**PROBLEMAS EVOLUTIVOS EN FISICA E INGENIERIA: TECNICAS DE RESOLUCION ANALITICAS Y NUMERICAS**

MTM2017-83490-P

•

**Nombre agencia financiadora**Agencia Estatal de Investigación

**Acrónimo agencia financiadora**AEI

**Programa**Programa Estatal de Fomento de la Investigación Científica y Técnica de Excelencia

**Subprograma**Subprograma Estatal de Generación de Conocimiento

**Convocatoria**Proyectos I+D

**Año convocatoria**2017

**Unidad de gestión**Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016

**Centro beneficiario**UNIVERSIDAD PUBLICA DE NAVARRA

**Identificador persistente**http://dx.doi.org/10.13039/501100011033

## Publicaciones

Found(s) 28 result(s)

Found(s) 1 page(s)

Found(s) 1 page(s)

#### Uniformly convergent additive schemes for 2d singularly perturbed parabolic systems of reaction-diffusion type

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza

- Clavero, C.
- Gracia, J.L.

In this work, we consider parabolic 2D singularly perturbed systems of reaction-diffusion type on a rectangle, in the simplest case that the diffusion parameter is the same for all equations of the system. The solution is approximated on a Shishkin mesh with two splitting or additive methods in time and standard central differences in space. It is proved that they are first-order in time and almost second-order in space uniformly convergent schemes. The additive schemes decouple the components of the vector solution at each time level of the discretization which makes the computation more efficient. Moreover, a multigrid algorithm is used to solve the resulting linear systems. Numerical results for some test problems are showed, which illustrate the theoretical results and the efficiency of the splitting and multigrid techniques.

#### Uniform representations of the incomplete beta function in terms of elementary functions

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza

- Ferreira, Chelo
- López, José L.
- Pérez Sinusía, Ester

We consider the incomplete beta function $B_{z}(a,b)$ in the maximum domain ofanalyticity of its three variables: $a,b,z\\in\\mathbb{C}$, $-a\\notin\\mathbb{N}$,$z\\notin[1,\\infty)$. For $\\Re b\\le 1$ we derive a convergent expansion of$z^{-a}B_{z}(a,b)$ in terms of the function $(1-z)^b$ and of rational functionsof $z$ that is uniformly valid for $z$ in any compact set in$\\mathbb{C}\\setminus[1,\\infty)$. When $-b\\in \\mathbb{N}\\cup\\{0\\}$, the expansionalso contains a logarithmic term of the form $\\log(1-z)$. For $\\Re b\\ge 1$ wederive a convergent expansion of $z^{-a}(1-z)^bB_{z}(a,b)$ in terms of thefunction $(1-z)^b$ and of rational functions of $z$ that is uniformly valid for$z$ in any compact set in the exterior of the circle $\\vert z-1\\vert=r$ forarbitrary $r>0$. The expansions are accompanied by realistic error bounds. Somenumerical experiments show the accuracy of the approximations.

#### A uniformly convergent scheme to solve two-dimensional parabolic singularly perturbed systems of reaction-diffusion type with multiple diffusion parameters

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza

- Clavero, C.
- Jorge, J.C.

In this work, we deal with solving two-dimensional parabolic singularly perturbed systems of reaction-diffusion type where the diffusion parameters at each equation of the system can be small and of different scale. In such case, in general, overlapping boundary layers appear at the boundary of the spatial domain and, because of this, special meshes are required to resolve them. The numerical scheme combines the central difference scheme to discretize in space and the fractional implicit Euler method together with a splitting by components to discretize in time. If the fully discrete scheme is defined on an adequate piecewise uniform Shishkin mesh in space then it is uniformly convergent of first order in time and of almost second order in space. Some numerical results illustrate the theoretical results. © 2020 John Wiley & Sons, Ltd.

#### A multi-splitting method to solve 2D parabolic reaction-diffusion singularly perturbed systems

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza

- Clavero, C.
- Jorge, J.C.

In this paper we design and analyze a numerical method to solve a type of reaction–diffusion 2D parabolic singularly perturbed systems. The method combines the central finite difference scheme on an appropriate piecewise uniform mesh of Shishkin type to discretize in space, and the fractional implicit Euler method together with a splitting by directions and components of the reaction–diffusion operator to integrate in time. We prove that the method is uniformly convergent of first order in time and almost second order in space. The use of this time integration technique has the advantage that only tridiagonal linear systems must be solved to obtain the numerical solution at each time step; because of this, our method provides a remarkable reduction of computational cost, in comparison with other implicit methods which have been previously proposed for the same type of problems. Full details of the uniform convergence are given only for systems with two equations; nevertheless, our ideas can be easily extended to systems with an arbitrary number of equations as it is shown in the numerical experiences performed. The numerical results show in practice the qualities of our proposal.

