Nonlinear Differential Equations Research Articles

The spatial Whitham equation

The Whitham equation is a non-local, nonlinear partial differential equation that models the temporal evolution of spatial profiles of surface displacement of water waves. However, many laboratory and field measurements record time series at fixed spatial locations. To directly model data of this type, it is desirable to have equations that model the spatial evolution of time series. The spatial Whitham (sWhitham) equation, proposed as the spatial generalization of the Whitham equation, fills this need. In this paper, we study this equation and apply it to water-wave experiments on shallow and deep water. We compute periodic travelling-wave solutions to the sWhitham equation and examine their properties, including their stability. Results for small-amplitude solutions align with known results for the Whitham equation. This suggests that the systems are consistent in the weakly nonlinear regime. At larger amplitudes, there are some discrepancies. Notably, the sWhitham equation does not appear to admit cusped solutions of maximal wave height. In the second part, we compare predictions from the temporal and spatial Korteweg–de Vries and Whitham equations with measurements from laboratory experiments. We show that the sWhitham equation accurately models measurements of tsunami-like waves of depression, waves of elevation, and solitary waves on shallow water. Its predictions also compare favourably with experimental measurements of waves of depression and elevation on deep water. Accuracy is increased by adding a phenomenological damping term. Finally, we show that neither the sWhitham nor the temporal Whitham equation accurately models the evolution of wave packets on deep water.

Features of microorganism and two-phase nanofluid in a tangent hyperbolic Darcy-Forchhiemer flow induced by a stretching sheet with Lorentz forces

This work examines the two-dimensional tangent hyperbolic flow over a stretching sheet with a uniform magnetic field. The Buongiorno model is utilized to analyze and explain the spread of uneven coefficients in the presence of gyrotactic microorganisms. The concept of microorganisms and the resulting bioconvection enhance the stability of the nanoparticles. The impacts of thermal radiation, heat sources, convective heating, and chemical reactions are also evaluated. The suggested mathematical problem results in a nonlinear set of partial differential equations (PDEs), which are subsequently reduced to ordinary differential equations (ODEs) by applying the appropriate transformation. The resultant highly nonlinear ordinary differential equations (ODEs) are numerically solved using MATLAB's built-in package known as bvp4c. An in-depth investigation into the changes in the velocity field, the temperature profile, the concentration of nanoparticles profile, and the motile density profile is analyzed through graphs against various influencing parameters. Additionally, computations of engineering interest quantities such as skin friction, local Nusselt number, local Sherwood number, and molecular density are presented in both graphical and tabular formats for further examination. It has been explored that with greater values of the Weissenberg number, the fluid velocity upsurges when n<1, whereas the opposite behavior is noticed when n>1. It is also noted that an increment in the Peclet number decreases motile density for both dilatants n>1 and pseudoplastic n<1 fluids. The computed results are compared with existing literature in limiting cases and found good agreement.

Numerical simulation of an effective transform mechanism with convergence analysis of the fractional diffusion-wave equations

In the current study, we solve two very important mathematical models, such as the time fractional-order space-fractional telegraph and diffusion-wave equations using a reliable technique called the Adomian decomposition natural method (ADNM), which combines Adomian decomposition and natural transform. The diffusion wave equation describes the flood wave propagation, which is used in solving overland and open channel flow problems. For this reason, it is critical to fully understand and effectively solve the diffusion wave equations. Because telegraph equations are crucial for modeling and developing voltage or frequency transmission, they are widely used in physics and engineering. Furthermore, the designing process is greatly impacted by the uncertainty in the system parameters. For nonlinear ordinary differential equations based on the theorem of Banach fixed point, we provide existence and uniqueness theorem proofs. The present approach has been successfully used to explore exact solutions for time fractional-order and space fractional-order applications. The results show how effective and valuable the ADNM. This paper presents a methodology that will be used in future work to address similar nonlinear problems related to fractional calculus.

On the nonlinear dynamics of in-contact rigid bodies experiencing stick–slip and wear phenomena

In this paper, the dynamic behavior of one degree-of-freedom oscillator subject to stick–slip and wear phenomena at the contact interface with a rigid substrate is investigated. The motion of the oscillator, induced by a harmonic excitation, depends on the tangential contact forces, exchanged with the rigid soil, which are modeled through piecewise nonlinear constitutive laws, accounting for stick–slip phenomena due to friction as well as wear due to abrasion, already developed by the authors in a previous work. The nonlinear ordinary differential equations governing the problem are derived, whose solution is numerically obtained via a typical Runge–Kutta-based algorithm. The main target of this study is to analyze and discuss the strong nonlinear behavior, descending from the presence of stick–slip and wear phenomena, thus investigating the effect of the different interface modeling. In this framework, the analysis is carried out considering the whole evolution of non-smooth contact laws, starting from the virgin interface.

