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1 12 4 ; S ; "This is the peer reviewed version of the following article: Calatayud, J, Cortés, J-;C, Jornet, M. On the Legendre differential equation with uncertainties at the regular-singular point 1: Lp random power series solution and approximation of its statistical moments. Comp and Math Methods. 2019; 1:e1045. https://doi.org/10.1002/cmm4.1045 , which has been published in final form at https://doi.org/10.1002/cmm4.1045. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving." [EN] In this paper, we construct two linearly independent response processes to the random Legendre differential equation on (-1,1)U(1,3), consisting of Lp(omega) convergent random power series around the regular¿singular point 1. A theorem on the existence and uniqueness of Lp(omega) solution to the random Legendre differential equation on the intervals (-1,1) and (1,3) is obtained. The hypotheses assumed are simple: initial conditions in Lp(omega) and random input A in L infinite(omega) (this is equivalent to A having absolute moments that grow at most exponentially). Thus, this paper extends the deterministic theory to a random framework. Uncertainty quantification for the solution stochastic process is performed by truncating the random series and taking limits in Lp(omega). In the numerical experiments, we approximate its expectation and variance for certain forms of the differential equation. The reliability of our approach is compared with Monte Carlo simulations and generalized polynomial chaos expansions. Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). On the Legendre differential equation with uncertainties at the regular-singular point 1: Lp random power series solution and approximation of its statistical moments. Computational and Mathematical Methods. 1(4):1-12. https://doi.org/10.1002/cmm4.1045 Calbo, G., Cortés, J.-C., Jódar, L., & Villafuerte, L. (2011). Solving the random Legendre differential equation: Mean square power series solution ...

[EN] This paper is aimed to extend, the non-autonomous case, the results recently given in the paper [1] for solving autonomous linear and quadratic random matrix differential equations. With this goal, important deterministic results like the Abel-Liouville-Jacobi's formula, are extended to the random scenario using the so-called $\mathrm{L}_{p}$-random matrix calculus. In a first step, random time-dependent matrix linear differential equations are studied and, in a second step, random non-autonomous Riccati matrix differential equations are solved using the hamiltonian approach based on dealing with the extended underlying linear system. Illustrative numerical examples are also included. [1] M.-C. Casabán, J.-C. Cortés, L. Jódar, Solving linear and quadratic random matrix differential equations: A mean square approach, Applied Mathematical Modelling 40 (2016) 9362-9377. ; This work has been partially supported by the Spanish Ministerio de Economia y Competitividad grant MTM2013-41765-P and by the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement no. 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance). ; Casabán, M.; Cortés, J.; Jódar Sánchez, LA. (2018). Solving linear and quadratic random matrix differential equations using: A mean square approach. The non-autonomous case. Journal of Computational and Applied Mathematics. 330:937-954. https://doi.org/10.1016/j.cam.2016.11.049 ; S ; 937 ; 954 ; 330

937 954 330 ; S ; [EN] This paper is aimed to extend, the non-autonomous case, the results recently given in the paper [1] for solving autonomous linear and quadratic random matrix differential equations. With this goal, important deterministic results like the Abel-Liouville-Jacobi's formula, are extended to the random scenario using the so-called $\mathrm{L}_{p}$-random matrix calculus. In a first step, random time-dependent matrix linear differential equations are studied and, in a second step, random non-autonomous Riccati matrix differential equations are solved using the hamiltonian approach based on dealing with the extended underlying linear system. Illustrative numerical examples are also included. [1] M.-C. Casabán, J.-C. Cortés, L. Jódar, Solving linear and quadratic random matrix differential equations: A mean square approach, Applied Mathematical Modelling 40 (2016) 9362-9377. This work has been partially supported by the Spanish Ministerio de Economia y Competitividad grant MTM2013-41765-P and by the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement no. 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance). Casabán, M.; Cortés, J.; Jódar Sánchez, LA. (2018). Solving linear and quadratic random matrix differential equations using: A mean square approach. The non-autonomous case. Journal of Computational and Applied Mathematics. 330:937-954. https://doi.org/10.1016/j.cam.2016.11.049

[EN] This paper deals with the study of a Bessel-type differential equation where input parameters (coefficient and initial conditions) are assumed to be random variables. Using the so-called Lp-random calculus and assuming moment conditions on the random variables in the equation, a mean square convergent generalized power series solution is constructed. As a result of this convergence, the sequences of the mean and standard deviation obtained from the truncated power series solution are convergent as well. The results obtained in the random framework extend their deterministic counterpart. The theory is illustrated in two examples in which several distributions on the random inputs are assumed. Finally, we show through examples that the proposed method is computationally faster than Monte Carlo method. ; This work has been partially supported by the Spanish Ministerio de Economía y Competitividad grant MTM2013-41765-P and by the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement No. 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance) and Mexican Conacyt. ; Cortés, J.; Jódar Sánchez, LA.; Villafuerte, L. (2017). Mean square solution of Bessel differential equation with uncertainties. Journal of Computational and Applied Mathematics. 309:383-395. https://doi.org/10.1016/j.cam.2016.01.034 ; S ; 383 ; 395 ; 309

