BUSCADOR RECOLECTA

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., This research was supported by the Spanish Ministerio de Ciencia, Innovación y Universidades, project PID2022-136441NB-I00.

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., This research was supported by the Spanish Ministerio de Ciencia, Innovación Universidades, project PID2022-136441NB-I00

In this work we solve initial-boundary value problems associated to coupled 2D parabolic singularly perturbed systems of convection-diffusion type. The analysis is focused on the cases where the diffusion parameters are small, distinct and also they may have different order of magnitude. In such cases, overlapping regular boundary layers appear at the outflow boundary of the spatial domain. The fully discrete scheme combines the classical upwind scheme defined on an appropriate Shishkin mesh to discretize the spatial variables, and the fractional implicit Euler method joins to a decomposition of the difference operator in directions and components to integrate in time. We prove that the resulting method is uniformly convergent of first order in time and of almost first order in space. Moreover, as only small tridiagonal linear systems must be solved to advance in time, the computational cost of our method is remarkably smaller than the corresponding ones to other implicit methods considered in the previous literature for the same type of problems. The numerical results, obtained for some test problems, corroborate in practice the good behavior and the advantages of the algorithm., This research was partially supported by the projects PID2022-136441NB-I00 and TED2021-130884B-I00, the Aragón Government and European Social Fund (group E24-17R) and the Public University of Navarra.

In this work, we propose and study a numerical method to solve efficiently one-dimensional parabolic singularly perturbed systems of convection-diffusion type, for which the convection coefficient is zero at an interior point of the spatial domain. We focus our attention on the case of having the same diffusion parameter in both equations; as well we assume adequate signs on the convective coefficients in order to the interior turning point is of repulsive type. Under these conditions, if the data of the problem are composed by continuous functions, the exact evolutionary solution, in general, has regular boundary layers at the end points of the spatial domain. To solve this type of problems, we combine the fractional implicit Euler method and the classical upwind scheme, defined on a special mesh of Shishkin type. The resulting numerical method reach uniform convergence of first order in time and almost first order in space. Numerical results obtained for different test problems are shown which corroborate in practice the uniform convergence of the numerical algorithm and also their computational efficiency in comparison with classical numerical methods used for the same type of problems., This research was partially supported by the project PID2022-136441NB-I00, the Aragón Government and the European Social Fund (group E24-17R).

Helmholtz decompositions of elastic fields is a common approach for the solution of Navier scattering problems. Used in the context of boundary integral equations (BIE), this approach affords solutions of Navier problems via the simpler Helmholtz boundary integral operators (BIOs). Approximations of Helmholtz Dirichlet-to-Neumann (DtN) can be employed within a regularizing combined field strategy to deliver BIE formulations of the second kind for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions, at least in the case of smooth boundaries. Unlike the case of scattering and transmission Helmholtz problems, the approximations of the DtN maps we use in the Helmholtz decomposition BIE in the Navier case require incorporation of lower order terms in their pseudodifferential asymptotic expansions. The presence of these lower order terms in the Navier regularized BIE formulations complicates the stability analysis of their Nyström discretizations in the framework of global trigonometric interpolation and the Kussmaul–Martensen kernel singularity splitting strategy. The main difficulty stems from compositions of pseudodifferential operators of opposite orders, whose Nyström discretization must be performed with care via pseudodifferential expansions beyond the principal symbol. The error analysis is significantly simpler in the case of arclength boundary parametrizations and considerably more involved in the case of general smooth parametrizations that are typically encountered in the description of one-dimensional closed curves., Projects “Funciones especiales y métodos numéricos avanzados” from Universidad Pública de Navarra, Spain and ‘Técnicas innovadoras para la resolución de problemas evolutivos’, ref. PID2022-136441NB-I00 from Ministerio de Ciencia e Innovación, Gobierno de España, Spain (to V.D.); NSF through contract “A new class of high-order integral solvers for wave propagation problems in composite media” ref. NSF DMS 2408635 (to C.T.).

We continue the program initiated in [López & all, 2009–2011] to simplify asymptotic methods for integrals: in this paper we revise the uniform method ‘‘saddle point near an end point’’. We obtain a more systematic version of this uniform asymptotic method where the computation of the coefficients of the asymptotic expansion is remarkably simpler than in the classical method. On the other hand, as in the standard method, the asymptotic sequence is given in terms of parabolic cylinder functions. New asymptotic expansions of the confluent hypergeometric functions 𝑀(𝑐, 𝑥∕𝛼 + 𝑐 + 1, 𝑥) and 𝑈(𝑐, 𝛼𝑥 + 𝑐 + 1, 𝑥) for large 𝑥, 𝑐 fixed, uniformly valid for 𝛼 ∈ (0, ∞), are given as an illustration., This research was supported by the Ministerio de Ciencia, Innovación y Universidades, grant PID2022-136441NB-I00. Open access funding provided by Universidad Pública de Navarra.

