Some results about randomized binary Markov chains: Theory, computing and applications

[EN] This paper is addressed to give a generalization of the classical Markov methodology allowing the treatment of the entries of the transition matrix and initial condition as random variables instead of deterministic values lying in the interval [0,1]. This permits the computation of the first probability density function (1-PDF) of the solution stochastic process taking advantage of the so-called Random Variable Transformation technique. From the 1-PDF relevant probabilistic information about the evolution of Markov models can be calculated including all one-dimensional statistical moments. We are also interested in determining the computation of distribution of some important quantities related to randomized Markov chains (steady state, hitting times, etc.). All theoretical results are established under general assumptions and they are illustrated by modelling the spread of a technology using real data.This work has been partially supported by the Ministerio de Economía y Competitividad [grant MTM2017-89664-P]. Ana Navarro Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de ValènciaCortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M. (2020). Some results about randomized binary Markov chains: Theory, computing and applications. International Journal of Computer Mathematics. 97(1-2):141-156. https://doi.org/10.1080/00207160.2018.1440290S141156971-

Full solution of random autonomous first-order linear systems of difference equations. Application to construct random phase portrait for planar systems

[EN] This paper deals with the explicit determination of the first probability density function of the solution stochastic process to random autonomous first-order linear systems of difference equations under very general hypotheses. This finding is applied to extend the classical stability classification of the zero-equilibrium point based on phase portrait to the random scenario. An example illustrates the potentiality of the theoretical results established and their connection with their deterministic counterpart.This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2013-41765-P. Ana Navarro Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigation y Desarrollo (PAID), Universitat Politecnica de Valencia.Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M. (2017). Full solution of random autonomous first-order linear systems of difference equations. Application to construct random phase portrait for planar systems. Applied Mathematics Letters. 68:150-156. https://doi.org/10.1016/j.aml.2016.12.0151501566

Introducing randomness in the analysis of chemical reactions: An analysis based on random differential equations and probability density function

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 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. (2021). Introducing randomness in the analysis of chemical reactions: An analysis based on random differential equations and probability density function. Computational and Mathematical Methods. 3(6):1-10. https://doi.org/10.1002/cmm4.1141S1103

Randomizing the parameters of a Markov chain to model the stroke disease: A technical generalization of established computational methodologies towards improving real applications

[EN] Classical Markov models are defined through a stochastic transition matrix, i.e., a matrix whose columns (or rows) are deterministic values representing transition probabilities. However, in practice these quantities could often not be known in a deterministic manner, therefore, it is more realistic to consider them as random variables. Following this approach, this paper is aimed to give a technical generalization of classical Markov methodology in order to improve modelling of stroke disease when dealing with real data. With this goal, we randomize the entries of the transition matrix of a Markov chain with three states (susceptible, reliant and deceased) that has been previously proposed to model the stroke disease. This randomization of the classical Markov model permits the computation of the first probability density function of the solution stochastic process taking advantage of the so-called Random Variable Transformation technique. Afterwards, punctual and probabilistic predictions are computed from the first probability density function. In addition, the probability density functions of the time instants until a certain proportion of the total population remains susceptible, reliant and deceased are also computed. The study is completed showing the usefulness of our computational approach to determine, from a probabilistic point of view, key quantities in medical decision making, such as the cost-effectiveness ratio.This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2013-41765-P. Ana Navarro-Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia. Authors would like to thank Prof. Dr. Javier Mar for providing them medical data about stroke disease from his research.Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M. (2017). Randomizing the parameters of a Markov chain to model the stroke disease: A technical generalization of established computational methodologies towards improving real applications. Journal of Computational and Applied Mathematics. 324:225-240. https://doi.org/10.1016/j.cam.2017.04.040S22524032

Solving fully randomized higher-order linear control differential equations: Application to study the dynamics of an oscillator

