On liouvillian solutions of linear differential equations

This paper deals with the problem of finding liouvillian solutions of a homogeneous linear differential equationL(y)=0 of ordern with coefficients in a differential fieldk. For second order linear differential equations with coefficients ink _o(x), wherek _o is a finite algebraic extension ofQ, such an algorithm has been given by J. Kovacic and implemented. A general decision procedure for finding liouvillian solutions of a differential equation of ordern has been given by M.F. Singer, but the resulting algorithm, although constructive, is not in implementable form even for second order equations. Both algorithms use the fact that, ifL(y)=0 has a liouvillian solution, then,L(y)=0 has a solutionz such thatu=z/z is algebraic overk. Using the action of the differential galois group onu and the theory of projective representation we get sharp bounds (n) for the algebraic degree ofu for differential equations of arbitrary ordern. For second order differential equations we get the bound (2)=12 used in the algorithm of J. Kovacic and for third order differential equation we improve the bound given by M.F. Singer from 360 to (3)36. We also show that not all values less than or equal to (n) are possible values for the algebraic degree ofu. For second order differential equations we rediscover the values 2, 4, 6, and 12 used in the Kovacic Algorithm and for third order differential equations we get the possibilities 3,4, 6, 7, 9, 12, 21, and 36. We prove that if the differential Galois group ofL(y)=0 is a primitive unimodular linear group, then all liouvillian solutions are algebraic. From this it follows that, if a third order differential equationL(y)=0 is not of Fuchsian type, then the logarithmic derivative of some liouvillian solution ofL(y)=0 is algebraic of degree 3. We also derive an upper bound for the minimal numberN(n) of possible degreesm of the minimal polynomial of an algebraic solution of the riccati equation associated withL(y)=0.Supported by Deutsche Forschungsgemeinschaft while visiting North Carolina State University     

Fundamental solutions for second order linear elliptic partial differential equations

This paper is concerned with outlining some fundamental solutions and Green's functions for a system of second order linear elliptic partial differential equations in two independent variables. The fundamental solution and a number of Green's functions are given in relatively elementary closed form for some cases when the coefficients in the equations are constant. When the coefficients are variable the fundamental solution is obtained for some particular classes of equations.     

Necessary conditions for liouvillian solutions of (third order) linear differential equations

In this paper we show how group theoretic information can be used to derive a set of necessary conditions on the coefficients ofL(y) forL(y=0 to have a liouvillian solution. The method is used to derive (and improve in one case) the necessary conditions of the Kovacic algorithm and to derive an explicit set of necessary conditions for third order differential equations.A weaker version of these results were announced inLiouvillian Solutions of Third Order Linear Differential Equations; New Bounds and Necessary Conditions, Proceedings of the 1992 International Symposium on Symbolic and Algebraic Computation, ACM PressPartially supported by NSF Grant 90-24624Partially supported by Deutsche Forschungsgemeinschaft, while on leave from Universität Karlsruhe. This paper was written during the two year visit of the second author at North Carolina State University. The second author would like to thank North Carolina State University for its hospitality and support during the preparation of this paper     

Parallel multisplitting,block Jacobi type solutions of linear systems of equations

Three parallel iterative schemes to solve banded systems of equations are presented in this study. The techniques are special implementations of the theory of matrix multisplitting. The resulting algorithms are implemented on a Multiple Instruction Multiple Data (MIMD) grid architecture using 16 Transputer processors. Parametric analyses are performed to develop convergence criteria and examine the speed-up of solution. It is concluded that the algorithms make very efficient use of the parallel computer architecture, especially if this consists of a large number of nodes.     

Exponential stability of linear impulsive differential equations

In the present paper the notion of exponential stability for linear impulsive differential equations at fixed moments is made precise     

Electromodeling of solutions of fourier differential equations

An examination is made of some aspects of the modeling of heat conduction or diffusion process described by Fourier equations on analog computers. A scheme is given for reproducing a sum of exponential functions with the aid of a single amplifier and a group of switching circuits.     

Stochastic uniform observability of general linear differential equations

The aim of this article is to discuss the uniform observability property of general linear differential equations with multiplicative white noise in Hilbert spaces. Based on perturbation theory for evolution operators on Hilbert Schmidt spaces and on the space of nuclear operators, new representations of the covariance operators associated to the mild solutions of the investigated stochastic differential equations are given. Using these results we obtain deterministic characterizations of the stochastic uniform observability property. We also identify an entire class of stochastic differential equations which are never stochastic uniformly observable. Some examples will illustrate the theory.     