[EN] A fractional forward Euler-like method is developed to solve initial value problems with uncertainties formulated via the Caputo fractional derivative. The analysis is conducted by using the so-called random mean square calculus. Under mild conditions on the data, the mean square convergence of the numerical method is proved. This type of stochastic convergence guarantees the approximations of the mean and the variance of the solution stochastic process, computed via the aforementioned numerical scheme, will converge to their corresponding exact values. Furthermore, from this probability information, we calculate reliable approximations to the first probability density function of the solution by taking advantage of the Maximum Entropy Principle. The theoretical analysis is illustrated by two examples. ; This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2017-89664-P and by the European Union through the Operational Program of the European Regional Development Fund (ERDF)/European Social Fund (ESF) of the Valencian Community 20142020, grants GJIDI/2018/A/009 and GJIDI/2018/A/010. ; Burgos, C.; Cortés, J.; Villafuerte, L.; Villanueva Micó, RJ. (2020). Mean square convergent numerical solutions of random fractional differential equations: Approximations of moments and density. Journal of Computational and Applied Mathematics. 378:1-14. https://doi.org/10.1016/j.cam.2020.112925 ; S ; 1 ; 14 ; 378 ; Slama, H., El-Bedwhey, N. A., El-Depsy, A., & Selim, M. M. (2017). Solution of the finite Milne problem in stochastic media with RVT Technique. The European Physical Journal Plus, 132(12). doi:10.1140/epjp/i2017-11763-6 ; Golmankhaneh, A. K., Arefi, R., & Baleanu, D. (2013). Synchronization in a nonidentical fractional order of a proposed modified system. Journal of Vibration and Control, 21(6), 1154-1161. doi:10.1177/1077546313494953 ; Lupulescu, V., O'Regan, D., & ur Rahman, G. (2014). Existence results for random fractional differential equations. Opuscula ...

[EN] In this paper we propose a stochastic mathematical model with distributed delay in order to describe the transmission dynamics of cocaine consumption in Spain. We investigate conditions to guarantee the stability in probability of the equilibrium points under stochastic perturbations via the white noise processes. The results are applied to the model cocaine consumption using data retrieved from the Spanish Drug National Plan, http://www.pnsd.mscbs.gob.es/. The obtained results may be useful for policy health authorities in order to improve the strategies against the drug consumption in the long-run. ; This work has been supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P. Computations have been carried thanks to the collaboration of Raul San Julian Garces and Elena Lopez Navarro granted by European Union through the Operational Program of the European Regional Development Fund (ERDF)/European Social Fund (ESF) of the Valencian Community 2014-2020, grants GJIDI/2018/A/009 and GJIDI/2018/A/010, respectively ; Burgos-Simon, C.; Cortés, J.; Shaikhet, L.; Villanueva Micó, RJ. (2021). A delayed nonlinear stochastic model for cocaine consumption: Stability analysis and simulation using real data. Discrete and Continuous Dynamical Systems. Series S. 14(4):1233-1244. https://doi.org/10.3934/dcdss.2020356 ; S ; 1233 ; 1244 ; 14 ; 4

[EN] We combine the stochastic perturbation method with the maximum entropy principle to construct approximations of the first probability density function of the steady-state solution of a class of nonlinear oscillators subject to small perturbations in the nonlinear term and driven by a stochastic excitation. The nonlinearity depends both upon position and velocity, and the excitation is given by a stationary Gaussian stochastic process with certain additional properties. Furthermore, we approximate higher-order moments, the variance, and the correlation functions of the solution. The theoretical findings are illustrated via some numerical experiments that confirm that our approximations are reliable. ; This work has been supported by the Spanish Ministerio de Economía, Industria y Competitividad (MINECO), the Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017¿89664¿P. Elena López-Navarro has been supported by the European Union through the Operational Program of the [European Regional Development Fund (ERDF)/European Social Fund (ESF)] of the Valencian Community 2014-2020 (GJIDI/2018/A/010) ; Cortés, J.; López-Navarro, E.; Romero, J.; Roselló, M. (2021). Approximating the Density of Random Differential Equations with Weak Nonlinearities via Perturbation Techniques. Mathematics. 9(3):1-17. https://doi.org/10.3390/math9030204 ; S ; 1 ; 17 ; 9 ; 3

