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Chara Pantazi - One of the best experts on this subject based on the ideXlab platform.

  • differential galois theory and non integrability of planar polynomial vector fields
    Journal of Differential Equations, 2018
    Co-Authors: Primitivo B Acostahumanez, Tomas J Lazaro, Juan J Moralesruiz, Chara Pantazi
    Abstract:

    Abstract We study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential Galois Theory. More concretely, by means of the variational Equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational Equation. We illustrate this result with several families of examples. A key point is to check whether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order Linear Equation, the Risch Equation. This is a classical problem studied by Risch in 1969, and the solution is given by the “Risch algorithm”. In this way we point out the connection of the non integrability with some higher transcendent functions, like the error function.

  • differential galois theory and non integrability of planar polynomial vector fields
    arXiv: Dynamical Systems, 2017
    Co-Authors: Primitivo B Acostahumanez, Tomas J Lazaro, Juan J Moralesruiz, Chara Pantazi
    Abstract:

    We study a necessary condition for the integrability of the polynomials fields in the plane by means of the differential Galois theory. More concretely, by means of the variational Equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational Equation. We illustrate this result with several families of examples. A key point is to check wether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order Linear Equation, the Risch Equation. This is a classical problem studied by Risch in 1969, and the solution is given by the "Risch algorithm". In this way we point out the connection of the non integrablity with some higher transcendent functions, like the error function.

Primitivo B Acostahumanez - One of the best experts on this subject based on the ideXlab platform.

  • differential galois theory and non integrability of planar polynomial vector fields
    Journal of Differential Equations, 2018
    Co-Authors: Primitivo B Acostahumanez, Tomas J Lazaro, Juan J Moralesruiz, Chara Pantazi
    Abstract:

    Abstract We study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential Galois Theory. More concretely, by means of the variational Equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational Equation. We illustrate this result with several families of examples. A key point is to check whether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order Linear Equation, the Risch Equation. This is a classical problem studied by Risch in 1969, and the solution is given by the “Risch algorithm”. In this way we point out the connection of the non integrability with some higher transcendent functions, like the error function.

  • differential galois theory and non integrability of planar polynomial vector fields
    arXiv: Dynamical Systems, 2017
    Co-Authors: Primitivo B Acostahumanez, Tomas J Lazaro, Juan J Moralesruiz, Chara Pantazi
    Abstract:

    We study a necessary condition for the integrability of the polynomials fields in the plane by means of the differential Galois theory. More concretely, by means of the variational Equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational Equation. We illustrate this result with several families of examples. A key point is to check wether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order Linear Equation, the Risch Equation. This is a classical problem studied by Risch in 1969, and the solution is given by the "Risch algorithm". In this way we point out the connection of the non integrablity with some higher transcendent functions, like the error function.

Alberto Cabada - One of the best experts on this subject based on the ideXlab platform.

  • Solutions and Green's function of the first order Linear Equation with reflection and initial conditions
    arXiv: Classical Analysis and ODEs, 2017
    Co-Authors: Alberto Cabada, F. Adrián F. Tojo
    Abstract:

    This work is devoted to the study of the existence and sign of Green's functions for first order Linear problems with constant coefficients and initial (one point) conditions. We first prove a result on the existence of solutions of $n$-th order Linear Equations with involutions via some auxiliary functions to later prove a uniqueness result in the first order case. We study then different situations for which a Green's function can be obtained explicitly and derive several results in order to obtain information about the sign of the Green's function. Once the sign is known, optimal maximum and anti-maximum principles follow.

  • Solutions and Green’s function of the first order Linear Equation with reflection and initial conditions
    Boundary Value Problems, 2014
    Co-Authors: Alberto Cabada, Fernando Adrián Fernández Tojo
    Abstract:

    This work is devoted to the study of the existence and sign of Green’s functions for first order Linear problems with constant coefficients and initial (one point) conditions. We first prove a result on the existence of solutions of n th order Linear Equations with involutions via some auxiliary functions to later prove a uniqueness result in the first order case. We study then different situations for which a Green’s function can be obtained explicitly and derive several results in order to obtain information as regards the sign of the Green’s function. Once the sign is known, optimal maximum and anti-maximum principles follow.