#### A splitting uniformly convergent method for one-dimensional parabolic singularly perturbed convection-diffusion systems

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza

- Clavero Gracia, Carmelo
- Jorge Ulecia, Juan Carlos

In this paper we deal with solving robustly and efficiently one-dimensional linear parabolic singularly perturbed systems of convection-diffusion type, where the diffusion parameters can be different at each equation and even they can have different orders of magnitude. The numerical algorithm combines the classical upwind finite difference scheme to discretize in space and the fractional implicit Euler method together with an appropriate splitting by components to discretize in time. We prove that if the spatial discretization is defined on an adequate piecewise uniform Shishkin mesh, the fully discrete scheme is uniformly convergent of first order in time and of almost first order in space. The technique used to discretize in time produces only tridiagonal linear systems to be solved at each time level; thus, from the computational cost point of view, the method we propose is more efficient than other numerical algorithms which have been used for these problems. Numerical results for several test problems are shown, which corroborate in practice both the uniform convergence and the efficiency of the algorithm.

#### New recurrence relations for several classical families of polynomials

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza

- Ferreira, Chelo
- López, José
- Pérez Sinusía, Ester

In this paper, we derive new recurrence relations for the following families of polynomials: Nørlund 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.

**Proyecto:**ES/MINECO/MTM2017-83490-P

#### Numerical solution of singularly perturbed 2-D convection-diffusion elliptic interface PDEs with Robin-type boundary conditions

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza

- Clavero, Carmelo
- Shiromani, Ram
- Shanthi, Vembu

We consider a singularly perturbed two-dimensional convection-diffusion elliptic interface problem with Robin boundary conditions, where the source term is a discontinuous function. The coefficient of the highest-order terms in the differential equation and in the boundary conditions, denoted by ε, is a positive parameter which can be arbitrarily small. Due to the discontinuity in the source term and the presence of the diffusion parameter, the solutions to such problems have, in general, boundary, corner and weak-interior layers. In this work, a numerical approach is carried out using a finite-difference technique defined on an appropriated layer-adapted piecewise uniform Shishkin mesh to provide a good estimate of the error. We show some numerical results which corroborate in practice that these results are sharp.

#### A numerical approach for a two-parameter singularly perturbed weakly-coupled system of 2-D elliptic convection–reaction–diffusion PDEs

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza

- Clavero, Carmelo
- Shiromani, Ram
- Shanthi, Vembu

In this work, we consider the numerical approximation of a two dimensional elliptic singularly perturbed weakly-coupled system of convection–reaction–diffusion type, which has two different parameters affecting the diffusion and the convection terms, respectively. The solution of such problems has, in general, exponential boundary layers as well as corner layers. To solve the continuous problem, we construct a numerical method which uses a finite difference scheme defined on an appropriate layer-adapted Bakhvalov–Shishkin mesh. Then, the numerical scheme is a first order uniformly convergent method with respect both convection and diffusion parameters. Numerical results obtained with the algorithm for some test problems are presented, which show the best performance of the proposed method, and they also corroborate in practice the theoretical analysis.

#### A computational approach for 2D elliptic singularly perturbed weakly coupled systems of convection–diffusion type with multiple scales and parameters in the diffusion and the convection terms

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza

- Shiromani, Ram
- Clavero, Carmelo
- Shanthi, Vembu

In this work, we consider the efficient resolution of a 2D elliptic singularly perturbed weakly coupled system of convection–diffusion type, which has small parameters at both the diffusion and the convection terms. We assume that the diffusion parameters can be distinct at each equation of the system, and also, they can have different orders of magnitude, but the convection parameter is the same at both equations of the system. Then, in general, overlapping regular or parabolic boundary layers can appear in the exact solution. The continuous problem is approximated by using the standard upwind finite difference scheme, which is constructed on a special piecewise uniform Shishkin mesh. Then, the numerical scheme is an almost first‐order uniformly convergent method with respect to all the perturbation parameters. Some numerical results, obtained with the numerical algorithm for one test problem, are presented, which show the good performance of the proposed numerical method and also corroborate in practice the theoretical results.

