Solutions Of Algebraic Differential Equations Research Articles

Adomian decomposition method for solution of differential-algebraic equations

Solutions of differential algebraic equations is considered by Adomian decomposition method. In E. Babolian, M.M. Hosseini [Reducing index and spectral methods for differential-algebraic equations, J. Appl. Math. Comput. 140 (2003) 77] and M.M. Hosseini [An index reduction method for linear Hessenberg systems, J. Appl. Math. Comput., in press], an efficient technique to reduce index of semi-explicit differential algebraic equations has been presented. In this paper, Adomian decomposition method is applied to reduced index problems. The scheme is tested for some examples and the results demonstrate reliability and efficiency of the proposed methods.

Note on algebraic solutions of differential equations with known finite Galois group

Given a linear differential equation with known finite differential Galois group, we discuss methods to construct the minimal polynomial of a solution. We first outline a well known general method involving a basis transformation of the basis of formal solutions at a singular point. In the second part we construct directly the minimal polynomial of an eigenvector of the monodromy matrix at a singular point. The method is very efficient for irreducible second and third order linear differential equations where a one dimensional eigenspace of some monodromy matrix always exists.

Algebraic solutions of algebraic differential equations

Using value distribution theory and techniques, the problem of the algebroid solutions of second order algebraic differential equation is investigated. Examples show that the results are sharp.

Pseudospectral method for numerical solution of DAEs with an error estimation

In [E. Babolian, M.M. Hosseini, Reducing index, and pseudospectral methods for differential-algebraic equations, Appl. Math. Comput. 140 (2003) 77–90], numerical solution of linear differential-algebraic equations (DAEs) has been presented by pseudospectral method. In this paper, a new error estimation technique is proposed to pseudospectral method such that is well done for linear semi-explicit DAEs. When the DAEs has index 2, the obtained approximate solution can be improved by using the proposed error estimation technique. Furthermore, with providing some examples the proposed error estimation method is dealt with numerically.

On the growth of entire solutions of algebraic differential equations

This is the opening memorial plenary lecture on memorial meeting in honor of Professor Shlomo Strelitz on Conference of Differential Equations and Complex Analysis, University of Haifa, Israel, December 2000. Shlomo Strelitz: 7.1.1923–27.9.1999, teached at Vilnius University 1946–1973, professor 1967–1973.

Order of growth of solutions to algebraic differential equations in the unit disk

S. B. Bank has shown that there is no uniform growth estimate for meromorphic solutions of algebraic differential equations with meromorphic coefficients in the unit disk. We give conditions under which such solutions must have a finite order of growth.

Linear relations among algebraic solutions of differential equations

For any univariate polynomial with coefficients in a differential field of characteristic zero and any integer, q, there exists an associated nonzero linear ordinary differential equation (LODE) with the following two properties. Each term of the LODE lies in the differential field generated by the rational numbers and the coefficients of the polynomial, and the qth power of each root of the polynomial is a solution of this LODE. This LODE is called a qth power resolvent of the polynomial. We will show how one can get a resolvent for the logarithmic derivative of the roots of a polynomial from the αth power resolvent of the polynomial, where α is an indeterminate that takes the place of q. We will demonstrate some simple relations among the algebraic and differential equations for the roots and their logarithmic derivatives. We will also prove several theorems regarding linear relations of roots of a polynomial over constants or the coefficient field of the polynomial depending upon the (nondifferential) Galois group. Finally, we will use a differential resolvent to solve the Riccati equation.

Verification of invertibility of complicated functions over large domains

A new method to decide the invertibility of a given high-dimensional function over a domain is presented. The problem arises in the field of verified solution of differential algebraic equations (DAEs) related to the need to perform projections of certain constraint manifolds over large domains. The question of invertibility is reduced to a verified linear algebra problem involving first partials of the function under consideration. Different from conventional approaches, the elements of the resulting matrices are Taylor models for the derivatives of the functions. The linear algebra problem is solved based on Taylor model methods, and it will be shown the method is able to decide invertibility with a conciseness that often goes substantially beyond what can be obtained with other interval methods. The theory of the approach is presented. Comparisons with three other interval-based methods are performed for practical examples, illustrating the applicability of the new method.

Locations of values of meromorphic solutions of algebraic differential equations

Meromorphic solutions of algebraic differential equations has been intensively investigated during the last three decades. These studies were mainly concerned with the growth of solutions and numbers of their a-points. In the present article we transfer to study location of a-points of meromorphic solutions for some important classes of algebraic differential equations. To this end, we apply the properties of proximitly and comparability of a-points. These properties offer additional information to the classical value distribution theory by describing locations of a-points of arbitrary meromorphic functions. In particular, we show for first-order algebraic differential equations that mutual locations of different a-points of the solutions are completely determined in terms of the equation.