This is the peer reviewed version of the following article: Cortés, J-C, Navarro-Quiles, A, Romero, J-V, Roselló, M-D. Solving fully randomized higher-order linear control differential equations: Application to study the dynamics of an oscillator. Comp and Math Methods. 2021; 3:e1163, which has been published in final form at https://doi.org/10.1002/cmm4.1163. 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 control problems represented by a linear differential equation assuming that all the coefficients are random variables and with an additive control that is a stochastic process. Specifically, we will work with controllable problems in which the initial condition and the final target are random variables. The probability density function of the solution and the control has been calculated. The theoretical results have been applied to study, from a probabilistic standpoint, a damped oscillator.European Social Fund, Grant/Award Numbers: GJIDI/2018/A/009, GJIDI/2018/A/010; Spanish Ministerio de Economía, Industria y Competitividad (MINECO), the Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE), Grant/Award Number: MTM2017-89664-P.Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M. (2021). Solving fully randomized higher-order linear control differential equations: Application to study the dynamics of an oscillator. Computational and Mathematical Methods. 3(6):1-15. https://doi.org/10.1002/cmm4.1163S1153

Solving second-order linear differential equations with random analytic coefficients about regular-singular points

[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, respectivelyCortés, 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/math8020230S12082Hussein, 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.2011.12.088Dorini, 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, 160-173. doi:10.1016/j.cnsns.2015.09.009Santos, L. T., Dorini, F. A., & Cunha, M. C. C. (2010). The probability density function to the random linear transport equation. Applied Mathematics and Computation, 216(5), 1524-1530. doi:10.1016/j.amc.2010.03.001Hussein, A., & Selim, M. M. (2019). A complete probabilistic solution for a stochastic Milne problem of radiative transfer using KLE-RVT technique. Journal of Quantitative Spectroscopy and Radiative Transfer, 232, 54-65. doi:10.1016/j.jqsrt.2019.04.034Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2018). Solving second-order linear differential equations with random analytic coefficients about ordinary points: A full probabilistic solution by the first probability density function. Applied Mathematics and Computation, 331, 33-45. doi:10.1016/j.amc.2018.02.051Cortés, J.-C., Jódar, L., Camacho, F., & Villafuerte, L. (2010). Random Airy type differential equations: Mean square exact and numerical solutions. Computers & Mathematics with Applications, 60(5), 1237-1244. doi:10.1016/j.camwa.2010.05.046Calbo, G., Cortés, J.-C., & Jódar, L. (2011). Random Hermite differential equations: Mean square power series solutions and statistical properties. Applied Mathematics and Computation, 218(7), 3654-3666. doi:10.1016/j.amc.2011.09.008Calbo, G., Cortés, J.-C., Jódar, L., & Villafuerte, L. (2011). Solving the random Legendre differential equation: Mean square power series solution and its statistical functions. Computers & Mathematics with Applications, 61(9), 2782-2792. doi:10.1016/j.camwa.2011.03.045Cortés, J.-C., Villafuerte, L., & Burgos, C. (2017). A Mean Square Chain Rule and its Application in Solving the Random Chebyshev Differential Equation. Mediterranean Journal of Mathematics, 14(1). doi:10.1007/s00009-017-0853-6Cortés, J.-C., Jódar, L., & Villafuerte, L. (2017). Mean square solution of Bessel differential equation with uncertainties. Journal of Computational and Applied Mathematics, 309, 383-395. doi:10.1016/j.cam.2016.01.034Khudair, A. R., Haddad, S. A. M., & Khalaf, S. L. (2016). Mean Square Solutions of Second-Order Random Differential Equations by Using the Differential Transformation Method. Open Journal of Applied Sciences, 06(04), 287-297. doi:10.4236/ojapps.2016.64028Qi, Y. (2018). A Very Brief Introduction to Nonnegative Tensors from the Geometric Viewpoint. Mathematics, 6(11), 230. doi:10.3390/math6110230Ragusa, M. A., & Tachikawa, A. (2016). Boundary regularity of minimizers of p(x)-energy functionals. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 33(2), 451-476. doi:10.1016/j.anihpc.2014.11.003Ragusa, M. A., & Tachikawa, A. (2019). Regularity for minimizers for functionals of double phase with variable exponents. Advances in Nonlinear Analysis, 9(1), 710-728. doi:10.1515/anona-2020-0022Braumann, C. A., Cortés, J.-C., Jódar, L., & Villafuerte, L. (2018). On the random gamma function: Theory and computing. Journal of Computational and Applied Mathematics, 335, 142-155. doi:10.1016/j.cam.2017.11.04

