Computing Closed Form Solutions of First Order ODEs Using the Prelle-Singer Procedure

AbstractThe Prelle-Singer procedure is an important method for formal solution of first order ODEs. Two different REDUCE implementations (PSODE versions 1 & 2) of this procedure are presented in this paper. The aim is to investigate which implementation is more efficient in solving different types of ODEs (such as exact, linear, separable, linear in coefficients, homogeneous or Bernoulli equations). The test pool is based on Kamke's collection of first order and first degree ODEs. Experimental results, timings and comparison of efficiency and solvability with the present REDUCE differential equation solver (ODESOLVE) and a MACSYMA implementation (ODEFI) of the Prelle-Singer procedure are provided. Discussion of technical difficulties and some illustrative examples are also included

Finding Liouvillian first integrals of rational ODEs of any order in finite terms

It is known, due to Mordukhai-Boltovski, Ritt, Prelle, Singer, Christopher and others, that if a given rational ODE has a Liouvillian first integral then the corresponding integrating factor of the ODE must be of a very special form of a product of powers and exponents of irreducible polynomials. These results lead to a partial algorithm for finding Liouvillian first integrals. However, there are two main complications on the way to obtaining polynomials in the integrating factor form. First of all, one has to find an upper bound for the degrees of the polynomials in the product above, an unsolved problem, and then the set of coefficients for each of the polynomials by the computationally-intensive method of undetermined parameters. As a result, this approach was implemented in CAS only for first and relatively simple second order ODEs. We propose an algebraic method for finding polynomials of the integrating factors for rational ODEs of any order, based on examination of the resultants of the polynomials in the numerator and the denominator of the right-hand side of such equation. If both the numerator and the denominator of the right-hand side of such ODE are not constants, the method can determine in finite terms an explicit expression of an integrating factor if the ODE permits integrating factors of the above mentioned form and then the Liouvillian first integral. The tests of this procedure based on the proposed method, implemented in Maple in the case of rational integrating factors, confirm the consistence and efficiency of the method.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Qualitative Analysis to a Nonlinear System

In this thesis, we first present a qualitative analysis to a nonlinear system under certain parametric conditions. Then for a special case, we make a series of variable transformation and apply the Prelle-Singer Method to find the first integrals of the simplified equations without complicated calculations. Through the inverse transformations we get the first integrals of the original equation. Finally, we use the same Prelle-Singer method to get the first integral for an extended nonlinear system

Finding Liouvillian First Integrals of Rational ODEs of Any Order in Finite Terms

It is known, due to Mordukhai-Boltovski, Ritt, Prelle, Singer, Christopher and others, that if a given rational ODE has a Liouvillian first integral then the corresponding integrating factor of the ODE must be of a very special form of a product of powers and exponents of irreducible polynomials. These results lead to a partial algorithm for finding Liouvillian first integrals. However, there are two main complications on the way to obtaining polynomials in the integrating factor form. First of all, one has to find an upper bound for the degrees of the polynomials in the product above, an unsolved problem, and then the set of coefficients for each of the polynomials by the computationally-intensive method of undetermined parameters. As a result, this approach was implemented in CAS only for first and relatively simple second order ODEs. We propose an algebraic method for finding polynomials of the integrating factors for rational ODEs of any order, based on examination of the resultants of the polynomials in the numerator and the denominator of the right-hand side of such equation. If both the numerator and the denominator of the right-hand side of such ODE are not constants, the method can determine in finite terms an explicit expression of an integrating factor if the ODE permits integrating factors of the above mentioned form and then the Liouvillian first integral. The tests of this procedure based on the proposed method, implemented in Maple in the case of rational integrating factors, confirm the consistence and efficiency of the method

Reaction-diffusion systems with a nonlinear rate of growth

In the literature there are quite a few elegant approaches which have been proposed to find (he first integrals of nonlinear differential equations. Recently, the modified Prelle-Singer method for finding the first integrals of second-order nonlinear ordinary differential equations (ODEs) has attracted considerable attention. Many researchers used this method to derive the first integrals to various systems. In this thesis, we are concerned with the first integrals for reaction-diffusion systems with a nonlinear rate of growth. Under certain parametric conditions we express the first integrals explicitly by applying an analytical method as well as the modified Prelle-Singer method

A computational approach to polynomial conservation laws

For polynomial ODE models, we introduce and discuss the concepts of exact and approximate conservation laws, which are the first integrals of the full and truncated sets of ODEs. For fast-slow systems, truncated ODEs describe the fast dynamics. We define compatibility classes as subsets of the state space, obtained by equating the conservation laws to constants. A set of conservation laws is complete when the corresponding compatibility classes contain a finite number of steady states. Complete sets of conservation laws can be used for model order reduction and for studying the multistationarity of the model. We provide algorithmic methods for computing linear, monomial, and polynomial conservation laws of polynomial ODE models and for testing their completeness. The resulting conservation laws and their completeness are either independent or dependent on the parameters. In the latter case, we provide parametric case distinctions. In particular, we propose a new method to compute polynomial conservation laws by comprehensive Gröbner systems and syzygies

A computer algebra approach to rational general solutions of algebraic ordinary differential equations

In this thesis, I approach to algebraic ODEs from Differential Algebra's point of view. I look for rational solutions of AODE, I present an algebro-geometric method to decide the existence of rational solutions of a first-order algebraic ODE and if they exist an algorithm to compute them. This method depends heavily on rational parametrizations, in particular for autonomous equations on the parametrization of algebraic curves, and for non-autonomous equations on the parametrization of algebraic surfaces. In the last case, I prove the correspondence between rational solutions of a parametrizable algebraic ODE and rational solutions of a first-order linear autonomous differential system of two equations in two variables. I provide an algorithm to compute rational solutions of such system based on its invariant algebraic curves. I also study a group of affine transformations which preserves the rational solvability, in order to reduce, when possible, an algebraic ODE to an easier one. Moreover I present the results of the implementation of all these algorithms in two computer algebra system: CoCoA and Singular