Applying the Random Variable Transformation method to solve a class of random linear differential equation with discrete delay

[EN] We randomize the following class of linear differential equations with delay, x(tau)' (t) = ax(tau) (t) bx(tau) (t -tau), t> 0, and initial condition, x(tau )(t) = g(t), -tau 0(+), to the first probability density function, say f(x, t), of the corresponding associated random linear problem without delay (tau = 0). The paper concludes with several numerical experiments illustrating our theoretical findings.This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2017-89664-P and MTM2015-63723-P and Junta de Andalucia under Proyecto de Excelencia P12-FQM-1492. Ana Navarro Quiles acknowledges the Fundacio Ferran Sunyer i Balaguer and the Instituto de Estudios Catalanes for its contribution through the Borsa Ferran Sunyer i Balaguer. Ana Navarro Quiles acknowledges the postdoctoral contract financed by DyCon project funding from the European Research Council(ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No 694126-DYCON). The authors express their deepest thanks and respect to the editors and reviewers for their valuable comments.Caraballo, T.; Cortés, J.; Navarro-Quiles, A. (2019). Applying the Random Variable Transformation method to solve a class of random linear differential equation with discrete delay. Applied Mathematics and Computation. 356:198-218. https://doi.org/10.1016/j.amc.2019.03.048S19821835

A full probabilistic solution of the random linear fractional differential equation via the Random Variable Transformation technique

[EN] This paper provides a full probabilistic solution of the randomized fractional linear nonhomogeneous differential equation with a random initial condition via the computation of the first probability density function of the solution stochastic process. To account for most generality in our analysis, we assume that uncertainty appears in all input parameters (diffusion coefficient, source term, and initial condition) and that a wide range of probabilistic distributions can be assigned to these parameters. Throughout our study, we will consider that the fractional order of Caputo derivative lies in] 0,1], that corresponds to the main standard case. To conduct our analysis, we take advantage of the random variable transformation technique to construct approximations of the first probability density function of the solution process from a suitable infinite series representation. We then prove these approximations do converge to the exact density assuming mild conditions on random input parameters. Our theoretical findings are illustrated through 2 numerical examples.Ministerio de Economia y Competitividad, Grant/Award Number: MTM2017-89664-P; Programa de Ayudas de Investigacion y Desarrollo, Grant/Award Number: PAID-2014; UNiversitat Politecncia de ValenciaBurgos-Simon, C.; Calatayud-Gregori, J.; Cortés, J.; Navarro-Quiles, A. (2018). A full probabilistic solution of the random linear fractional differential equation via the Random Variable Transformation technique. Mathematical Methods in the Applied Sciences. 41(18):9037-9047. https://doi.org/10.1002/mma.4881S90379047411

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

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 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

Uncertainty quantification analysis of the biological Gompertz model subject to random fluctuations in all its parameters

