Computational methods for random differential equations: probability density function and estimation of the parameters

[EN] Mathematical models based on deterministic differential equations do not take into account the inherent uncertainty of the physical phenomenon (in a wide sense) under study. In addition, inaccuracies in the collected data often arise due to errors in the measurements. It thus becomes necessary to treat the input parameters of the model as random quantities, in the form of random variables or stochastic processes. This gives rise to the study of random ordinary and partial differential equations. The computation of the probability density function of the stochastic solution is important for uncertainty quantification of the model output. Although such computation is a difficult objective in general, certain stochastic expansions for the model coefficients allow faithful representations for the stochastic solution, which permits approximating its density function. In this regard, Karhunen-Loève and generalized polynomial chaos expansions become powerful tools for the density approximation. Also, methods based on discretizations from finite difference numerical schemes permit approximating the stochastic solution, therefore its probability density function. The main part of this dissertation aims at approximating the probability density function of important mathematical models with uncertainties in their formulation. Specifically, in this thesis we study, in the stochastic sense, the following models that arise in different scientific areas: in Physics, the model for the damped pendulum; in Biology and Epidemiology, the models for logistic growth and Bertalanffy, as well as epidemiological models; and in Thermodynamics, the heat partial differential equation. We rely on Karhunen-Loève and generalized polynomial chaos expansions and on finite difference schemes for the density approximation of the solution. These techniques are only applicable when we have a forward model in which the input parameters have certain probability distributions already set. When the model coefficients are estimated from collected data, we have an inverse problem. The Bayesian inference approach allows estimating the probability distribution of the model parameters from their prior probability distribution and the likelihood of the data. Uncertainty quantification for the model output is then carried out using the posterior predictive distribution. In this regard, the last part of the thesis shows the estimation of the distributions of the model parameters from experimental data on bacteria growth. To do so, a hybrid method that combines Bayesian parameter estimation and generalized polynomial chaos expansions is used.[ES] Los modelos matemáticos basados en ecuaciones diferenciales deterministas no tienen en cuenta la incertidumbre inherente del fenómeno físico (en un sentido amplio) bajo estudio. Además, a menudo se producen inexactitudes en los datos recopilados debido a errores en las mediciones. Por lo tanto, es necesario tratar los parámetros de entrada del modelo como cantidades aleatorias, en forma de variables aleatorias o procesos estocásticos. Esto da lugar al estudio de las ecuaciones diferenciales aleatorias. El cálculo de la función de densidad de probabilidad de la solución estocástica es importante en la cuantificación de la incertidumbre de la respuesta del modelo. Aunque dicho cálculo es un objetivo difícil en general, ciertas expansiones estocásticas para los coeficientes del modelo dan lugar a representaciones fieles de la solución estocástica, lo que permite aproximar su función de densidad. En este sentido, las expansiones de Karhunen-Loève y de caos polinomial generalizado constituyen herramientas para dicha aproximación de la densidad. Además, los métodos basados en discretizaciones de esquemas numéricos de diferencias finitas permiten aproximar la solución estocástica, por lo tanto, su función de densidad de probabilidad. La parte principal de esta disertación tiene como objetivo aproximar la función de densidad de probabilidad de modelos matemáticos importantes con incertidumbre en su formulación. Concretamente, en esta memoria se estudian, en un sentido estocástico, los siguientes modelos que aparecen en diferentes áreas científicas: en Física, el modelo del péndulo amortiguado; en Biología y Epidemiología, los modelos de crecimiento logístico y de Bertalanffy, así como modelos de tipo epidemiológico; y en Termodinámica, la ecuación en derivadas parciales del calor. Utilizamos expansiones de Karhunen-Loève y de caos polinomial generalizado y esquemas de diferencias finitas para la aproximación de la densidad de la solución. Estas técnicas solo son aplicables cuando tenemos un modelo directo en el que los parámetros de entrada ya tienen determinadas distribuciones de probabilidad establecidas. Cuando los coeficientes del modelo se estiman a partir de los datos recopilados, tenemos un problema inverso. El enfoque de inferencia Bayesiana permite estimar la distribución de probabilidad de los parámetros del modelo a partir de su distribución de probabilidad previa y la verosimilitud de los datos. La cuantificación de la incertidumbre para la respuesta del modelo se lleva a cabo utilizando la distribución predictiva a posteriori. En este sentido, la última parte de la tesis muestra la estimación de las distribuciones de los parámetros del modelo a partir de datos experimentales sobre el crecimiento de bacterias. Para hacerlo, se utiliza un método híbrido que combina la estimación de parámetros Bayesianos y los desarrollos de caos polinomial generalizado.[CA] Els models matemàtics basats en equacions diferencials deterministes no tenen en compte la incertesa inherent al fenomen físic (en un sentit ampli) sota estudi. A més a més, sovint es produeixen inexactituds en les dades recollides a causa d'errors de mesurament. Es fa així necessari tractar els paràmetres d'entrada del model com a quantitats aleatòries, en forma de variables aleatòries o processos estocàstics. Açò dóna lloc a l'estudi de les equacions diferencials aleatòries. El càlcul de la funció de densitat de probabilitat de la solució estocàstica és important per a quantificar la incertesa de la sortida del model. Tot i que, en general, aquest càlcul és un objectiu difícil d'assolir, certes expansions estocàstiques dels coeficients del model donen lloc a representacions fidels de la solució estocàstica, el que permet aproximar la seua funció de densitat. En aquest sentit, les expansions de Karhunen-Loève i de caos polinomial generalitzat esdevenen eines per a l'esmentada aproximació de la densitat. A més a més, els mètodes basats en discretitzacions mitjançant esquemes numèrics de diferències finites permeten aproximar la solució estocàstica, per tant la seua funció de densitat de probabilitat. La part principal d'aquesta dissertació té com a objectiu aproximar la funció de densitat de probabilitat d'importants models matemàtics amb incerteses en la seua formulació. Concretament, en aquesta memòria s'estudien, en un sentit estocàstic, els següents models que apareixen en diferents àrees científiques: en Física, el model del pèndol amortit; en Biologia i Epidemiologia, els models de creixement logístic i de Bertalanffy, així com models de tipus epidemiològic; i en Termodinàmica, l'equació en derivades parcials de la calor. Per a l'aproximació de la densitat de la solució, ens basem en expansions de Karhunen-Loève i de caos polinomial generalitzat i en esquemes de diferències finites. Aquestes tècniques només són aplicables quan tenim un model cap avant en què els paràmetres d'entrada tenen ja determinades distribucions de probabilitat. Quan els coeficients del model s'estimen a partir de les dades recollides, tenim un problema invers. L'enfocament de la inferència Bayesiana permet estimar la distribució de probabilitat dels paràmetres del model a partir de la seua distribució de probabilitat prèvia i la versemblança de les dades. La quantificació de la incertesa per a la resposta del model es fa mitjançant la distribució predictiva a posteriori. En aquest sentit, l'última part de la tesi mostra l'estimació de les distribucions dels paràmetres del model a partir de dades experimentals sobre el creixement de bacteris. Per a fer-ho, s'utilitza un mètode híbrid que combina l'estimació de paràmetres Bayesiana i els desenvolupaments de caos polinomial generalitzat.This work has been supported by the Spanish Ministerio de Economía y Competitividad grant MTM2017–89664–P.Calatayud Gregori, J. (2020). Computational methods for random differential equations: probability density function and estimation of the parameters [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/138396TESISPremios Extraordinarios de tesis doctorale

