Nonlinear Ordinary Differential Equations Research Articles

Comparative analysis of thermal behaviour in functionally graded material fin under convective-radiative heat transfer: Homotopy perturbation method

In this study, the thermal characteristics of a fully wet, moving longitudinal fin made from linear Functionally Graded Material (FGM) is explored. The fin response to convective-radiative heat transfer is examined through three distinct FGM scenarios namely, a Homogeneous material (HM), Functionally Graded Material I (FGM I) having greater thermal grading near the base, and Functionally Graded Material II (FGM II) having increased thermal grading towards the tip. The derived energy equation is a second-order nonlinear ordinary differential equation is solved using the Homotopy Perturbation Method (HPM) compared with results found in the existing literature. The fin thermal profile has been graphically analysed for the thermal conductivity grading parameter (β), Peclet number (Pe) and other pertinent parameters. It is found that the temperature distribution along the fin is highest for the FGM II fin and lowest for the HM fin, with the FGM I fin displaying intermediate values. Efficiency of the fin (η) is discovered to be lowest in the case of HM fin structures followed by FGM I and FGM II fin structures at the base and vice versa at the tip. The results show significant potential of HPM which benefits the manufacture and design of FGM fin structures.

Dynamics of propagation patterns: An analytical investigation into fractional systems

Recent years have seen a growing interest in fractional differential equations, particularly the fractional Chaffee-Infante ([Formula: see text]) equation, pivotal for understanding dynamics governed by fractional orders in specific physical systems. Exploring solitary wave solutions, this study employs the extended Khater method and truncated Mittag-Leffler function properties to formulate tailored solutions for the ([Formula: see text]) model. Through a traveling wave ansatz, the equation transforms into a nonlinear ordinary differential equation, revealing intricate propagation patterns of solitary waves. Visual representations aid comprehension, while rigorous validation ensures solution precision, ultimately providing a comprehensive understanding of system responses to external stimuli. This study effectively integrates analytical and numerical methodologies to derive precise solitary wave solutions, with significant implications for advancing comprehension of complex phenomena in various disciplines governed by fractional-order dynamics.

Tangent hyperbolic nanofluid flow through a vertical cone: Unraveling thermal conductivity and Darcy–Forchheimer effects

Purpose: This paper demonstrates the way tangent hyperbolic nanofluid flow through a vertical cone is influenced by varying viscosity and varying thermal conductivity. This study also seeks to illustrate the impact of convective boundary conditions on a fluid. The mathematical modeling also takes the Darcy–Forchheimer effect into account. Methodology: Using the appropriate similarity transformation, the fluid problem is reduced into a set of nonlinear ordinary differential equations. These systems of differential equations are evaluated numerically by applying the Optimal Homotopy Asymptotic Method. Findings: The nature of emergent parameters is examined in relation to the temperature distribution, nanoparticle concentration profile, and velocity profile. An increase in variable viscosity corresponds to a decrease in fluid velocity, while enhanced thermal conductivity results in elevated fluid temperature. The skin friction coefficient, Sherwood, and Nusselt numbers are numerically examined for active concerned parameters. These findings can be put into practice in a variety of fields such as polymer cooling systems and medication. Originality: Existing literature has yet to explore the combination of tangent hyperbolic nanofluids with varying viscosity and thermal conductivity under convective boundary conditions.

Analog quantum simulation of partial differential equations

Abstract Quantum simulators were originally proposed for simulating one partial differential equation in particular -- Schrodinger's equation. Can quantum simulators also efficiently simulate other partial differential equations? While most computational methods for partial differential equations -- both classical and quantum -- are digital (they must be discretised first), partial differential equations have continuous degrees of freedom. This suggests that an analog representation can be more natural. While digital quantum degrees of freedom are usually described by qubits, the analog or continuous quantum degrees of freedom can be captured by qumodes. Based on a method called Schrodingerisation, we show how to directly map D-dimensional linear partial differential equations onto a (D+1)-qumode quantum system where analog or continuous-variable Hamiltonian simulation on D+1 qumodes can be used. This very simple methodology does not require one to discretise partial differential equations first, and it is not only applicable to linear partial differential equations but also to some nonlinear partial differential equations and systems of nonlinear ordinary differential equations. We show some examples using this method, including the Liouville equation, heat equation, Fokker-Planck equation, Black-Scholes equations, wave equation and Maxwell's equations. We also devise new protocols for linear partial differential equations with random coefficients, important in uncertainty quantification, where it is clear how the analog or continuous-variable framework is most natural. This also raises the possibility that some partial differential equations may be simulated directly on analog quantum systems by using Hamiltonians natural for those quantum systems.

