INAMAT2 - Institute for Advanced Materials and Mathematics
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Browsing INAMAT2 - Institute for Advanced Materials and Mathematics by Department/Institute "Ingeniería Matemática e Informática"
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Publication Open Access Convergent and asymptotic methods for second-order difference equations with a large parameter(Springer, 2018) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Matematika eta Informatika Ingeniaritza; Institute for Advanced Materials and Mathematics - INAMAT2; Ingeniería Matemática e Informática; Universidad Pública de Navarra / Nafarroako Unibertsitate PublikoaWe consider the second-order linear difference equation y(n+2)−2ay(n+1)−Λ2y(n)=g(n)y(n)+f(n)y(n+1) , where Λ is a large complex parameter, a≥0 and g and f are sequences of complex numbers. Two methods are proposed to find the asymptotic behavior for large |Λ|of the solutions of this equation: (i) an iterative method based on a fixed point method and (ii) a discrete version of Olver’s method for second-order linear differential equations. Both methods provide an asymptotic expansion of every solution of this equation. The expansion given by the first method is also convergent and may be applied to nonlinear problems. Bounds for the remainders are also given. We illustrate the accuracy of both methods for the modified Bessel functions and the associated Legendre functions of the first kind.Publication Open Access Convergent expansions of the Bessel functions in terms of elementary functions(Springer US, 2018) López García, José Luis; Pagola Martínez, Pedro Jesús; Matematika eta Informatika Ingeniaritza; Institute for Advanced Materials and Mathematics - INAMAT2; Ingeniería Matemática e Informática; Universidad Pública de Navarra / Nafarroako Unibertsitate PublikoaWe consider the Bessel functions Jν (z) and Yν (z) for ν > −1/2 and z ≥ 0. We derive a convergent expansion of Jν (z) in terms of the derivatives of (sin z)/z, and a convergent expansion of Yν (z) in terms of derivatives of (1−cos z)/z, derivatives of (1 − e−z)/z and (2ν, z). Both expansions hold uniformly in z in any fixed horizontal strip and are accompanied by error bounds. The accuracy of the approximations is illustrated with some numerical experiments.Publication Open Access Convergent expansions of the incomplete gamma functions in terms of elementary functions(World Scientific Publishing, 2017) Bujanda Cirauqui, Blanca; López García, José Luis; Pagola Martínez, Pedro Jesús; Matematika eta Informatika Ingeniaritza; Institute for Advanced Materials and Mathematics - INAMAT2; Ingeniería Matemática e Informática; Universidad Pública de Navarra / Nafarroako Unibertsitate PublikoaWe consider the incomplete gamma function γ(a,z) for Ra>0 and z∈C. We derive several convergent expansions of z−aγ(a,z) in terms of exponentials and rational functions of z that hold uniformly in z with Rz bounded from below. These expansions, multiplied by ez, are expansions of ezz−aγ(a,z) uniformly convergent in z with Rz bounded from above. The expansions are accompanied by realistic error bounds.Publication Open Access Dynamics of axially symmetric perturbed Hamiltonians in 1:1:1 resonance(Springer, 2018) Carrasco, Dante; Palacián Subiela, Jesús Francisco; Vidal Díaz, Claudio; Vidarte, Jhon; Yanguas Sayas, Patricia; Matematika eta Informatika Ingeniaritza; Institute for Advanced Materials and Mathematics - INAMAT2; Ingeniería Matemática e InformáticaWe study the dynamics of a family of perturbed three-degree-of-freedom Hamiltonian systems which are in 1:1:1 resonance. The perturbation consists of axially symmetric cubic and quartic arbitrary polynomials. Our analysis is performed by normalisation, reduction and KAM techniques. Firstly, the system is reduced by the axial symmetry, and then, periodic solutions and KAM 3-tori of the full system are determined from the relative equilibria. Next, the oscillator symmetry is extended by normalisation up to terms of degree 4 in rectangular coordinates; after truncation of higher orders and reduction to the orbit space, some relative equilibria are established and periodic solutions and KAM 3-tori of the original system are obtained. As a third step, the reduction in the two symmetries leads to a one-degree-of-freedom system that is completely analysed in the twice reduced space. All the relative equilibria together with the stability