Higher Order Ordinary Differential Equations Research Articles

Positive solutions of m-point boundary value problems for higher order ordinary differential equations

In this paper, we study m-point boundary value problems for higher order ordinary differential equation { y ( 2 n ) ( t ) = f ( t , y ( t ) , y ″ ( t ) , … , y ( 2 ( n − 1 ) ) ( t ) ) , 0 ≤ t ≤ 1 , y ( 2 i ) ( 0 ) = 0 , y ( 2 i ) ( 1 ) = ∑ j = 1 m − 2 k i j y ( 2 i ) ( ξ j ) , 0 ≤ i ≤ n − 1 , where f is allowed to change sign, and 0 = ξ 0 < ξ 1 < ξ 2 < ⋯ < ξ m − 2 < ξ m − 1 = 1 . We show sufficient conditions for the existence of at least two positive solutions by applying a new fixed point theorem in cones and the associated Green’s function. In particular, the second positive solutions for the above problem is not concave.

Exact solutions of generalized Zakharov and Ginzburg–Landau equations

Abstract By using the homogeneous balance principle, the exact solutions of the generalized Zakharov equations and generalized Ginzburg–Landau equation are obtained with the aid of a set of subsidiary higher-order ordinary differential equations (sub-equations for short).

Parallel two-Point Explicit Block method for solving high-order Ordinary Differential Equations

The derivation of a new parallel two-Point Explicit Block (2PEB) method for solving dth-order Ordinary Differential Equations (ODEs) directly is presented. The computational advantages of using this technique over the conventional one-point method are discussed. Numerical results suggest that the parallel 2PEB method is superior when used for solving second-order ODEs directly with small step size.

Exact solutions to two higher order nonlinear Schrödinger equations

Abstract Using the homogeneous balance principle and F-expansion method, the exact solutions to two higher order nonlinear Schrodinger equations which describe the propagation of femtosecond pulses in nonlinear fibres are obtained with the aid of a set of subsidiary higher order ordinary differential equations (sub-equations for short).

Subharmonic bifurcation from infinity

We are concerned with a subharmonic bifurcation from infinity for scalar higher order ordinary differential equations. The equations contain principal linear parts depending on a scalar parameter, 2 π-periodic forcing terms, and continuous nonlinearities with saturation. We suggest sufficient conditions for the existence of subharmonics (i.e., periodic solutions of multiple periods 2 π n ) with arbitrarily large amplitudes and periods. We prove that this type of the subharmonic bifurcation occurs whenever a pair of simple roots of the characteristic polynomial crosses the imaginary axis at the points ± α i with an irrational α. Under some further assumptions, we estimate asymptotically the parameter intervals, where large subharmonics of periods 2 π n exist. These assumptions relate the quality of the Diophantine approximations of α, the rate of convergence of the nonlinearity to its limits at infinity, and the smoothness of the forcing term.

WITHDRAWN: A Paley–Wiener proven uniqueness theorem in the inverse spectral theory of higher order ordinary differential equations

WITHDRAWN: A Paley–Wiener proven uniqueness theorem in the inverse spectral theory of higher order ordinary differential equations

A second order ODE is sufficient for modelling of many periodic signals

Which is the minimum order an autonomous non-linear ordinary differential equation (ODE) needs to have to be able to model a periodic signal? This question is motivated by recent research on periodic signal analysis, where non-linear ODEs are used as models. The results presented here show that a second order ODE is sufficient for a large class of periodic signals. More precisely, conditions on a periodic signal are established that imply the existence of an ODE that has the periodic signal as a solution. A criterion that characterizes the above class of periodic signals by means of the overtone contents of the signals is also presented. The reason why higher order ODEs are sometimes needed is illustrated with geometric arguments. Extensions of the theoretical analysis to cases with orders higher than two are developed using this insight.

