Asymptotic behaviour of the solutions of second order functional differential equations

This paper presents integral criteria to determine the asymptotic behaviour of the solutions of second order nonlinear differential equations of the type y^″(x)+q(x)f(y(x))=0, with q(x)>0 and f(y) odd and positive for y>0, as x tends to +∞. It also compares them with the results obtained by Chanturia (1975) in [11] for the same problem.     

Perturbation of solutions of ordinary linear homogeneous differential equations of the second order

In this paper we give a method for peturbation of solutions of linear homogeneous differential equation of the second order.     

Liouvillian solutions of third order differential equations

The Kovacic algorithm and its improvements give explicit formulae for the Liouvillian solutions of second order linear differential equations. Algorithms for third order differential equations also exist, but the tools they use are more sophisticated and the computations more involved. In this paper we refine parts of the algorithm to find Liouvillian solutions of third order equations. We show that, except for four finite groups and a reduction to the second order case, it is possible to give a formula in the imprimitive case. We also give necessary conditions and several simplifications for the computation of the minimal polynomial for the remaining finite set of finite groups (or any known finite group) by extracting ramification information from the character table. Several examples have been constructed, illustrating the possibilities and limitations.     

Eigenvalue problems of second order impulsive differential equations

This paper is concerned with an eigenvalue problem for second order differential equations with impulse. The existence of a countably infinite set of eigenvalues and eigenfunctions is proved.     

Computer-assisted reduction of second order linear differential equations

So as to explore the possibilities of representing its solutions in terms of special functions, and using factorization techniques, a process is defined to decide whether a second order linear differential equation with polynomial coefficients can be brought to the hypergeometric or the confluent hypergeometric equation by a rational change of variable. In the first case, an upper bound has to be provided for the degree of the numerator of the rational function which defines the change of variable.     

Polynomial solutions to second order linear homogeneous ordinary differential equations. Properties and approximation

Second order linear homogeneous ODEs which generalize those considered in the orthogonal polynomial theory are studied. A method and an algorithm to approach their polynomial solutions are also given.     

Optimal existence conditions for second order periodic solutions of delay differential equations with upper and lower solutions in the reverse order

In this paper we show that the monotone iterative technique provides two monotone sequences that converge uniformly to extremal (periodic) solutions of second order delay differential equations without assuming properties of monotonicity in the nonlinear part. Moreover, we obtain optimal existence conditions with upper and lower solutions in the reverse order. Our results are new even for ordinary differential equations.     

Anti-periodic boundary value problems of second order impulsive differential equations

This paper discusses anti-periodic boundary value problems of second order impulsive differential equations. By using the method of upper and lower solutions coupled with the monotone iterative technique, new existence results of coupled solutions and uniqueness of problems are obtained.     

Riccati equations for second order spatially invariant partial differential systems

Recently, the class of spatially invariant systems was introduced with motivating examples of partial differential equations on an infinite domain. For these it was shown that by taking Fourier transforms, one obtains infinitely many finite-dimensional systems with a scalar parameter. The idea is that, for the LQR controller design for these systems, one can solve the parameterized LQR-Riccati equation pointwise. While for simple first order systems like the heat equations this approach works, for second order systems like wave or beam equations it is easy to construct examples for which this approach fails. Here we give a correct formulation for second order partial differential systems including wave and beam type equations.     

Stability of linear systems governed by second order vector differential equations

We consider linear non-autonomous systems governed by second order vector ordinary differential equations. Explicit stability conditions are derived.     

Multiple solutions to semipositone Dirichlet boundary value problems with singular dependent nonlinearities for second order three-point differential equations

In this paper we present some new existence results for a singular semipositone Dirichlet boundary value problem for second order three-point differential equations by using the upper and lower solutions method and the fixed point theorem in cones.     

Iterative solutions to non-linear fourth order differential equations through multi-integral methods

Implicit solution to a certain class of non-linear fourth order differential equation is presented. Bounds on sup norm for the derivative of a certain function f involved in the equation with different boundary values, are computed. These bounds provide the rate of convergence of the iterative sequence of approximate solutions obtained by Picard method, to the exact solution.