Open Access
On a modifcation of Olver's method: a special case
(Springer US, 2016) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza
We consider the asymptotic method designed by Olver (Asymptotics and special functions. Academic Press, New York, 1974) for linear differential equations of the second order containing a large (asymptotic) parameter : xm y −2 y = g(x)y, with m ∈ Z and g continuous. Olver studies in detail the cases m = 2, especially the cases m = 0, ±1, giving the Poincaré-type asymptotic expansions of two independent solutions of the equation. The case m = 2 is different, as the behavior of the solutions for large is not of exponential type, but of power type. In this case, Olver’s theory does not give many details. We consider here the special case m = 2. We propose two different techniques to handle the problem: (1) a modification of Olver’s method that replaces the role of the exponential approximations by power approximations, and (2) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter.
Open Access
Analysis of singular one-dimensional linear boundary value problems using two-point Taylor expansions
(University of Szeged (Hungría), 2020) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
We consider the second-order linear differential equation (x2 − 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, g and h are analytic in a Cassini disk Dr with foci at x = ±1 containing the interval [−1, 1]. Then, the two end points of the interval may be regular singular points 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 appro-ximation of the analytic solutions when they exist.
Open Access
Orthogonal basis with a conicoid first mode for shape specification of optical surfaces
(Optical Society of America, 2016) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Navarro, Rafael; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza
A rigorous and powerful theoretical framework is proposed to obtain systems of orthogonal functions (or shape modes) to represent optical surfaces. The method is general so it can be applied to different initial shapes and different polynomials. Here we present results for surfaces with circular apertures when the first basis function (mode) is a conicoid. The system for aspheres with rotational symmetry is obtained applying an appropriate change of variables to Legendre polynomials, whereas the system for general freeform case is obtained applying a similar procedure to spherical harmonics. Numerical comparisons with standard systems, such as Forbes and Zernike polynomials, are performed and discussed.
Open Access
The Pearcey integral in the highly oscillatory region II
(Elsevier, 2025-08-01) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2
We consider the Pearcey integral P(x, y) for large values of |x| and bounded values of |y|. The standard saddle point analysis is difficult to apply because the Pearcey integral is highly oscillating in this region. To overcome this problem we use the modified saddle point method introduced in López et al. (2009). A complete asymptotic analysis is possible with this method, and we derive a complete asymptotic expansion of P(x, y) for large |x|, accompanied by the exact location of the Stokes lines. There are two Stokes lines that divide the complex x−plane in two different sectors in which P(x, y) behaves differently when |x| is large. The asymptotic approximation is the sum of two asymptotic series whose terms are elementary functions of x and y. Both of them are of Poincaré type; one of them is given in terms of inverse powers of x; the other one in terms of inverse powers of x 1/2 , and it is multiplied by an exponential factor that behaves differently in the two mentioned sectors. Some numerical experiments illustrate the accuracy of the approximation.
Open Access
Asymptotic approximation of a highly oscillatory integral with application to the canonical catastrophe integrals
(Wiley, 2023) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
We consider the highly oscillatory integral 𝐹(𝑤) ∶= ∫ ∞ −∞ 𝑒𝑖𝑤(𝑡𝐾+2+𝑒𝑖𝜃𝑡𝑝) 𝑔(𝑡)𝑑𝑡 for large positive values of 𝑤, −𝜋 < 𝜃 ≤ 𝜋, 𝐾 and 𝑝 positive integers with 1 ≤ 𝑝 ≤ 𝐾, and 𝑔(𝑡) an entire function. The standard saddle point method is complicated and we use here a simplified version of this method introduced by López et al. We derive an asymptotic approximation of this integral when 𝑤 → +∞ for general values of 𝐾 and 𝑝 in terms of elementary functions, and determine the Stokes lines. For 𝑝 ≠ 1, the asymptotic behavior of this integral may be classified in four different regions according to the even/odd character of the couple of parameters 𝐾 and 𝑝; the special case 𝑝=1 requires a separate analysis. As an important application, we consider the family of canonical catastrophe integrals Ψ𝐾(𝑥1, 𝑥2,…,𝑥𝐾) for large values of one of its variables, say 𝑥𝑝, and bounded values of the remaining ones. This family of integrals may be written in the form 𝐹(𝑤) for appropriate values of the parameters 𝑤, 𝜃 and the function 𝑔(𝑡). Then, we derive an asymptotic approximation of the family of canonical catastrophe integrals for large |𝑥𝑝|. The approximations are accompanied by several numerical experiments. The asymptotic formulas presented here fill up a gap in the NIST Handbook of Mathematical Functions by Olver et al.
Open Access
New recurrence relations for several classical families of polynomials
(Taylor and Francis, 2021) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
In this paper, we derive new recurrence relations for the following families of polynomials: nörlund polynomials, generalized Bernoulli polynomials, generalized Euler polynomials, Bernoulli polynomials of the second kind, Buchholz polynomials, generalized Bessel polynomials and generalized Apostol–Euler polynomials. The recurrence relations are derived from a differential equation of first order and a Cauchy integral representation obtained from the generating function of these polynomials.
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 Publikoa
We 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.
Embargo
Convergent and asymptotic expansions of the displacement elastodynamic integral in terms of known functions
(Elsevier, 2025-05-01) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2
The integral [Formula presented] plays an essential role in the study of several phenomena in the theory of elastodynamics (Ceballos and Prato, 2014). But an exact evaluation of this integral in terms of known functions is not possible. In this paper, we derive an analytic representation of this integral in the form of convergent series of elementary functions and hypergeometric functions. This series have an asymptotic character for either, small values of the variable s, or for small values of the variables r and R. It is derived by using the asymptotic technique designed in Lopez (2008) for Mellin convolution integrals. Some numerical experiments show the accuracy of the approximation supplied by the first few terms of the expansion.
Open Access
The swallowtail integral in the highly oscillatory region III
(Taylor & Francis, 2021) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
We consider the swallowtail integral Ψ(x,y,z):=∫∞−∞ei(t5+xt3+yt2+zt)dt for large values of |z| and bounded values of |x| and |y|. The integrand of the swallowtail integral oscillates wildly in this region and the asymptotic analysis is subtle. The standard saddle point method is complicated and then we use the modified saddle point method introduced in López et al., A systematization of the saddle point method application to the Airy and Hankel functions. J Math Anal Appl. 2009;354:347–359. The analysis is more straightforward with this method and it is possible to derive complete asymptotic expansions of Ψ(x,y,z) for large |z| and fixed x and y. The asymptotic analysis requires the study of three different regions for argz separated by three Stokes lines in the sector −π
Open Access
The swallowtail integral in the highly oscillatory region II
(Kent State University, 2020) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
We analyze the asymptotic behavior of the swallowtail integral R ∞ −∞ e i(t 5+xt3+yt2+zt)dt for large values of |y| and bounded values of |x| and |z|. We use the simpli ed saddle point method introduced in [López et al., 2009]. With this method, the analysis is more straightforward than with the standard saddle point method and it is possible to derive complete asymptotic expansions of the integral for large |y| and xed x and z. There are four Stokes lines in the sector (−π, π] that divide the complex y−plane in four sectors in which the swallowtail integral behaves di erently when |y| is large. The asymptotic approximation is the sum of two asymptotic series whose terms are elementary functions of x, y and z. One of them is of Poincaré type and is given in terms of inverse powers of y 1/2 . The other one is given in terms of an asymptotic sequence of the order O(y −n/9 ) when |y| → ∞, and it is multiplied by an exponential factor that behaves di erently in the four mentioned sectors. Some numerical experiments illustrate the accuracy of the approximation.