Recent Trends in Special Functions and Analysis of Differential Equations
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Juan L.G. Guirao, Zulqurnain Sabir, Tareq Saeed, "Design and Numerical Solutions of a Novel ThirdOrder Nonlinear Emden–Fowler Delay Differential Model", Mathematical Problems in Engineering, vol. 2020, Article ID 7359242, 9 pages, 2020. https://doi.org/10.1155/2020/7359242
Design and Numerical Solutions of a Novel ThirdOrder Nonlinear Emden–Fowler Delay Differential Model
Abstract
In this study, the design of a novel model based on nonlinear thirdorder Emden–Fowler delay differential (EFDD) equations is presented along with two types using the sense of delay differential and standard form of the secondorder EF equation. The singularity at = 0 at single or multiple points of each type of the designed EFDD model are discussed. The detail of shape factors and delayed points is provided for both types of the designed thirdorder EFDD model. For the verification and validation of the model, two numerical examples are presented of each case and numerical results have been performed using the artificial neural network along with the hybrid of global and local capabilities. The comparison of the obtained numerical results with the exact solutions shows the perfection and correctness of the designed thirdorder EFDD model.
1. Introduction
The delay differential (DD) equation is known as one of the historical and important equations. Recently, DD equation has attained much attention of the researcher’s community due to its vast applications in many biological models, as well as scientific phenomena such as communication system model, dynamical population model, economical systems, engineering system, and transport and propagation model [1–5]. It is always interested to find the solution of DD equations and many researchers have applied different numerical/analytical techniques. Brunner et al. [6] solved DD equation by applying a discontinuous Galerkin numerical scheme. Hsiao and Wu [7] applied Haar wavelet to solve DD equations, while Wang [8] presented the solution of DD equations using Legendre wavelet. Adomian and Rach [9] solved DD equation using the Adomian decomposition scheme. Shakeri and Dehghan [10] found the solutions of DD initial value problems using the homotopy perturbation scheme. Erdogan et al. [11] implemented finite difference approach on layeradapted mesh using the singularly perturbed DD equations. The general form of the DD model is written as [12, 13]where h shows the linear/nonlinear function and is the delayed term, whereas A, B, and C are the constants.
The singular study has become very significant in the modern era due to the variety of applications in technology, engineering, and biological and physical sciences. The singular nature models are always difficult, grim, and challengeable to solve for the research community. One of the important, famous, historical, and singular models is Emden–Fowler (EF) model that shows the singularity at the origin. Since its invention, this model has been solved by various analytical and numerical schemes, and it has a number of applications in the study of relativistic mechanics, fluid dynamics, population growth model, pattern creation, and the study of chemical reactor models. The literature form of the EF model is written as [14–16]where is the shape vector. The EF model (1) becomes the Lane–Emden model by taking and is written as follows:
The above singular models have been achieved from the work of Homer Lane and Robert Emden. These models designate inner construction of polytropic stars, gas cloud model, cluster galaxies, and radiative cooling. Due to the worth of these models, no one can deny the value and importance of such models, which has vast applications in the physical science field [17], isotropic continuous media [18], density of gaseous star [19], morphogenesis [20], dusty fluid models [21], stellar structure models [22], reactions based on catalytic diffusion [23], oscillating magnetic systems [24], isothermal gas sphere models [25], mathematical physics [26], catalytic diffusion reactions [23], classical/quantum mechanics [27], and electromagnetic theory [28].
Due to the fame of these models, the researcher’s community is interested to solve these models and only a few methods are available in the literature that has been investigated. One of the wellknown methods used to solve these models is the Adomian decomposition method, which is proposed by Shawagfeh and Wazwaz [29, 30]. Parand and Razzaghi [31] implemented a famous numerical scheme to solve singular equations. Liao [32] applied an analytic technique to avoid the difficulty of singular points. Bender et al. [33] proposed a perturbative scheme to solve the singular models. Nouh [34] presented two techniques’ power series and Pade approximation to solve the singular models.
The aim of this study is to design a novel thirdorder Emden–Fowler delay differential (EFDD) model along with two types. Two examples of the designed thirdorder EFDD model have been presented for both of the types. For the correctness of the model, the numerical investigations have been performed by using an artificial neural network along with its global/local competences. The singular ordinary differential equations are much important and have many applications in engineering as well as scientific applications, e.g., optimization and control theory, reactant application in the area of chemical reactor, theory of boundary layer, and biological sciences.