  • Solutions of the first order Linear Equation with reflection and general Linear conditions
    2013
    Co-Authors: Alberto Cabada
    Abstract:

    This work is devoted to the study of first order Linear problems with involution and general Linear conditions. We first study the problem in the case of antiperiodic boundary conditions, giving an explicit Green's function for it. Then we move forward to more general Linear boundary conditions, focusing on sufficient conditions for existence and uniqueness of solution. At the end of the paper we give estimates that ensure the positivity of the solution in the general problems and illustrate these applications with examples.

Morales Ruiz, Juan José - One of the best experts on this subject based on the ideXlab platform.

  • Differential galois theory and non-integrability of planar polynomial vector fields
    'Elsevier BV', 2018
    Co-Authors: Lázaro Ochoa, José Tomás, Pantazi Chara, Acosta Humanez Primitivo, Morales Ruiz, Juan José
    Abstract:

    We study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential Galois Theory. More concretely, by means of the variational Equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational Equation. We illustrate this result with several families of examples. A key point is to check whether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order Linear Equation, the Risch Equation. This is a classical problem studied by Risch in 1969, and the solution is given by the “Risch algorithm”. In this way we point out the connection of the non integrability with some higher transcendent functions, like the error functionPeer Reviewe

Roland Masson - One of the best experts on this subject based on the ideXlab platform.

  • Convergence of a Numerical Scheme for Stratigraphic Modeling
    SIAM Journal on Numerical Analysis, 2005
    Co-Authors: Robert Eymard, Thierry Gallouët, Véronique Gervais, Roland Masson
    Abstract:

    In this paper, we consider a multilithology diffusion model used in the field of stratigraphic basin simulations to simulate large scale depositional transport processes of sediments described as a mixture of L lithologies. This model is a simplified one for which the surficial fluxes are proportional to the slope of the topography and to a lithology fraction with unitary diffusion coefficients. The main variables of the system are the sediment thickness h, the L surface concentrations cis in lithology i of the sediments at the top of the basin, and the L concentrations ci in lithology i in the sediments inside the basin. For this simplified model, the sediment thickness decouples from the other unknowns and satisfies a Linear parabolic Equation. The remaining Equations account for the mass conservation of the lithologies, and couple, for each lithology, a first order Linear Equation for cis with a Linear advection Equation for ci for which cis appears as an input boundary condition. For this coupled system, a weak formulation is introduced. The system is discretized by an implicit time integration and a cell centered finite volume method. This numerical scheme is shown to satisfy stability estimates and to converge, up to a subsequence, to a weak solution of the problem.

  • Mathematical and numerical analysis of a stratigraphic model
    ESAIM: Mathematical Modelling and Numerical Analysis, 2004
    Co-Authors: Véronique Gervais, Roland Masson
    Abstract:

    In this paper, we consider a multi-lithology diffusion model used in stratigraphic modelling to simulate large scale transport processes of sediments described as a mixture of L lithologies. This model is a simplified one for which the surficial fluxes are proportional to the slope of the topography and to a lithology fraction with unitary diffusion coefficients. The main unknowns of the system are the sediment thickness h , the L surface concentrations in lithology i of the sediments at the top of the basin, and the L concentrations c i in lithology i of the sediments inside the basin. For this simplified model, the sediment thickness decouples from the other unknowns and satisfies a Linear parabolic Equation. The remaining Equations account for the mass conservation of the lithologies, and couple, for each lithology, a first order Linear Equation for with a Linear advection Equation for c i for which appears as an input boundary condition. For this coupled system, a weak formulation is introduced which is shown to have a unique solution. An implicit finite volume scheme is derived for which we show stability estimates and the convergence to the weak solution of the problem.

  • Existence and Uniqueness of a Weak Solution to a Stratigraphic Model
    Numerical Mathematics and Advanced Applications, 2004
    Co-Authors: Robert Eymard, Thierry Gallouët, Véronique Gervais, Roland Masson
    Abstract:

    In this paper, we study a multi-lithology diffusion model used to simulate the evolution through time of a sedimentary basin composed of several lithologies such as sand or shale. It is a simplified model for which the surficial flux in lithology i is taken proportional to the slope and to a lithology fraction c i s in lithology i at the top of the basin with a unitary diffusion coefficient. Thus, the sediment thickness variable satisfies a Linear parabolic problem and decouples from the other unknowns. The remaining Equations couple, for each lithology, a first order Linear Equation for the surface concentration c i s with a Linear advection Equation for the basin concentration, for which c i s appears as an input boundary condition at the top of the basin in case of sedimentation. The existence and uniqueness of a weak solution in L ∞ is proved for this problem.