Order Differential Equations Research Articles

Precession motions of a gyrostat, having a fixed point, in three homogeneous force fields

The subject of investigation is the problem on precession motions of a gyrostat with a fixed point in three homogeneous force fields. The class of precessions under consideration is characterized by the constancy of the precession angle and by the commensurability of the precession and proper rotation velocities. Equations of a gyrostat motion are reduced to a system of three second order differential equations with respect to velocities of precession and proper rotation. Integration of these equations was conducted in the case of precessionally isoconic motions (the precession velocity equals to the proper rotation velocity) and in the case of 2:1 resonance, when the precession velocity is two times more, than the proper rotation velocity. It was proved that the obtained solutions can be described by elementary functions of time.

Dilatonic dynamics of baryonic crystals, branes, and spheres

We systematically analyze the impact of dilatonic dynamics on Skyrme spheres, crystals, and branes. The effects of the dilatonic model parameters, encompassing different underlying near-conformal dynamics, on the macroscopic properties of skyrmions such as their mass and radius are discussed. For spheres and crystals we identify special values of the ratio of the decay constants for which the second order differential equations reduce to a solvable first order system. Additionally, in the case of the crystals, the dilaton presence spatially separates the baryon and isospin charge distributions. For branes, we show how the dilaton smooths out their configurations. Our results are expected to have wide implications from the study of near-conformal dynamics stemming from QCD-like theories to phenomenological investigations of nuclear matter in extreme regimes. Published by the American Physical Society 2024

Impact of High-Power Cosh-Gaussian Beam on Second Harmonic Generation in Collisionless Magnetoplasma

The impact of high power Cosh-Gaussian (ChG) beam on Second harmonic generation (SHG) in Collisionless magnetoplasma is explored in present work. Whenever the input beam propagates along external magnetic magnetic field direction, then there are two propagation modes viz. extraordinary mode and ordinary mode. The modification in magnetic field strength causes redistribution of carriers. The density gradients get established in plasma in a normal direction to input wave due to ponderomotive force. Further, there is production of electron plasma wave (EPW) at input wave frequency due to density gradients. EPW nonlinearly interacts with pump wave causing generation of 2nd harmonics. The 2nd order differential equation (ODE) for beam width of and efficiency of 2nd harmonics are derived through well-known paraxial theory approach. RK4 method is employed for carrying out numerical calculation of nonlinear ODE along with efficiency of 2nd harmonics. Impact of change in selective laserplasma parameters and externally applied magnetic field on beam waist of input wave and efficiency of SHG are also explored.

Numerical Technique Based on Operational Matrices of Fractional Integration Using ψ$$ \psi $$‐Shifted Chebyshev Polynomials

ABSTRACTIn this paper, we present a numerical scheme based on a modified form of shifted Chebyshev polynomials to find the numerical solution of a class of fractional differential equations. For this purpose, we work out operational matrices of fractional integration of ‐shifted Chebyshev polynomials obtained from shifted Chebyshev polynomials. Finally, the solution to the problem under consideration is obtained by solving a system of algebraic equations that results from the use of operational matrices of integration. The analysis of integer and non‐integer order differential equations is presented to show the convergence of the solution of fractional order differential equation to the corresponding solution of the integer order differential equation. At the end, we present some linear and non‐linear examples to validate the theoretical analysis. Non‐linear examples are solved using Quasilinearization and proposed numerical technique.

A model for micro-scale propulsion using flexible rotating flagella

Micro-scale propulsion by rotating helical flagella is of interest for the study of bacteria and robotic micro-swimmers. The propulsive thrust and torque produced by the rotating flagella are usually estimated assuming that they are rigid. In this paper we assume the flagella to be deformable elastic rods and compute propulsive forces and torques by enforcing local equilibrium of the rod within the context of resistive force theory. The torque-speed characteristics of the flagellar motor driving the rotation are taken into account. We show that the problem can be cast as a system of algebraic equations if the flagella are assumed to be helical before and after deformation when no spontaneous curvature is included. If the assumption of helical shape is dropped then we show that the propulsion problem can be cast as a system of first order differential equations that can be solved numerically. Our results in both cases agree reasonably well with experimental observations of bacterial propulsion and deviate from the predictions of Purcell depending on the mechanical properties of the flagellum.

On the Periodicity Solutions of Five Systems of Rational Systems of Difference Equations of Order Five

The primary aim of this paper is to explore the behavior of several nonlinear systems of difference equations following T_{n+1}=(E_{n}E_{n-4})/(T_{n-3}(\pm 1 \pm E_{n}E_{n-4})), E_{n+1}=(T_{n}T_{n-4})/(E_{n-3}(pm 1 \pm T_{n}T_{n-4})), and obtain solution expressions for them. Moreover, we utilize MATLAB programming to simulare the dynamics and validate our results.