The structure of remaining paper is summarized as follows. Section 2 defines the construction of the thirdorder EFDD model along with two types. Methodology and the detail of the results for solving the thirdorder EFDD equations are provided in the Section 3. The conclusions along with future research directions are drawn in the Section 4.
2. Construction of ThirdOrder EFDD Model
In this section, two different types are presented based on the thirdorder EFDD model. The construction of the thirdorder EFDD model along with the singular points, delayed points, and shape factors for both of the types is discussed. The initial conditions of the designed thirdorder EFDD model are achieved using the standard form of the Lane–Emden. To derive the thirdorder EFDD model system of Emden–Fowler equations, the mathematical form is used as follows:where k is real positive number. To determine the thirdorder DDEF model, the values of p and q should be designated as follows:
The following two possibilities satisfy equation (5) as follows:
2.1. Type 1
Using equations (6), the updated form of equation (4) is
The derivative part of the above equation is obtained as follows:
Using the above expression in equation (8), the thirdorder EFDD equation becomeswhere the singular point at = 0 appears two times as . The shape factors expressed in equation (10) are , respectively. The multiple delays have been noticed in the first, second, and third term of equation (10). Moreover, the third expression vanishes for k = 1 and the shape factor reduces to 2.
2.2. Type 2
Equation (4) by putting takes the form as follows:
The derivative part of the above equation is obtained as follows:
Using the above value in equation (11), the thirdorder EFDD model becomes as follows:
The single singularity at = 0 has been noticed in the above equation (13). The shape factor is k and delayed expression appears twice in the above equation.
Some prime features of the designed model are presented as follows: The design of thirdorder Emden–Fowler delay differential model is presented by using the sense of standard Emden–Fowler equation and delaydifferential equation Two types of the designed model are presented and two numerical nonlinear examples of each type are designed based on the designed model The shape factors, delay expressions, and singularities are discussed in both of the types The artificial neural network is used to check the perfection and correctness of the designed thirdorder Emden–Fowler model
3. Methodology and Numerical Examples
Two numerical examples based on the EFDD novel model are presented in this section. The numerical investigations of the examples are performed using the artificial neural network. The error function is provided by using the sense of the differential equations and initial conditions. The optimization of the error function is performed using the hybrid of global and local search captaincies, which are genetic algorithm (GA) and activeset method (ASM). The artificial neural network is famous and widely applied in many wellknown recent applications, see [35–41]. To approximate the results, feedforward ANN system along with its respective derivatives is used as follows:where , , and are the ith components of , and vectors, while n is the order of derivative. An activation logsigmoid function, i.e., along with its third derivative is used as follows:
The thirdorder derivative is provided as follows:
The fitness function is given as follows:where and are the respective error functions related to differential equation and initial conditions.
3.1. EFDD Equation of Type 1
In this type, two different thirdorder EFDDbased equations will be discussed. The updated form of equation (10) using k = 2 is given as follows.
Example 1. Consider the nonlinear thirdorder EFDD equation having multiple singularities is shown as follows:The exact solution of equation (20) is .
Example 2. Consider the nonlinear thirdorder EFDD equation having multiple singularities and trigonometric functions is written as follows:The exact solution of equation (21) is
3.2. EFDD Equation of Type 2
In this type, two different thirdorder EFDDbased equations will be discussed. The updated form of equation (13) using k = 1 is given in the form of two examples.
Example 3. Consider the nonlinear thirdorder EFDD equation having exponential function is given as follows:The exact solution of equation (22) is
Example 4. Consider the nonlinear thirdorder EFDD equation having multi trigonometric function is given as follows:The exact solution of equation (23) is .
Figures 1 and 2 represent the current point and function values using 10 neurons based on the hybrid combination of GAAS scheme for both of the examples of types 1 and 2. The current function values (CFVs) are 10^{−09} and 10^{−08} for both of the examples of type 1 and 10^{−07} and 10^{−09} for both of the examples of 2 using 10 numbers of neurons. The comparison of results is presented in the rest of the figures for both examples of types 1 and 2. The overlapping of the exact and obtained results shows the correctness and the perfection of the novel thirdorder nonlinear EFDD model.