Kink Wave Phenomena in the Nonlinear Partial Differential Equation Representing the Transmission Line Model of Microtubules for Nanoionic Currents

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.

Nonlinear geometrically exact dynamics of hyperelastic pipes conveying fluid: Comparative study of different hyperelastic models

This work presents a comparative analysis of the nonlinear dynamic responses of fluid-conveying pipes modelled by four different phenomenological hyperelastic constitutive models, namely the Mooney-Rivlin model, the Neo-Hookean model, the Yeoh model, and the Ogden model. Based on the strain energy density functions and the exact geometric relation, the geometrically exact dynamic models corresponding to different hyperelastic contitutive models are established. Then the Galerkin's truncation is adopted to transfer the obtained nonlinear integral-partial differential equations into the nonlinear ordinary differential equations. Subsequently, linear dynamic study and finite difference method are taken to explore the natural frequencies and stabilities and nonlinear dynamic responses of the pipe, respectively. And the fitting coefficients for various models obtained from Treloar's experiment data are utilized in the numerical calculation. It is found the Hopf bifurcation and the limit cycle oscillations in the post-flutter status are influenced by the choice of the hyperelastic models, particularly under large flow velocity. However, hyperelastic constitutive models do not exert a significantly extreme influence on the results, which can be attributed to the relatively small strains even though large deformations occur in the pipe. Mathematically, it is proven that the symmetry observed in the pipe's displacement response stems from the symmetry of its cross-section and the inextensibility assumption of the centerline, with hyperelasticity having no effect on this symmetry.

Uncovering Hidden Patterns: Approximate Resurgent Resummation from Truncated Series

We analyze truncated series generated as divergent formal solutions of non-linear ordinary differential equations. Motivating the study is a specific non-linear, first-order differential equation, which is the basis of the resurgent formulation of renormalized perturbation theory in quantum field theory. We use the Borel–Padé approximant and classical analysis to determine the analytic structure of the solution using the first few terms of its asymptotic series. Afterward, we build an approximant, consistent with the resurgent properties of the equation. The procedure gives an approximate expression for the Borel–Ecalle resummation of the solution useful for practical applications. Connections with other physical applications are also discussed.

Effects of Arrhenius activation energy and melting heat transfer on Carreau fluid through a stretching sheet

ABSTRACT This article investigates the effect of activation energy on Carreau fluid flow in the presence of variable thermal conductivity, inclined magnetic field, melting heat transfer, and Soret and Dufour phenomena. A set of suitable similarity transformations is applied for transforming the governing partial differential equations (PDEs) considered for the physical phenomena into non-linear ordinary differential equations (ODEs). We obtained the results by solving the non-linear ODEs numerically which describe the behavior of the velocity, temperature, and concentration profiles of the fluid against the governing parameters and illustrated these graphically. The velocity distribution decreases for angle of inclination while it is increasing against higher melting parameter. The temperature distribution enhances with higher modified Dufour parameter and Prandtl number while it declines for thermal conductivity, activation energy, and melting parameters. The concentration distribution increases for melting parameter, Soret number, and activation energy parameter while decreasing for temperature difference parameter, Schmidt number, and reaction rate parameter. Skin friction, Nusselt and Sherwood numbers are evaluated numerically against the various governing parameters. Accuracy of the present work is illustrated and established through comparison carried out for skin friction versus magnetic parameter with existing results in the literature.

Significant statistical model of heat transfer rate in radiative Carreau tri-hybrid nanofluid with entropy analysis using response surface methodology used in solar aircraft

The recent advancement of technologies focuses on the use of solar-based heat radiation and nanotechnology. The primary source of heat is solar energy, which is obtained by absorbing sunlight. In addition, solar thermal planes, photovoltaic cells, and sun-based hybrid nanofluids all help to harness solar energy. Therefore, main concern of the study is to explored the two-dimensional Engine oil-based Carreau tri-hybrid nanofluid (SiO2 (silica dioxide), MOS2 (Molybdenum disulfide), Fe3O4(black iron oxide)) flow via a stretching sheet within solar wings. The flow phenomena of the non-Newtonian fluid enrich for the effects of radiant energy and the external heat source. By adopting similarity transfiguration, the dimensional partial main flow equations are changed into nonlinear dimensionless ordinary differential equations. The most effective and powerful tool of the semi-analytical method ADM (Adomain Decomposition Method) is utilized to solve the nanofluid flow problem. The uniqueness of the proposed study arises due to the analysis of entropy generation obtained by various irreversibility processes. The graphical illustration of outcome of various governing parameters on velocity, temperature, entropy terms and engineering quantities have been presented and theoretically analyzed. The findings reveal that The Carreau ternary hybrid nanofluids enhance the system's thermal conductivity for optimal temperature regulation in solar aircraft and better heat dissipation. The use of Carreau ternary hybrid nanofluids significantly reduces entropy generation, which implies a more reversible process, enhancing the overall efficiency of the energy conversion systems in solar aircraft. The heat flow of the system is accurately reflected by the chosen variables and their interactions, as demonstrated by the constructed model's strong statistical significance. For real-world applications, this improves the model's predictability and dependability.