#### A splitting uniformly convergent method for one-dimensional parabolic singularly perturbed convection-diffusion systems

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Clavero, Carmelo
- Jorge Ulecia, Juan Carlos

In this paper we deal with solving robustly and efficiently one-dimensional linear parabolic

singularly perturbed systems of convection-diffusion type, where the diffusion parameters

can be different at each equation and even they can have different orders of magnitude.

The numerical algorithm combines the classical upwind finite difference scheme to

discretize in space and the fractional implicit Euler method together with an appropriate

splitting by components to discretize in time. We prove that if the spatial discretization

is defined on an adequate piecewise uniform Shishkin mesh, the fully discrete scheme is

uniformly convergent of first order in time and of almost first order in space. The technique

used to discretize in time produces only tridiagonal linear systems to be solved at each

time level; thus, from the computational cost point of view, the method we propose is

more efficient than other numerical algorithms which have been used for these problems.

Numerical results for several test problems are shown, which corroborate in practice both

the uniform convergence and the efficiency of the algorithm., The authors thank the referees for their valuable suggestions which have helped to improve the presentation of this paper. This research was partially supported by the project MTM2017-83490-P, the Aragon Government and European Social Fund (group E24-17R ) and the Public University of Navarre, project PRO-UPNA 6158 .

singularly perturbed systems of convection-diffusion type, where the diffusion parameters

can be different at each equation and even they can have different orders of magnitude.

The numerical algorithm combines the classical upwind finite difference scheme to

discretize in space and the fractional implicit Euler method together with an appropriate

splitting by components to discretize in time. We prove that if the spatial discretization

is defined on an adequate piecewise uniform Shishkin mesh, the fully discrete scheme is

uniformly convergent of first order in time and of almost first order in space. The technique

used to discretize in time produces only tridiagonal linear systems to be solved at each

time level; thus, from the computational cost point of view, the method we propose is

more efficient than other numerical algorithms which have been used for these problems.

Numerical results for several test problems are shown, which corroborate in practice both

the uniform convergence and the efficiency of the algorithm., The authors thank the referees for their valuable suggestions which have helped to improve the presentation of this paper. This research was partially supported by the project MTM2017-83490-P, the Aragon Government and European Social Fund (group E24-17R ) and the Public University of Navarre, project PRO-UPNA 6158 .

**Proyecto:**AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-83490-P

#### An overlapping decomposition framework for wave propagation in heterogeneous and unbounded media: formulation, analysis, algorithm, and simulation

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Domínguez Baguena, Víctor
- Ganesh, M.
- Sayas, Francisco Javier

A natural medium for wave propagation comprises a coupled bounded heterogeneous region and an unbounded homogeneous free-space. Frequency-domain wave propagation models in the medium, such as the variable coefficient Helmholtz equation, include a faraway decay radiation condition (RC). It is desirable to develop algorithms that incorporate the full physics of the heterogeneous and unbounded medium wave propagation model, and avoid an approximation of the RC. In this work we first present and analyze an overlapping decomposition framework that is equivalent to the full-space heterogeneous-homogenous continuous model, governed by the Helmholtz equation with a spatially dependent refractive index and the RC. Our novel overlapping framework allows the user to choose two free boundaries, and gain the advantage of applying established high-order finite and boundary element methods (FEM and BEM) to simulate an equivalent coupled model. The coupled model comprises auxiliary interior bounded heterogeneous and exterior unbounded homogeneous Helmholtz problems. A smooth boundary can be chosen for simulating the exterior problem using a spectrally accurate BEM, and a simple boundary can be used to develop a high-order FEM for the interior problem. Thanks to the spectral accuracy of the exterior computational model, the resulting coupled system in the overlapping region is relatively very small. Using the decomposed equivalent framework, we develop a novel overlapping FEM-BEM algorithm for simulating the acoustic or electromagnetic wave propagation in two dimensions. Our FEM-BEM algorithm for the full-space model incorporates the RC exactly. Numerical experiments demonstrate the efficiency of the FEM-BEM approach for simulating smooth and non-smooth wave fields, with the latter induced by a complex heterogeneous medium and a discontinuous refractive index., Víctor Domínguez thanks the support of the project MTM2017-83490-P. Francisco-Javier Sayas was partially supported by the NSF grant DMS-1818867.

