  Algorithmic Linearization of Ordinary Differential Equations: a New Approach

# Algorithmic Linearization of Ordinary Differential Equations: a New Approach

Mathematical modeling of many processes in Biology, Chemistry, Economics, Physics, and other sciences is impossible without solving ordinary differential equations (ODE). Such equations most often contain a nonlinear dependence on an unknown function and its derivatives of various orders (n>1), which makes it very difficult to study and solve them. The problem of checking the linearity of nonlinear ODEs consists of finding out whether there is an invertible transformation of the desired function and its argument to new variables, in the way that the transformed equation becomes linear. Finding a solution to a linear equation is incomparably easier. If such a transformation exists and can be found, the solution of the resulting linear equation can be converted for the solution of the original equation. For many decades, the problem of checking the linearity of ordinary differential equations above the fourth order remained unsolved.

An algorithmic solution to the problem of checking linearity, and in the case of a positive result, performing a linearization transformation with a software implementation in Maple language was first created by Vladimir Gerdt, the Director of the Research Center for Computational Methods in Applied Mathematics at the Russian Academy of Sciences, and his colleagues.

The Background of the Issue

In the late XIX century, Sophus Lie, the famous Norwegian mathematician, developed the most general mathematical approach to the study and solution of differential equations using group (symmetry) analysis. It is based on a mathematical apparatus which uses the concept of groups introduced by Lie. One of the fundamental tasks set by S. Lie was to develop a test for the linearity of nonlinear general differential equations by point invertible transformations. A positive test result makes it much easier to build an analytical solution. Sophus Lie himself established the general form of second-order equations (n = 2) that allow linearization. In the XXI century, with the leading contribution of the Russian mathematical school, S. Lie's results on finding the general form of linearized equations were generalized to equations of order of n = 3 and 4. However, the complete problem of linearization, including finding the linearization transformation even for equations of orders of 2 to 4, remained algorithmically unsolved.

Vladimir Gerdt, Doctor of Physical and Mathematical Sciences at RUDN, provided more details concerning the new solution.

You propose two fundamentally new algorithms for linearization of nonlinear ODEs. How do they work?

Yes, both algorithms are fundamentally new. Prior to our work, no algorithms had been developed that can answer, for a sufficiently wide range of nonlinear ODEs, the question of the linearizability of such equations by invertible point transformations, I. e., transformations of dependent and independent variables.

Are the algorithms interconnected? Do they need to be applied in sequence?

The algorithms have different mathematical nature. The first is based on group analysis of the original equation, and the computation it involves is fast. It only answers the question whether a linearizing transformation exists. However, this algorithm does not assist in finding such a transformation, even if it exists.

The second algorithm is more consuming in terms of the calculations required, but it not only answers the question stated, but also builds equations for the linearization transformation.

In practice, it is best to first apply the first algorithm, and if it shows the existence of a linearizing transformation, the second algorithm is used to find it.

How can one describe the scope of the practical application of the research findings? Where can they be used?

These algorithms can be used in any scientific and technical problems where a nonlinear ODE to be solved has a higher derivative (of the second order or higher) in the left part, and in the right part, there is a rational expression (the ratio of polynomials) for the lower derivatives of an unknown function, including this function itself.

In 2017, your work was granted some high-profile awards: the best work award at the world's largest forum on symbolic and algebraic computing (Kaiserslautern, Germany) and ISSAC Distinguished Paper Award by ACM SIGSAM. Can you share your secret of success, please? Is it about innovative approach?

Our work made a strong impression on the international community, since the problem to which we first provided a solution was set back in 1883 by the outstanding Norwegian mathematician Sophus Lie. Of course, the approach was innovative, and the second of the algorithms, which not only checks the existence of the linearization transformation, but also builds it, is based on a new and completely algorithmic method of differential Thomas decomposition. This is the most universal algebraic method for studying and solving polynomial-nonlinear systems of partial differential equations. It is such a system of differential equations that defines the linearization transformation.

• V.P.Gerdt, D.A.Lyakhov, D.L.Michels (both KAUST, Thuwal, Kingdom of Saudi Arabia). Proceedings of ISSAC 2017, ACM Press, 2017, PP.285-292
• D.A. Lyakhov, V.P. Gerdt, D.L. Michels. On the algorithmic linearizability of nonlinear ordinary differential equations, Journal of Symbolic Computation, V. 98, 2020, PP. 3-22)
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