Second-order Nonlinear Ordinary Differential Equations Research Articles

A new way to use nonlocal symmetries to determine first integrals of second-order nonlinear ordinary differential equations

Finding first integrals of second-order nonlinear ordinary differential equations (nonlinear 2ODEs) is a very difficult task. In very complicated cases, where we cannot find Darboux polynomials (to construct an integrating factor) or a Lie symmetry (that allows us to simplify the equations), we sometimes can solve the problem by using a nonlocal symmetry. In [1–3] we developed (and improved) a method (S-function method) that is successful in finding nonlocal Lie symmetries to a large class of nonlinear rational 2ODEs. However, even with the nonlocal symmetry, we still need to solve a 1ODE (which can be very difficult to solve) to find the first integral. In this work we present a novel way of using the nonlocal symmetry to compute the first integral with a very efficient linear procedure. New version program summaryProgram Title: InSyDE – Invariants and Symmetries of (rational second order ordinary) Differential Equations.CPC Library link to program files:https://doi.org/10.17632/4ytft6zgk7.3Licensing provisions: CC by NC 3.0Programming language: MapleSupplementary material: Theoretical results and revision of the S-function method.Journal reference of previous version: Comput. Phys. Comm. Volume 234, January 2019, Pages 302-314 - https://doi.org/10.1016/j.cpc.2018.05.009Does the new version supersede the previous version?: Yes.Nature of problem: Determining first integrals of rational second order ordinary differential equations.Solution method: The method is explained in the Summary of revisions and Supplementary material.Reasons for the new version: The InSyDE package after determining the S-function still needs to solve a first-order ordinary differential equation (1ODE) associated with the nonlocal symmetry (the so-called associated 1ODE – see [2]). The problem is that, for very complicated 1ODEs, this may not be practically feasible. We have developed an new and more efficient method that uses the nonlocal symmetry to (for a large class of 1ODEs) determine the first integral in a linear way.Summary of revisions: In order to implement the new method just mentioned above we have made modifications to the command (Sfunction) and introduced a new one: command (Darlin).

Periodic and subharmonic solutions in the motion of a bead on a rotating circular hoop

We establish the necessary conditions for the existence and multiplicity of periodic and subharmonic solutions to a second-order nonlinear ordinary differential equation (ODE). This ODE describes the motion of a bead on a rotating circular hoop subjected to a constant angular velocity ω and a T-periodic forcing. Our approach involves estimating bounds for the angular velocity and period using upper and lower solution methods.

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.

Self-similar solutions to a multidimensional singular heat equation with power nonlinearity

The paper deals with the construction of exact solutions to a singular heat equation with power nonlinearity in the case of numerous independent variables with spatial (e.g. axial or central) symmetry. A new class of self-similar solutions is proposed, which reduce to solving the Cauchy problem for a second-order nonlinear ordinary differential equation having singularities at the higher derivative with respect to the required function and/or the independent variable. The ordinary differential equation is studied in two ways: analytically and numerically. The analytical study uses a truncated Taylor series with recurrently computed coefficients, for which explicit formulas are obtained. The numerical solution to the problem uses an iteration algorithm based on the collocation method and radial basis functions. The numerical analysis shows the convergence of the proposed numerical algorithm and its sufficient accuracy enabling one to use the found self-similar solutions to verify approximate solutions to the original heat equation. Besides, the numerical analysis has allowed the radius of convergence of the constructed Taylor series to be evaluated. The form of the constructed self-similar solutions, namely their unboundedness near the symmetry center (axis), enables us to study the behavior and exactness of the numerical solutions to the nonlinear singular parabolic-type equation that have been obtained by the stepwise solution method proposed by us earlier and possess the same property.

On the Solvability of a Periodic Problem for a System of Second-Order Nonlinear Ordinary Differential Equations

On the Solvability of a Periodic Problem for a System of Second-Order Nonlinear Ordinary Differential Equations

Stability analysis of nonlinear convective boundary layer flow on a shrinking surface in the presence of viscous dissipation

This study investigates analytically the stability analysis of a nonlinear convective boundary layer (BL) flow of the nanofluids GO-EG and GO-W on a shrinking surface in the presence of a magnetic field and viscous dissipation. We provide a second-order nonlinear ordinary differential equation (NODE) for temperature distribution (TD) and a third-order NODE for velocity profile (VP) coupling based on the thermophysical characteristics of the base fluid and nanofluid as well as similarity transformations (STs) in the fundamental governing equations of momentum and energy. The analytical method HAM is used to answer the equations that have been gathered. In many different industries, such as manufacturing, automotive, microelectronics, and defense, cooling of various types of equipment and devices has very important challenges. To fulfill the demands of the engineering and industrial industries, it is hoped that this issue would significantly improve the heat transformation ratio.

