Proceedings ArticleDOI
Solutions of linear ordinary differential equations in terms of special functions
Manuel Bronstein,Sébastien Lafaille +1 more
- pp 23-28
TLDR
The algorithm is able to find all rational transformations for a large class of functions, in particular the special functions of mathematical physics, such as Airy, Bessel, Kummer and Whittaker functions, and can be generalized to equations of higher order.Abstract:
We describe a new algorithm for computing special function solutions of the form y(x) = m(x)F(ξ(x)) of second order linear ordinary differential equations, where m(x) is an arbitrary Liouvillian function, ξ(x) is an arbitrary rational function, and F satisfies a given second order linear ordinary differential equation. Our algorithm, which is based on finding an appropriate point transformation between the equation defining F and the one to solve, is able to find all rational transformations for a large class of functions F, in particular (but not only) the 0F1 and 1F1 special functions of mathematical physics, such as Airy, Bessel, Kummer and Whittaker functions. It is also able to identify the values of the parameters entering those special functions, and can be generalized to equations of higher order.read more
Citations
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Journal ArticleDOI
A polynomial time algorithm for finding rational general solutions of first order autonomous ODEs
Ruyong Feng,Xiao-Shan Gao +1 more
TL;DR: A polynomial time algorithm is given for computing a rational general solution if it exists based on the computation of Laurent series solutions and Pade approximants and experimental results show that the algorithm is quite efficient.
Proceedings ArticleDOI
Rational general solutions of algebraic ordinary differential equations
Ruyong Feng,Xiao-Shan Gao +1 more
TL;DR: An algorithm to compute a rational general solution if it exists is given based on the relation between rational solutions of the first order ODE and rational parametrizations of the plane algebraic curve defined by thefirst order ODR and Padé approximants.
Proceedings ArticleDOI
Solving second order linear differential equations with Klein's theorem
M. van Hoeij,J.-A. Weil +1 more
TL;DR: A variation on the earlier formulas, namely the formulas will base the formulas on invariants of the differential Galois group instead of semi-invariants, to make the algorithm more easy to implement for various differential fields k.
Analytical solution of linear ordinary differential equations by differential transfer matrix method
Sina Khorasani,Ali Adibi +1 more
TL;DR: In this paper, a new analytical method for finding the exact solution of homogeneous linear ordinary dierential equations with arbitrary order and variable coecients is presented. But the important feature of the presented method is that it deals with the evolution of independent solutions, rather than its derivatives.
Proceedings ArticleDOI
Algebraic general solutions of algebraic ordinary differential equations
TL;DR: For a first order autonomous ODE, the optimal bound for the degree of its algebraic general solutions is given and a polynomial-time algorithm to compute an algebraicgeneral solution if it exists is given.
References
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BookDOI
Differentialgleichungen : Lösungsmethoden und Lösungen
TL;DR: A molded plastic bowling pin incorporating a core extending the length of the pin substantially from its base to its head and constituted of a plurality of plastic core elements of varying sizes and shapes such that the composite pin incorporating the same meets the weight distribution standards set by the American Bowling Congress, and a shell having a substantially uniform thickness about the core elements and being integrally molded on and firmly bonded thereto.
Journal ArticleDOI
Note on Kovacic's algorithm
Felix Ulmer,Jacques-Arthur Weil +1 more
TL;DR: It is shown how, by carefully combining the techniques of those algorithms, one can find the Liouvillian solutions of an irreducible second order linear differential equation by computing only rational solutions of some associated linear differential equations.
Journal ArticleDOI
Abstract differential algebra and the analytic case. II
TL;DR: Seidenberg et al. as discussed by the authors used the JSTOR archive to digitize, preserve, and extend access to Proceedings of the American Mathematical Society (AMS) for the first time.
Journal ArticleDOI
Liouvillian solutions of linear differential equations of order three and higher
TL;DR: This paper extends the algorithm in van Hoeij and Weil (1997) to compute semi-invariants and a theorem in Singer and Ulmer ( 1997) in such a way that, by computing one semi-Invariant that factors into linear forms, one gets all coefficients of the minimal polynomial of an algebraic solution of the Riccati equation, instead of only one coefficient.