**Proyecto:**AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-83490-P

#### New recurrence relations for several classical families of polynomials

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Ferreira González, Chelo
- López García, José Luis
- Pérez Sinusía, Ester

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., This research was supported by the Ministerio de Economía y Competitividad, Secretaría de Estado de Investigación, Desarrollo e Innovación (MTM2017-83490-P) and the Universidad Pública de Navarra.

**Proyecto:**AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-83490-P

#### An extension of the multiple Erdélyi-Kober operator and representations of the generalized hypergeometric functions

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Karp, D. B.
- López García, José Luis

In this paper we investigate the extension of the multiple Erd elyi-Kober

fractional integral operator of Kiryakova to arbitrary complex values of parameters

by the way of regularization. The regularization involves derivatives

of the function in question and the integration with respect to a kernel

expressed in terms of special case of Meijer's G function. An action of the

regularized multiple Erd elyi-Kober operator on some simple kernels leads

to decomposition formulas for the generalized hypergeometric functions. In

the ultimate section, we de ne an alternative regularization better suited

for representing the Bessel type generalized hypergeometric function p1Fp.

A particular case of this regularization is then used to identify some new

facts about the positivity and reality of zeros of this function., The research of the first author has been supported by the Russian Science Foundation under the project 14-11-00022. The research of the second author has been supported by the Spanish Ministry of "Economía y Competitividad" under the project MTM2017-83490-P and by the Universidad Pública de Navarra.

fractional integral operator of Kiryakova to arbitrary complex values of parameters

by the way of regularization. The regularization involves derivatives

of the function in question and the integration with respect to a kernel

expressed in terms of special case of Meijer's G function. An action of the

regularized multiple Erd elyi-Kober operator on some simple kernels leads

to decomposition formulas for the generalized hypergeometric functions. In

the ultimate section, we de ne an alternative regularization better suited

for representing the Bessel type generalized hypergeometric function p1Fp.

A particular case of this regularization is then used to identify some new

facts about the positivity and reality of zeros of this function., The research of the first author has been supported by the Russian Science Foundation under the project 14-11-00022. The research of the second author has been supported by the Spanish Ministry of "Economía y Competitividad" under the project MTM2017-83490-P and by the Universidad Pública de Navarra.

**Proyecto:**AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-83490-P

#### Series representations of the Volterra function and the Fransén–Robinson constant

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- López García, José Luis
- Pagola Martínez, Pedro Jesús
- Palacios Herrero, Pablo

The Volterra function μ(t,β,α) was introduced by Vito Volterra in 1916 as the solution to certain integral equations with a logarithmic kernel. Despite the large number of applications of the Volterra function, the only known analytic representations of this function are given in terms of integrals. In this paper we derive several convergent expansion of μ(t,β,α) in terms of incomplete gamma functions. These expansions may be used to implement numerical evaluation techniques for this function. As a particular application, we derive a numerical series representation of the Fransén–Robinson constant F := µ(1, 1, 0) = R ∞ 0 1 Γ(x) dx. Some numerical examples illustrate the accuracy of the approximations, This research was supported by the Ministerio de Economía y Competitividad (MTM2017-83490-P) and the Universidad Pública de Navarra, Spain.

**Proyecto:**AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-83490-P

#### Uniform convergent expansions of integral transforms

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- López García, José Luis
- Palacios Herrero, Pablo
- Pagola Martínez, Pedro Jesús