Matrix factorizations for the generalized Charlier and Meixner orthogonal polynomials

The Cholesky factorization of the moment matrix is considered for the generalized Charlier and generalized Meixner discrete orthogonal polynomials. For the generalized Charlier, we present an alternative derivation of the Laguerre–Freud relations found by Smet and Van Assche. Third-order and second-order nonlinear ordinary differential equations are found for the recursion coefficient γn, that happen to be forms of the Painlevé deg-PV in disguise. New Laguerre–Freud relations are also found for the generalized Meixner case, which are compared with those of Smet and Van Assche.

Qualitative analysis on the electrohydrodynamic flow equation

<abstract><p>In this paper, we present a comprehensive analysis of the lower and upper bounds of solutions for a nonlinear second-order ordinary differential equation governing the electrohydrodynamic flow of a conducting fluid in cylindrical conduits. The equation describes the radial distribution of the flow velocity in an "ion drag" configuration, which is affected by an externally applied electric field. Our study involves the establishment of rigorous analytical bounds on the radial distribution, taking into account the Hartmann number $ H $ and a parameter $ \alpha. $ An analytic approximate solution is obtained as an improvement of the a priori estimates and it is found to exhibit strong agreement with numerical solutions, particularly when considering small Hartmann numbers. Further, estimates for the central velocity $ w(0) $ of the fluid occurring at the center of the cylindrical conduit were also established, and some interesting relationships were found between $ H $ and $ \alpha. $ These findings establish a framework that illuminates the potential range of values for the physical parameter within the conduit.</p></abstract>

Interval of the existence of positive solutions for a boundary value problem for system of three second-order differential equations

The aim of this paper is to estimate an interval of the existence of positive solutions for a boundary value problem for the system of three nonlinear second-order ordinary differential equations. Krasnosel'skiĭ–Precup fixed point theorem is used to determine this interval theoretically. For Dirichlet boundary conditions theoretical result is compared with result obtained numerically.

On special solutions to the Ermakov–Painlevé XXV equation

In this paper, we study a nonlinear second-order ordinary differential equation which we call the Ermakov–Painlevé XXV equation since under certain restrictions on its coefficients it can be reduced either to the Ermakov or the Painlevé XXV equation. The Ermakov–Painlevé XXV equation arises from a generalized Riccati equation and the related third-order linear differential equation via the Schwarzian derivative. The generalized Riccati equation has two families of Riccati solutions and we study the corresponding solutions to the Ermakov–Painlevé XXV equation. We show that one of these families appears only in the Ermakov case. We give examples of the Ermakov–Painlevé XXV equations and show how to construct their solutions expressed in terms of elementary or in terms of the classical special functions.

Linearization of Second-Order Non-Linear Ordinary Differential Equations: A Geometric Approach

Using the coefficients of a system semilinear cubic in the first derivative second order differential equations one defines a connection in the space of the independent and dependent variables, which is specified modulo two free parameters. In this way, to any such equation one associates an affine space which is not necessarily Riemannian, that is, a metric is not required. If such a metric exists, then under the Cartan parametrization the geodesic equations of the metric coincide with the system of the considered semilinear equations. In the present work, we consider semilinear cubic in the first derivative second order differential equations whose Lie symmetry algebra is the sl(3,R). The covariant condition for these equations is the vanishing of the curvature tensor. We demonstrate the method in the solution of the Painlevé-Ince equation and in a system of two equations. Because the approach is geometric, the number of equations in the system is not important besides the complication in the calculations. It is shown that it is possible to linearize an equation in this form using a different covariant condition, for example, assuming the space to be of constant non-vanishing curvature. Finally, it is shown that one computes the associated metric to a semilinear cubic in the first derivatives differential equation using the inverse transformation derived from the transformation of the connection.

