Differential Equations Of Arbitrary Order Research Articles

On the Asymptotic and Numerical Solution of Linear Ordinary Differential Equations

An investigation is made of the nature of asymptotic solutions of homogeneous linear differential equations of arbitrary order in the neighborhood of a singularity of unit rank. We introduce a classification of the solutions into two types, explicit and implicit. For the former there exists a sector on which the solution is dominated by all independent solutions as the singularity is approached. No such sector exists for implicit solutions. In consequence, the two types of solution have different uniqueness properties. Another difference is that error bounds for the asymptotic expansions of explicit solutions are generally stronger than those for implicit solutions. We also investigate the computation of the solutions by numerical integration of the differential equation. It is shown that it is the exception rather than the rule for the integration process to be stable, particularly so for implicit solutions. To overcome these instabilities we develop boundary-value methods, complete with error analysis. Numerical examples illustrate the computation of both explicit and implicit solutions, and also the associated Stokes multipliers.

New ray (characteristic) equations for the equations of physics; Hamilton - Jacobi theory in more-dimensional space

The geometrical optical concept of rays is generalized for fields generated by any system of coupled non-linear partial differential equations of arbitrary order like the vector wave equations for (non)linear media, the Maxwell equations, the equations of magneto-hydrodynamics, the system of continuity (transport) equations for an inhomogeneous semiconductor, etc. It is shown that the pertinent solutions of such a system of partial differential equations can be obtained exactly from a set of ordinary non-linear differential equations, the so-called characteristic or ray equations. These characteristic (ray) equations are connected with the key equation of the theory, the generalized Hamilton - Jacobi equation. The generalized Hamilton - Jacobi equation is a first-order non-linear partial differential equation, a special case of which is known to be of great importance for classical mechanics. This equivalence between partial and ordinary differential equations implies an enormous simplification for the numerical evaluation of the original problem, as solving partial differential equations is much more difficult than solving ordinary differential equations. From the theoretical point of view this equivalence between partial and ordinary differential equations is also interesting for various reasons. We mention, for example, that interesting properties like stability, soliton behaviour, decay, growth, etc of the solutions of the field equations of physics can be directly deduced from the relatively easily obtained behaviour of the solutions of the ordinary non-linear differential equations for the `rays'. As an example of the potential powers of this new method we analyse the non-linear Schrodinger equation and derive its soliton solutions.

Algorithms and methods in differential algebra

Founded by J.F. Ritt, Differential Algebra is a true part of Algebra so that constructive and algorithmic problems and methods appear in this field. In this talk, I do not intend to give an exhaustive survey of algorithmic aspects of Differential Algebra but I only propose some examples to give an insight of the state of knowledge in this domain. Some problems are known to have an effective solution, others have an efficient effective solution which is implemented in recent computer algebra systems, and the decidability of some others is still an open question, which does not prevent computations from leading to interesting results. Liouville's theory of integration in finite terms and Risch's theorem are examples of problems that computer algebra systems now deal with very efficiently (implementation work by M Bronstein). In what concerns linear differential equations of arbitrary order, a basis for the vector space of all liouvillian solutions can “in principle” be computed effectively thanks to a theorem of Singer's [17, 22]; the complexity bound is actually awful and a lot of work is done or in progress, especially by M. Singer and F. Ulmer, to give realistic algorithms [20, 21] for third-order linear differential equations. Existence of liouvillian first integrals is a way to make precise the notion of integrability of vector fields. Even in the simplest case of three-dimensional polynomial vector fields, no decision procedure is known for this existence. Nevertheless, explicit computations with computer algebra yield interesting solutions for special examples. In this case, the process of looking for so-called Darboux curves can only be called a method but not an algorithm; for a given degree, this search is a classical algebraic elimination process but no bound is known on the degree of the candidate polynomials. This paper insists on the search of liouvillian first integrals of polynomial vector fields and a new result is given: the generic absence of such liouvillian first integrals for factorisable polynomial vector fields in three variables.

Resonance and non-resonance in a problem of boundedness

This paper studies the existence of bounded solutions of a forced non-linear differential equation of arbitrary order. Necessary and sufficient conditions for the existence of such solutions are obtained. These results are inspired by classical results on the periodic problem, both in the resonant and non-resonant cases.

Theory of second- and higher-order stochastic processes

This paper presents a general approach to linear stochastic processes driven by various random noises. Mathematically, such processes are described by linear stochastic differential equations of arbitrary order [the simplest nontrivial example is x\ifmmode\ddot\else\textasciidieresis\fi{}=R(t), where R(t) is not a Gaussian white noise]. The stochastic process is discretized into n time steps, all possible realizations are summed up, and the continuum limit is taken. This procedure often yields closed form formulas for the joint probability distributions. Completely worked out examples include all Gaussian random forces and a large class of Markovian (non-Gaussian) forces. This approach is also useful for deriving Fokker-Planck equations for the probability distribution functions. This is worked out for Gaussian noises and for the Markovian dichotomous noise.