Several convergent expansions are available for most of the special functions of the mathematical physics, as well as some asymptotic expansions [NIST Handbook of Mathematical Functions, 2010]. Usually, both type of expansions are given in terms of elementary functions; the convergent expansions provide a good approximation for small values of a certain variable, whereas the asymptotic expansions provide a good approximation for large values of that variable. Also, quite often, those expansions are not uniform: the convergent expansions fail for large values of the variable and the asymptotic expansions fail for small values. In recent papers [Bujanda & all, 2018-2019] we have designed new expansions of certain special functions, given in terms of elementary functions, that are uniform in certain variables, providing good approximations of those special functions in large regions of the variables, in particular for large and small values of the variables. The technique used in [Bujanda & all, 2018-2019] is based on a suitable integral representation of the special function. In this paper we face the problem of designing a general theory of uniform approximations of special functions based on their integral representations. Then, we consider the following integral transform of a function g(t) with kernel h(t, z), F(z) := 1 0 h(t, z)g(t)dt. We require for the function h(t, z) to be uniformly bounded for z ∈D⊂ C by a function H(t) integrable in t ∈ [0, 1], and for the function g(t) to be analytic in an open region Ω that contains the open interval (0, 1). Then, we derive expansions of F(z) in terms of the moments of the function h, M[h(·, z), n] := 1 0 h(t, z)tndt, that are uniformly convergent for z ∈ D. The convergence of the expansion is of exponential order O(a−n), a > 1, when [0, 1] ∈ Ω and of power order O(n−b), b > 0, when [0, 1] ∈/ Ω. Most of the special functions F(z) having an integral representation may be cast in this form, possibly after an appropriate change of the integration variable. Then, special interest has the case when the moments M[h(·, z), n] are elementary functions of z, because in that case the uniformly convergent expansion derived for F(z) is given in terms of elementary functions. We illustrate the theory with several examples of special functions different from those considered in [Bujanda & all, 2018-2019]., This research was supported by the Spanish Ministry of Economía y Competitividad, project MTM2017-83490-P. The Universidad Pública de Navarra is acknowledged for its financial support.

**Proyecto:**AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-83490-P

#### On a particular class of Meijer's G functions appearing in fractional calculus

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Karp, D. B.
- López García, José Luis

In this paper we investigate the Meijer G-function G p+1,p+1 p,1 which, for certain parameter values, represents the Riemann-Liouville fractional integral of the Meijer-Nørlund function G p,p. p,0 The properties of this function play an important role in extending the multiple Erdélyi-Kober fractional integral operator to arbitrary values of the parameters which is investigated in a separate work, in Fract. Calc. Appl. Anal., Vol. 21, No 5 (2018). Our results for G p+1,p+1 p,1 include: a regularization formula for overlapping poles, a connection formula with the Meijer-Nørlund function, asymptotic formulas around the origin and unity, formulas for the moments, a hypergeometric transform and a sign stabilization theorem for growing parameters., The research of the first author has been supported by the Russian Science Foundation under the project 14-11-00022. The research of the second author has been supported by the Spanish Ministry of Economía y Competitividad under the project MTM2017-83490-P and by the Universidad Pública de Navarra.

**Proyecto:**AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-83490-P

#### Analysis of singular one-dimensional linear boundary value problems using two-point Taylor expansions

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Ferreira González, Chelo
- López García, José Luis
- Pérez Sinusía, Ester

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., The Ministerio de Economía y Competitividad (project MTM2017-83490-P) and Gobierno de Aragón (project E24_17R) are acknowledged by their financial support.

**Proyecto:**AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-83490-P

#### A multi-splitting method to solve 2D parabolic reaction-diffusion singularly perturbed systems

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Clavero, Carmelo
- Jorge Ulecia, Juan Carlos

In this paper we design and analyze a numerical method to solve a type of reaction-diffusion 2D parabolic singularly perturbed systems. The method combines the central finite difference scheme on an appropriate piecewise uniform mesh of Shishkin type to discretize in space, and the fractional implicit Euler method together with a splitting by directions and components of the reaction-diffusion operator to integrate in time. We prove that the method is uniformly convergent of first order in time and almost second order in space. The use of this time integration technique has the advantage that only tridiagonal linear systems must be solved to obtain the numerical solution at each time step; because of this, our method provides a remarkable reduction of computational cost, in comparison with other implicit methods which have been previously proposed for the same type of problems. Full details of the uniform convergence are given only for systems with two equations; nevertheless, our ideas can be easily extended to systems with an arbitrary number of equations as it is shown in the numerical experiences performed. The numerical results show in practice the qualities of our proposal., This research was partially supported by the project MTM2017-83490-P and the Aragón Government and European Social Fund (group E24-17R ).

**Proyecto:**AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-83490-P

#### Uniformly convergent expansions for the generalized hypergeometric functions p –1Fp and pFp

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- López García, José Luis
- Pagola Martínez, Pedro Jesús
- Karp, D. B.

We derive a convergent expansion of the generalized hypergeometric function p−1 F p in terms of the Bessel functions 0 F 1 that holds uniformly with respect to the argument in any horizontal strip of the complex plane. We further obtain a convergent expansion of the generalized hypergeometric function p F p in terms of the confluent hypergeometric functions 1 F 1 that holds uniformly in any right half-plane. For both functions, we make a further step forward and give convergent expansions in terms of trigonometric, exponential and rational functions that hold uniformly in the same domains. For all four expansions we present explicit error bounds. The accuracy of the approximations is illustrated by some numerical experiments., The authors López and Pagola acknowledge the Spanish Ministry of Economía, Industria y Competitividad, project MTM2017-83490-P (REF. MTM2017-83490-P) for its financial support.