On the construction of various soliton solutions of two space-time fractional nonlinear models

Abstract In this article, we investigate a couple of nonlinear fractional models of eminent interests subsequently the conformable derivative sense is used to designate the fractional order derivatives. The given structures are transformed into nonlinear ordinary differential equations of integer order, and the extended simple equation technique is then employed to solve the resulting equations. Initially, the nonlinear space time fractional Klein–Gordon equation is considered emerging from quantum and classical relativistic mechanics, which have application in plasma physics, dispersive wave phenomena, quantum field theory, and optical fibres. Later, the (2 + 1)-dimensional time fractional Zoomeron equation is analysed which is convenient to explore the innovative phenomena related to boomerons and trappons. As a result, various new soliton solutions are successfully established. The reported results offer a key implementation for analysing the soliton solutions of nonlinear fractional models which are extremely encouraging arising in the recent era of science and engineering. The 3D simulations have been carried out to demonstrate dynamics of the various soliton solutions for a given set of parameters.

Approximate closed‐form solution of free convection flow past a vertical cone

AbstractThis paper presents a technique to find the closed‐form solution of the boundary‐value problem representing free convection flow past a vertical cone. Partial differential equations that govern the flow are converted into nonlinear ordinary differential equations with the aid of similarity transformations. First, a numerical solution is obtained with the help of the bvp4c package of MATLAB, and then utilizing the numerical solution, the closed‐form solution is obtained. The accuracy of the closed‐form solution is analyzed by the residual technique. Graphs of the residuals of the governing equations indicate that our approximate closed‐form solution closely matches the exact solution.

On the solutions of space-time fractional CBS and CBS-BK equations describing the dynamics of Riemann wave interaction

In this paper, Kudryashov and modified Kudryashov methods are implemented for the first time to compute new exact traveling wave solutions of the space-time fractional [Formula: see text]-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation and Calogero–Bogoyavlenskii–Schiff and Bogoyavlensky Konopelchenko (CBS-BK) equation. With the help of wave transformation, the aforementioned fractional differential equations are converted into nonlinear ordinary differential equations. The purpose of this paper is to devise novel exact solutions for the space-time-fractional [Formula: see text]-dimensional CBS and the space-time-fractional CBS-BK equations by utilizing the Kudryashov and modified Kudryashov techniques. The solutions, thus, acquired are demonstrated in figures by choosing appropriate values for the parameters. The solutions derived take the form of various wave patterns, including the kink type, the anti-kink type and the singular kink wave solutions. The obtained solutions are indeed beneficial to analyze the dynamic behavior of fractional CBS and CBS-BK equations in describing the interesting physical phenomena and mechanisms. The obtained solutions are entirely new and can be considered as a generalization of the existing results in the ordinary derivative case. The techniques presented here are very simple, efficacious and plausible and hence can be employed to attain new exact solutions for fractional PDEs.

Simulation of hybrid boiling nano fluid flow with convective boundary conditions through a porous stretching sheet through Levenberg Marquardt artificial neural networks approach