This is the peer reviewed version of the following article: Cortés, J-C, Navarro-Quiles, A, Romero, J-V, Roselló, M-D. Introducing randomness in the analysis of chemical reactions: An analysis based on random differential equations and probability density functions. Comp and Math Methods. 2021; 3:e1141, which has been published in final form at https://doi.org/10.1002/cmm4.1141. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. ; [EN] In this work we consider a particular randomized kinetic model for reaction-deactivation of hydrogen peroxide decomposition. We apply the Random Variable Transformation technique to obtain the first probability density function of the solution stochastic process under general conditions. From the rst probability density function, we can obtain fundamental statistical information, such as the mean and the variance of the solution, at every instant time. The transformation considered in the application of the Random Variable Transformation technique is not unique. Then, the first probability density function can take different expressions, although essentially equivalent in terms of computing probabilistic information. To motivate this fact, we consider in our analysis two different mappings. Several numerical examples show the capability of our approach and of the obtained results as well. We show, through simulations, that the choice of the transformation, that permits computing the first probability density function, is a crucial issue regarding the computational time. ; This work has been supported by the Spanish Ministerio de Economía, Industria y Competitividad (MINECO), the Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P. Computations have been carried thanks to the collaboration of Raúl San Julián Garcés and Elena López Navarro granted by European Union through the Operational Program of the European Regional Development Fund (ERDF)/European ...

[EN] The aim of this paper is to perform a stochastic analysis of nonlinear oscillators subject to stationary Gaussian forcing sources using the random perturbation technique along with the Maximum Entropy Principle. By combining these two stochastic techniques, we construct reliable approximations of the probability density function of the solution, which is a stochastic process. ; This work has been supported by the Spanish Ministerio de Economía, Industria y Competitividad (MINECO), the Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P. Elena Lopez Navarro has been supported by the European Union through the Operational Program of the [European Regional Development Fund (ERDF)/European Social Fund (ESF)] of the Valencian Community 2014-2020 (GJIDI/2018/A/010). ; Cortés, J.; López-Navarro, E.; Romero, J.; Roselló, M. (2020). Analysing nonlinear oscillators subject to Gaussian inputs via the random perturbation technique. 82-86. http://hdl.handle.net/10251/178194 ; S ; 82 ; 86

[EN] In this contribution, we construct approximations for the density associated with the solution of second-order linear differential equations whose coefficients are analytic stochastic processes about regular-singular points. Our analysis is based on the combination of a random Fröbenius technique together with the random variable transformation technique assuming mild probabilistic conditions on the initial conditions and coefficients. The new results complete the ones recently established by the authors for the same class of stochastic differential equations, but about regular points. In this way, this new contribution allows us to study, for example, the important randomized Bessel differential equation. ; This work was partially funded by the Ministerio de Economia y Competitividad Grant MTM2017-89664-P. Ana Navarro Quiles acknowledges the funding received from Generalitat Valenciana through a postdoctoral contract (APOSTD/2019/128). Computations were carried out thanks to the collaboration of Raul San Julian Garces and Elena Lopez Navarro granted by the European Union through the Operational Program of the European Regional Development Fund (ERDF)/European Social Fund (ESF) of the Valencian Community 2014-2020, Grants GJIDI/2018/A/009 and GJIDI/2018/A/010, respectively ; Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M. (2020). Solving second-order linear differential equations with random analytic coefficients about regular-singular points. Mathematics. 8(2):1-20. https://doi.org/10.3390/math8020230 ; S ; 1 ; 20 ; 8 ; 2 ; Hussein, A., & Selim, M. M. (2012). Solution of the stochastic radiative transfer equation with Rayleigh scattering using RVT technique. Applied Mathematics and Computation, 218(13), 7193-7203. doi:10.1016/j.amc.2011.12.088 ; Dorini, F. A., Cecconello, M. S., & Dorini, L. B. (2016). On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density. Communications in Nonlinear Science and Numerical Simulation, 33, ...

[EN] We provide a full stochastic description, via the first probability density function, of the solution of linear-quadratic logistic-type differential equation whose parameters involve both continuous and discrete random variables with arbitrary distributions. For the sake of generality, the initial condition is assumed to be a random variable too. We use the Dirac delta function to unify the treatment of hybrid (discrete-continuous) uncertainty. Under general hypotheses, we also compute the density of time until a certain value (usually representing the population) of the linear-quadratic logistic model is reached. The theoretical results are illustrated by means of several examples, including an application to modelling the number of users of Spotify using real data. We apply the Principle Maximum Entropy to assign plausible distributions to model parameters ; This work has been supported by the Spanish Ministerio de Economa, Industria y Competitividad (MINECO) , the Agencia Estatal de Investigaci on (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM201789664P. Computations have been carried thanks to the collaboration of Raul San Julian Garces and Elena Lopez Navarro granted by European Union through the Operational Program of the European Regional Development Fund (ERDF) /European Social Fund (ESF) of the Valencian Community 2014-2020, grants GJIDI/2018/A/009 and GJIDI/2018/A/010, respectively ; Burgos-Simon, C.; Cortés, J.; López-Navarro, E.; Villanueva Micó, RJ. (2021). Probabilistic analysis of linear-quadratic logistic-type models with hybrid uncertainties via probability density functions. AIMS Mathematics. 6(5):4938-4957. https://doi.org/10.3934/math.2021290 ; S ; 4938 ; 4957 ; 6 ; 5 ; R. Goebel, R. G. Sanfelice, A. R. Teel, i>Hybrid Dynamical Systems: Modeling, Stability, and Robustness /i>, Princeton University Press, 2012. ; J. Lunze, F. Lamnabhi-Lagarrigue, i>Handbook of Hybrid Systems Control: Theory, Tools, Applications /i>, ser. Nonlinear Systems and Complexity, ...