A ninth-order first derivative method for numerical integration

In this paper, we present a ninth–order block hybrid method for the numerical solution of stiff and non–stiff systems of first–order differential equations. The method is based on an interpolation and collocation approach which results in a single continuous formulation; from which eight discrete schemes that make the block method were obtained. A convergence analysis of our method illustrated that it is A–stable, consistent, and convergent. We applied our method to some numerical examples which showed that the new method not only outperformed a second derivative method of order fourteen in the literature but also compared well with the exact solution.

Nonlinear vibro-acoustic analysis of an axially moving submerged circular cylindrical shell

In this study, the nonlinear vibro-acoustic dynamics and stability of doubly-clamped axially moving cylindrical shell are investigated. The exterior surface of the shell is in contact with fluid and subjected to oblique incident plane sound wave. Donnell’s nonlinear shallow shell theory is used to derive the nonlinear partial differential equation of the cylindrical shell for the radial motion. The Galerkin method is employed to discretize the equations of motion into the set of coupled nonlinear nonhomogeneous ordinary second order differential equations. Considering both driven and companion modes, Multiple Scales Method is used to obtain the response of the system. The effects of sound level, incident angle and axially velocity on the frequency response of the system are studied. The results show that, with increase in the shell velocity transmission loss of the circular shell increases. In addition, at low shell axial velocities simultaneous excitation of the driven and companion modes occurs.

The dynamics and behavior of logarithmic type fuzzy difference equation of order two.

Fuzzy difference equations are becoming increasingly popular in fields like engineering, ecology, and social science. Difference equations find numerous applications in real-life problems. Our study demonstrates that the logarithmic-type fuzzy difference equation of order two possesses a nonnegative solution, and an equilibrium point, and exhibits asymptotic behavior. [Formula: see text] Where, xi represents the sequence of fuzzy numbers, and the parameters α, β, A, along with the initial conditions x-1 and x0, are positive fuzzy numbers. The characterization theorem is employed to convert each single logarithmic fuzzy difference equation into a set of two crisp logarithmic difference equations within a fuzzy environment. We evaluated the stability of the equilibrium point of the fuzzy system. Utilizing variational iteration techniques, the method of g-division, inequality skills, and a theory of comparison for logarithmic fuzzy difference equations, we investigated the governing equation dynamics, including its boundedness, existence, and both local and global stability analysis. Additionally, we provided some numerical solutions for the equation describing the system to verify our results.

On a Poincaré-Perron problem for high order differential equations

On a Poincaré-Perron problem for high order differential equations

Applications of Laguerre transform to solve Schrödinger-type equations and differential equations of order four

The finite Laguerre transform is applied to solve differential equations problems of order higher than two and a one-dimensional steady-state Schrödinger equation, by using elementary Linear Algebra methods.

Entropy generation of Williamson hydromagnetic non-linear radiative nanofluid flow near a stagnation point over a porous stretching sheet

Purpose This study aims to analyze the multi-slip effects of entropy generation in steady non-linear magnetohydrodynamics thermal radiation with Williamson nanofluid flow across a porous stretched sheet near a stagnation point. Also, the qualities of viscous dissipation, Cattaneo–Christove heat flux and Arrhenius activation energy are taken into account. Thermophoresis, Brownian motion and Joule heating are also considered. Design/methodology/approach The Navier–Stokes equation, the thermal energy equation and the Solutal concentration equations are the governing mathematical equations that describe the flow and heat and mass transfer phenomena for fluid domains. By using the proper similarity transformations, a set of ordinary differential equationss are retrieved from boundary flow equations. The classical Runge–Kutta fifth-order algorithm along with the shooting technique is implemented to solve the obtained first order differential equations. Findings The study concludes that the temperature distribution boosting for thermal radiation, magnetic field and Eckert number where as the velocity and entropy generation escalate for the Williamson parameter, diffusion parameter and Brinkman number. The skin-friction and heat and mass transfer rate increases with the fluid injection. In addition, tabulated values of friction drag and rate of heat and mass transfer for various values of constraints are provided. Originality/value The comparison of the present results is carried out with the published results and noted a good agreement.

