The expansion problems associated with regular differential systems of the second order

The highly fruitful conception of adjoint pairs of linear systems, each consisting of a homogeneous linear differential equation of the nth order and of n homogeneous linear boundary conditions, was first introduced in an explicit and general manner by BIRKHOFF.t That we have here a conception of fundamental importance is hardly yet generally appreciated. It is my purpose in the present paper to make an application of this idea to an important question in the theory of linear boundary problems which has so far been treated only in very special cases.+ I will mention in passing that the results may also be obtained by the use of integral equations; though, if we wish to include all cases, not perhaps so simply or obviously as might at first sight appear. In § 1 the conception of adjoint system and a few other fundamental matters concerning linear systems are explained and a few preliminary questions disposed of. In § 2 come the main results of the paper, namely Theorems I and II, which apply to the case of complex as well as to that of real independent variNble. These theorems are then applied in some detail to the equation of the second order, whereby the relation to MASON'S results becomes apparent. In § 3 an extension to systems of equations of any order is briefly indicated; and a somewhat generalized type of differential equation is also considered. Finally, ill § 4, the conception of adjoint system is extended to the case in which the number of boundary conditions is not the same as the order of the differential equation.

TABLE OF CONTENTS INTRODUCTION. Historical background. Summary. Notation and terminology. CHAPTER I. ALGEBRAIC MATRIC GROUPS. 1. Reducibility of sets of matrices. 2. Algebraic matric groups. 3. Jordan-Holder-Schreier theorem. 4. Commutator groups. 5. Solvable algebraic matric groups. 6. Anticompact and quasicompact algebraic matric groups. 7. Reducibility to triangular form. 8. Algebraic matric groups with certain types of normal chains. CHAPTER II. SOME P.ESULTS FROM THE THEORY OF ALGEBRAIC DIFFERENTIAL EQUATIONS. 9. Differential rings, fields, and ideals. 10. Differential polynomials. 11. Solutions. 12. Relative isomorphisms. 13. Order. 14. Dependence. 15. Homogeneous linear ordinary differential equations. CHAPTER III. NORMAL DIFFERENTIAL EXTENSION FIELDS. 16. Normal differential extension fields. CHAPTER IV. PICCARD-VESSIOT EXTENSIONS. 17. Picard-Vessiot extensions and their isomorphisms. 18. Normality. 19. Characterization of G. 20. Dimension. 21. Adjunction of new elements. 22. Linear reducibility of L(y). CHAPTER V. LIOUVILLIAN EXTENSIONS. 23. Integrals and exponentials of intgrals. 24. Liouvillian extensions. 25. The principal theorem and some consequences. 26. The proof, first half. 27. The proof, second half. REFERENCES INTRODUCTION

One the Solutions of Ordinary Linear Homogeneous Differential Equations of the Third Order

A method for the numerical solution of the n -point boundary value problem for homogeneous linear ordinary differential equations is developed. The method requires two Runge-Kutta integrations over the interval under consideration and the solution of a linear system of equations with n -1 unknowns.

Recently much work has been devoted to nonlocal problems. However, very little has been accomplished in the literature for nonlocal initial problems with nonlinear boundary conditions. It is the purpose of this paper to prove the existence results for solutions to a semilinear parabolic PDE with linear homogeneous boundary conditions, and to other ones with nonlinear boundary conditions, provided the ordered upper and lower solutions are given. Semigroup, fractional order function spaces and generalized Poincare operators play an important role in proving the existence of solutions.

High regularity of the solutions of the telegraph system subjected to nonlinear boundary conditions

Let be given a homogeneous linear ordinary differential equation with coefficients which are polynomial over a field K. The well known Frobenius’ algorithm [7] computes the fundamental solutions system in the neighborhood of regular singularities. These solutions are from \(\mathop K\limits^ \sim \) [[x]][ln(x)] where \(\mathop K\limits^ \sim \) is an algebraic extension of K. We present a new method based on the Frobenius’ algorithm. This method allows to build solutions with hypergeometric coefficients of Laurent or Puiseux series.

