The Inverse Problem of the Calculus of Variations for Sixth- and Eighth-order Scalar Ordinary Differential Equations

On the equation manifold of the 2nth-order scalar ordinary differential equation, n3,

Helmholtz's inverse problem of the discrete calculus of variations

We derive the discrete version of the classical Helmholtz's condition. Precisely, we state a theorem characterizing second-order finite difference equations admitting a Lagrangian formulation. Moreover, in the affirmative case, we provide the class of all possible Lagrangian formulations.     

On the self-adjoint nonlinear eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities

Certain properties of the nonlinear self-adjoint eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities are examined. Under certain assumptions on the way in which the matrix of the system and the matrix specifying the boundary condition at a regular point depend on the spectral parameter, a numerical method is proposed for determining the number of eigenvalues lying on a prescribed interval of the spectral parameter.     

General nonlinear self-adjoint eigenvalue problem for systems of ordinary differential equations

The general nonlinear self-adjoint eigenvalue problem for systems of ordinary differential equations is considered. A method is proposed for reducing the problem to one for a Hamiltonian system. Results for Hamiltonian systems previously obtained by the authors are extended to this system.     

Computation of local symmetries of second-order ordinary differential equations by the Cartan equivalence method

The Cartan equivalence method is used to find out if a given equation has a nontrivial Lie group of point symmetries. In particular, we compute invariants that permit one to recognize equations with a three-dimensional symmetry group. An effective method to transform the Lie system (the system of partial differential equations to be satisfied by the infinitesimal point symmetries) into a formally integrable form is given. For equations with a three-dimensional symmetry group, the formally integrable form of the Lie system is found explicitly. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 75–91, July, 1996.     

A new HPM for ordinary differential equations

In this paper, we introduce a new version of the homotopy perturbation method (NHPM) that efficiently solves linear and non‐linear ordinary differential equations. Several examples, including Euler‐Lagrange, Bernoulli and Ricatti differential equations, are given to demonstrate the efficiency of the new method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

On certain properties of a nonlinear eigenvalue problem for Hamiltonian systems of ordinary differential equations

Properties of the eigenvalues are examined in a nonlinear self-adjoint eigenvalue problem for linear Hamiltonian systems of ordinary differential equations. In particular, it is proved that, under certain assumptions, every eigenvalue is isolated and there exists an eigenvalue with any prescribed index.     

Moving averages of ordinary differential equations via convolution

We introduce an averaging framework, where the solution of a time-varying equation with a small amplitude is approximated by the solution of a slowly-varying auxiliary system, generated by convolving the original equation with a kernel function. The effect of the convolution is smoothing of the equation, thus, making it more amenable to numerical computations. We present tight results on the approximation error for general classes of vector fields and kernels.     

Nondegeneracy and uniqueness of positive solutions for Robin problem of second order ordinary differential equations and its applications

This article studies positive solutions of Robin problem for semi-linear second order ordinary differential equations. Nondegeneracy and uniqueness results are proven for homogeneous differential equations. Necessary and sufficient conditions for the existence of one or two positive solutions for inhomogeneous differential equations or differential equations with concave-convex nonlinearities are obtained by making use of the nondegeneracy and uniqueness results for positive solutions of homogeneous differential equations.     

Smoothness of generalized solutions of boundary value problems for certain degenerate nonlinear ordinary differential equations

The variational method is applied to the study of a boundary value problem of the first kind for a class of nonlinear ordinary differential equations of order 2r with strong degeneracy at the endpoints of the interval (a, b). An inequality is obtained in which the norm of the solutionU of the problem under study in the sense ofW _p, ^r (a, b) is estimated from above by the norms of the given functions (x) andF(x).Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 882–890, June, 1998.     

On the index of the boundary value problem for a homogeneous Hamiltonian system of differential equations

The index of the homogeneous self-adjoint boundary value problem for the Hamiltonian systems of ordinary differential equations is introduced. It is assumed that the system has a nontrivial solution. The relationship between the index of an eigenvalue of the nonlinear eigenvalue problem and the index of the corresponding homogeneous problem is established. Properties of the index of the problem and those of the eigenvalue are examined.     

Numerical methods based on multipoint Hermite interpolating polynomials for solving the Cauchy problem for stiff systems of ordinary differential equations

Families of A-, L-, and L(δ)-stable methods are constructed for solving the Cauchy problem for a system of ordinary differential equations (ODEs). The L(δ)-stability of a method with a parameter δ ∈ (0, 1) is defined. The methods are based on the representation of the right-hand sides of an ODE system at the step h in terms of two-or three-point Hermite interpolating polynomials. Comparative results are reported for some test problems. The multipoint Hermite interpolating polynomials are used to derive formulas for evaluating definite integrals. Error estimates are given.     

Invariant characterization of scalar third‐order ODEs that admit the maximal point symmetry Lie algebra

The Cartan equivalence method is used to deduce an invariant characterization of the scalar third‐order ordinary differential equation , which admits the maximal 7‐dimensional point symmetry Lie algebra. The method provides auxiliary functions that can be used to efficiently obtain the point transformation that does the reduction to the simplest linear equation . Moreover, examples are given to illustrate the method.     

A block-by-block method for the impulsive fractional ordinary differential equations

In this paper, a block-by-block numerical method is constructed for the impulsive fractional ordinary differential equations (IFODEs). Firstly, the stability and convergence analysis of the scheme are established. Secondly, the numerical solution which converges to the exact solution with order $3+\gamma$ for $0<\gamma<1$ is proved, where $\gamma$ is the order of the fractional derivative. Finally, a series of numerical examples are carried out to verify the correctness of the theoretical analysis.     

The inverse problem of the calculus of variations: The use of geometrical calculus in Douglas's analysis

The main objective of this paper is to work out a full-scale application of the integrability analysis of the inverse problem of the calculus of variations, as developed in recent papers by Sarlet and Crampin. For this purpose, the celebrated work of Douglas on systems with two degrees of freedom is taken as the reference model. It is shown that the coordinate-free, geometrical calculus used in Sarlet and Crampin's general theoretical developments provides effective tools also to do the practical calculations. The result is not only that all subcases distinguished by Douglas can be given a more intrinsic characterization, but also that in most of the cases, the calculations can be carried out in a more efficient way and often lead to sharper conclusions.

     

Conditional linearizability criteria for a system of third-order ordinary differential equations

We provide linearizability criteria for a class of systems of two third-order ordinary differential equations that is cubically nonlinear in the first derivative, by differentiating a system of second-order quadratically nonlinear ordinary differential equations and using the original system to replace the second derivatives. The procedure developed splits into two cases: those for which the coefficients are constant and those for which they are variables. Both cases are discussed and examples given.     

The construction of order 4 DIMSIMs for ordinary differential equations

Diagonally Implicit Multistage Integration Methods (DIMSIMs) of type 1 and 2 have considerable potential as numerical algorithms for ordinary differential equations. The aim of this paper is to construct such methods of order 4 of type 1 and 2, which completes the set for orders 1–8.