Nonlinear dynamics and chaos control of circular dielectric energy generator

The current study focuses on the nonlinear dynamics of the dielectric elastomer generator (DEG). To begin with, we will go through DEG’s operating principles, parameters, materials, and deformation modes. On this basis, the dynamical modeling for the nonlinear behavior of the DEG system is described, and the second-order nonlinear ordinary differential equation representing radially symmetric motion is derived using Hamilton’s variational principle. A static applied electric field allows the DEG system to attain two equilibrium states: thick and thin. The effects of the stiffening phenomenon on the dynamics of the DEG system are investigated. The system converges to a stable equilibrium point under constant loads, and the convergence position and velocity are closely related to both the initial states. Subjected to periodic loads, certain phenomena such as quasiperiodicity and chaos are detected, and the chaotic phenomena are explored using the bifurcation diagram and 0-1 chaos test. Furthermore, parametric studies are carried out using numerical analyses to demonstrate the influence of load amplitude and external frequency on the dynamics of the DEG system. Once the DEG enters the chaotic regime, the chaotic zone is transformed into a stable manifold using established chaos control techniques such as system parameter change and parametric perturbation. However, because these techniques require changes in system design, we proposed a chaos control by using a non-linear controller, specifically, adaptive integral backstepping sliding mode control (AIBSMC) as one demonstrative candidate. The controller also enables the DEG system to follow motion along a predefined trajectory. The current work may be expected to substantially impact the future design and performance of energy harvesters.

Exact Solutions of Nonlinear Second-Order Autonomous Ordinary Differential Equations: Application to Mechanical Systems

Many physical processes can be described via nonlinear second-order ordinary differential equations and so, exact solutions to these equations are of interest as, aside from their accuracy, they may reveal beforehand key properties of the system’s response. This work presents a method for computing exact solutions of second-order nonlinear autonomous undamped ordinary differential equations. The solutions are divided into nine cases, each depending on the initial conditions and the system’s first integral. The exact solutions are constructed via a suitable parametrization of the unknown function into a class of functions capable of representing its behavior. The solution is shown to exist and be well-defined in all cases for a general nonlinear form of the differential equation. Practical properties of the solution, such as its period, time to reach an extreme value or long-term behavior, are obtained without the need of computing the solution in advance. Illustrative examples considering different types of nonlinearity present in classical physical systems are used to further validate the obtained exact solutions.

Thermal performance of stretching/shrinking fully wet porous cylindrical and conical pin fin structures by differential transformation approach

In this paper, the thermal behavior of cylindrical and conical pin fin structures operating in a convective–radiative environment has been examined. The fully wet porous fin structures are assumed to be mounted with a stretching/shrinking mechanism similar to a conveyer belt. The conical pin fin has been comparatively analyzed with the cylindrical one for three distinct cases of stretching, stagnant, and shrinking. Further, the Darcy model has been employed to analyze the fluid–solid interactions. The differential transformation method (DTM) has been used to solve the resulting second-order nonlinear ordinary differential equations. On the thermal performance of conical and cylindrical pin fin structures, the effects of the stretching/shrinking parameter, Peclet number, wet porous parameter, convective parameter, and radiative parameter have been established graphically. The conical pin fin is found to aid the heat transmission process better than the cylindrical one. The work is also helpful in the realm of microelectronics, particularly in the development of micro-pin-fin structures.

Finite Deformations of Internally Pressurized Synthetic Compressible Cylindrical Rubber-Like Material

The finite deformation of internally pressurized isotropic compressible synthetic rubber-like material governed by Levinson and Burgess strain energy function is analysed. A second-order nonlinear ordinary differential (Lane-Emden) equationwith shooting boundary value was derivedfor the determination of displacements distributions. Several analytical methods were employed to solve the resulting boundary value problem but no closed form solution was obtained at the moment. Fortunately, a lot of software have been developed to handle such highly nonlinear second order ordinary differential equations with specific values of parameters. Also, the stresses acting on the material were determined. We obtained numerical solution by applying shooting method and validated the result using collocation method on mathematica (ode45 solver). The simulation of the system is made forρ = 14N/m², and the cylindrical symmetric deformation attained its maximum displacements and stresses atr(1) = 1.16638m and σ_rr = (-1.2973e-05)kg/m/s². We were able to develop numerical schemes using shooting and Collocation methods which made it easier to determine position of maximum stresses and pressure in a cylindrical material of Levinson-Burgess strain energy function. These numerical schemes can solve any nonlinear second-order ordinary differential equations with any given boundary conditions on Mathematica Software. The results of the two schemes were statistically compared using t-test and results obtained showed, the two methods have no significant difference which validates the solutions.

