Nonlinear Second-order ODE Research Articles

Solving Nonlinear Second-Order ODEs via the Eisenhart Lift and Linearization

The linearization of nonlinear differential equations represents a robust approach to solution derivation, typically achieved through Lie symmetry analysis. This study adopts a geometric methodology grounded in the Eisenhart lift, revealing transformative techniques that linearize a set of second-order ordinary differential equations. The research underscores the effectiveness of this geometric approach in the linearization of a class of Newtonian systems that cannot be linearized through symmetry analysis.

The Galerkin Method for Solving Strongly Nonlinear Oscillators.

In this paper, we make use of the Galerkin method for solving nonlinear second-order ODEs that are related to some strongly nonlinear oscillators arising in physics and engineering. We derive the iterative schemes for finding the coefficients that appear in the linear Galerkin hat combination in the ansatz form solution. These coefficients may be found iteratively by solving either a quadratic or a higher degree algebraic equation. Examples are presented to illustrate the obtained results. Some exact solutions are given, and they are compared with both the Runge–Kutta numerical solution and the solution obtained using the Galerkin finite element method.

A characterization of constant p-mean curvature surfaces in the Heisenberg group H1

In Euclidean 3-space, it is well known that the Sine-Gordon equation was considered in the nineteenth century in the course of investigations of surfaces of constant Gaussian curvature K=−1. Such a surface can be constructed from a solution to the Sine-Gordon equation, and vice versa. With this as motivation, employing the fundamental theorem of surfaces in the Heisenberg group H1, we show in this paper that the existence of a constant p-mean curvature surface (without singular points) is equivalent to the existence of a solution to a nonlinear second-order ODE (1.2), which is a kind of Liénard equations. Therefore, we turn to investigate this equation. It is a surprise that we give a complete set of solutions to (1.2) (or (1.5)), and hence use the types of the solution to divide constant p-mean curvature surfaces into several classes. As a result, after a kind of normalization, we obtain a representation of constant p-mean curvature surfaces and classify further all constant p-mean curvature surfaces. In Section 9, we provide an approach to construct p-minimal surfaces. It turns out that, in some sense, generic p-minimal surfaces can be constructed via this approach. Finally, as a derivation, we recover the Bernstein-type theorem which was first shown in [14] (or see [19,20]).

λ-Symmetry and Integrating Factor For x ̈(f(t,x)+g(t,x)x ̇)e^x

In this paper, we will calculate an integrating factor, first integral, and reduce the order of the non-Linear second-order ODEs , through the λ-symmetry method. Moreover, we compute an integrating factor, first integral and reduce the order for particular cases of this equation.

Modified iteration method for numerical solution of nonlinear differential equations arising in science and engineering

In this paper, a modified iteration method (MIM) has been proposed to solve nonlinear second-order ODEs. Convergence analysis and error estimate of the proposed method are also discussed. Computational efficiency of this method is illustrated through numerical examples.

On the Linearization of Second-Order Ordinary Differential Equations to the Laguerre Form via Generalized Sundman Transformations

The linearization problem for nonlinear second-order ODEs to the Laguerre form by means of generalized Sundman transformations (S-transformations) is considered, which has been investigated by Duarte et al. earlier. A characterization of these S-linearizable equations in terms of first integral and procedure for construction of linearizing S-transformations has been given recently by Muriel and Romero. Here we give a new characterization of S-linearizable equations in terms of the coefficients of ODE and one auxiliary function. This new criterion is used to obtain the general solutions for the first integral explicitly, providing a direct alternative procedure for constructing the first integrals and Sundman transformations. The effectiveness of this approach is demonstrated by applying it to find the general solution for geodesics on surfaces of revolution of constant curvature in a unified manner.

Limits of solutions to a nonlinear second-order ODE

In this paper, the existence of solutions to the equation ẍ+2f(t)ẋ+β(t)x+g(t,x)=0, t≥0, is discussed. Our approach allows us achieve extension to the case of the whole real line, for which the existence of homoclinic solutions having zero limit at ±∞ is deduced. The result is obtained through the method of the Lyapunov function and differential inequalities.