Borel control and efficient numerical techniques to solve the Allen–Cahn equation governed by temporal multiplicative noise

In this manuscript, we investigate a stochastic version of the Allen–Cahn equation, which is a nonlinear partial differential equation from mathematical physics. The stochastic model considers temporal multiplicative noise in the form of the derivative of the standard Wiener process. The existence and the uniqueness of solutions for this stochastic model is established rigorously using the theory of distributions. As a corollary from these analytical results, some a priori optimal estimates for the solutions of this model are constructed. In this work, we develop reliable numerical schemes which possess similar features as those of the solutions for the analytical model. Moreover, we establish mathematically the von Neumann stability and the consistency for the schemes proposed in this paper. Both the analytical and numerical results derived in this work are computationally verified through some simulations.

Numerical Heat Transfer Analysis of Carreau Nanofluid Flow over Curved Stretched Surface with Nonlinear Effects and Viscous Dissipation.

The article presents a comprehensive study that thoroughly assesses the effects of Carreau fluid flow over a curved stretch surface. The study meticulously considers various factors, including nonlinear thermal radiation, convective boundary conditions, viscous dissipation, thermophoresis, Brownian motion, and nonlinear mixed convection. By using similarity variables, the problem's governing equations are transformed from nonlinear partial differential equations to nonlinear ordinary differential equations, and then the shooting method is applied to obtain the results. The study also rigorously examines the effect of modifying flow factors on velocity, temperature, and concentration in shear-thickening situations through graphical assessment. Furthermore, according to the corresponding values, a table is provided with data on skin friction, Nusselt, and Sherwood numbers. This study offers new perspectives on the complex mechanisms of Carreau fluids flowing over curved surfaces, with potential applications in materials engineering and fluid mechanics.

Stability and boundedness criteria for certain second-order nonlinear neutral stochastic functional differential equations

This paper presents stochastic stability and stochastic boundedness to certain second-order nonlinear neutral stochastic differential equations. The second-order differential equation is weakened to a neutral stochastic system of first-order equations and used together with a second-order quadratic function to obtain perfect Lyapunov-Krasovskii functional. This functional is adapted and applied to obtain criteria on the nonlinear functions to ensure novel results on stochastic stability and stochastic asymptotic stability of the zero solution. Furthermore, when the forcing term is nonzero, fresh results on stochastic boundedness and uniform stochastic boundedness of solutions are obtained. The results of this paper are original, new, essentially improving, complementing, and simplifying several related ones in the literature. Two special cases of the theoretical results are supplied to demonstrate the applicability of the hypothetical results.

Phase trajectories, chaotic behavior, and solitary wave solutions for (3+1)-dimensional integrable Kadomtsev–Petviashvili equation in fluid dynamics

This paper focuses on finding exact soliton solutions and further examining the qualitative characteristics of these solutions for a nonlinear partial differential equation known as the (3+1)-dimensional integrable Kadomtsev–Petviashvili equation that portrays a unique dispersion effect. The (3+1)-dimensional integrable Kadomtsev–Petviashvili equation models the evolution of weakly nonlinear dispersive waves in three spatial dimensions “x,y,z” and time “t”. It balances nonlinearity and dispersion, leading to solitary wave solutions that are stable and localized. This equation is crucial in understanding complex wave phenomena in fields like fluid dynamics, plasma physics, and optics. Initially, the traveling wave transformation is employed to convert the nonlinear partial differential equations into the non-linear ordinary differential equations. Then, we use the Riccati equation approach to find solitary wave solutions for the resulting ordinary differential equation. To better comprehend the physical significance of these derived solutions, we illustrate them using various visual representations. Diverse novel collections of solitary wave solutions are obtained such as bright solitons, dark solitons, and dark singular solitons. Second, we convert the nonlinear ordinary differential equation into a linear dynamical system using the planar dynamical transformation. Then, we undertake a qualitative analysis of the dynamical system to study its chaotic characteristics and bifurcation phenomena. Phase portraits of bifurcation are observed at the equilibrium points of the planner dynamical system. We discuss various tools to identify the chaos (random and unpredictable behavior) in dynamical systems. Graphical visualization of qualitative analysis of the system is also depicted in 3-D phase portraits, 2-D phase portraits, Poincaré mapping, and time series. Additionally, the Runge–Kutta technique is used to do a thorough sensitivity analysis of the dynamical system. Using this analytical procedure, it is verified that the stability of the solution has negligible impact while there is little change in the beginning circumstances. This paper aims to provide vital views to scientists and researchers seeking to advance their experimental operations. Furthermore, this research might be expanded by incorporating solitons and situations and investigating wave dynamics. The methods employed provide a direct and uncomplicated approximation of all solutions when compared to the previously established techniques. Various combinations and values of the physical parameters are utilized to explore the optical soliton solutions of the resultant system.