and parametric bifurcations are determined. Moreover, the invariant 2-tori (related to the critical points of the twice reduced space), some periodic solutions and the KAM3-tori, all corresponding to the full system, are established. Additionally, the bifurcations of equilibria occurring in the twice reduced space are reconstructed as quasi-periodic bifurcations involving 2-tori and periodic solutions of the full system.Publication Open Access Oscillatory motions in restricted N-body problems(Elsevier, 2018) Álvarez-Ramírez, Martha; Rodríguez García, Antonio; Palacián Subiela, Jesús Francisco; Yanguas Sayas, Patricia; Matematika eta Informatika Ingeniaritza; Institute for Advanced Materials and Mathematics - INAMAT2; Ingeniería Matemática e InformáticaWe consider the planar restricted N-body problem where the N−1 primaries are assumed to be in a central configuration whereas the infinitesimal particle escapes to infinity in a parabolic orbit. We prove the existence of transversal intersections between the stable and unstable manifolds of the parabolic orbits at infinity which guarantee the existence of a Smale’s horseshoe. This implies the occurrence of chaotic mo-tions, namely the oscillatory motions, that is, orbits for which the massless particle leaves every bounded region but it returns infinitely often to some fixed bounded region. Our achievement is based in an adequate scaling of the variables which allows us to write the Hamiltonian function as the Hamiltonian of the Kepler problem plus higher-order terms that depend on the chosen configuration. We compute the Melnikov function related to the first non-null perturbative term and characterize the cases where it has simple zeroes. Concretely, for some combinations of the configuration parameters, i.e. mass values and positions of the primaries, and for a specific value of a parameter related to the angular momentum vector, the Melnikov function vanishes, otherwise it has simple zeroes and the transversality condition is satisfied. When the Melnikov function corresponding to the principal part of the perturbation is zero we compute the next non-zero Melnikov function proving that it has simple zeroes. The theory is illustrated for various cases of restricted N-body problems, including the circular restricted three-body problem. No restrictions on the mass parameters are assumed.Publication Open Access The Pearcey integral in the highly oscillatory region(Elsevier, 2016) López García, José Luis; Pagola Martínez, Pedro Jesús; Matematika eta Informatika Ingeniaritza; Institute for Advanced Materials and Mathematics - INAMAT2; Ingeniería Matemática e Informática; Universidad Pública de Navarra / Nafarroako Unibertsitate PublikoaWe consider the Pearcey integral P(x, y) for large values of |y| and bounded values of |x|. The integrand of the Pearcey integral oscillates wildly in this region and the asymptotic saddle point analysis is complicated. Then we consider here the modified saddle point method introduced in [Lopez, Pérez and Pagola, 2009] [4]. With this method, the analysis is simpler and it is possible to derive a complete asymptotic expansion of P(x, y) for large |y|. The asymptotic analysis requires the study of three different regions for separately. In the three regions, the expansion is given in terms of inverse powers of y2/3 and the coefficients are elementary functions of x. The accuracy of the approximation is illustrated with some numerical experiments.Publication Open Access A simplification of the stationary phase method: application to the Anger and Weber functions(Kent State University, 2017) López García, José Luis; Matematika eta Informatika Ingeniaritza; Institute for Advanced Materials and Mathematics - INAMAT2; Ingeniería Matemática e Informática; Universidad Pública de Navarra / Nafarroako Unibertsitate PublikoaThe main difficulty in the practical use of the stationary phase method in asymptotic expansions of integrals is originated by a change of variables. The coefficients of the asymptotic expansion are the coefficients of the Taylor expansion of a certain function implicitly defined by that change of variables. In general, this function is not explicitly known, and then the computation of those coefficients is cumbersome. Using the factorization of the exponential factor used in previous works of [Tricomi, 1950], [Erdélyi and Wyman, 1963], and [Dingle, 1973], we obtain a variant of the method that avoids that