Evaluation of thermal and concentration slip effects on heat and mass transmission of nanofluid over a moving wedge surface using Keller box scheme

The current research is based on the impact of thermal and solutal slip in the boundary layer nanofluid flow through a moving accelerating wedge. The present investigation is considered with the influence of Brownian motion and thermophoresis. Thermal insulation, geothermal engineering, crude oil extraction, and heat exchangers are very important applications of nanofluid movement over a wedge surface with thermal and concentration slip. The suggested mathematical analysis is expressed in terms of partial differential equations (PDEs). These PDEs are transformed into ordinary differential equations via similarity transformation. The Keller Box technique is used to integrate the resultant non-similar equations. The set of discretized and first order differential equations is formed with the help of central difference and the Newton–Raphson technique. The graphical and numerical results are extracted with the help of MATLAB. The numerical results with the influence of the Prandtl factor (Pr), constant moving factor (λ), thermal slip factor (S2), and concentration slip parameter (S2) are interpreted visually and numerically. Graphical representations of velocity, thermal, and mass concentration profiles are analyzed in depth. The solution for skin friction coefficient, heat transport rate, and mass transport rate is calculated. The moving velocity function increases as Pr increases. The rate of slip temperature and slip concentration rate is enhanced for a lower Prandtl factor. The maximum slip behavior in temperature function and fluid concentration slip is deduced for each value of thermal-slip and concentration-slip factors. For high Prandtl and Brownian motion factors, the rate of Nusselt number is enhanced significantly.

Curvature form of Raychaudhuri equation and its consequences: A geometric approach

The paper aims at deriving a curvature form of the famous Raychaudhuri equation (RE) and the associated criteria for focusing of a hyper-surface orthogonal congruence of time-like geodesic. Moreover, the paper identifies a transformation of variable related to the metric scalar of the hyper-surface which converts the first order RE into a second order differential equation that resembles an equation of a Harmonic oscillator and also gives a first integral that yields the analytic solution of the RE and Lagrangian of the dynamical system representing the congruence.

Three-dimensional elasticity solutions for doubly-curved composite shells by extended differential quadrature method

This paper presents the three-dimensional (3D) stress analysis of doubly-curved composite shells with general boundary conditions using the strong sampling surfaces (SaS) formulation. The SaS method is based on the choice of SaS parallel to the middle surface and located at Chebyshev polynomial nodes to introduce the displacements of these surfaces as unknown functions that leads to a non-conventional shell formulation, in which strain–displacement and stress–strain relationships are represented in terms of SaS displacements. This is due to the use of Lagrange polynomials in approximating displacements, strains and stresses in the thickness direction. The outer surfaces are not included into a set of SaS that makes it possible to uniformly minimize the error ca used by Lagrange interpolation. The strong SaS formulation, based on direct integration of elasticity equilibrium equations in the thickness direction by the extended differential quadrature (EDQ) method, can be applied efficiently for high-precision calculations of doubly-curved composite shells with clamped and free edges. This is because in the SaS/EDQ formulation, displacements, strains and stresses of SaS are interpolated in a rectangular domain specified in a curvilinear coordinate system using a Chebyshev-Gauss-Lobatto grid and Lagrange polynomials are also used as basis functions. The proposed approach deals with equilibrium equations in terms of SaS stresses, avoiding the integration of second order differential equations in terms of SaS displacements, that greatly simplifies the implementation of the EDQ method.

A New Methodology for the Development of Efficient Multistep Methods for First–Order IVPs with Oscillating Solutions IV: The Case of the Backward Differentiation Formulae

A theory for the calculation of the phase–lag and amplification–factor for explicit and implicit multistep techniques for first–order differential equations was recently established by the author. His presentation also covered how the approaches’ efficacy is affected by the elimination of the phase–lag and amplification–factor derivatives. This paper will apply the theory for computing the phase–lag and amplification–factor, originally developed for implicit multistep methods, to a subset of implicit methods, called backward differentiation formulae (BDF), and will examine the impact of the phase–lag and amplification–factor derivatives on the efficiency of these strategies. Next, we will show you the stability zones of these brand-new approaches. Lastly, we will discuss the results of numerical experiments and draw some conclusions about the established approaches.