**Proyecto:**AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-83490-P

#### Uniform approximations of the first symmetric elliptic integral in terms of elementary functions

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Bujanda Cirauqui, Blanca
- López García, José Luis
- Pagola Martínez, Pedro Jesús
- Palacios Herrero, Pablo

We consider the standard symmetric elliptic integral RF(x, y, z) for complex x, y, z. We derive convergent expansions of RF(x, y, z) in terms of elementary functions that hold uniformly for one of the three variables x, y or z in closed subsets (possibly unbounded) of C\ (−∞, 0]. The expansions are accompanied by error bounds. The accuracy of the expansions and their uniform features are illustrated by means of some numerical examples., The financial support of the following entities is acknowledged: Ministerio de Economía y Competitividad, project MTM2017-83490-P; Ministerio de Ciencia, Innovación y Universidades, research grant RTI2018-095499-B-C31 IoTrain; Gobierno de Navarra, Grant 0011-1365-2019-000083 - SIAGUS; the European Union-European Regional Development Fund ERDF-FEDER.

#### The swallowtail integral in the highly oscillatory region III

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Ferreira González, Chelo
- López García, José Luis
- Pérez Sinusía, Ester

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<argz≤π. The asymptotic approximation is a certain combination of two asymptotic series whose terms are elementary functions of x, y and z. They are given in terms of an asymptotic sequence of the order O(z−n/12) when |z|→∞, and it is multiplied by an exponential factor that behaves differently in the three mentioned sectors. The accuracy and the asymptotic character of the approximations are illustrated with some numerical experiments., This research was supported by the Ministerio de Economía y Competitividad (MTM2017-83490-P) and the Universidad Pública de Navarra.

**Proyecto:**AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-83490-P

#### The swallowtail integral in the highly oscillatory region II

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Ferreira González, Chelo
- López García, José Luis
- Pérez Sinusía, Ester

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., This research was supported by the Ministerio de Economia y Competitividad (MTM2017-83490-P). The Universidad Publica de Navarra is acknowledged by its financial support.

**Proyecto:**AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-83490-P

#### Analytic approximations of integral transforms in terms of elementary functions: application to special functions

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Palacios Herrero, Pablo

This thesis focuses on the study of new analytical methods for the approximation of integral transforms and, in particular, of special functions that admit an integral representation. The importance of these functions lies in the fact that they are solutions to a great variety of functional equations that model speciﬁc physical phenomena. Moreover, they play an important role in pure and applied mathematics, as well as in other branches of science such as chemistry, statistics or economics. Usually, the integrals deﬁning these special functions depend on various parameters that have a speciﬁc physical meaning. For this reason, it is important to have analytical techniques that allow their computation in a quick and easy manner. The most commonly used analytical methods are based on series expansions of local validity: Taylor series and asymptotic (divergent) expansions that are, respectively, valid for small or large values of the physically relevant variable. However, neither of them is, in general, simultaneously valid for large and small values of the variable. In this thesis we seek new methods for the computation of analytic expansions of integral transforms satisfying the following three properties: (a) The expansions are uniformly valid in a large region of the complex plane. Ideally, these regions should be unbounded and contain the point 0 in their interior. (b) The expansions are convergent. Therefore, it is not necessary to obtain error bounds or to study the optimal term to truncate the expansion: the more terms considered, the smaller the error committed.