Riccati-parameter solutions of nonlinear second-order ODEs

It has been proven by Rosu and Cornejo-Pérez (Rosu and Cornejo-Pérez 2005 Phys. Rev. E 71 046607, Cornejo-Pérez and Rosu 2005 Prog. Theor. Phys. 114 533) that for some nonlinear second-order ODEs it is a very simple task to find one particular solution once the nonlinear equation is factorized with the use of two first-order differential operators. Here, it is shown that an interesting class of parametric solutions is easy to obtain if the proposed factorization has a particular form, which happily turns out to be the case in many problems of physical interest. The method that we exemplify with a few explicitly solved cases consists in using the general solution of the Riccati equation, which contributes with one parameter to this class of parametric solutions. For these nonlinear cases, the Riccati parameter serves as a ‘growth’ parameter from the trivial null solution up to the particular solution found through the factorization procedure.

A nonlocal phase-field system with inertial term

We study a phase-field system where the energy balance equation has the standard (parabolic) form, while the kinetic equation ruling the evolution of the order parameter X is a nonlocal and nonlinear second-order ODE. The main features of the latter equation are a space convolution term which models long-range interactions of particles and a singular configuration potential that forces X to take values in (-1,1). We first prove the global existence and uniqueness of a regular solution to a suitable initial and boundary value problem associated with the system. Then, we investigate its long time behavior from the point of view of ω-limits. In particular, using a nonsmooth version of the Lojasiewicz-Simon inequality, we show that the ω-limit of any trajectory contains one and only one stationary solution, provided that the configuration potential in the kinetic equation is convex and analytic.

An existence result for homoclinic solutions to a nonlinear second-order ODE through differential inequalities

In this paper an existence result of homoclinic solutions to a nonlinear second-order ODE is presented. To this end, a method based on differential inequalities is used.

A Two-Point Boundary-Value Problem for the Axial Shear of Hardening Isotropic Incompressible Nonlinearly Elastic Materials

The purpose of this research is to investigate a pure axial shear problem for the annular region between two concentric rigid cylinders occupied by an incompressible isotropic nonlinearly elastic material. Our main concern is with the subclass of these materials that exhibits hardening at large deformations. Such hardening at large strains is observed experimentally but is often not predicted by commonly used strain-energy densities. The particular axial shear problem that we investigate arises when the elastic material is perfectly bonded to an inner rigid circular core and to an outer rigid circular container. The deformation is driven by an axial pressure gradient. The problem is formulated as a two-point boundary-value problem for a second-order nonlinear ODE. For a broad class of incompressible isotropic hyperelastic materials, existence of smooth solutions is established on conversion to an initial value problem. We then specialize the material class to be of generalized neo-Hookean type, that is, where the strain-energy density depends only on the first invariant of the strain tensor. Two classes of such materials that exhibit hardening at large deformations are considered. The first class models limiting chain extensibility at the molecular level, and the second class is of power-law type. An interesting limiting case of the latter leads to an exponential strain-energy function commonly used in modeling biological materials. Some specific strain-energy densities from each class are examined in detail. Numerical boundary-value methods with quasilinearization for nonlinear second-order ODEs are used to illustrate a special feature of the solutions. It was previously shown by other authors that, for softening power-law materials, a boundary layer behavior is exhibited near the bonded surfaces: the axial displacement undergoes a sharp increase from zero at the endpoints and is slowly varying in the interior. We provide further numerical results regarding this phenomenon here. Our main concern is, however, with hardening materials. For such materials, the numerical results exhibit an interior localization at a location which is almost the midpoint between the boundaries. The axial displacement has a sharp change of slope which, in the limit of infinitely large pressure gradients, gives rise to a cusp in the displacement profile. The results highlight the contrasting behavior between softening and hardening rubber-like or biological materials.

Unidirectional steady flow of a viscoelastic fluid with a free surface

We study the steady flow of a second grade fluid down an open inclined channel. We formulate the mathematical problem, a mixed boundary value problem for the Laplacian with an unknown free boundary described by a nonlinear second-order ODE, and prove existence of a unique solution for small data using a contraction argument.