Dynamic characteristics and prediction and control models of subsonic exhaust from a container

This study combines experimental measurements, numerical simulations, and theoretical analysis to investigate the subsonic discharge process in a container under normal temperature and pressure conditions. Experimental data captured the internal pressure dynamics during exhaust at the atmospheric environmental pressure. Numerical simulations using OpenFOAM validated the isothermal exhaust model against the experimental results. Under the assumption of ideal gas and isothermal processes, a nonlinear differential equation was derived to describe the evolution of the container's internal pressure. This equation was simplified for a specific range of pressure ratios, yielding analytical solutions for the internal pressure of the container under both constant and variable external pressures. The effectiveness of the expressions of pressure inside the container was verified by comparing them with experimental and numerical simulation data. We further developed a formula for predicting exhaust mass flow rate, with a prediction error within 9%. An improved formula was subsequently proposed to reduce the error to below 0.4%, enhancing prediction accuracy. For containers with variable external pressure, a method for controlling the exhaust mass flow rate by predicting external pressure changes was proposed, demonstrating its effectiveness in achieving precise control.

An exact solution of the lubrication equations for the Oldroyd-B model in a hyperbolic pipe

An exact analytical solution of the lubrication equations for the steady, isothermal, incompressible flow of a viscoelastic Oldroyd-B fluid in a hyperbolic cylindrical contracting pipe is derived. The solution is valid for values of the Deborah number, De, up to order unity (De is defined as the ratio of the longest relaxation time of the polymer to the characteristic residence time of the fluid in the pipe), all values of the ratio of the polymer viscosity to the total viscosity of the fluid, η, and typical values of the contraction ratio, Λ, encountered in experiments and practical applications. It is provided in terms of the streamfunction only and is used in the momentum balance to derive a strongly non-linear ordinary differential equation of second order with unknown a function which corresponds to a modified fluid velocity along the main flow direction. The final equation is solved semi-numerically using a fully spectral (Legendre)-Galerkin approach to resolve the unknown function almost down to machine accuracy. The exact solution for the polymer extra-stresses, which is emphasized is not the full solution of the complete lubrication equations, allows for the derivation of a variety of theoretical expressions for the average pressure-drop along the pipe. In all cases, a decrease in the pressure drop compared to the Newtonian value with increasing De, η and/or Λ is predicted. The differences between the corresponding analytical solution for the planar geometrical configuration are also identified and discussed.

Tangent hyperbolic MHD nanoliquids on non-isothermal stretched sheets: Analyzing the impact of transport parameters, variable fluid properties and convective boundary conditions

The key focus of this research is to look at how tangent hyperbolic magnetohydrodynamic (MHD) nano-liquids move in a non-isothermal coagulated stretched sheet. In this study, we intend to explore how fluid behavior responds to alternations in transport physical parameters. The coagulated sheet is subjected to two distinct boundary conditions to inspect the heat and mass transport rate of the nano liquid. The convective heat boundary condition (CBC) and mass-convective boundary condition (MCBC) are employed to specify the physical conditions at the boundaries of the problem domain. The boundary conditions, responsible for regulating heat flow and mass transfer rates, play a crucial role in shaping the liquid’s behavior. Using a similarity solution, the partial differential equation describing the flow of mass and heat within the system transforms into an interconnected network of non-linear ordinary differential equations. These mathematical equations are subsequently figured out using a numerical finite difference method. This study additionally examines the correlation between thermophoresis and Brownian motion, two fundamental concepts in the dynamics of colloidal suspensions. The study’s findings indicate that the temperature profile increases in all scenarios when the variable thermal conductivity and variable viscosity parameters are increased. In contrast, for the same parameters, the velocity profile is decreased. Further, increasing wall thickness reduces heat dissipation; consequently, the temperature profile similarly affects the velocity power index. The Biot number improves the rates of temperature transfer. These results underscore the significant influence of these parameters in predicting the behavior of MHD tangent hyperbolic nanofluids. This study elucidates the intricate interaction among different physical parameters in the dynamics of nano liquids, which holds significant implications for various industrial applications.