change of variables and simplifies the computations. On the one hand, the calculation of the coefficients of the asymptotic expansion is remarkably simpler and explicit. On the other hand, the asymptotic sequence is as simple as in the standard stationary phase method: inverse powers of the asymptotic variable. New asymptotic expansions of the Anger and Weber functions Jλx(x) and Eλx(x) for large positive x and real parameter λ 6= 0 are given as an illustration.Publication Open Access Singular reduction of resonant Hamiltonians(IOP Publishing, 2018) Meyer, Kenneth Ray; Palacián Subiela, Jesús Francisco; Yanguas Sayas, Patricia; Matematika eta Informatika Ingeniaritza; Institute for Advanced Materials and Mathematics - INAMAT2; Ingeniería Matemática e InformáticaWe investigate the dynamics of resonant Hamiltonians with n degrees of freedom to which we attach a small perturbation. Our study is based on the geometric interpretation of singular reduction theory. The flow of the Hamiltonian vector field is reconstructed from the cross sections corresponding to an approximation of this vector field in an energy surface. This approximate system is also built using normal forms and applying reduction theory obtaining the reduced Hamiltonian that is defined on the orbit space. Generically, the reduction is of singular character and we classify the singularities in the orbit space, getting three different types of singular points. A critical point of the reduced Hamiltonian corresponds to a family of periodic solutions in the full system whose characteristic multipliers are approximated accordingly to the nature of the critical point.Publication Open Access Uniform convergent expansions of the Gauss hypergeometric function in terms of elementary functions(Taylor & Francis, 2018) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Matematika eta Informatika Ingeniaritza; Institute for Advanced Materials and Mathematics - INAMAT2; Ingeniería Matemática e InformáticaWe consider the hypergeometric function 2F1(a, b; c; z) for z ∈ C \ [1,∞). For Ra ≥ 0, we derive a convergent expansion of 2F1(a, b; c; z) in terms of the function (1 − z)−a and of rational functions of z that is uniformly valid for z in any compact in C \ [1,∞). When a ∈ N, the expansion also contains a logarithmic term of the form log(1 − z). For Ra ≤ 0, we derive a convergent expansion of (1 − z)a 2F1(a, b; c; z) in terms of the function (1 − z)−a and of rational functions of z that is uniformly valid for z in any compact in C \ [1,∞) in the exterior of the circle |z − 1| = r for arbitrary r > 0. The expansions are accompanied by realistic error bounds. Some numerical experiments show the accuracy of the approximation.Publication Open Access Uniform representation of the incomplete beta function in terms of elementary functions(Kent State University, 2018) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Matematika eta Informatika Ingeniaritza; Institute for Advanced Materials and Mathematics - INAMAT2; Ingeniería Matemática e InformáticaWe consider the incomplete beta function Bz(a, b) in the maximum domain of analyticity of its three variables: a, b, z ∈ C, −a /∈ N, z /∈ [1, ∞). For 0. The expansions are accompanied by realistic error bounds. Some numerical experiments show the accuracy of the approximations.Publication Open Access The use of two-point Taylor expansions in singular one-dimensional boundary value problems I(Elsevier, 2018) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Matematika eta Informatika Ingeniaritza; Institute for Advanced Materials and Mathematics - INAMAT2; Ingeniería Matemática e InformáticaWe consider the second-order linear differential equation (x + 1)y′′ + f(x)y′ + g(x)y = h(x) in the interval (−1, 1) with initial conditions or boundary conditions (Dirichlet, Neumann or mixed Dirichlet-Neumann). The functions f(x), g(x) and h(x) are analytic in a Cassini disk Dr with foci at x = ±1 containing the interval [−1, 1]. Then, the end point of the interval x = −1 may be a regular singular point of the differential equation. The two-point Taylor expansion of the solution y(x) at the end points ±1 is used to study the space of analytic solutions in Dr of the differential equation, and to give a criterion for the existence and uniqueness of analytic solutions of the boundary value problem. This method is constructive and provides the two-point Taylor approximation of the analytic solutions when they exist.