow frequency ambient vibration using piezoelectric materials.

On the Existence and Asymptotic Behavior for a Strongly Damped Nonlinear Coupled Petrovsky-Wave System

In this paper, we consider the initial-boundary value problem for a class of nonlinear coupled wave equation and Petrovesky system in a bounded domain. The strong damping is nonlinear. First, we prove the existence of global weak solutions by using the energy method combined with Faedo-Galarkin method and the multiplier method. In addition, under suitable conditions on functions gi(⋅), i = 1, 2 and a(⋅), we obtain both exponential and polynomial decay estimates. The method of proofs is direct and based on the energy method combined with the multipliers technique, on some integral inequalities due to Haraux and Komornik.

Blow-up dynamics in nonlinear coupled wave equations with fractional damping and external source

<p>This study focused on a coupled nonlinear wave equation system featuring fractional damping and polynomial source terms within a bounded domain. We demonstrated that, under specific conditions, the solutions exhibited blow-up in a finite time-frame.</p>

On Two Types of Stability of Solutions to a Pair of Damped Coupled Nonlinear Evolution Equations

On Two Types of Stability of Solutions to a Pair of Damped Coupled Nonlinear Evolution Equations

Long-Time Limit of Nonlinearly Coupled Measure-Valued Equations that Model Many-Server Queues with Reneging

Long-Time Limit of Nonlinearly Coupled Measure-Valued Equations that Model Many-Server Queues with Reneging

Self-similar collapse in a circular magnetic field and electron jets by hybrid transverse plasmon

Based on a set of nonlinear coupling equations describing the interaction of the HF field, self-generated magnetic field, and ion-acoustic wave, the dispersion relation of hybrid transverse plasmon under the circular self-generated magnetic field is obtained. The analysis of magnetic modulation instability shows that the circular self-generated magnetic fields have the tendency to self-similar collapse which makes the electron escape along the axial region and form a collimated jet. In addition, the velocity of the electron jets is calculated, and the result is consistent with experimental observation. The present research may be applied to understand the dynamic process of electron jets produced in laser plasma.

Coupled Fixed Point and Hybrid Generalized Integral Transform Approach to Analyze Fractal Fractional Nonlinear Coupled Burgers Equation

In this manuscript, the nonlinear Burgers equations are studied via a fractal fractional (FF) Caputo operator. The results of coupled fixed point theorems in cone metric space are used to discuss the uniqueness of solution to the proposed coupled equations. The solution of the proposed equation is computed via Natural transform associated with the Adomian decomposition method (NADM). The acquired results are graphically presented for some values of fractional order and fractal dimensions. The accuracy and consistency of the applied method is verified through error analysis.

Heat and mass transfer analysis of non-Newtonian radiative nanofluid flow driven by the combined action of peristalsis and electroosmosis

The present numerical investigation uncovers the flow, heat, and mass transfer of ethylene glycol-based nanofluid (BN-EG) led by the combined impacts of peristaltic pumping and electroosmosis. The thermal conductivity of carrier liquid is deemed to change with temperature. Further, the features of electric field, Ohmic heating, radiative heat flux, mixed convection, and magnetic field are taken. Mathematical modeling is conducted under the assumption of lubrication theory. The nonlinear-coupled equations are addressed numerically by implementing Shooting method. The effective results of all physical constraints linked with the flow model are vigilantly studied and highlighted through various curves. The observations obtained from current analysis reveal that temperature augments with higher electroosmotic parameter. However, the Joule heating parameter increases the rate of heat transmission, while a lower rate of heat transfer is perceived for the radiation parameter. An increment in Helmholtz–Smoluchowski velocity increases the velocity profile. Pressure gradient rises in a forward way when electrical field favors peristaltic flow. Concentration of nanomaterial considerably declines when concentration Grashoff number is assigned higher values.

