Addition theorems for solutions to linear homogeneous constant coefficient ordinary differential equations

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

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.

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.

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

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.

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.

In this paper the First Order Linear Fuzzy Ordinary Differential Equations are described. Here coefficients and /or initial condition of said differential equation are taken as the Generalized Triangular Fuzzy Numbers (GTFNs).The solution procedure of this Fuzzy Differential Equation is developed by Lagrange Multiplier Method. An imprecise barometric pressure problem is described.

The simulation problem for high order linear differential systems

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.

In the case of n functions of a single variable, the vanishing of the wronskian is the most familiar criterion for their linear dependence. The wronskian also plays an important part in connection with the theory of a single ordinary homogeneous linear differential equation of the nth order, even in questions which do not concern the linear dependence of solutions of the equation. It is the purpose of the present paper to generalize the fundamental facts connected with these topics, to the case of functions of several variables. The characteristic property of an ordinary homogeneous linear differential equation is that any solution is expressible linearly, with constant coefficients, in terms of a fundamental set of solutions. The natural generalization to several independent variables is afforded by the completely integrable system of homogeneous linear partial differential equations, in one dependent variable, any solution of which is likewise linearly dependent on a finite number of solutions of the system. It is such systems which are discussed in the latter part of this paper; the subject of the linear dependence of functions of several variables is developed in the first part, chiefly with a view to its subsequent application to completely integrable systems. In our discussion of linear dependence, we consider throughout n functions Yl, Y2, Y n,, of p independent variables u1, u2, * up . Functions and variables may be either real or complex; if any of the variables be complex, we suppose the functions to be analytic in those variables. All of the main theorems, however, are stated and proved under the supposition that the independent variables are real; the modifications usually simplifications and omissions-are easily supplied for cases in which one or more of the independent variables is complex. The n functions, then, of the real variables

The underlying structure inherent in the classical method of undetermined coefficients, which is used to obtain particular solutions to linear constant coefficient (LCC) difference or differential equations, is investigated. A system of equations of the form B=M/sup -1/ A is obtained, where B and A are vectors whose elements are the coefficients of the terms in the expressions for the input and solutions, respectively, to the LCC equations. The structure of M that arises for both LCC difference and differential equations, as well as moving-average (FIR) systems, are investigated. It is shown that M is always a lower triangular matrix of order (r-1)*(r+1), where r is the degree of the expressions for the input and solutions. Furthermore, M is characterized by r+1 unique elements, each one defining the diagonal and off-diagonal elements, and is a member of an infinite set of matrices, all of order r+1, which form a group. M can be obtained whenever A and B are given. As a result, if one desires an FIR filter whose output is some linear operation, then the computation of M from A and B imposes a set of necessary and sufficient conditions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

This document describes how one can derive the solutions to a linear constant coefficient homogeneous differential equation with repeated roots in the characteristic equation with Abel's Theorem. y (n) (t) + p1(t)y (n−1) (t) + p2(t)y (n−2) (t) + · · · + pn−1(t)y ' (t) + pn(t)y(t) = 0 (1.1) is encountered by every differential equations student. We most often study second order equations, the case when n = 2, due to Newton's laws of motion and with ease one can generalize the theory to higher order. We consider the case when the coefficient functions pi(t) are constant. Under these conditions, one makes the ansatz y = e rt which transforms (1.1) into an algebraic equation, which is more easily solved. In particular, if pi = ai, with the above ansatz, we have the characteristic equation r n + a1r n−1 + a2r n−2 + · · · + an−1r + an = 0. (1.2) By factoring out the e rt , one finds that solving an algebraic equation yields solutions to the original differential equation. Most textbooks first handle distinct real roots and later come back to discuss when r = r1 is a repeated solution to (1.2). We show an approach that relies on Abel's Theorem and the idea of reduction of order can constructively establish that the repeated root r = r1 with multiplicity m gives rise to m linearly independent solutions of the form yi(t) = t i−1 e r1t for 1 ≤ i ≤ m.

In 1908 Birkhoff defined a class of series generalizing the Fourier and Sturm-Liouville series which arise from ordinary linear homogeneous differential equations of the second order with linear homogeneous boundary conditions. He starts from a differential system of order n with boundary conditions restricted to be regular and proves that an arbitrary function with a sufficient number of derivatives can be represented in terms of the solutions of this differential system.t Subsequently, Tamarkin extended the discussion to apply to functions integrable in the sense of Lebesgue by the device of comparing the formal Fourier and Birkhoff series for the same function.t Recently the author has published further researches on the problem. ? It is now proposed to complete in some respects our knowledge of regular differential systems of the second order. In our paper cited above, we left certain necessary conditions unstudied and now desire to fill this gap in the theory. 11 In addition, the attempt to discuss the formal expansions for functions integrable in accordance with definitions of integration more extended than that of Lebesgue has led us to results quite different from those hitherto established. We shall indicate the facts for functions integrable in the sense of Denjoy. The situation may be described roughly by saying that, whereas all the regular expansions associated with an arbitrary function integrable in the sense of Lebesgue have virtually the same behavior, the series formed for an arbitrary function integrable in the sense of Denjoy or non-absolutely integrable in the sense of Riemann fall into a non-denumerable infinity of distinct types exhibiting individual behavior. The result is due essentially to the fact that the Riemann-Lebesgue theorem does not hold true for nonabsolutely integrable functions. On the other hand, if the series for an arbi-

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