We consider the one-dimensional linear free space Schrödinger equation on a bounded interval subject to homogeneous linear boundary conditions. We prove that, in the case of pseudoperiodic boundary conditions, the solution of the initial-boundary value problem exhibits the phenomenon of revival at specific (“rational”) times, meaning that it is a linear combination of a certain number of copies of the initial datum. Equivalently, the fundamental solution at these times is a finite linear combination of delta functions. At other (“irrational”) times, for suitably rough initial data, e.g., a step or more general piecewise constant function, the solution exhibits a continuous but fractal-like profile. Further, we express the solution for general homogenous linear boundary conditions in terms of numerically computable eigenfunctions. Alternative solution formulas are derived using the Unified Transform Method (UTM), that can prove useful in more general situations. We then investigate the effects of general linear boundary conditions, including Robin, and find novel “dissipative” revivals in the case of energy decreasing conditions.

Complex-valued functions defined on a finite interval [a, b] generalizing power functions of the type (x−x0)n for n ≥ 0 are studied. These functions called Φ-generalized powers, Φ being a given nonzero complex-valued function on the interval, were considered to construct a general solution representation of the Sturm–Liouville equation in terms of the spectral parameter [V. V. Kravchenko and R. M. Porter, Math. Methods Appl. Sci. 33(4), 459–468 (2010)]. The Φ-generalized powers can be considered as natural basis functions for the one-dimensional supersymmetric quantum mechanics systems taking Φ=ψ02, where the function ψ0(x) is the ground state wave function of one of the supersymmetric scalar Hamiltonians. Several properties are obtained such as Φ-symmetric conjugate and antisymmetry of the Φ-generalized powers, a supersymmetric binomial identity for these functions, a supersymmetric Pythagorean elliptic (hyperbolic) identity involving four Φ-trigonometric (Φ-hyperbolic) functions, as well as a supersymmetric Taylor series expressed in terms of the Φ-derivatives. We show that the first n Φ-generalized powers are a fundamental set of solutions associated with nonconstant coefficient homogeneous linear ordinary differential equations of order n + 1. Finally, we present a general solution representation of the stationary Schrödinger equation in terms of geometric series where the Volterra compositions of the first type are considered.

We explain how the space of linearly recursive sequences over a field can be considered as a Hopf algebra. The algebra structure is that of divided-power sequences, so we concentrate on the perhaps lesser-known coalgebra (diagonalization) structure. Such a sequence satisfies a minimal recursive relation, whose solution space is the subcoalgebra generated by the sequence. We discuss possible bases for the solution space from the point of view of diagonalization. In particular, we give an algorithm for diagonalizing a sequence in terms of the basis of the coalgebra it generates formed by its images under the difference-operator shift. The computation involves inverting the Hankel matrix of the sequence. We stress the classical connection (say over the real or complex numbers) with formal power series and the theory of linear homogeneous ordinary differential equations. It is hoped that this exposition will encourage the use of Hopf algebraic ideas in the study of certain combinatorial areas of mathematics.

Hurwitz series as formal functions

In the framework of instantaneous approximations to the Bethe-Salpeter formalism for the description of bound states within quantum field theories, depending on the Lorentz structure of the Bethe-Salpeter interaction kernel the solutions of the full Salpeter with some confining interaction may exhibit certain instabilities, which are possibly related to the Klein paradox and signal the decay of states assumed to be bound by the confining interactions. They are observed in numerical (variational) studies of the Salpeter equation. The presumably simplest scenario allowing for the fully analytic investigation of this problem is set by the reduced Salpeter with interaction. In this case, Salpeter's integral simplifies to either an algebraic relation or a second-order homogeneous linear ordinary differential equation, immediately accessible to standard techniques. There one may hope to be able to decide unambiguously whether this setting poses a well-defined eigenvalue problem the solutions of which correspond to stable bound states associated to real energy eigenvalues bounded from below. By analytical spectral analysis the bound-state solutions of this harmonic-oscillator reduced Salpeter equation can be shown to be free of such instabilities.

The eigentensors of an arbitrary second-order tensor and their quality analyses

An alternative linear flow model and boundary integral flow representation for ship motions in regular waves

CHAPTER II - LINEAR DIFFERENTIAL EQUATIONS. SUPPLEMENTARY REMARKS ON THE THEORY OF DIFFERENTIAL EQUATIONS