A theoretical study of a short rate model

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2016, Director: José Manuel Corcuera ValverdeThe goal of this project is to do a theoretical study of a short interest rate model under the risk neutral probability, which is able to represent long range dependence. In order to do this, it will be explained the necessary literature to understand the model. Furthermore, we will expose the consequences of adapting this model for evaluating bonds and derivatives. In order to do this, we will use ambit processes which in general are not semimartingales. Our purpose is to see if these new models can capture the features of the bond market by extending popular models like the Vasicek model

Markov chains and applications in the generation of combinatorial designs

This Thesis deals with discrete Markov chains and their applications in the generation of combinatorial designs. A conjecture on the generation of proper edge colorings of the complete graph K_n, for n even, is tackled. A proper edge coloring is an edge coloring such that no two adjacent edges have the same color. Proper edge colorings are characterized by minimizing the potential and maximizing the entropy. We implement an algorithm in the software R to generate proper colorings from any arbitrary coloring of K_n, by identifying the colorings as nodes in a Markov chain, where transition probabilities are defined so that the potential decreases or, alternatively, the entropy increases. The conjecture states that the algorithm converges in polynomial time. We give original proofs of the conjecture in K_4 and K_6, and we provide new results and ideas that could be used to prove the conjecture in the general case K_n

Lp-calculus approach to the random autonomous linear differential equation with discrete delay