In López, Pagola and Perez (2009) we introduced a modification of the Laplace's method for deriving asymptotic expansions of Laplace integrals which simplifies the computations, giving explicit formulas for the coefficients of the expansion. On the other hand, motivated by the approximation of special functions with two asymptotic parameters, Nemes has generalized Laplace's method by considering Laplace integrals with two asymptotic parameters of a different asymptotic order. Nemes considers a linear dependence of the phase function on the two asymptotic parameters. In this paper, we investigate if the simplifying ideas introduced in López, Pagola and Perez (2009) for Laplace integrals with one large parameter may be also applied to the more general Laplace integrals considered in Nemes's theory. We show in this paper that the answer is yes, but moreover, we show that those simplifying ideas can be applied to more general Laplace integrals where the phase function depends on the large variable in a more general way, not necessarily in a linear form. We derive new asymptotic expansions for this more general kind of integrals with simple and explicit formulas for the coefficients of the expansion. Our theory can be applied to special functions with two or more large parameters of a different asymptotic order. We give some examples of special functions that illustrate the theory., The Universidad Pública de Navarra, plan de promoción de grupos de investigación, and the Ministerio de Ciencia, Innovación y Universidades, project PID2022-136441NB-I00, are acknowledged by their financial support.

We present three high-order Nyström discretization strategies of various boundary
integral equation formulations of the impenetrable time-harmonic Navier equations
in two dimensions. One class of such formulations is based on the four classical
Boundary Integral Operators (BIOs) associated with the Green’s function of the Navier
operator. We consider two types of Nyström discretizations of these operators, one that
relies on Kussmaul–Martensen logarithmic splittings (Chapko et al., 2000; Domínguez
and Turc, 2000), and the other on Alpert quadratures (Alpert, 1999). In addition, we
consider an alternative formulation of Navier scattering problems based on Helmholtz
decompositions of the elastic fields (Dong et al., 2021), which can be solved via a
system of boundary integral equations that feature integral operators associated with
the Helmholtz equation. Owing to the fact that some of the BIOs that are featured in
those formulations are non-standard, we use Quadrature by Expansion (QBX) methods
for their high order Nyström discretization. Alternatively, we use Maue integration by
parts techniques to recast those non-standard operators in terms of single and double
layer Helmholtz BIOs whose Nyström discretizations is amenable to the Kussmaul–
Martensen methodology. We present a variety of numerical results concerning the high
order accuracy that our Nyström discretization elastic scattering solvers achieve for both
smooth and Lipschitz boundaries. We also present extensive comparisons regarding the
iterative behavior of solvers based on different integral equations in the high frequency
regime. Finally, we illustrate how some of the Nyström discretizations we considered
can be incorporated seamlessly into the Convolution Quadrature (CQ) methodology to
deliver high-order solutions of the time domain elastic scattering problems., Catalin Turc gratefully acknowledges support from NSF through contract DMS-1908602. Víctor Domínguez is partially supported by projects “Adquisición de conocimiento
minería de datos, funciones especiales
métodos numéricos avanzados” from Universidad Pública de Navarra, Spain and “Técnicas innovadoras para la resolución de problemas evolutivos”, ref. PID2022-136441NB-I00 from Ministerio de Ciencia e Innovación, Gobierno de España, Spain .

Helmholtz decompositions of the elastic fields open up new avenues for the solution of linear elastic scattering problems via boundary integral equations (BIE) (Dong et al. (2021) [20]). The main appeal of this approach is that the ensuing systems of BIE feature only integral operators associated with the Helmholtz equation. However, these BIE involve non standard boundary integral operators that do not result after the application of either the Dirichlet or the Neumann trace to Helmholtz single and double layer potentials. Rather, the Helmholtz decomposition approach leads to BIE formulations of elastic scattering problems with Neumann boundary conditions that involve boundary traces of the Hessians of Helmholtz layer potential. As a consequence, the classical combined field approach applied in the framework of the Helmholtz decompositions leads to BIE formulations which, although robust, are not of the second kind. Following the regularizing methodology introduced in Boubendir et al. (2015) [6] we design and analyze novel robust Helmholtz decomposition BIE for the solution of elastic scattering that are of the second kind in the case of smooth scatterers in two dimensions. We present a variety of numerical results based on Nyström discretizations that illustrate the good performance of the second kind regularized formulations in connections to iterative solvers., Catalin Turc gratefully acknowledges support from NSF through contract DMS-1908602. Víctor Domínguez is partially supported by project
"Adquisición de conocimiento y minería de datos, funciones especiales y métodos numéricos avanzados" from Universidad Pública de Navarra and
"Técnicas innovadoras para la resolución de problemas evolutivos", ref. PID2022-136441NB-I00 from Ministerio de Ciencia e Innovación, Gobierno
de España, Spain.