[EN] In spite of its simple formulation via a nonlinear differential equation, the Gompertz model has been widely applied to describe the dynamics of biological and biophysical parts of complex systems (growth of living organisms, number of bacteria, volume of infected cells, etc.). Its parameters or coefficients and the initial condition represent biological quantities (usually, rates and number of individual/particles, respectively) whose nature is random rather than deterministic. In this paper, we present a complete uncertainty quantification analysis of the randomized Gomperz model via the computation of an explicit expression to the first probability density function of its solution stochastic process taking advantage of the Liouville-Gibbs theorem for dynamical systems. The stochastic analysis is completed by computing other important probabilistic information of the model like the distribution of the time until the solution reaches an arbitrary value of specific interest and the stationary distribution of the solution. Finally, we apply all our theoretical findings to two examples, the first of numerical nature and the second to model the dynamics of weight of a species using real data.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.Bevia, V.; Burgos, C.; Cortés, J.; Navarro-Quiles, A.; Villanueva Micó, RJ. (2020). Uncertainty quantification analysis of the biological Gompertz model subject to random fluctuations in all its parameters. Chaos, Solitons and Fractals. 138:1-12. https://doi.org/10.1016/j.chaos.2020.109908S112138Golec, J., & Sathananthan, S. (2003). Stability analysis of a stochastic logistic model. Mathematical and Computer Modelling, 38(5-6), 585-593. doi:10.1016/s0895-7177(03)90029-xCortés, J. C., Jódar, L., & Villafuerte, L. (2009). Random linear-quadratic mathematical models: Computing explicit solutions and applications. Mathematics and Computers in Simulation, 79(7), 2076-2090. doi:10.1016/j.matcom.2008.11.008Dorini, 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.2015.09.009Dorini, F. A., Bobko, N., & Dorini, L. B. (2016). A note on the logistic equation subject to uncertainties in parameters. Computational and Applied Mathematics, 37(2), 1496-1506. doi:10.1007/s40314-016-0409-6Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2019). Analysis of random non-autonomous logistic-type differential equations via the Karhunen–Loève expansion and the Random Variable Transformation technique. Communications in Nonlinear Science and Numerical Simulation, 72, 121-138. doi:10.1016/j.cnsns.2018.12.013Calatayud, J., Cortés, J. C., & Jornet, M. (2019). Improving the approximation of the probability density function of random nonautonomous logistic‐type differential equations. Mathematical Methods in the Applied Sciences, 42(18), 7259-7267. doi:10.1002/mma.5834Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (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. doi:10.1016/j.cam.2014.11.028Hesam, S., Nazemi, A. R., & Haghbin, A. (2012). Analytical solution for the Fokker–Planck equation by differential transform method. Scientia Iranica, 19(4), 1140-1145. doi:10.1016/j.scient.2012.06.018Lakestani, M., & Dehghan, M. (2009). Numerical solution of Fokker-Planck equation using the cubic B-spline scaling functions. Numerical Methods for Partial Differential Equations, 25(2), 418-429. doi:10.1002/num.20352Mao, X., Yuan, C., & Yin, G. (2005). Numerical method for stationary distribution of stochastic differential equations with Markovian switching. Journal of Computational and Applied Mathematics, 174(1), 1-27. doi:10.1016/j.cam.2004.03.016Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (2017). Computing probabilistic solutions of the Bernoulli random differential equation. Journal of Computational and Applied Mathematics, 309, 396-407. doi:10.1016/j.cam.2016.02.034Kegan, B., & West, R. W. (2005). Modeling the simple epidemic with deterministic differential equations and random initial conditions. Mathematical Biosciences, 195(2), 179-193. doi:10.1016/j.mbs.2005.02.004Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (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. doi:10.1016/j.aml.2016.12.015Cortés, J. C., Navarro‐Quiles, A., Romero, J., & Roselló, M. (2019). (CMMSE2018 paper) Solving the random Pielou logistic equation with the random variable transformation technique: Theory and applications. Mathematical Methods in the Applied Sciences, 42(17), 5708-5717. doi:10.1002/mma.5440Dorini, F. A., & Cunha, M. C. C. (2011). On the linear advection equation subject to random velocity fields. Mathematics and Computers in Simulation, 82(4), 679-690. doi:10.1016/j.matcom.2011.10.008Slama, 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/i2017-11763-6Hussein, A., & Selim, M. M. (2013). A general analytical solution for the stochastic Milne problem using Karhunen–Loeve (K–L) expansion. Journal of Quantitative Spectroscopy and Radiative Transfer, 125, 84-92. doi:10.1016/j.jqsrt.2013.03.018Hussein, 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., 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.2010.05.046Bekiryazici, Z., Merdan, M., & Kesemen, T. (2020). Modification of the random differential transformation method and its applications to compartmental models. Communications in Statistics - Theory and Methods, 50(18), 4271-4292. doi:10.1080/03610926.2020.1713372Calatayud, J., Cortés, J.-C., Díaz, J. A., & Jornet, M. (2020). Constructing reliable approximations of the probability density function to the random heat PDE via a finite difference scheme. Applied Numerical Mathematics, 151, 413-424. doi:10.1016/j.apnum.2020.01.012Laird, A. K. (1965). Dynamics of Tumour Growth: Comparison of Growth Rates and Extrapolation of Growth Curve to One Cell. British Journal of Cancer, 19(2), 278-291. doi:10.1038/bjc.1965.32Nahashon, S. N., Aggrey, S. E., Adefope, N. A., Amenyenu, A., & Wright, D. (2006). Growth Characteristics of Pearl Gray Guinea Fowl as Predicted by the Richards, Gompertz, and Logistic Models. Poultry Science, 85(2), 359-363. doi:10.1093/ps/85.2.35