[EN] In this paper, we provide a full probabilistic study of the random autonomous linear differential equation with discrete delay , with initial condition x(t)=g(t), -t0. The coefficients a and b are assumed to be random variables, while the initial condition g(t) is taken as a stochastic process. Using Lp-calculus, we prove that, under certain conditions, the deterministic solution constructed with the method of steps that involves the delayed exponential function is an Lp-solution too. An analysis of Lp-convergence when the delay tends to 0 is also performed in detail.This work has been supported by the Spanish Ministerio de Economia y Competitividad Grant MTM2017-89664-P. The author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). Lp-calculus approach to the random autonomous linear differential equation with discrete delay. Mediterranean Journal of Mathematics. 16(4):1-17. https://doi.org/10.1007/s00009-019-1370-6S117164Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics. Springer, New York (2011)Driver, Y.: Ordinary and Delay Differential Equations. Applied Mathematical Science Series. Springer, New York (1977)Kuang, Y.: Delay Differential Equations: with Applications in Population Dynamics. Academic Press, Cambridge (2012)Bocharov, G.A., Rihan, F.A.: Numerical modelling in biosciences using delay differential equations. J. Comput. Appl. Math. 125, 183–199 (2000). https://doi.org/10.1016/S0377-0427(00)00468-4Jackson, M., Chen-Charpentier, B.M.: Modeling plant virus propagation with delays. J. Comput. Appl. Math. 309, 611–621 (2017). https://doi.org/10.1016/j.cam.2016.04.024Chen-Charpentier, B.M., Diakite, I.: A mathematical model of bone remodeling with delays. J. Comput. Appl. Math. 291, 76–84 (2016). https://doi.org/10.1016/j.cam.2017.01.005Erneux, T.: Applied Delay Differential Equations, Surveys and Tutorials in the Applied Mathematical Sciences Series. Springer, New York (2009)Kyrychko, Y.N., Hogan, S.J.: On the Use of delay equations in engineering applications. J. Vib. Control 16(7–8), 943–960 (2017). https://doi.org/10.1177/1077546309341100Matsumoto, A., Szidarovszky, F.: Delay Differential Nonlinear Economic Models (in Nonlinear Dynamics in Economics, Finance and the Social Sciences), 195–214. Springer-Verlag, Berlin Heidelberg (2010)Harding, L., Neamtu, M.: A dynamic model of unemployment with migration and delayed policy intervention. Comput. Econ. 51(3), 427–462 (2018). https://doi.org/10.1007/s10614-016-9610-3Oksendal, B.: Stochastic Differential Equations. Springer, New York (1998)Shaikhet, L.: Lyapunov Functionals and Stability of Stochastic Functional Differential Equations. Springer, New York (2013)Hartung, F., Pituk, M.: Recent Advances in Delay Differential and Differences Equations. Springer-Verlag, Berlin Heidelberg (2014)Shaikhet, L.: Stability of equilibrium states of a nonlinear delay differential equation with stochastic perturbations. Int. J. Robust Nonlinear Control 27(6), 915–924 (2016). https://doi.org/10.1002/rnc.3605Shaikhet, L.: About some asymptotic properties of solution of stochastic delay differential equation with a logarithmic nonlinearity. Funct. Differ. Equ. 4(1–2), 57–67 (2017)Fridman, E., Shaikhet, L.: Delay-induced stability of vector second-order systems via simple Lyapunov functionals. Automatica 74, 288–296 (2016). https://doi.org/10.1016/j.automatica.2016.07.034Benhadri, M., Zeghdoudi, H.: Mean square asymptotic stability in nonlinear stochastic neutral Volterra-Levin equations with Poisson jumps and variable delays. Functiones et Approximatio Commentarii Mathematici 58(2), 157–176 (2018). https://doi.org/10.7169/facm/1657Nouri, K., Ranjbar, H.: Improved Euler-Maruyama method for numerical solution of the Itô stochastic differential systems by composite previous-current-step idea. Mediterr. J. Math. 15, 140 (2018). https://doi.org/10.1007/s00009-018-1187-8Santonja, F., Shaikhet, L.: Probabilistic stability analysis of social obesity epidemic by a delayed stochastic model. Nonlinear Anal. Real World Appl. 17, 114–125 (2014). https://doi.org/10.1016/j.nonrwa.2013.10.010Santonja, F., Shaikhet, L.: Analysing social epidemics by delayed stochastic models. Discret. Dyn. Nat. Soc. 2012, 13 (2012). https://doi.org/10.1155/2012/530472 . (Article ID 530472)Liu, L., Caraballo, T.: Analysis of a stochastic 2D-Navier-Stokes model with infinite delay. J. Dyn. Differ. Equ. pp 1–26 (2018). https://doi.org/10.1007/s10884-018-9703-xCaraballo, T., Colucci, R., Guerrini, L.: On a predator prey model with nonlinear harvesting and distributed delay. Commun. Pure Appl. Anal. 17(6), 2703–2727 (2018). https://doi.org/10.3934/cpaa.2018128Smith, R.C.: Uncertainty Quantification. Theory, Implementation and Applications. SIAM, Philadelphia (2014)Soong, T.T.: Random Differential Equations in Science and Engineering. Academic Press, New York (1973)Nouri, K., Ranjbar, H.: Mean square convergence of the numerical solution of random differential equations. Mediterr. J. Math. 12(3), 1123–1140 (2015). https://doi.org/10.1007/s00009-014-0452-8Zhou, T.: A stochastic collocation method for delay differential equations with random input. Adv. Appl. Math. Mech. 6(4), 403–418 (2014). https://doi.org/10.4208/aamm.2012.m38Shi, W., Zhang, C.: Generalized polynomial chaos for nonlinear random delay differential equations. Appl. Numer. Math. 115, 16–31 (2017). https://doi.org/10.1016/j.apnum.2016.12.004Lupulescu, V., Abbas, U.: Fuzzy delay differential equations. Fuzzy Optim. Decis. Mak. 11(1), 91–111 (2012). https://doi.org/10.1007/s10700-011-9112-7Liu, S., Debbouche, A., Wang, J.R.: Fuzzy delay differential equations. On the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths. J. Comput. Appl. Math. 312, 47–57 (2017). https://doi.org/10.1016/j.cam.2015.10.028Krapivsky, P.L., Luck, J.L., Mallick, K.: On stochastic differential equations with random delay. J. Stat. Mech. Theory Exp. (2011). https://doi.org/10.1088/1742-5468/2011/10/P10008Garrido-Atienza, M.J., Ogrowsky, A., Schmalfuss, B.: Random differential equations with random delays. Stoch. Dyn. 11(2–3), 369–388 (2011). https://doi.org/10.1142/S0219493711003358Khusainov, D.Y., Ivanov, A.F., Kovarzh, I.V.: Solution of one heat equation with delay. Nonlinear Oscil. 12, 260–282 (2009). https://doi.org/10.1007/s11072-009-0075-3Asl, F.M., Ulsoy, A.G.: Analysis of a system of linear delay differential equations. J. Dyn. Syst. Meas. Control 125, 215–223 (2003). https://doi.org/10.1115/1.1568121Kyrychko, Y.N., Hogan, S.J.: On the use of delay equations in engineering applications. J. Vib. Control 16(7–8), 943–960 (2010). https://doi.org/10.1177/1077546309341100Villafuerte, L., Braumann, C.A., Cortés, J.C., Jódar, L.: Random differential operational calculus: theory and applications. Comput. Math. Appl. 59(1), 115–125 (2010). https://doi.org/10.1016/j.camwa.2009.08.061Strand, J.L.: Random ordinary differential equations. J. Diff. Equ. 7(3), 538–553 (1970). https://doi.org/10.1016/0022-0396(70)90100-2Khusainov, D.Y., Pokojovy, M.: Solving the linear 1d thermoelasticity equations with pure delay. Int. J. Math. Math. Sci. 2015, 1–11 (2015). https://doi.org/10.1155/2015/47926

