Integrating Factors and ODE Patterns
نویسندگان
چکیده
A systematic algorithm for building integrating factors of the form μ(x, y′) or μ(y, y′) for nonlinear second order ODEs is presented. When such an integrating factor exists, the scheme returns the integrating factor itself, without solving any differential equations. The scheme was implemented in Maple, in the framework of the ODEtools package and its ODE-solver. A comparison between this implementation and other computer algebra ODE-solvers in solving non-linear examples from Kamke’s book is shown. (Submitted for publication in Journal of Symbolic Computation) 1Department of Computer Science, Faculty of Mathematics, University of Waterloo, Ontario, Canada 2Symbolic Computation Group, Departamento de F́ısica Teórica, IF-UERJ. Available as http://dft.if.uerj.br/preprint/e7-2.tex; also as http://lie.uwaterloo.ca/odetools/e7-2.tex
منابع مشابه
Integrating Factors for Second-order ODEs
A systematic algorithm for building integrating factors of the form μ(x, y), μ(x, y) or μ(y, y) for second order ODEs is presented. The algorithm can determine the existence and explicit form of the integrating factors themselves without solving any differential equations, except for a linear ODE in one subcase of the μ(x, y) problem. Examples of ODEs not having point symmetries are shown to be...
متن کاملTHE USE OF A RUNGE-KUTTA SCHEME FOR AN ODE-PDE MODEL OF SUPPLY CHAINS
Integrating various suppliers to satisfy market demand is of great importance for e ective supply chain management. In this paper, we consider the ODE-PDE model of supply chain and apply a classical explicit fourth-order Runge-Kutta scheme for the related ODE model of suppliers. Also, the convergence of the proposed method is proved. Finally a numerical example is studied to demonstrate the acc...
متن کاملFinding 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 a...
متن کاملNew solutions for ordinary differential equations
This paper introduces a new method for solving ordinary differential equations (ODEs) that enhances existing methods that are primarily based on finding integrating factors and/or point symmetries. The starting point of the new method is to find a non-invertible mapping that maps a given ODE to a related higher-order ODE that has an easily obtained integrating factor. As a consequence, the rela...
متن کاملEN 202 : Problem Set 3
At this point, we require an explicit form of the characteristics. First, consider the solution to the ODE ȳξ(ξ, η) = 1. Integrating with respect to ξ, we find ȳ = ξ + φ(η). Applying the initial value on Γ, where y0(η) = 0 and ξ = 0, we find φ(η) = 0. Next, consider the solution to the ODE x̄ξ(ξ, η) = η . Integrating with respect to ξ, we find x̄ = ξη + ψ(η). Using the initial value on Γ, where x...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1997