In the domain of artificial neural networks, the Levenberg-Marquardt technique stands out for its innovative approach and convergent stability. It generates a numerical approach for the progression of a hybrid Cu–Al2CO3/water nanofluid on a porous stretched sheet (HNF-PSS), employing regression plots, state transition measures, histogram representations, and mean squared errors. This work explores the analysis of the fluid flow problem associated with HNF-PSS, introducing a novel application of an intelligent computing system that utilizes the effectiveness of neural networks trained by the Levenberg-Marquardt algorithm (NN-TLMA). The initial mathematical expression, expressed in terms of partial differential equations (PDEs) for HNF-PSS, undergoes transformation into dimensionless nonlinear ordinary differential equations (ODEs). The collection of data for the envisioned NN-TLMA involves parameters influencing the velocity of the HNF-PSS system model, generated using the Lobatto IIIa formula. The efficacy of the suggested NN-TLMA for solving the HNF-PSS is robustly verified by examinations of state transition dynamics, regression analyses, mean square error, and error histogram studies. The consistent alignment of the obtained results with corresponding solutions attests to the legitimacy of the structure, achieving a high level of accuracy within the range of 10–6 to 10–8. Insightful findings demonstrate that the thermal Biot number contributes significantly to an upsurge in fluid temperature. Furthermore, an augmentation in nanofluid concentration and thermal profile is found to increase the porosity strength of the medium. Nevertheless, an attenuation in the velocity profile is noted under these conditions. The application of this simulation work can be found in optimizing the designs of heat exchanger, in biomedical application and in renewable energy processes.

Gluon mass generation from renormalons and resurgence

We establish a link between the concepts of infrared renormalons, infrared fixed point, and dynamical nonperturbative mass generation of gluons in pure Yang-Mills theories. By utilizing recent results in the resurgent analysis of renormalons through non-linear ordinary differential equations, we develop a new description for the gluon propagator, thereby realizing the Schwinger mechanism. Specifically, this approach leads to a nonperturbative, dynamic mass generation for Yang-Mills gauge bosons in the deep infrared region, a phenomenon closely associated with color confinement. Furthermore, we present arguments about the limit of applicability of the Borel-Ecalle resummation of the renormalons by comparing it with the Kallen-Lehman representation of the gluon propagator.

Numerical analysis of the MHD Williamson nanofluid flow over a nonlinear stretching sheet through a Darcy porous medium: Modeling and simulation

Abstract In the current study, we delve into examining the movement of a nanofluid within a Williamson boundary layer, focusing on the analysis of heat and mass transfer (HMT) processes. This particular flow occurs over a sheet that undergoes nonlinear stretching. A significant facet of this investigation involves the incorporation of both the magnetic field and the influence of viscous dissipation within the model. The sheet is situated within a porous medium, and this medium conforms to the Darcy model. Since more precise outcomes are still required, the model assumes that both fluid conductivity and viscosity change with temperature. In this research, we encounter a system of extremely nonlinear ordinary differential equations that are treated through a numerical technique, specifically by employing the spectral collocation method. Graphical representations are used to illustrate how the relevant parameters impact the nanoparticle volume fraction, velocity, and temperature profiles. The study involves the computation and analysis of the effect of physical parameters on the local Sherwood number, skin friction coefficient, and local Nusselt number. Specific significant findings emerging from the present study highlight that the rate of mass transfer is particularly influenced by the thermophoresis factor, porous parameter, and Williamson parameter, showing heightened effects, while conversely, the Brownian motion parameter demonstrates an opposing pattern. The results were computed and subjected to a comparison with earlier research, indicating a notable degree of conformity and accord.

Mixed Convection Flow of a Casson Fluid Past a Thin Needle

Mixed Convection Flow of a Casson Fluid Past a Thin Needle

A new version of trial equation method for a complex nonlinear system arising in optical fibers

In this study, the dissipation problem of nonlinear pulse in mono mode optical fibers which is governed by the Fokas system (FS) is considered. The solutions of this system have an important role in comprehending the different wave structures in physical settings. Therefore, a new version of the trial equation method (NVTEM) is employed to present the new exact wave solutions of the FS. The advantage of the NVTEM is to use different root possibilities of a polynomial which shape the solutions of the related model. Primarily this system is converted to a nonlinear ordinary differential equation (NODE) via the traveling wave transform to apply the proposed method. Various exact wave solutions to the FS are obtained such as rational function, exponential function, hyperbolic function, and Jacobi elliptic function solutions. Thus, we have revealed solutions featly which are unlike the wave solutions previously found by other analytical methods. The present results depict the formation and development of such waves and their interactions. The exhibition of the solutions is given by 3D plots together with the corresponding 2D plots. The outcomes have shown that the proposed technique is abundant in achieving different wave solutions of many nonlinear differential equations in the field of optics.