The integral ∫∞0()()/ (−) 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.

This work is the continuation of [11], where a two-dimensional elliptic singularly perturbed weakly system, for which small parameters affected both the diffusion and the convection terms, was solved; moreover, all perturbation parameters could have different orders of magnitude, which is the most interesting and difficult case for this type of problem. It is well known that then, in general, overlapping regular or parabolic boundary layers appear in the solution of the continuous problem. To solve numerically the problem, the classical upwind finite difference scheme, defined on special piecewise uniform Shsihkin meshes, was used, proving its uniform convergence, with respect to all parameters, for four different ratios between them. In this paper, we complete the previous analysis, considering the two cases for these possible ratios, that were not considered in [11]. To see in practice the efficiency of the numerical method, we show the numerical results obtained with our algorithm for a test problem, when the cases analyzed in this work are fixed; from those results, the uniform convergence of the numerical algorithm follows, in agreement with the theoretical results.

In this work we are interested in constructing a uniformly convergent method to solve a 2D elliptic singularly perturbed weakly system of convection-diffusion type. We assume that small positive parameters appear at both the diffusion and the convection terms of the partial differential equation. Moreover, we suppose that both the diffusion and the convection parameters can be distinct and also they can have a different order of magnitude. Then, the nature of the overlapping regular or parabolic boundary layers, which, in general, appear in the exact solution, is much more complicated. To solve the continuous problem, we use the classical upwind finite difference scheme, which is defined on piecewise uniform Shishkin meshes, which are given in a different way depending on the value and the ratio between the four singular perturbation parameters which appear in the continuous problem. So, the numerical algorithm is an almost first order uniformly convergent method. The numerical results obtained with our algorithm for a test problem are presented; these results corroborate in practice the good behavior and the uniform convergence of the algorithm, aligning with the theoretical results.

We consider the Pearcey integral for large values of and bounded values of . 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 for large , accompanied by the exact location of the Stokes lines. There are two Stokes lines that divide the complex plane in two different sectors in which behaves differently when is large. The asymptotic approximation is the sum of two asymptotic series whose terms are elementary functions of and . Both of them are of Poincaré type; one of them is given in terms of inverse powers of ; the other one in terms of inverse powers of , 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.

In this work, we propose and study a numerical method to solve efficiently one-dimensional parabolic singularly perturbed systems of convection–diffusion type, for which the convection coefficient is zero at an interior point of the spatial domain. We focus our attention on the case of having the same diffusion parameter in both equations; as well we assume adequate signs on the convective coefficients in order to the interior turning point is of repulsive type. Under these conditions, if the data of the problem are composed by continuous functions, the exact evolutionary solution, in general, has regular boundary layers at the end points of the spatial domain. To solve this type of problems, we combine the fractional implicit Euler method and the classical upwind scheme, defined on a special mesh of Shishkin type. The resulting numerical method reach uniform convergence of first order in time and almost first order in space. Numerical results obtained for different test problems are shown which corroborate in practice the uniform convergence of the numerical algorithm and also their computational efficiency in comparison with classical numerical methods used for the same type of problems.

We consider an integral representation of the Lerch transcendent function (z, s, a) of the form (z, s, a) = 1 0h(t, z)g(t, s, a)dt, and two different analytical methods for the approximation of this integral transform to obtain new convergent expansions of the Lerch transcendent in the variable z. The first method uses multi-point Taylor expansions of h(t, z) at certain appropriately selected base points that provides convergent expansions of the Lerch transcendent in terms of elementary functions of z uniformly valid in compact sets of the complex z−plane. The second method expands g(t, s, a) in a Taylor series at a selected point in [0, 1] giving a uniform convergent expansion of (z, s, a) in terms of elementary functions of z valid in a large unbounded region of the complex plane. We provide explicit and/or recursive algorithms for the computation of the coefficients of the expansions. Numerical experiments illustrate the accuracy of the new approximations.