The plots of the absolute error (AE) for both types of examples 1 and 2 based on the thirdorder nonlinear EFDD model are provided in Figure 3. It is clear that most of the values lie around 10^{−04} to 10^{−05} for both types of examples 1 and 2, which indicates the exactness of the designed model. These witnesses prove the correctness of the designed thirdorder nonlinear EFDD model. Comparison of the obtained results from GAASM for solving the nonlinear EFDD model based on both problems of both types is tabulated in Tables 1 and 2. The exact solution, proposed results from GAASM, and the AE are provided in these tables. One can conclude on the behalf of AE the exactness and accurateness of the proposed model, as well as designed scheme.
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
(a)
(b)


4. Conclusion
In the present study, a novel design of thirdorder Emden–Fowler delay differential model is presented. The designed model is obtained by using the sense of fundamental Emden–Fowler model. The details of singular points, delay expressions, and the shape factors are also provided of the modeled equations of each type. The singularity at = 0 appears twice in the first type, while single singularity is noticed in the second type. Similarly, the shape factor is unique in the standard form of the Emden–Fowler model, while the occurrence of shape factor is noticed twice in the type 1; however, single shape factor is noticed in type 2. For the perfection of the designed model, two nonlinear examples are presented of each type and numerical investigations have been performed using the powerful artificial neural networks. The comparison of the results is also plotted and overlapping of the proposed and exact solution enhanced more satisfaction of the model. The graphs of absolute error show that most of the values are found in good ranges for all examples of both types, which shows the exactness, worth, and the precision of the designed thirdorder Emden–Fowler delay differential model.
In the future, the proposed scheme ANNGAASM can be applied as an accurate and efficient stochastic numerical solver for nonlinear singular models [42–44], computational models of fluid dynamics [45–48], fractional models [49–52], and biological models [53–57].
Data Availability
No data were used to support the findings of the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
References
 T. Zhao, “Global periodicsolutions for a differential delay system modeling a microbial population in the chemostat,” Journal of Mathematical Analysis and Applications, vol. 193, no. 1, pp. 329–352, 1995. View at: Publisher Site  Google Scholar
 Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, Cambridge, MA, USA, 1993.
 D. S. Li and M. Z. Liu, “Exact solution properties of a multipantograph delay differential equation,” Journal of Harbin Institute of Technology, vol. 32, no. 3, pp. 1–3, 2000. View at: Google Scholar
 W. Li, B. Chen, C. Meng et al., “Ultrafast alloptical graphene modulator,” Nano Letters, vol. 14, no. 2, pp. 955–959, 2014. View at: Publisher Site  Google Scholar
 S. I. Niculescu, Delay Effects On Stability: A Robust Control Approach, vol. 269, Springer Science & Business Media, Berlin, Germany, 2001.
 H. Brunner, Q. Huang, and H. Xie, “Discontinuous Galerkin methods for delay differential equations of pantograph type,” SIAM Journal on Numerical Analysis, vol. 48, no. 5, pp. 1944–1967, 2010. View at: Publisher Site  Google Scholar
 C. H. Hsiao and S. P. Wu, “Numerical solution of timevarying functional differential equations via Haar wavelets,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 1049–1058, 2007. View at: Publisher Site  Google Scholar
 X. T. Wang, “Numerical solution of timevarying systems with a stretch by general Legendre wavelets,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 613–620, 2008. View at: Publisher Site  Google Scholar