Exploring gradual material rigidity degradation effects on the nonlinear coupled axial-transversal nonlinear response of a capacitive micro-beam

The accumulation of damage in materials due to crack propagation can lead them to unpredicted failures. This phenomenon is particularly relevant in materials science, engineering, and other fields where structures or components are subjected to repeated loading, cyclic stresses, or environmental factors. In this paper, the mechanical behavior of a microelectromechanical system subjected to cyclic loads has been modeled, when the stiffness of the material gradually is decreased. As a case study the dynamic response of a micro-beam to DC bias voltage as a static load and AC harmonic voltage as a cyclic load has been studied. The nonlinear coupled equations have been formed and converted to time-dependent ones using a spatial discretization method. Due to long-term integration over time, to achieve more accurate results besides direct 4th order Runge-Kutta integration method, a temporal decomposition method has been used to solve the coupled nonlinear ordinary differential equations. To realize this method, a temporal decomposed solution based on the physical nature of the problem is presented in the form of slow-growing terms and fast-altering harmonic terms with gradually-growing amplitudes. To find the coefficient of these terms a least square method has been utilized to minimize a cost function in a given dynamic time-interval, where the independency of the results to this interval has been assessed. Also, the nonlinear frequency behavior of the proposed model has been studied when the system has suffered various damages. Due to considering gradual rigidity degradation of the material interesting dynamic responses have been observed, that can help to expand the insight in the design of structures.

On the solution of MHD Jeffery–Hamel problem involving flow between two nonparallel plates with a blood flow application

AbstractThe Jeffery–Hamel flow phenomenon appears in a variety of real‐world applications involving the flow of two nonparallel plates. BY using a similarity transformation derived from the equation of continuity, partial differential equations determining flow characteristics are translated into nonlinear ordinary differential equations. The problem involves the flow of a specific type of fluid, namely, an incompressible and electrically conducting fluid, between two nonparallel plates. The flow is assumed to be steady, two‐dimensional, and subject to certain boundary conditions. Specifically, the plates are impermeable, and the fluid adheres to a no‐slip condition, resulting in zero fluid velocity at the plates' surfaces. Moreover, the problem incorporates the effects of magnetic fields and pressure fluctuations, making it highly applicable to scenarios, such as blood flow through arteries in the human body, which can be modeled as a special case of the magnetohydrodynamic (MHD) Jeffery–Hamel problem referred to as the (MHD) blood pressure equation. This work compares two numerical approaches for solving the MHDs Jeffery–Hamel problem: B‐spline and Bernstein polynomial collocation. The given approaches are used to discretize and transform the equation into a system of algebraic equations. Matrix algebra techniques are then used to solve the resultant system. A complete error analysis and convergence rates for different grid sizes are derived for both methods and are used to compare the accuracy and efficiency of the two approaches. Both approaches produce correct solutions, according to the numerical findings, although the Bernstein polynomial collocation method is more efficient and accurate than the B‐spline collocation.

Bioconvective flow of thermally radiative Casson nanofluid with aerobic microorganisms over a stretching surface

A mathematical model is proposed to analyze the flow characteristics of a thermally radiative, chemically reacting Casson nanofluid incorporating aerobic microorganisms over a stretching sheet. Nonlinear dimensionless ordinary differential equations are formulated through suitable similarity transformations. The resulting coupled system of nonlinear ordinary differential equations is numerically solved using the finite difference method with the bvp4c routine in MATLAB. The verification of the numerical solution’s accuracy is conducted by comparing it with previously established results under specific conditions pertinent to the present investigation. Skin friction coefficients, local Nusselt number, and Sherwood number are computed for various parameters and their implications for engineering applications are discussed. The investigation also explores the impact of different dimensionless parameters on velocity, temperature, mass concentration, motile microorganism density, and oxygen concentration profiles for both Casson and Newtonian nanofluids, offering a comprehensive analysis. The study highlights that suspending non-Newtonian nanofluids with fast-moving aerobic microorganisms and nanoparticles results in enhanced heat transfer rates, thus presenting potential benefits for practical applications such as the manufacturing of biofilms.