Traveling wave solutions of nonlinear partial differential equations

Elementary transformations are utilized to obtain traveling wave solutions of some diffusion and wave equations, including long wave equations and wave equations the nonlinearity of which consists of a linear combination of periodic functions, either trigonometric or elliptic. In particular, a theorem is established relating the solutions of a single cosine equation and a double sine-cosine equation. It is shown that the latter admits a Bäcklund Transformation.     

Stochastic semi-discretization for linear stochastic delay differential equations

An efficient numerical method is presented to analyze the moment stability and stationary behavior of linear stochastic delay differential equations. The method is based on a special kind of discretization technique with respect to the past effects. The resulting approximate system is a high dimensional linear discrete stochastic mapping. The convergence properties of the method is demonstrated with the help of the stochastic Hayes equation and the stochastic delayed oscillator.     

Almost periodic solutions for impulsive fractional differential equations

In this paper, we consider the existence of almost periodic solutions for impulsive fractional evolution equations involving Caputo fractional derivative. The main results are obtained by means of the theory of operators semi-group, probability density functions, fixed point theorems and the techniques based on fractional calculus. An example is also discussed to illustrate the theory. Some known results are improved and generalized.     

A class of piecewise linear differential equations arising in biological models

We investigate the properties of the solutions of a class of piecewise-linear differential equations. The equations are appropriate to model biological systems (e.g. genetic networks) in which there are switch-like interactions between the elements. The analysis uses the concept of Filippov solutions of differential equations with a discontinuous right-hand side. It gives an insight into the so-called singular solutions which lie on the surfaces of discontinuity. We show that this notion clarifies the study of several examples studied in the literature.     

Lipschitz stability of impulsive systems of differential equations

In the present paper notions of Lipschitz stability of the zero solution of impulsive systems of differential equations with fixed moments of impulse effect arc introduced. Sufficient conditions for various types of uniform Lipschitz stability are obtained and the relations between these options are investigated The results obtained are used for the investigation of the uniform Lipschitz stability of the zero solution of linear impulsive systems of differential equations.     

Variations on the fundamental principle for linear systems of partial differential and difference equations with constant coefficients

New and known spaces of locally finite or polynomial exponential multivariate sequences and functions are constructed by means of substantial theorems from Commutative Algebra. They satisfy Ehrenpreis'fundamental principle and hence permit the solution of linear systems of partial differential or difference equations with constant coefficients. On the one hand this paper thus continues the author's work on multidimensional linear systems, on the other hand it generalizes and improves related work in approximation theory.     

Solution of linear equations for small computer systems

A solution technique, based on Gauss elimination, is described which can solve symmetric or unsymmetric matrices on computers with small core and disk requirement capabilities. The method is related to frontal techniques in that renumbering of the nodes, such as in a finite element mesh, is not required, and the elimination is performed immediately after the equations for a particular node have been fully summed. Only two rows of the matrix need be on core at any step of the solution, but for more efficiency, the program presented here requires all the equations associated with two nodes to be on core. Minimum disk storage is achieved by storing only nonzero entries of the matrix, a single pointing vector for each node, regardless of the number of degrees-of-freedom, and the use of a single sequential file. Special care is taken of the boundary nodes where only the diagonal and the right-hand-side vector are stored. Assembly and elimination for these nodes are avoided completely. The performance of the program is compared with both symmetric and nonsymmetric frontal routines and is shown to be acceptable. The major merit of the method lies in the fact that it can be implemented on small minicomputers. The reduction of core and disk storage inevitably increases the solution time, but the decrease in the output file size also makes the back-substitution and resolution processes more efficient. In some cases, the total solution time can be shorter than for the frontal method due to this property.     

Periodic solutions and their bifurcations in a non-smooth second-order delay differential equation

We consider a non-smooth second order delay differential equation (DDE) that was previously studied as a model of the pupil light reflex. It can also be viewed as a prototype model for a system operated under delayed relay control. We use the explicit construction of solutions of the non-smooth DDE hand-in-hand with a numerical continuation study of a related smoothed system. This allows us to produce a comprehensive global picture of the dynamics and bifurcations, which extends and completes previous results. Specifically, we find a rich combinatorial structure consisting of solution branches connected at resonance points. All new solutions of the smoothed system were subsequently constructed as solutions of the non-smooth system. Furthermore, we show an example of the unfolding in the smoothed system of a non-smooth bifurcation point, from which infinitely many solution branches emanate. This shows that smoothing of the DDE may provide insight even into bifurcations that can only occur in non-smooth systems.