(c) The expansions are given in terms of elementary functions. We develop a theory of uniform expansions that shows the necessary and suﬃcient conditions to obtain expansions of integral transforms fulﬁlling the three conditions (a),(b) and (c) above. This theory is applied to obtain new series approximations satisfying (a), (b) and (c) of a large number of special functions. The new expansions are compared with other known representations that we may ﬁnd in the literature to show their ad-vantages and drawbacks. In contrast to the Taylor and asymptotic expansions, the main beneﬁt of the uniform expansions is that they are valid in a large region of the complex plane. For this reason, they may be used to replace the function they approximate (which is often diﬃcult to work with) when it appears in certain calculations, such as a factor of an integral or in a diﬀerential equation. Since these developments are also given in terms of elementary functions, such calculations may be carried out easily. Next, we consider a particularly important case: when the kernel of the integral transform is given by an exponential. We develop a new Laplace’s method for integrals that produces asymptotic and convergent expansions, in contrast to the classical Laplace method which produces divergent developments. The expansions obtained with this new method satisfy (a) and (b) but not (c), since the asymptotic sequence is given in terms of incomplete beta functions. Finally, we develop a new uniform asymptotic method 'saddle point near an end point' which does not satisfy (b) and (c) but, unlike the classical 'saddle point near an end point' method, allows us to calculate the coeﬃcients of the expansion by means of a simple and systematic formula., This work was supported by the Ministerio de Economa y Competitividad (MTM2014-52859-P and MTM2017-83490-P) and Universidad Publica de Navarra PRO-UPNA (6158)., Programa de Doctorado en Matemáticas y Estadística (RD 99/2011), Matematikako eta Estatistikako Doktoretza Programa (ED 99/2011)

(c) The expansions are given in terms of elementary functions. We develop a theory of uniform expansions that shows the necessary and suﬃcient conditions to obtain expansions of integral transforms fulﬁlling the three conditions (a),(b) and (c) above. This theory is applied to obtain new series approximations satisfying (a), (b) and (c) of a large number of special functions. The new expansions are compared with other known representations that we may ﬁnd in the literature to show their ad-vantages and drawbacks. In contrast to the Taylor and asymptotic expansions, the main beneﬁt of the uniform expansions is that they are valid in a large region of the complex plane. For this reason, they may be used to replace the function they approximate (which is often diﬃcult to work with) when it appears in certain calculations, such as a factor of an integral or in a diﬀerential equation. Since these developments are also given in terms of elementary functions, such calculations may be carried out easily. Next, we consider a particularly important case: when the kernel of the integral transform is given by an exponential. We develop a new Laplace’s method for integrals that produces asymptotic and convergent expansions, in contrast to the classical Laplace method which produces divergent developments. The expansions obtained with this new method satisfy (a) and (b) but not (c), since the asymptotic sequence is given in terms of incomplete beta functions. Finally, we develop a new uniform asymptotic method 'saddle point near an end point' which does not satisfy (b) and (c) but, unlike the classical 'saddle point near an end point' method, allows us to calculate the coeﬃcients of the expansion by means of a simple and systematic formula., This work was supported by the Ministerio de Economa y Competitividad (MTM2014-52859-P and MTM2017-83490-P) and Universidad Publica de Navarra PRO-UPNA (6158)., Programa de Doctorado en Matemáticas y Estadística (RD 99/2011), Matematikako eta Estatistikako Doktoretza Programa (ED 99/2011)

#### A note on the asymptotic expansion of the Lerch’s transcendent

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Cai, Xing Shi
- López García, José Luis

This is an accepted manuscript of an article published by Taylor & Francis in Integral Transforms and Special Functions on 10 Jun 2019, available online: https://doi.org/10.1080/10652469.2019.1627530, In Ferreira and López [Asymptotic expansions of the Hurwitz–Lerch zeta function. J Math Anal Appl. 2004;298(1):210–224], the authors derived an asymptotic expansion of the Lerch's transcendent Φ(z,s,a) for large |a|, valid for Ra>0, Rs>0 and z∈C∖[1,∞). In this paper, we study the special case z≥1 not covered in Ferreira and López [Asymptotic expansions of the Hurwitz–Lerch zeta function. J Math Anal Appl. 2004;298(1):210–224], deriving a complete asymptotic expansion of the Lerch's transcendent Φ(z,s,a) for z>1 and Rs>0 as Ra goes to infinity. We also show that when a is a positive integer, this expansion is convergent for Rz≥1. As a corollary, we get a full asymptotic expansion for the sum ∑mn=1zn/ns for fixed z>1 as m→∞. Some numerical results show the accuracy of the approximation., This work is supported by the Knut and Alice Wallenberg Foundation and the Ministerio de Economía y Competitividad of the spanish government (MTM2017-83490-P).

**Proyecto:**AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-83490-P

#### Uniform convergent expansions of the Gauss hypergeometric function in terms of elementary functions

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Ferreira González, Chelo
- López García, José Luis
- Pérez Sinusía, Ester

This is an accepted manuscript of an article published by Taylor & Francis in Integral Transforms and Special Functions on 2018-09-28, available online: https://doi.org/10.1080/10652469.2018.1525369, We 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., This research was supported by Ministerio de Economía, Industria y Competitividad, Gobierno de España (MTM2017-83490-P).