 G. Adomian and R. Rach, “A nonlinear differential delay equation,” Journal of Mathematical Analysis and Applications, vol. 91, no. 2, pp. 301–304, 1983. View at: Publisher Site  Google Scholar
 F. Shakeri and M. Dehghan, “Solution of delay differential equations via a homotopy perturbation method,” Mathematical and Computer Modelling, vol. 48, no. 34, pp. 486–498, 2008. View at: Publisher Site  Google Scholar
 F. Erdogan, M. G. Sakar, and O. Saldır, “A finite difference method on layeradapted mesh for singularly perturbed delay differential equations,” Applied Mathematics and Nonlinear Sciences, vol. 5, no. 1, pp. 425–436, 2020. View at: Publisher Site  Google Scholar
 A. Shvets and A. Makaseyev, “Deterministic chaos in pendulum systems with delay,” Applied Mathematics and Nonlinear Sciences, vol. 4, no. 1, pp. 1–8, 2019. View at: Publisher Site  Google Scholar
 W. Adel and Z. Sabir, “Solving a new design of nonlinear secondorder Lane–Emden pantograph delay differential model via Bernoulli collocation method,” The European Physical Journal Plus, vol. 135, no. 6, p. 427, 2020. View at: Publisher Site  Google Scholar
 R. H. Fowler, “Further studies of emden's and similar differential equations,” The Quarterly Journal of Mathematics, vol. 2, no. 1, pp. 259–288, 1931. View at: Publisher Site  Google Scholar
 Z. Sabir, F. Amin, D. Pohl, and J. L. Guirao, “Intelligence computing approach for solving second order system of Emden–Fowler model,” Journal of Intelligent & Fuzzy Systems, vol. 2020, pp. 1–16, 2020. View at: Google Scholar
 Z. Sabir, H. A. Wahab, M. Umar, M. G. Sakar, and M. A. Z. Raja, “Novel design of morlet wavelet neural network for solving second order laneemden equation,” Mathematics and Computers in Simulation, vol. 172, pp. 1–14, 2020. View at: Publisher Site  Google Scholar
 V. B. Mandelzweig and F. Tabakin, “Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs,” Computer Physics Communications, vol. 141, no. 2, pp. 268–281, 2001. View at: Publisher Site  Google Scholar
 V. Radulescu and D. Repovs, “Combined effects in nonlinear problems arising in the study of anisotropic continuous media,” Nonlinear Analysis: Theory, Methods and Applications, vol. 75, no. 3, pp. 1524–1530, 2012. View at: Publisher Site  Google Scholar
 T. Luo, Z. Xin, and H. Zeng, “Nonlinear asymptotic stability of the LaneEmden solutions for the viscous gaseous star problem with degenerate density dependent viscosities,” Communications in Mathematical Physics, vol. 347, no. 3, pp. 657–702, 2016. View at: Publisher Site  Google Scholar
 M. Ghergu and V. Rădulescu, “On a class of singular GiererMeinhardt systems arising in morphogenesis,” Comptes Rendus Mathematique, vol. 344, no. 3, pp. 163–168, 2007. View at: Publisher Site  Google Scholar
 D. Flockerzi and K. Sundmacher, “On coupled LaneEmden equations arising in dusty fluid models,” Journal of Physics: Conference Series, vol. 268, no. No. 1, Article ID 012006, 2011. View at: Publisher Site  Google Scholar
 A. Taghavi and S. Pearce, “A solution to the LaneEmden equation in the theory of stellar structure utilizing the Tau method,” Mathematical Methods in the Applied Sciences, vol. 36, no. 10, pp. 1240–1247, 2013. View at: Publisher Site  Google Scholar
 R. Rach, J.S. Duan, and A.M. Wazwaz, “Solving coupled LaneEmden boundary value problems in catalytic diffusion reactions by the adomian decomposition method,” Journal of Mathematical Chemistry, vol. 52, no. 1, pp. 255–267, 2014. View at: Publisher Site  Google Scholar
 M. Dehghan and F. Shakeri, “Solution of an integrodifferential equation arising in oscillating magnetic fields using Hes homotopy perturbation method,” Progress in Electromagnetics Research, vol. 78, pp. 361–376, 2008. View at: Publisher Site  Google Scholar
 K. Boubaker and R. A. Van Gorder, “Application of the BPES to LaneEmden equations governing polytropic and isothermal gas spheres,” New Astronomy, vol. 17, no. 6, pp. 565–569, 2012. View at: Publisher Site  Google Scholar
