On the algorithmic linearizability of nonlinear ordinary differential equations
KAUST DepartmentVisual Computing Center (VCC)
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Computer Science Program
Online Publication Date2019-07-15
Print Publication Date2020-05
Embargo End Date2021-07-15
Permanent link to this recordhttp://hdl.handle.net/10754/660360
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AbstractSolving nonlinear ordinary differential equations is one of the fundamental and practically important research challenges in mathematics. However, the problem of their algorithmic linearizability so far remained unsolved. In this contribution, we propose a solution of this problem for a wide class of nonlinear ordinary differential equation of arbitrary order. We develop two algorithms to check if a nonlinear differential equation can be reduced to a linear one by a point transformation of the dependent and independent variables. In this regard, we have restricted ourselves to quasi-linear equations with a rational dependence on the occurring variables and to point transformations. While the first algorithm is based on a construction of the Lie point symmetry algebra and on the computation of its derived algebra, the second algorithm exploits the differential Thomas decomposition and allows not only to test the linearizability, but also to generate a system of nonlinear partial differential equations that determines the point transformation and the coefficients of the linearized equation. The implementation of our algorithms is discussed and evaluated using several examples.
CitationLyakhov, D. A., Gerdt, V. P., & Michels, D. L. (2020). On the algorithmic linearizability of nonlinear ordinary differential equations. Journal of Symbolic Computation, 98, 3–22. doi:10.1016/j.jsc.2019.07.004
SponsorsThe authors are grateful to Daniel Robertz and Greg Reid for helpful discussions and suggestions. This work has been partially supported by KAUST Baseline Funding (D. A. Lyakhov and D. L. Michels), by the RUDN University Program (5-100; V. P. Gerdt), and by Stanford University (D. L. Michels).
JournalJournal of Symbolic Computation