Loewy decomposition of linear differential equations

Schwarz, Fritz

doi:10.1007/s13373-012-0026-7

Cited by 8 publications

(14 citation statements)

References 47 publications

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“…We stress the fact that our application (Theorem 3) relies on the factorization of some nonlinear differential operator which is an approach that belongs to an active domain of research in symbolic computation these last years, see for instance [6], [7], [12], [28], [29], [33].…”

Section: Introductionmentioning

confidence: 99%

On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems

Lastra¹,

Malek²

2015

Adv Differ Equ

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We study a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter whose coefficients depend holomorphically on ( , t) near the origin in C 2 and are bounded holomorphic on some horizontal strip in C w.r.t. the space variable. In our previous contribution (Lastra and Malek in Parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems, arXiv:1403.2350), we assumed the forcing term of the Cauchy problem to be analytic near 0. Presently, we consider a family of forcing terms that are holomorphic on a common sector in time t and on sectors w.r.t. the parameter whose union form a covering of some neighborhood of 0 in C * , which are asked to share a common formal power series asymptotic expansion of some Gevrey order as tends to 0. We construct a family of actual holomorphic solutions to our Cauchy problem defined on the sector in time and on the sectors in mentioned above. These solutions are achieved by means of a version of the so-called accelero-summation method in the time variable and by Fourier inverse transform in space. It appears that these functions share a common formal asymptotic expansion in the perturbation parameter. Furthermore, this formal series expansion can be written as a sum of two formal series with a corresponding decomposition for the actual solutions which possess two different asymptotic Gevrey orders, one stemming from the shape of the equation and the other originating from the forcing terms. The special case of multisummability in is also analyzed thoroughly. The proof leans on a version of the so-called Ramis-Sibuya theorem which entails two distinct Gevrey orders. Finally, we give an application to the study of parametric multi-level Gevrey solutions for some nonlinear initial value Cauchy problems with holomorphic coefficients and forcing term in ( , t) near 0 and bounded holomorphic on a strip in the complex space variable. MSC: 35C10; 35C20

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Section: Introductionmentioning

confidence: 99%

On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems

Lastra¹,

Malek²

2015

Adv Differ Equ

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“…Now we compare the ODE (31) with the equation (8). One can check that the compatibility conditions (32) hold if and only if c = 0 or…”

Section: Remark 47mentioning

confidence: 99%

“…As we will see later, the study of ODE (2) is motivated by the consideration of Lowey decomposition (Theorem 1.2) to linear ODEs (see [32,33] and the references therein) and it also covers some interesting well-known ODEs. Another reason to consider it is that the particular meromorphic solutions of (2) can have a tower structure because a solution of [D − f k (u)] · · · [D − f 1 (u)](u − α) = 0 will also be a solution of [D − f k+1 (u)][D − f k (u)] · · · [D − f 1 (u)](u − α) = 0 for k = 1, ..., n − 1 and it seems that one can get meromorphic solutions which grow faster and faster and eventually produce solutions which show that the estimate in (1) is sharp.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Loewy factorizable algebraic ODEs and Hayman’s conjecture

2018

Isr. J. Math.

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In this paper, we introduce certain n-th order nonlinear Loewy factorizable algebraic ordinary differential equations for the first time and study the growth of their meromorphic solutions in terms of the Nevanlinna characteristic function. It is shown that for generic cases all their meromorphic solutions are elliptic functions or their degenerations and hence their order of growth are at most two. Moreover, for the second order factorizable algebraic ODEs, all the meromorphic solutions of them (except for one case) are found explicitly. This allows us to show that a conjecture proposed by Hayman in 1996 holds for these second order ODEs.

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“…If g contains a constant C and may be integrated, the general solution of F = 0 may be obtained in this way. If g does not contain a constant and f does not contain y explicitly it may be possible to proceed with the solution procedure by solving g = z 1 where z 1 is a solution of f = 0; this resembles the case of solving a linear ode by decomposition, details are given in Chapters 4 and 5 of Schwarz [20]. If f does contain the dependent variable y this leads to an integro-differential equation; in this case only the right component may be applied for the solution procedure; it may allow determining special solutions.…”

Section: Decomposing Quasilinear Equations Of Second Ordermentioning

confidence: 99%

“…For general differential equations this goal is out of reach at present. Only for linear equations a fairly detailed solution scheme along these lines is available [20]. It is based on the decomposition of the differential operator that is associated with the differential equation.…”

Section: Introduction: Description Of the Problemmentioning

confidence: 99%

Decomposition of ordinary differential equations

Schwarz

2017

Bull. Math. Sci.

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Decompositions of linear ordinary differential equations (ode's) into components of lower order have successfully been employed for determining their solutions. Here this approach is generalized to nonlinear ode's. It is not based on the existence of Lie symmetries, in that it is a genuine extension of the usual solution algorithms. If an equation allows a Lie symmetry, the proposed decompositions are usually more efficient and often lead to simpler expressions for the solution. For the vast majority of equations without a Lie symmetry decomposition is the only available systematic solution procedure. Criteria for the existence of diverse decomposition types and algorithms for applying them are discussed in detail and many examples are given. The collection of Kamke of solved equations, and a tremendeous compilation of random equations are applied as a benchmark test for comparison of various solution procedures. Extensions of these proceedings for more general types of ode's and also partial differential equations are suggested.

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Loewy decomposition of linear differential equations

Cited by 8 publications

References 47 publications

On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems

On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems

Nonlinear Loewy factorizable algebraic ODEs and Hayman’s conjecture

Decomposition of ordinary differential equations

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