 A. H. Bhrawy, A. S. Alofi, and R. A. Van Gorder, “An efficient collocation method for a class of boundary value problems arising in mathematical physics and geometry,” Abstract and Applied Analysis, vol. 2014, 2014. View at: Publisher Site  Google Scholar
 J. I. Ramos, “Linearization methods in classical and quantum mechanics,” Computer Physics Communications, vol. 153, no. 2, pp. 199–208, 2003. View at: Publisher Site  Google Scholar
 J. A. Khan, M. A. Z. Raja, M. M. Rashidi, M. I. Syam, and A. M. Wazwaz, “Natureinspired computing approach for solving nonlinear singular EmdenFowler problem arising in electromagnetic theory,” Connection Science, vol. 27, no. 4, pp. 377–396, 2015. View at: Publisher Site  Google Scholar
 N. T. Shawagfeh, “Nonperturbative approximate solution for LaneEmden equation,” Journal of Mathematical Physics, vol. 34, no. 9, pp. 4364–4369, 1993. View at: Publisher Site  Google Scholar
 A.M. Wazwaz, “A new algorithm for solving differential equations of LaneEmden type,” Applied Mathematics and Computation, vol. 118, no. 23, pp. 287–310, 2001. View at: Publisher Site  Google Scholar
 K. Parand and M. Razzaghi, “Rational Legendre approximation for solving some physical problems on semiinfinite intervals,” Physica Scripta, vol. 69, no. 5, p. 353, 2004. View at: Publisher Site  Google Scholar
 S. Liao, “A new analytic algorithm of LaneEmden type equations,” Applied Mathematics and Computation, vol. 142, no. 1, pp. 1–16, 2003. View at: Publisher Site  Google Scholar
 C. M. Bender, K. A. Milton, S. S. Pinsky, and L. M. Simmons, “A new perturbative approach to nonlinear problems,” Journal of Mathematical Physics, vol. 30, no. 7, pp. 1447–1455, 1989. View at: Publisher Site  Google Scholar
 M. Nouh, “Accelerated power series solution of polytropic and isothermal gas spheres,” New Astronomy, vol. 9, no. 6, pp. 467–473, 2004. View at: Publisher Site  Google Scholar
 Z. Sabir, H. A. Wahab, M. Umar, and F. Erdoğan, “Stochastic numerical approach for solving second order nonlinear singular functional differential equation,” Applied Mathematics and Computation, vol. 363, p. 124605, 2019. View at: Publisher Site  Google Scholar
 M. Umar, F. Amin, H. A. Wahab, and D. Baleanu, “Unsupervised constrained neural network modeling of boundary value corneal model for eye surgery,” Applied Soft Computing, vol. 85, Article ID 105826, 2019. View at: Publisher Site  Google Scholar
 M. Umar, Z. Sabir, and M. A. Z. Raja, “Intelligent computing for numerical treatment of nonlinear preypredator models,” Applied Soft Computing, vol. 80, pp. 506–524, 2019. View at: Publisher Site  Google Scholar
 M. A. Z. Raja, M. Umar, Z. Sabir, J. A. Khan, and D. Baleanu, “A new stochastic computing paradigm for the dynamics of nonlinear singular heat conduction model of the human head,” The European Physical Journal Plus, vol. 133, no. 9, p. 364, 2018. View at: Publisher Site  Google Scholar
 Z. Sabir, M. A. Manzar, M. A. Z. Raja, M. Sheraz, and A. M. Wazwaz, “Neuroheuristics for nonlinear singular ThomasFermi systems,” Applied Soft Computing, vol. 65, pp. 152–169, 2018. View at: Publisher Site  Google Scholar
 M. A. Z. Raja, J. Mehmood, Z. Sabir, A. K. Nasab, and M. A. Manzar, “Numerical solution of doubly singular nonlinear systems using neural networksbased integrated intelligent computing,” Neural Computing and Applications, vol. 31, no. 3, pp. 793–812, 2019. View at: Publisher Site  Google Scholar
 M. A. Z. Raja, Z. Sabir, N. Mehmood, E. S. AlAidarous, and J. A. Khan, “Design of stochastic solvers based on genetic algorithms for solving nonlinear equations,” Neural Computing and Applications, vol. 26, no. 1, pp. 1–23, 2015. View at: Publisher Site  Google Scholar
 Z. Sabir, “On a new model based on third order nonlinear multisingular functional differential equations,” Mathematical Problems in Engineering, vol. 2020, 2020. View at: Publisher Site  Google Scholar