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

Aleatorizando el modelo de crecimiento malthusiano

[EN] This paper deals with the randomization of the classical malthusian model using a markovian approach. We show that the solution stochastic process, usually referred to as birth process, corresponds to the shift negative binomial distribution. A simulation procedure is included. A brief discussion regarding alternative ways to consider randomness into the malthusian model are also included.[ES] Este trabajo ilustra, mediante el modelo de crecimiento clásico malthusiano, cómo introducir la aleatoriedad utilizando el enfoque markoviano. Se prueba que la función generatriz de probabilidad del proceso estocástico solución, denominado de nacimiento, es una distribución binomial negativa desplazada. Se detalla una técnica para simular el proceso estocástico solución. También se realiza una breve digresión acerca de otras formas posibles de introducir la incertidumbre en el modelo determinista malthusiano.Cortés, J.; García Moreno, E.; Navarro-Quiles, A. (2017). Aleatorizando el modelo de crecimiento malthusiano. Sociedad Puig Adam de profesores de matemáticas. 103:52-65. http://hdl.handle.net/10251/148893S526510

Approximating the solution stochastic process of the random Cauchy one-dimensional heat model

[EN] This paper deals with the numerical solution of the random Cauchy one-dimensional heat model. We propose a random finite difference numerical scheme to construct numerical approximations to the solution stochastic process. We establish sufficient conditions in order to guarantee the consistency and stability of the proposed random numerical scheme.The theoretical results are illustrated by means of an example where reliable approximations of the mean and standard deviation to the solution stochastic process are given.This work has been partially supported by the Ministerio de Economía y Competitividad Grant MTM2013-41765-P. Ana Navarro Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València. M. A. Sohaly is also indebted to Egypt Ministry of Higher Education, Cultural Affairs, for its financial support [mohe-casem (2016)].Navarro Quiles, A.; Romero, J.; Roselló, M.; Sohaly, M. (2016). Approximating the solution stochastic process of the random Cauchy one-dimensional heat model. Abstract and Applied Analysis. 2016:1-7. https://doi.org/10.1155/2016/5391368S172016Logan, J. D. (2004). Partial Differential Equations on Bounded Domains. Undergraduate Texts in Mathematics, 121-171. doi:10.1007/978-1-4419-8879-9_4Wang, J. (1994). A Model of Competitive Stock Trading Volume. Journal of Political Economy, 102(1), 127-168. doi:10.1086/261924Tsynkov, S. V. (1998). Numerical solution of problems on unbounded domains. A review. Applied Numerical Mathematics, 27(4), 465-532. doi:10.1016/s0168-9274(98)00025-7Koleva, M. N. (2006). Numerical Solution of the Heat Equation in Unbounded Domains Using Quasi-uniform Grids. Lecture Notes in Computer Science, 509-517. doi:10.1007/11666806_58Han, H., & Huang, Z. (2002). A class of artificial boundary conditions for heat equation in unbounded domains. Computers & Mathematics with Applications, 43(6-7), 889-900. doi:10.1016/s0898-1221(01)00329-7Wu, X., & Sun, Z.-Z. (2004). Convergence of difference scheme for heat equation in unbounded domains using artificial boundary conditions. Applied Numerical Mathematics, 50(2), 261-277. doi:10.1016/j.apnum.2004.01.001Cortés, J. C., Sevilla-Peris, P., & Jódar, L. (2005). Analytic-numerical approximating processes of diffusion equation with data uncertainty. Computers & Mathematics with Applications, 49(7-8), 1255-1266. doi:10.1016/j.camwa.2004.05.015Casabán, M.-C., Cortés, J.-C., García-Mora, B., & Jódar, L. (2013). Analytic-Numerical Solution of Random Boundary Value Heat Problems in a Semi-Infinite Bar. Abstract and Applied Analysis, 2013, 1-9. doi:10.1155/2013/676372Casabán, M.-C., Company, R., Cortés, J.-C., & Jódar, L. (2014). Solving the random diffusion model in an infinite medium: A mean square approach. Applied Mathematical Modelling, 38(24), 5922-5933. doi:10.1016/j.apm.2014.04.063Villafuerte, L., Braumann, C. A., Cortés, J.-C., & Jódar, L. (2010). Random differential operational calculus: Theory and applications. Computers & Mathematics with Applications, 59(1), 115-125. doi:10.1016/j.camwa.2009.08.061Øksendal, B. (2003). Stochastic Differential Equations. Universitext. doi:10.1007/978-3-642-14394-6Kloeden, P. E., & Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. doi:10.1007/978-3-662-12616-5Holden, H., Øksendal, B., Ubøe, J., & Zhang, T. (2010). Stochastic Partial Differential Equations. doi:10.1007/978-0-387-89488-