[EN] In this paper we perform a complete probabilistic study of a finite dimensional linear control system with uncertainty. The controllability condition with random initial data and final target is analysed. To conduct this investigation we determine the first probability density function of the control and the solution of the random control problem under different scenarios. To achieve this objective, Random Variable Transformation technique is extensively applied. Several examples illustrate the theoretical results. (C) 2020 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. ; This work has been partially supported by the European Research Council (ERC) under the European Unions's Horizon 2020 research and innovation programme (Grant agreement no. 694126-DyCon) and the Ministerio de Economía y Competitividad grant MTM2017-89664-P. Enrique Zuazua has also been funded by the Alexander von Humboldt-Professorship program, the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant agreement Nos. 765579-ConFlex, MTM2017-92996 of MINECO (Spain), ELKARTEK project KK-2018/00083 ROAD2DC of the Basque Government, ICON-ANR-16-ACHN-0014 of the French ANR and Nonlocal PDEs: Analysis, Control and Beyond, AFOSR Grant FA9550-18-1-0242. Ana Navarro Quiles acknowledges the postdoctoral contract financed by DyCon project funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 694126-DYCON) and the funding received from Generalitat Valenciana through a postdoctoral contract (APOSTD/2019/128). Authors acknowledge professor Martin Lazar for fruitful discussions. The authors express their deepest thanks and respect to the four reviewers for their valuable comments that have improve the final version of the paper. ; Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M.; Zuazua, E. (2020). Full probabilistic solution of a finite dimensional linear control system with random ...

[EN] Electronic commerce (EC) has numerous advantages. It allows saving time when we purchase an item, offers the possibility of review without depending on the schedules of traditional stores, access to a wider variety and quantity of articles, in many cases, with lower prices, etc. Based upon mathematical epidemiology tenets strongly related to social behavior able to describe the influence of peers, in this paper we propose an age-group dynamic model with population varying size based on a system of difference equations to study the evolution of the frequent users of EC over time in Spain. Using data from surveys retrieved from the Spanish National Statistics Institute, we use and design computational algorithms to perform a probabilistic estimation of the model parameters that allow the model output to capture the data uncertainty. Then, we will be able to perform a precise prediction with uncertainty. ; This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2017-89664-P and by the European Union through the Operational Program of the European Regional Development Fund (ERDF)/European Social Fund (ESF) of the Valencian Community 2014-2020, grants GJIDI/2018/A/009 and GJIDI/2018/A/010. ; Burgos-Simon, C.; Cortés, J.; Martínez-Rodríguez, D.; Villanueva Micó, RJ. (2019). Computational modelling with uncertainty of frequent users of e-commerce in Spain using an age-group dynamic nonlinear model with varying size population. Advances in Complex Systems. 22(4):1950009-1-1950009-17. https://doi.org/10.1142/S0219525919500097 ; S ; 1950009-1 ; 1950009-17 ; 22 ; 4 ; Bettencourt, L. (1997). Customer voluntary performance: Customers as partners in service delivery. Journal of Retailing, 73(3), 383-406. doi:10.1016/s0022-4359(97)90024-5 ; Brauer, F., & Castillo-Chávez, C. (2001). Mathematical Models in Population Biology and Epidemiology. Texts in Applied Mathematics. doi:10.1007/978-1-4757-3516-1 ; Cortés, J.-C., Lombana, I.-C., & Villanueva, R.-J. (2010). Age-structured ...

1 8 ; S ; Copyright © 2015 J.-C. Cortés 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. [EN] We introduce infinite matrix products including some of their main properties and convergence results.We apply them in order to extend to the matrix scenario the definition of the scalar gamma function given by an infinite product due to Weierstrass. A limit representation of the matrix gamma function is also provided. This work has been supported by Ministerio de Economía y Competitividad Grant MTM2013-41765-P. Cortés, J.; Jódar Sánchez, LA.; Solis, FJ.; Ku Carrillo, R. (2015). Infinite matrix products and the representation the gamma matrix function. Abstract and Applied Analysis. 1-8. https://doi.org/10.1155/2015/564287