Fluid-Fluid Interaction Problems at High Reynolds Numbers: Reducing the Modeling Error with LES-C

We consider a fluid-fluid interaction problem, where two flows (with high Reynolds numbers for one or both of these flows) are coupled through a joint interface. A nonlinear coupling equation, known as the rigid lid condition, creates an extra level of difficulty, typical for atmosphere-ocean problems. We propose a novel turbulence model, NS- -C, from the recently introduced family of LES-C (large eddy simulation with correction) models. Combining it with the so-called geometric averaging (GA) partitioning method, we obtain the NS- -C-GA model that is shown to possess several key properties. First, the preexisting solvers for the subdomains can be used, which is critical, e.g., for atmosphere-ocean applications. Second, the LES-C turbulence models use defect correction to efficiently reduce the modeling error of the corresponding LES models; we demonstrate numerically that the NS- -C model outperforms its LES counterpart, the NS- model. It has also been shown recently that it is favorable for an LES model to have the nonfiltered velocity in the interface terms. The NS- -C-GA model possesses this important property; we also show it to be stable and have optimal convergence properties.

Global well-posedness and exponential decay rates of the strong solutions to the two-dimensional full compressible magnetohydrodynamics equations with vacuum in some class of large initial data

In this paper, the initial-boundary value problem for the two-dimensional viscous, compressible, and heat conducting magnetohydrodynamics equations with vacuum is considered. When all the coefficients of viscosity, heat conductivity and magnetic diffusivity are constants, first, we show that for general initial data and any time T>0, the local strong solution in the sense of Sobolev norms will never blow-up provided that the density is bounded from above and the temperature belongs to space L∞(0,T;L2)∩L2(0,T;L∞). Second, based on the above blow-up criterion and the delicate analysis of the nonlinear strong coupling equations, we establish the global existence of strong solutions when the initial mass of the fluid and the initial energy of the magnetic field (i.e., ‖ρ0‖L1 and ‖H0‖L2) are suitably small, where the refined Zlotnik inequality and the decoupling of the full compressible MHD equations play an essential role in proving the uniform upper bound of the density and the higher-order a priori uniform in time estimates, respectively. It is worth mentioning that the initial velocity and temperature can be large. Moreover, we obtain exponential decay under the H1-norm of (u,θ,H) and L2-norm of (ρu˙,ρθ˙,Ht), where the long-time behaviors of the higher-order derivative of velocity are given and it is different from the known results.

Dynamic Analysis of the High Speed Train–Track Spatial Nonlinear Coupling System Under Track Irregularity Excitation

Track irregularity is the main factor that affects the safety and comfort of high-speed railway. It is important to study the influence of the track irregularity on the safety service of high speed train–track coupling system. In this paper, a vehicle–track spatial nonlinear coupled system dynamics model is developed with FEM. The model includes vehicle subsystem and track subsystem, which are coupled by wheel–rail interface. The vehicle subsystem is a 31-DOF spatial vehicle model, and the track subsystem is a three-dimensional slab track model composed of the rail, rail pad and track slab. Considering wheel tread profile and rail head profile, the wheel–rail spatial contact geometry relationship is established, and the “Trace method” and the “Minimum distance method” are used to search for wheel–rail contact points. Considering the nonlinear contact between the wheel and the rail, a numerical approach is presented, where the “Trace method” is embedded into the algorithm for solving the vehicle–track nonlinear coupling dynamics equation by cross iteration, realizing searching the wheel–rail contact points and solving the vehicle–track nonlinear coupling equation simultaneously, which greatly improves the efficiency of numerical analysis. As application examples, the dynamic responses of a high speed train–track spatial nonlinear coupled system excited by the track V-shaped local profile irregularity and the track torsional irregularity are analyzed. The results show that the algorithm is effective in solving the vehicle–track spatial nonlinear coupled system dynamics problem, and the track irregularity has significant influence on the dynamic response of vehicle and track structure.

Unconditional Superconvergence Analysis of Energy Conserving Finite Element Methods for the Nonlinear Coupled Klein-Gordon Equations

Unconditional Superconvergence Analysis of Energy Conserving Finite Element Methods for the Nonlinear Coupled Klein-Gordon Equations