Solving second-order linear differential equations with random analytic coefficients about ordinary points: A full probabilistic solution by the first probability density function

[EN] This paper deals with the approximate computation of the first probability density function of the solution stochastic process to second-order linear differential equations with random analytic coefficients about ordinary points under very general hypotheses. Our approach is based on considering approximations of the solution stochastic process via truncated power series solution obtained from the random regular power series method together with the so-called Random Variable Transformation technique. The validity of the proposed method is shown through several illustrative examples.This work has been partially supported by the Ministerio de Econom ia y Competitividad grant MTM2013-41765-P. Ana Navarro Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M. (2018). Solving second-order linear differential equations with random analytic coefficients about ordinary points: A full probabilistic solution by the first probability density function. Applied Mathematics and Computation. 331:33-45. https://doi.org/10.1016/j.amc.2018.02.051S334533

Solving the random Cauchy one-dimensional advection-diffusion equation: Numerical analysis and computing

[EN] In this paper, a random finite difference scheme to solve numerically the random Cauchy one-dimensional advection-diffusion partial differential equation is proposed and studied. Throughout our analysis both the advection and diffusion coefficients are assumed to be random variables while the deterministic initial condition is assumed to possess a discrete Fourier transform. For the sake of generality in our study, we consider that the advection and diffusion coefficients are statistical dependent random variables. Under mild conditions on the data, it is demonstrated that the proposed random numerical scheme is mean square consistent and stable. Finally, the theoretical results are illustrated by means of two numerical examples.This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2013-41765-P. Ana Navarro Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia. M.A. Sohaly is also indebted to Egypt Ministry of Higher Education Cultural Affairs for its financial support [mohe-casem(2016)].Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M.; Sohaly, M. (2018). Solving the random Cauchy one-dimensional advection-diffusion equation: Numerical analysis and computing. Journal of Computational and Applied Mathematics. 330:920-936. https://doi.org/10.1016/j.cam.2017.02.001S92093633

Probabilistic solution of the homogeneous Riccati differential equation: A case-study by using linearization and transformation techniques

[EN] This paper deals with the determination of the first probability density function of the solution stochastic process to the homogeneous Riccati differential equation taking advantage of both linearization and Random Variable Transformation techniques. The study is split in all possible casuistries regarding the deterministic/random character of the involved input parameters. An illustrative example is provided for each one of the considered cases.This work has been partially supported by the Ministerio de Economia y Competitividad grants MTM2013-41765-P and TRA2012-36932.Casabán, MC.; Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M.; Villanueva Micó, RJ. (2016). Probabilistic solution of the homogeneous Riccati differential equation: A case-study by using linearization and transformation techniques. Journal of Computational and Applied Mathematics. 291:20-35. https://doi.org/10.1016/j.cam.2014.11.028S203529

A comprehensive probabilistic solution of random SIS-type epidemiological models using the Random Variable Transformation technique

[EN] This paper provides a complete probabilistic description of SIS-type epidemiological models where all the input parameters (contagion rate, recovery rate and initial conditions) are assumed to be random variables. By applying the Random Variable Transformation technique, the first probability density function, the mean and the variance functions as well as confidence intervals associated to the solution of SIS-type epidemiological models are determined under the general hypothesis that the random inputs have any joint probability density function. The distributions to describe the time until a given proportion of the population remains susceptible and infected are also determined. Lastly, a probabilistic description of the so-called basic reproductive number is included. The theoretical results are applied to an illustrative example showing good fitting.This work has been partially supported by the Ministerio de Economia y Competitividad grants MTM2013-41765-P and TRA2012-36932. Ana Navarro Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Casabán, M.; Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M.; Villanueva Micó, RJ. (2016). A comprehensive probabilistic solution of random SIS-type epidemiological models using the Random Variable Transformation technique. Communications in Nonlinear Science and Numerical Simulation. 32:199-210. https://doi.org/10.1016/j.cnsns.2015.08.009S1992103