**Proyecto:**AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-83490-P

#### Uniform representation of the incomplete beta function in terms of elementary functions

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Ferreira González, Chelo
- López García, José Luis
- Pérez Sinusía, Ester

We 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 <b ≤ 1 we derive a convergent expansion of z−aBz(a, b) in terms of the function (1 − z) b

and of rational functions of z that is uniformly valid for z in any compact set in C \ [1, ∞). When −b ∈ N ∪ {0}, the expansion also contains a logarithmic term of the form log(1 − z). For <b ≥ 1 we derive a convergent expansion of z−a(1 − z) bBz(a, b) in terms of the function (1 − z) b and of rational functions of z that is uniformly valid for z in any compact set 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 approximations., This research was supported by Ministerio de Economía, Industria

y Competitividad, Gobierno de España, project MTM2017-83490-P, Gobierno de Aragón and

European Social Fund (group E24-17R).

and of rational functions of z that is uniformly valid for z in any compact set in C \ [1, ∞). When −b ∈ N ∪ {0}, the expansion also contains a logarithmic term of the form log(1 − z). For <b ≥ 1 we derive a convergent expansion of z−a(1 − z) bBz(a, b) in terms of the function (1 − z) b and of rational functions of z that is uniformly valid for z in any compact set 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 approximations., This research was supported by Ministerio de Economía, Industria

y Competitividad, Gobierno de España, project MTM2017-83490-P, Gobierno de Aragón and

European Social Fund (group E24-17R).

**Proyecto:**AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-83490-P

#### New fractional step Runge-Kutta-Nyström methods up to order three

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Bujanda Cirauqui, Blanca
- Jorge Ulecia, Juan Carlos
- Moreta, M. Jesús

Fractional Step Runge–Kutta–Nyströ (FSRKN) methods have been revealed to be an excellent option to integrate numerically many multidimensional evolution models governed by second order in time partial differential equations. These methods, combined with suitable spatial discretizations, lead to strong computational cost reductions respect to many classical implicit time integrators. In this paper, we present the construction process of several implicit FSRKN methods of two and three levels which attain orders up to three and satisfy adequate stability properties. We have also performed some numerical experiments in order to show the unconditionally convergent behavior of these schemes as well as their computational advantages., B. Bujanda had financial support from VOLTEADOR 0011-1365-2018-000178, PERISEIS 4.0 PC054-055 and IoTrain RTI2018-095499-B-C31. M. J. Moreta had financial support from MTM 2015-66837-P. and PGC2018-101443-B-I00 J. C. Jorge had financial support from MTM2014-52859-P and MTM2017-83490-P.

#### Analysis and application of an overlapped FEM-BEM for wave propagation in unbounded and heterogeneous media

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Domínguez Baguena, Víctor
- Ganesh, M.

An overlapped continuous model framework, for the Helmholtz wave propagation problem in unbounded regions comprising bounded heterogeneous media, was recently introduced and analyzed by the authors (2020) [10]. The continuous Helmholtz system incorporates a radiation condition (RC) and our equivalent hybrid framework facilitates application of widely used finite element methods (FEM) and boundary element methods (BEM), and the resulting discrete systems retain the RC exactly. The FEM and BEM discretizations, respectively, applied to the designed interior heterogeneous and exterior homogeneous media Helmholtz systems include the FEM and BEM solutions matching in artificial interface domains, and allow for computations of the exact ansatz based far-fields. In this article we present rigorous numerical analysis of a discrete two-dimensional FEM-BEM overlapped coupling implementation of the algorithm. We also demonstrate the efficiency of our discrete FEM-BEM framework and analysis using numerical experiments, including applications to non-convex heterogeneous multiple particle Janus configurations. Simulations of the far-field induced differential scattering cross sections (DSCS) of heterogeneous configurations and orientation-averaged (OA) counterparts are important for several applications, including inverse wave problems. Our robust FEM-BEM framework facilitates computations of such quantities of interest, without boundedness or homogeneity or shape restrictions on the wave propagation model. © 2021 IMACS, The first author (Domínguez) is supported by the project MTM2017-83490-P. The second author (Ganesh) gratefully acknowledges the support of the Simons Foundation .

**Proyecto:**AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-83490-P