Consequences of thermal conductivity and thermal density on heat and mass transfer in the nanofluid boundary layer flow toward a stretching sheet in the presence of a magnetic field

The term “thermal conductance” is used to describe a material’s ability to transport or conduct heat. Materials with high thermal conductivity are employed as heating elements, while those with poor thermal conductivity are used for insulation purposes. It is known that the thermal conductivity of pure metals decreases as temperature increases. In this study, the primary focus is on the physical assessment of thermal conductivity, entropy, and the improvement rate of thermal density in a magnetic nanofluid. To achieve this, nonlinear partial differential equations are transformed into ordinary differential equations. These equations are further solved using a computational method known as the Keller box technique. Various flow parameters, such as the Eckert number, density parameter, magnetic-force parameter, thermophoretic number, buoyancy number, and Prandtl parameter, are examined for their impact on velocity, temperature distribution, and concentration distribution. For the asymptotic results, the appropriate range of parameters, such as 1.0 ≤ ξ ≤ 5.0, 0.0 ≤ n ≤ 0.9, 0.1 ≤ Ec ≤ 2.0, 0.7 ≤ Pr ≤ 7.0, 0.1 ≤ Nt ≤ 0.5, and 0.1 ≤ Nb ≤ 0.9, is utilized. The key findings of this study are related to the assessment of heat transfer in a magnetic nanofluid considering thermal conductivity, entropy generation, and temperature density. It is observed that the temperature distribution increases as entropy generation increases. From a physical perspective, thermal conductivity acts as a facilitating factor in enhancing heat transfer. The study concludes by emphasizing the consistency achieved through a comparison of the latest findings with previously reported analyses.

Existence Results for Differential Equations with Tempered Ψ–Caputo Fractional Derivatives

The method of the equivalent system of fractional integral equations is used to prove the existence results of a unique solution for initial value problems corresponding to various classes of nonlinear fractional differential equations involving the tempered Ψ–Caputo fractional derivative. These include equations with their right side depending on ordinary as well as fractional-order derivatives, or fractional integrals of the solution.

Convective MHD flow of Casson fluid through porous medium using the Darcy‐Forchheimer model under the impact of prescribed heat sources

AbstractBlood and other non‐Newtonian fluids are most appropriately described by a rheological model known as a Casson fluid. The presence of yield stress in Casson fluids makes them particularly relevant in the fields of biomechanics and polymer industries. In this analysis, a numerical simulation of flow of a Casson nanofluid over a stretching surface was estimated. The flow of fluid was subjected to a magnetic field of varying strength. The governing partial differential equations for momentum and heat transmission are converted into a set of nonlinear ordinary differential equations with the aid of similarity transformations. After that, numerical techniques are used to solve the equations. Figures and tables are utilized to illustrate how non‐dimensional parameters affect energy, concentration, and velocity trends.

Analysis of activation energy, chemical reaction, and variable density on magnetically driven heat transportation: Applications in nanofluid lubrication and machining

The importance of this investigation is to examine the heat and mass transportation of magneto nanofluid movement along a heated sheet with exponential temperature-dependent density, entropy optimization, thermal buoyancy, activation energy, and chemical reaction aspects. The influence of these factors in cutting tools by means of machining and nanofluid lubrication is a significant process in cutting zone, chip cleaning, lubricating, and cooling productivity in milling. The corresponding energy activation and chemical process are essential to understand the thermal behavior of nanofluid. The appropriate transformations are used to solve nonlinear partial differential equations within the framework of ordinary differential equations using stream functions and similarity variables. The Keller box method is employed to efficiently solve these equations computationally under the Newton–Raphson approach. Through tables and figures, the fluid velocity, temperature distribution, and concentration consequences are sketched using various controlling parameters. It is seen that the fluid temperature function increases with noticeable amplitude as the Eckert factor, variable density, chemical-reaction, and activation energy increase. It is found that the noticeable enhancement in heat and mass transportation is deduced for maximum Brownian motion and thermophoresis. This work is important in various applications such as cutting fluids, drilling, brake oil, engine oil, minimum quantity lubrication, enhanced oil recovery, and controlled friction between the tool-chip and tool-work during machining operations.