Analytic solutions to nonlinear ODEs via spectral power series

Analytic solutions to nonlinear ODEs via spectral power series

Analyzing the Effects of Polymeric Dielectric Materials on Micro Capacitive Pressure Sensors: A Model Incorporating Displacement-Dependent Porosity

Recently, the extensive utilization of porous polymeric materials to amplify the sensitivity of capacitive devices is noticeable. The absence of an effective mathematical model for studying these devices has spurred the development of a comprehensive mathematical model in the current work. This model is formulated to analyze the static and dynamic behavior of systems incorporating a porous polymer dielectric material within the gap between flexible and fixed microplates. The derived nonlinear governing equations encompass the effects of electrostatic force, von-Karman nonlinear strains, and displacement-dependent porosity. Employing spatial decomposition, the resulting nonlinear algebraic equations and ordinary differential equations are leveraged to study the static and transient dynamic behavior, as well as the frequency response of the sensor using a learning approach. Two scenarios are investigated to assess the impact of various geometrical and physical parameters on sensor sensitivity one with a polymeric material and another without, each with distinct parameter values. The results reveal that the inclusion of a polymeric dielectric material increases electrostatic force but concurrently elevates the equivalent stiffness of the structure. The effectiveness of using a polymeric dielectric material is contingent upon the specific geometrical and physical properties of the sensor. Moreover, the obtained results in simplified cases are compared to existing numerical and experimental data, demonstrating a high degree of agreement. This work significantly contributes to advancing the understanding of sensors incorporating porous polymer dielectric materials and underscores their potential for enhanced sensitivity across diverse applications.

Modeling and analyzing competitive epidemic diseases with partial and waning virus-specific and cross-immunity

In this paper, we consider a novel mathematical modeling framework for the spread of two competitive diseases in a well-mixed population. The proposed framework, which we term a bivirus SIRIS model, encapsulates key real-world features of natural immunity, accounting for different levels of (partial and waning) virus-specific and cross protection acquired after recovery. Formally, the proposed framework consists of a system of coupled nonlinear ordinary differential equations that builds on a classical bivirus susceptible–infected–susceptible model by means of the addition of further states to account for (temporarily) protected individuals. Through the analysis of the proposed framework and of two specializations, we offer analytical insight into how natural immunity can shape a wide range of complex emergent behaviors, including eradication of both diseases, survival of the fittest one, or even steady-state co-existence of the two diseases.

Novel dynamical group-invariant solutions and conserved vectors of the Gilson–Pickering equation with applications in plasma physics

This work delves into the analysis of the Gilson-Pickering equation which governs the waves propagation in plasma physics by invoking Lie symmetry analysis. We commence by identifying the Lie point symmetries associated with the equation. These symmetries are then leveraged on to compute the commutator table and subsequently the adjoint representation, ultimately leading to the establishment of an optimal system of one-dimensional subalgebras. Each subalgebra within this system is subsequently utilized to perform symmetry reductions. Through these reductions, various forms of nonlinear ordinary differential equations are obtained, which are subsequently solved using the power series method and Kudryashov’s technique. The resulting solutions are given in terms of hyperbolic functions. To gain deeper insights into the behavior of these solutions, three-dimensional and two-dimensional plots are presented. Furthermore, applying the Ibragimov’s theorem allows us to derive conserved vectors associated with the Gilson–Pickering equation.

Exact analysis of generalized degree-based percolation without memory

We study the problem of generalized degree-based percolation without memory, where the probability of node removal depends on a non-negative function of node degree within the remaining network. We derive a set of nonlinear ordinary differential equations describing the evolution of the degree distribution during the percolation process. Employing generating function methods, we calculate various quantities of interest such as the probability of a randomly selected node belonging to the giant component, the mean cluster size, and the critical point at which a network transitions from connected to fragmented. Validation of our approach is achieved through extensive Monte Carlo simulations and numerical solutions. Our results indicate that removing highly connected nodes with small preference probabilities significantly lowers the critical point compared to random removal. Moreover, our analytical framework is applicable to random networks generated by the configuration model with any bounded degree distribution. It provides a comprehensive and systematic approach to analyze network robustness against various degree-based attack strategies, offering valuable insights into network design and protection.