Linearisation of a second-order nonlinear ordinary differential equation

We analyse nonlinear second-order differential equations in terms of algebraic properties by reducing a nonlinear partial differential equation to a nonlinear second-order ordinary differential equation via the point symmetry f(v)∂v. The eight Lie point symmetries obtained for the second-order ordinary differential equation is of maximal number and a representation of the sl(3,R) algebra. We extend this analysis to a more general nonlinear second-order differential equation and we obtain similar interesting algebraic properties.

Bifurcation analysis, stationary optical solitons and exact solutions for generalized nonlinear Schrödinger equation with nonlinear chromatic dispersion and quintuple power-law of refractive index in optical fibers

In this paper, the bifurcation, stationary optical solitons and exact solutions in optical fiber propagation are studied. The propagation model is described by a generalized nonlinear Schrödinger equation with Kudryashov’s quintuple power law of refractive index. First, the equation is transformed into a second-order nonlinear ordinary differential equation by an appropriate transformation. Then, the bifurcation of the phase diagram of the obtained ordinary differential equation is established using planar dynamic system theory, and the types of optical solitons and exact solutions are determined. What is more, the stationary optical solitons and exact solutions are obtained by the polynomial complete discriminant system, and are demonstrated in 3D and 2D graphics by using the symbolic computation software Maple. Finally, the obtained stationary optical solitons and exact solutions include the Jacobi elliptic function solution, hyperbolic function solution, rational function solution, trigonometric function solution and exponential function solution. Moreover, the results provide a method to further study stationary optical solitons and exact solutions of propagation pulses in optical fibers.

Modeling and analysis of dynamic load factor in a fail-safe crane mechanism equipped with a hydraulic equalizing cylinder

In the design and manufacturing of sensitive crane mechanisms, in particular, those utilized in nuclear facilities, safety is of utmost importance. A few designs are based on a hydraulic equalizing cylinder where the purpose is mainly to dampen the dynamic load factor generated by the sudden failure of the wire rope system. To the best of the authors' knowledge, design equations and engineering models for such an intricate dynamic mechanism are not reported anywhere in the literature. Toward achieving this goal, a model is proposed for the dynamic behavior of a hydraulic equalizing cylinder when coupled to a single-failure-proof cross-reeved crane hoist. The constitutive equations involve the time-dependent incompressible Navier-Stokes equations. These equations are coupled with the dynamics of the lower block and the elastic wire rope. A nonlinear second-order ordinary differential equation is obtained, which is solved analytically. This yields a closed-form relation for the dynamic load factor. The validity of the new model is investigated using experimental studies, which are presented in this work. The normalized root mean square error for the dynamic load factor obtained by the model compares favorably to the experimental results, where the maximum observed error is 3.9 percent. The results confirm that the current model can be used as a design equation by practitioners.

On a Nonlinear Second-Order Ordinary Differential Equation

We consider a nonlinear second-order ordinary differential equation of a special form whose particular case arises when constructing exact solutions of the nonlinear heat equation with a power-law coefficient. Conditions are obtained for the parameters under which the equation admits a single integration. A number of examples of constructing exact solutions expressed in terms of elementary functions or in terms of the Lambert function are given.

Legendre-Petrov-Galerkin Chebyshev spectral collocation method for second-order nonlinear differential equations

<p style='text-indent:20px;'>We propose a Legendre-Petrov-Galerkin Chebyshev spectral collocation method for initial value problems (IVPs) of second-order nonlinear ordinary differential equations (ODEs). The Legendre-Petrov-Galerkin method is applied to time discretization and the nonlinear term is dealt with Chebyshev spectral collocation method. The scheme results in a simple algebraic system by choosing appropriate basis functions. Optimal error estimates in <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm for the single and multiple interval methods are given. As an application of the method, we construct the space-time spectral schemes for solving some nonlinear time-dependent partial differential equations (PDEs). Numerical experiments suggest the efficiency of the methods.</p>