A UNIQUENESS THEOREM IN THE INVERSE SPECTRAL THEORY OF A CERTAIN HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATION USING PALEY–WIENER METHODS

The paper examines a higher-order ordinary differential equation of the form P [ u ] : = ∑ j , k = 0 m D j a j k D k u = λ u , x ∈ [ 0 , b ) where D = i(d/dx), and where the coefficients ajk, j,k ∈ [0,m], with amm = 1, satisfy certain regularity conditions and are chosen so that the matrix (ajk) is hermitean. It is also assumed that m > 1. More precisely, it is proved, using Paley–Wiener methods, that the corresponding spectral measure determines the equation up to conjugation by a function of modulus 1. The paper also discusses under which additional conditions the spectral measure uniquely determines the coefficients ajk, j,k ∈ [0,m], j + k ≠ 2m, as well as b and the boundary conditions at 0 and at b (if any).

Deformable models of body organs for surgery simulation

Often the regression function appearing in fields like economics, engineering, and biomedical sciences obeys a system of higher-order ordinary differential equations (ODEs). The equations are usually not analytically solvable. We are interested in inferring on the unknown parameters appearing in such equations. Parameter estimation in first-order ODE models has been well investigated. Bhaumik and Ghosal (2015) considered a two-step Bayesian approach by putting a finite random series prior on the regression function using a B-spline basis. The posterior distribution of the parameter vector is induced from that of the regression function. Although this approach is computationally fast, the Bayes estimator is not asymptotically efficient. Bhaumik and Ghosal (2016) remedied this by directly considering the distance between the function in the nonparametric model and a Runge–Kutta (RK4) approximate solution of the ODE while inducing the posterior distribution on the parameter. They also studied the convergence properties of the Bayesian method based on the approximate likelihood obtained by the RK4 method. In this paper, we extend these ideas to the higher-order ODE model and establish Bernstein–von Mises theorems for the posterior distribution of the parameter vector for each method with n−1/2 contraction rate.

Virtual turning points and bifurcation of Stokes curves for higher order ordinary differential equations

For a higher order linear ordinary differential operator P, its Stokes curve bifurcates in general when it hits another turning point of P. This phenomenon is most neatly understandable by taking into account Stokes curves emanating from virtual turning points, together with those from ordinary turning points. This understanding of the bifurcation of a Stokes curve plays an important role in resolving a paradox recently found in the Noumi-Yamada system, a system of linear differential equations associated with the fourth Painleve equation.

Solving high order ordinary differential equations with radial basis function networks

This paper is concerned with the application of radial basis function networks (RBFNs) for numerical solution of high order ordinary differential equations (ODEs). Two unsymmetric RBF collocation schemes, namely the usual direct approach based on a differentiation process and the proposed indirect approach based on an integration process, are developed to solve high order ODEs directly and the latter is found to be considerably superior to the former. Good accuracy and high rate of convergence are obtained with the proposed indirect method. Copyright © 2004 John Wiley & Sons, Ltd.

Solvability of multipoint boundary value problems at resonance for higher-order ordinary differential equations

Let f : [0, 1]× R n → R be a continuous function and e ∈ L 1[0, 1]. Let β j (1 ≤ j ≤ m − 2) ∈ R, 0 < η 1 < η 2 < … < η m−2 < 1 be given. This paper is concerned with the existence of solutions for the following n th-order multipoint boundary value problems at resonance case x ( n ) ( t ) = f ( t , x ( t ) , x ′ ( t ) , … , x ( n − 1 ) ( t ) ) + e ( t ) , t ∈ ( 0 , 1 ) , x ( 0 ) = x ′ ( 0 ) = ⋯ = x ( n − 2 ) ( 0 ) = 0 , x ( 1 ) = ∑ j = 1 m − 2 β j x ( η j ) , and x ( n ) ( t ) = f ( t , x ( t ) , x ′ ( t ) , … , x ( n − 1 ) ( t ) ) + e ( t ) , t ∈ ( 0 , 1 ) , x ( 0 ) = x ′ ( 0 ) = ⋯ = x ( n − 2 ) ( 0 ) = 0 , x ′ ( 1 ) = ∑ j = 1 m − 2 β j x ′ ( η j ) . Some existence results are obtained by using the coincidence degree theory of Mawhin.