 Z. Sabir, S. Saoud, M. A. Z. Raja, H. A. Wahab, and A. Arbi, “Heuristic computing technique for numerical solutions of nonlinear fourth order Emden–Fowler equation,” Mathematics and Computers in Simulation, vol. 2020, 2020. View at: Google Scholar
 Z. Sabir, M. A. Z. Raja, J. L. Guirao, and M. Shoaib, “A NeuroSwarming Intelligence Based Computing for Second Order Singular Periodic Nonlinear Boundary Value Problems,” Frontiers in Physics, vol. 8, 2020. View at: Publisher Site  Google Scholar
 M. Umar, Z. Sabir, A. Imran, A. Wahab, M. Shoaib, and M. A. Raja, “Threedimensional flow of Casson nanofluid over a stretched sheet with chemical reactions, velocity slip, thermal radiation and Brownian motion,” Thermal Science, vol. 24, no. 5, pp. 2929–2939, 2019. View at: Publisher Site  Google Scholar
 Z. Sabir, R. Akhtar, Z. Zhiyu et al., “A computational analysis of twophase casson nanofluid passing a stretching sheet using chemical reactions and gyrotactic microorganisms,” Mathematical Problems in Engineering, vol. 2019, 2019. View at: Publisher Site  Google Scholar
 T. Sajid, S. Tanveer, Z. Sabir, and J. L. G. Guirao, “Impact of activation energy and temperaturedependent heat source/sink on maxwell–sutterby fluid,” Mathematical Problems in Engineering, vol. 2020, pp. 1–15, 2020. View at: Publisher Site  Google Scholar
 Y. Jiao and Q. Zheng, “Urea injection and uniformity of ammonia distribution in SCR system of diesel engine,” Applied Mathematics and Nonlinear Sciences, vol. 1, 2020. View at: Publisher Site  Google Scholar
 H. Dehestani, Y. Ordokhani, and M. Razzaghi, “A numerical technique for solving various kinds of fractional partial differential equations via Genocchi hybrid functions,” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, vol. 113, no. 4, pp. 3297–3321, 2019. View at: Publisher Site  Google Scholar
 H. Dehestani, Y. Ordokhani, and M. Razzaghi, “On the applicability of Genocchi wavelet method for different kinds of fractional‐order differential equations with delay,” Numerical Linear Algebra with Applications, vol. 26, no. 5, p. e2259, 2019. View at: Publisher Site  Google Scholar
 K. Rabiei and Y. Ordokhani, “Solving fractional pantograph delay differential equations via fractionalorder boubaker polynomials,” Engineering with Computers, vol. 35, no. 4, pp. 1431–1441, 2019. View at: Publisher Site  Google Scholar
 D. Kaur, P. Agarwal, M. Rakshit, and M. Chand, “Fractional calculus involving (p, q) mathieu type series,” Applied Mathematics and Nonlinear Sciences, vol. 5, no. 2, pp. 15–34, 2020. View at: Publisher Site  Google Scholar
 Y. G. Sánchez, Z. Sabir, H. Günerhan, and H. M. Baskonus, “Analytical and approximate solutions of a novel nervous stomach mathematical model,” Discrete Dynamics in Nature and Society, vol. 2020, pp. 1–9, 2020. View at: Publisher Site  Google Scholar
 M. Umar, “A stochastic computational intelligent solver for numerical treatment of mosquito dispersal model in a heterogeneous environment,” The European Physical Journal Plus, vol. 135, no. 7, pp. 1–23, 2020. View at: Publisher Site  Google Scholar
 Y. G. Sanchez, “Design of a nonlinear sitr fractal model based on the dynamics of a novel coronavirus (covid),” Fractals, Article ID 2040026, 2020, In press. View at: Publisher Site  Google Scholar
 F. Evirgen, S. Uçar, and N. Özdemir, “System analysis of HIV infection model with CD4+T under nonsingular kernel derivative,” Applied Mathematics and Nonlinear Sciences, vol. 5, no. 1, pp. 139–146, 2020. View at: Publisher Site  Google Scholar
 M. Umar, “Stochastic numerical technique for solving HIV infection model of CD4+ T cells,” The European Physical Journal Plus, vol. 135, no. 6, p. 403, 2020. View at: Publisher Site  Google Scholar
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Copyright © 2020 Juan L.G. Guirao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.