The diophantine approximation of general solutions of algebraic differential equations

Hoping to find bounds on the approximation by rational functions of all members of certain classes of power series, it is necessary to restrict to “reasonable” classes of functions. Algebraic functions are one such class. Another class is the set of irrational power series solutions of linear homogeneous differential equations with polynomial coefficients. Tight bounds have been found in both cases. The jump to nonlinear differential equations is another matter. The present paper obtains tight bounds, if one restricts to certain classes of general solutions. The presence of enough algebraically independent parameters permits the desired type of bound. It is to be hoped that the more general case will ultimately be settled; however, other methods will certainly be needed.

α-normal functions and yosida functions

α-normal functions, α ≥ 1, are meromorphic functions in the unit disk D with α-normal functions (α>1) are characterized by the normality of a family of functions on compact subsets of the finite complex plane . We prove that limit functions of converging sequences of functions {fn (ζ)} are Yosida's functions. We extract a subclass of the α-normal functions such that limit functions of converging sequences of functions are Yosida's functions of the first kind. Note that meromorphic solutions of algebraic differential equations of the first order with coefficients from the Hardy spaces Hp are a-normal functions.

A Time Adaptiv Finite‐Element Procedure Applied to Creep and Relaxation Processes

AbstractIn this paper we interpret the finite‐element method applied to constitutive models with internal variables as the solution of differential algebraic equations (DAE). With this interpretation we can utilize Diagonal Implicit Runge‐Kutta methods (DIRK) as well as a corresponding step‐size control and a special solution technique for block‐structured systems of equations. This proceeding doesn't affect already developed FE‐implementations which are based on the Backward Euler method. Furthermore, the meaning of the consistent tangent operator becomes more obvious. The paper ends with some examples in creep and relaxation tests, where step size control should always be used conditioned by the different time scales.

α-Yosida functions and meromorphic solutions of algebraic differential equations

α-Yosida functions and meromorphic solutions of algebraic differential equations

Pointwise Optimal Control of Dynamical Systems Described by Constrained Coordinates

A feedback control law is developed for dynamical systems described by constrained generalized coordinates. For certain complex dynamical systems, it is more desirable to develop the mathematical model using more general coordinates then degrees of freedom which leads to differential-algebraic equations of motion. Research in the last few decades has led to several advances in the treatment and in obtaining the solution of differential-algebraic equations. We take advantage of these advances and introduce the differential-algebraic equations and dependent generalized coordinate formulation to control. A tracking feedback control law is designed based on a pointwise-optimal formulation. The stability of pointwise optimal control law is examined.

On the order of growth of solutions of algebraic differential equations

Assume thatf is an integer transcendental solution of the differential equationP n (z, f, f′)=P n−1(z, f, f′, ... f (p)), whereP n andP n−1 are polynomials in all variables, the degree ofP n with respect tof andf′ is equal ton, and the degree ofP n−1 with respect tof, f′, ... f (p) is at mostn−1. We prove that the order ρ of growth off satisfies the relation 1/2≤ρ<∞. We also prove that if ρ=1/2, then, for a certain real ν, in the domain {z: ν<argz<ν+2π}/E *, whereE * is a certain set of disks with finite sum of radii, the estimate lnf(z)=z 1/2 (β+o(1)), β∈C, holds forz=re iϕ,r≥r(ϕ)≥0. Furthermore, on the ray {z: argz=ν}, the following relation is true: ln‖f(re iν)‖=o(r 1/2),r→+∞,r>0, $$r\bar \in \Delta $$ , where Δ is a certain set on the semiaxisr>0 with mes Δ<∞.

Solving Index-1 DAEs in MATLAB and Simulink

This paper describes mathematical and software developments needed for the effective solution of differential algebraic equations of index 1 in the integrated computing environment MATLAB and the dynamic simulation package Simulink. The developments are applicable to other problem-solving environments and some are applicable to general scientific computation.

On a theorem of gol'dberg concerning meromorphic solutions of algebraic differential equations

We give a new proof of a generalization of Gol'dberg's theorem that meromorphic solutions of first order algebraic differential equations have finite order of growth. The main tool is a result of Zalcman concerning normal families.

Criteria for the Trivial Solution of Differential Algebraic Equations with Small Nonlinearities to be Asymptotically Stable

Differential algebraic equations consisting of a constant coefficient linear part and a small nonlinearity are considered. Conditions that enable linearizations to work well are discussed. In particular, for index-2 differential algebraic equations, there results a kind of Perron Theorem that sounds as clear as its classical model.

On the meromorphic solutions of algebraic differential equations

On the meromorphic solutions of algebraic differential equations

Mixed symbolic-numerical computations with general DAEs I: System properties.

Differential-algebraic equations (DAEs) arise in many ways in many types of problems. In this expository paper we discuss a variety of situations where we have found mixed symbolic-numerical calculations to be essential. The paper is designed to both familiarize the reader with several fundamental DAE ideas and to present some applications. The situations discussed include the analysis of DAEs, the solution of DAEs, and applications which include DAEs. Both successes and challenges will be presented.