Standard forms of elliptic integrals and their applications to nonlinear evolution equations

Abstract Many nonlinear evolution equations can be converted into first- or high-order ordinary differential equations. In this paper, we illustrate a connection between the elliptic integrals and some nonlinear evolution equations such as the two-dimensional Boussinesq equation, and demonstrate that traveling wave solutions and periodic wave solutions to these nonlinear equations can be obtained by means of the standard forms of elliptic integrals directly.

Remark on the Hopf Bifurcation Theorem

A simple generalization of the Hopf Bifurcation Theorem for scalar higher order ordinary differential equations is suggested. We study the degenerate case where several roots of the characteristic polynomial cross the imaginary axis at the same point for some value λ0 of the parameter λ. The main result is that if N1 roots cross the imaginary axis from the left to the right and N2 roots cross it from the right to the left, then λ0 is a Hopf bifurcation point whenever N1 ≠ N2. In particular, in the classical Hopf Bifurcation Theorem the numbers Nj are 0 and 1. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Bäcklund transformations for higher order Painlevé equations

We present a new generalized algorithm which allows the construction of Backlund transformations (BTs) for higher order ordinary differential equations (ODEs). This algorithm is based on the idea of seeking transformations that preserve the Painleve property, and is applied here to ODEs of various orders in order to recover, amongst others, their auto-BTs. Of the ODEs considered here, one is seen to be of particular interest because it allows us to show that auto-BTs can be obtained in various ways, i.e. not only by using the severest of the possible restrictions of our algorithm.

Particular Solutions for a Class of ODE Related to the L-Exponential Functions

Abstract Particular solutions of a class of higher order ordinary differential equations, with non-constant coefficients, are determined by using the properties of the Laguerre exponentials functions introduced by G. Dattoli and P. E. Ricci in [Georgian Math. J. 10: 481–494, 2003]

Rational Chebyshev tau method for solving higher-order ordinary differential equations

An approximate method for solving higher-order ordinary differential equations is proposed. The approach is based on a rational Chebyshev (RC) tau method. The operational matrices of the derivative and product of RC functions are presented. These matrices together with the tau method are utilized to reduce the solution of the higher-order ordinary differential equations to the solution of a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Self-Invariant Contact Symmetries

Every smooth second-order scalar ordinary differential equation (ODE) that is solved for the highest derivative has an infinite-dimensional Lie group of contact symmetries. However, symmetries other than point symmetries are generally difficult to find and use. This paper deals with a class of one-parameter Lie groups of contact symmetries that can be found and used. These symmetry groups have a characteristic function that is invariant under the group action; for this reason, they are called ‘self-invariant.’ Once such symmetries have been found, they may be used for reduction of order; a straightforward method to accomplish this is described. For some ODEs with a one-parameter group of point symmetries, it is necessary to use self-invariant contact symmetries before the point symmetries (in order to take advantage of the solvability of the Lie algebra). The techniques presented here are suitable for use in computer algebra packages. They are also applicable to higher-order ODEs

The dynamic evolutionary modeling of HODEs for time series prediction

The prediction of future values of a time series generated by a chaotic dynamic system is an extremely challenging task. Besides some methods used in traditional time series analysis, a number of nonlinear prediction methods have been developed for time series prediction, especially the evolutionary algorithms. Many researchers have built various models by utilizing different evolutionary techniques. Different from those available models, this paper presents a new idea for modeling time series using higher-order ordinary differential equations (HODEs) models. Accordingly, a dynamic hybrid evolutionary modeling algorithm called DHEMA is proposed to approach this task. Its main idea is to embed a genetic algorithm (GA) into genetic programming (GP) where GP is employed to optimize the structure of a model, while a GA is employed to optimize its parameters. By running the DHEMA, the modeling and predicting processes can be carried on successively and dynamically with the renewing of observed data. Two practical examples are used to examine the effectiveness of the algorithm in performing the prediction task of time series whose experimental results are compared with those of standard GP.

First integrals and reduction of a class of nonlinear higher order ordinary differential equations

We propose a method for constructing first integrals of higher order ordinary differential equations. In particular third, fourth and fifth order equations of the form x (n)=h(x,x (n−1)) x ̇ are considered. The relation of the proposed method to local and nonlocal symmetries are discussed.