Person: López García, José Luis
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López García
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José Luis
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Estadística, Informática y Matemáticas
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InaMat2. Instituto de Investigación en Materiales Avanzados y Matemáticas
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0000-0002-6050-9015
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2369
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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 Asymptotic and convergent expansions for solutions of third-order linear differential equations with a large parameter(Shanghai Normal University, 2018) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Ingeniería Matemática e Informática; Matematika eta Informatika IngeniaritzaIn previous papers [6–8,10], we derived convergent and asymptotic expansions of solutions of second order linear differential equations with a large parameter. In those papers we generalized and developed special cases not considered in Olver’s theory [Olver, 1974]. In this paper we go one step forward and consider linear differential equations of the third order: y ′′′ +aΛ2y′ +bΛ3y = f(x)y′ +g(x)y, with a, b ∈ C fixed, f′ and g continuous, and Λ a large positive parameter. We propose two different techniques to handle the problem: (i) a generalization of Olver’s method and (ii) 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. As an application of the theory, we obtain new convergent and asymptotic expansions of the Pearcey integral P(x, y) for large |x|.Publication Open Access Analytic formulas for the evaluation of the Pearcey integral(American Mathematical Society, 2017) López García, José Luis; Pagola Martínez, Pedro Jesús; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza; Universidad Pública de Navarra / Nafarroako Unibertsitate PublikoaWe can find in the literature several convergent and/or asymptotic expansions of the Pearcey integral P(x, y) in different regions of the complex variables x and y, but they do not cover the whole complex x and y planes. The purpose of this paper is to complete this analysis giving new convergent and/or asymptotic expansions that, together with the known ones, cover the evaluation of the Pearcey integral in a large region of the complex x and y planes. The accuracy of the approximations derived in this paper is illustrated with some numerical experiments. Moreover, the expansions derived here are simpler compared with other known expansions, as they are derived from a simple manipulation of the integral definition of P(x, y).Publication Open Access Convergent and asymptotic expansions of the Pearcey integral(Elsevier, 2015) López García, José Luis; Pagola Martínez, Pedro Jesús; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza; Universidad Pública de Navarra / Nafarroako Unibertsitate PublikoaWe consider the Pearcey integral P(x; y) for large values of |x|, x, y ∈ C. We can find in the literature several convergent or asymptotic expansions in terms of elementary and special functions, with different levels of complexity. Most of them are based in analytic, in particular asymptotic, techniques applied to the integral definition of P(x; y). In this paper we consider a different method: the iterative technique used for differential equations in [Lopez, 2012]. Using this technique in a differential equation satisfied by P(x; y) we obtain a new convergent expansion analytically simple that is valid for any complex x and y and has an asymptotic property when |x|→ ∞ uniformly for y in bounded sets. The accuracy of the approximation is illustrated with some numerical experiments and compared with other expansions given in the literature.