Design of integrated evolutionary finite differences for nonlinear electrohydrodynamics ion drag flow in cylindrical conduit model

This research implements an evolutionary optimized finite differences scheme (FDS) for nonlinear electrohydrodynamics ion drag flow dynamics in a cylindrical conduit (EHD-IDFCC). In the scheme, finite differences are exploited to discretize governing expressions of EHD-IDFCC, represented with a nonlinear singular seconder order differential equation, into a nonlinear equations-based system. The fitness function based on residual error is constructed for the FDS-based discretized EHD-IDFCC model. Its optimization is conducted with the global search capability of genetic algorithms (GAs) aided by the local search efficacy of the interior-point method (IPM), i.e., FDS-GA-IPM. The performance of the designed stochastic numerical solver FDS-GA-IPM is evaluated for a variant of the EHD-IDFCC model for different scenarios to measure the effect of axial flow velocity by varying the electric Hartmann numbers, as well as the nonlinearity factor. Statistical interpretations based on histogram illustrations, probability plots, and boxplots in terms of the cost function, mean absolute error (MAE), Thai inequality coefficients (TIC), and coefficient of determination metrics (R2) are used to endorse the accuracy, convergence, stability, and strength of the FDS-GA-IPM.

Geometrical description and Faddeev-Jackiw quantization of electrical networks

In lumped-element electrical circuit theory, the problem of solving Maxwell's equations in the presence of media is reduced to two sets of equations, the constitutive equations encapsulating local geometry and dynamics of a confined energy density, and the Kirchhoff equations enforcing conservation of charge and energy in a larger, topological, scale. We develop a new geometric and systematic description of the dynamics of general lumped-element electrical circuits as first order differential equations, derivable from a Lagrangian and a Rayleigh dissipation function. Through the Faddeev-Jackiw method we identify and classify the singularities that arise in the search for Hamiltonian descriptions of general networks. The core of our solution relies on the correct identification of the reduced manifold in which the circuit state is expressible, e.g., a mix of flux and charge degrees of freedom, including the presence of compact ones. We apply our fully programmable method to obtain (canonically quantizable) Hamiltonian descriptions of nonlinear and nonreciprocal circuits which would be cumbersome/singular if pure node-flux or loop-charge variables were used as a starting configuration space. We also propose a specific assignment of topology for the branch variables of energetic elements, that when used as input to the procedure gives results consistent with classical descriptions as well as with spectra of more involved quantum circuits. This work unifies diverse existent geometrical pictures of electrical network theory, and will prove useful, for instance, to automatize the computation of exact Hamiltonian descriptions of superconducting quantum chips.

Influence of pressure anisotropy on mass-radius relation and stability of millisecond pulsars in f(Q) gravity

In this study we explore the astrophysical implications of pressure anisotropy on the physical characteristics of millisecond pulsars within the framework of f(Q) gravity, in particular f(Q) = - α Q - β,where α and β are constants. Starting off with the field equations for anisotropic matter configurations, we adopt the physically salient Durgapal-Fuloria ansatz together with a well-motivated anisotropic factor for the interior matter distribution. This leads to a nonlinear second order differential equation which is integrated to give the complete gravitational and thermodynamical properties of the stellar object. The resulting model is subjected to rigorous tests to ensure that it qualifies as a physically viable compact object within the f(Q)-gravity framework. We study in detail the impact of anisotropy on the mass, radius and stability of the star. Our analyses indicate that our models are well-behaved, singularity-free and can account for the existence of a wide range of observed pulsars with masses ranging from 2.08 to 2.67 M ⊙, with the upper value being in the so-called mass gap regime observed in gravitational events such as GW190814. A comparison of the so-called Symmetric Teleparallel Equivalent to GR (STEGR) models with classical General Relativity (GR) models reveal that the anisotropy parameter and the sign of β impact on the predicted radii of pulsars. In particular, STEGR models have larger radii than their GR counterparts.

Dynamics of quasi-periodic, bifurcation, sensitivity and three-wave solutions for (n + 1)-dimensional generalized Kadomtsev-Petviashvili equation.

This study endeavors to examine the dynamics of the generalized Kadomtsev-Petviashvili (gKP) equation in (n + 1) dimensions. Based on the comprehensive three-wave methodology and the Hirota's bilinear technique, the gKP equation is meticulously examined. By means of symbolic computation, a number of three-wave solutions are derived. Applying the Lie symmetry approach to the governing equation enables the determination of symmetry reduction, which aids in the reduction of the dimensionality of the said equation. Using symmetry reduction, we obtain the second order differential equation. By means of applying symmetry reduction, the second order differential equation is derived. The second order differential equation undergoes Galilean transformation to obtain a system of first order differential equations. The present study presents an analysis of bifurcation and sensitivity for a given dynamical system. Additionally, when an external force impacts the underlying dynamic system, its behavior resembles quasi-periodic phenomena. The presence of quasi-periodic patterns are identified using chaos detecting tools. These findings represent a novel contribution to the studied equation and significantly advance our understanding of dynamics in nonlinear wave models.