On rational approximations of values of a certain class of entire functions

A sharp lower estimate is proved for the measure of irrationality of the values of -functions satisfying a system of linear differential equations of arbitrary order at a rational point.

Asymptotic behavior of nonoscillatory solutions of neutral delay differential equations of arbitrary order

Asymptotic behavior of nonoscillatory solutions of neutral delay differential equations of arbitrary order

On variational principles for nonlinear partial differential equations in complex spaces

The necessary and sufficient conditions for an abstract equation in a complex Banach space to admit a variational principle, which one of us described in a previous paper, are here applied to a system of nonlinear partial differential equations of arbitrary order in a complexL2 space. More specific and helpful conditions are thus obtained. General consequences and special implications for some classes of nonlinear Schrondinger equations are also discussed and differences from an analogous result valid only in real spaces are put in evidence.

Biunitary transformations and ordinary differential equations.—I

We reformulate the theory of ordinary differential equations of arbitrary order with nonconstant coefficients, using the formalism of non-Hermitian operators. In particular, exploiting the technique of dissipative quantum mechanics, we show that the solution of the equations can be written in terms of a nonunitary evolution operator. Furthermore, we point out that the solution of the adjoint equations can be derived from an associated biunitary operator. We show that a number of invariants, not previously discussed, exhists. Finally, we prove that the method allows the search for approximate solutions that can be used in many physical problems.

Biunitary transformations and ordinary differential equations.—II

We use the results of a recent reformulation of the theory of arbitrary-order differential equations in terms of non-Hermitian operators to show that the invariant binorm is associated to a generalized Courant-Snyder invariant. Furthermore, we indicate the existence of higher-order invariants associated to the Casimir operators of the group, utilized to treat higher-order equations. We also discuss the intrinsic supersymmetric nature of the theory developed. Finally, we show the relevance of the proposed mathematical technique to the design of fiberoptics transport systems.

The Dirichlet Problem for Some Semilinear Elliptic Differential Equations of Arbitrary Order

The Dirichlet Problem for Some Semilinear Elliptic Differential Equations of Arbitrary Order

On the existence of local and global Lagrangians for ordinary differential equations

Necessary and sufficient conditions for the existence of local and global Lagrangians for ordinary differential equations of arbitrary order are described in terms of the geometry of higher-order tangent bundles. The results are applied to the study of gauge invariant differential equations and the second-order differential equation defined by the (2+1)-dimensional Yang-Mills Lagrangian with the Chern-Simons term is discussed.

On the oscillation theory of periodic linear differential equations

For equations in a broad class of linear differential equations of arbitrary order having periodic coefficients, we set forth a procedure for determining large regions in the plane in which no solution f(z) ≢ 0 can have infinitely many zeros. This permits us to determine locations in the plane where the zeros of a solution must be concentrated. Our results apply to higher-order analogues of the well-known Mathieu equation. The class of equations we treat has been investigated in several recent papers (e.g. [6, 7, 8, 9]) from the point of view of determining the frequency of zeros of the solutions

On Complex Oscillation Theory

In this paper we obtain new results on the location of complex Zeros of soulutions of linear differential equations of arbitrary order whose coofficients are rational funcitons.

Unbounded nonoscillatory solutions of nonlinear ordinary differential equations of arbitrary order

On considere des equations sous lineaires de la forme y (n) +σf(t,y,y',...y (n−1 )=0 ou n≥2, σ=±1, et f:[0,∞)×R n →R est une fonction continue telle que yof(t,y 0 ,y 1 ,…,y n−1 )≥0 pour (t,y 0 ,y 1 ,…,y n−1 )∈[0,∞)×R n

Zur Verallgemeinerung von W�rmepolynomen

The investigations ofRosenbloom-Widder [13] on expansions in terms of heat polynomials have been generalized for partial differential equations of second order with singular coefficients, equations of higher order etc. In the present paper we introduce a new method for the discussion of such problems. The technique is related to the treatment of the Cauchy problem on the base of the Ovcyannikov theorem. The method is used for the discussion of two classes of partial differential equations of arbitrary order.

Extendable and nonextendable solutions of a nonlinear differential equation of arbitrary order

Extendable and nonextendable solutions of a nonlinear differential equation of arbitrary order

Some comparison criteria in oscillation theory

AbstractThe purpose of this paper is to establish comparison criteria, by which the oscillatory and asymptotic behavior of linear retarded differential equations of arbitrary order is inherited from the oscillation of an associated second order linear ordinary differential equation. These criteria are new even in the case of ordinary differential equations.

Nonoscillation theorems for functional differential equations of arbitrary order

The authors give sufficient conditions for all oscillatory solutions of a sublinear forced higher order nonlinear functional differential equation to converge to zero. They then prove a nonoscillation theorem for such equations. A few intermediate results are also obtained.

Stability of solutions of linear differential equations of arbitrary orders

Stability of solutions of linear differential equations of arbitrary orders