Level spacing functions and connection formulas for Painlevé V transcendent

The connection formulas for Painlevé V transcendents are applied for calculations of the Fredholm determinant of the kernel sin π(x−y)/π(x−y) on the finite interval ( t,− t). The level spacing functions, arising in the theory of random matrices as well as for the correlators of one-dimensional XY-model, are expressed via this Fredholm determinant. Its asymptotics for large t come from nonlinear second-order ODE with known initial conditions at t=0. The Bäcklund transformations reduce this ODE to the Painlevé V equation, while the isomonodromic deformation method (IDM) provides the explicit connection formulas for the necessary PV transcendent. The result generalizes the well-known Dyson asymptotic expansion for the one-level spacing function to a case of arbitrary many levels.

Linearization criteria for a system of second-order ordinary differential equations

Firstly, we prove two linearization criteria for a system of two second-order ordinary differential equations (odes). The first states that a system of two non-linear second-order odes is reducible via a point transformation to a linear system if and only if it admits the four-dimensional abelian Lie algebra L 4,1 : [X i,X j]=0, i, j=1,…, 4 . The second states that in order for a system of two non-linear second-order odes to be linearizable, it is necessary and sufficient that it admits the four-dimensional Lie algebra L 4,2 : [X i,X j]=0, [X i,X 4]=X i, i, j=1, 2, 3 . The approach used is constructive and enables one to explicitly work out the transformation that leads to linearization. Secondly, we give conditions under which a system of two second-order non-linear odes is reducible to the free particle equations x″=0, y″=0 . These linearization criteria are then generalized to a system of n( n>2) second-order odes. Finally, we give examples of how one can effect linearization for a system.

Strongly Nonlinear Second-Order ODEs with Rapidly Growing Terms

Strongly Nonlinear Second-Order ODEs with Rapidly Growing Terms

Heteroclinic Solutions for a System of Strongly Coupled ODEs

We use the implicit function theorem to prove an existence of a heteroclinic orbit to a system of two non-linear second-order ODEs. The perturbation is carried out around infinite value of a ‘coupling parameter’. The form of the system which is considered in this paper is related to the system defining travelling wave solutions in a two temperature model of the laser sustained plasma.

On finite displacements of curved annular elastic membranes without wrinkling

Axisymmetric deformations of curved annular elastic membranes subjected to vertical surface loads and radial edge loads or displacements are considered within Reissner’s finite-rotation theory of thin shells of revolution, assuming linear stress-strain relations. The principal stresses in the membrane are determined by the solutions of a nonlinear second-order ODE, dependent on a geodesic variable. Analytical methods are used in order to determine the range of those boundary data for which the solutions of the differential equation are wrinkle-free in the sense that both the radial and the circumferential stress components are nonnegative everywhere.

Strongly nonlinear second-order ODEs with unilateral conditions

Strongly nonlinear second-order ODEs with unilateral conditions

Wrinkle-free solutions of circular membrane problems

Axisymmetric deformations of a plane circular membrane subjected to an axial surface load are studied, with prescribed radial stresses or radial displacements at the edge. Considering the small-finite-deflection theory of Foppl-Hencky as well as a simplified version of Reissner's nonlinear theory of thin shells of revolution, the determination of the principal stresses in the membrane is shown to reduce to the solution of a nonlinear second-order ODE. In the Foppl case, the basic equation becomes singular when the membrane edge is free of traction. Nevertheless, existence and uniqueness of nonnegative solutions for zero edge traction is proved in both Foppl and Reissner models. Thereby, a limit curve for the radial displacement at the boundary is induced in the Reissner case, which subdivides the actual parameter range into complementary domains of existence and nonexistence of tensile solutions. The limit curve is studied, analytically and numerically. Finally, a maximum principle is established in order to determine the more restricted subdomain of those parameters which admit wrinkle-free solutions, i.e. solutions governed by a nonnegative radial and circumferential stress component.

Solvability of Neumann BV problems for nonlinear second-order ODEs which need not be solvable for the highest-order derivative

Solvability of Neumann BV problems for nonlinear second-order ODEs which need not be solvable for the highest-order derivative