Approximate solutions of randomized non-autonomous complete linear differential equations via probability density functions

[EN] Solving a random differential equation means to obtain an exact or approximate expression for the solution stochastic process, and to compute its statistical properties, mainly the mean and the variance functions. However, a major challenge is the computation of the probability density function of the solution. In this article we construct reliable approximations of the probability density function to the randomized non-autonomous complete linear differential equation by assuming that the diffusion coefficient and the source term are stochastic processes and the initial condition is a random variable. The key tools to construct these approximations are the random variable transformation technique and Karhunen-Loeve expansions. The study is divided into a large number of cases with a double aim: firstly, to extend the available results in the extant literature and, secondly, to embrace as many practical situations as possible. Finally, a wide variety of numerical experiments illustrate the potentiality of our findings.This work has been supported by the Spanish Ministerio de Economía y Competitividad grant MTM2017-89664-P. The author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). Approximate solutions of randomized non-autonomous complete linear differential equations via probability density functions. Electronic Journal of Differential Equations. 2019:1-40. http://hdl.handle.net/10251/139661S140201

Random differential equations with discrete delay

[EN] In this article, we study random differential equations with discrete delay with initial condition The uncertainty in the problem is reflected via the outcome omega. The initial condition g(t) is a stochastic process. The term x(t) is a stochastic process that solves the random differential equation with delay in a probabilistic sense. In our case, we use the random calculus approach. We extend the classical Picard theorem for deterministic ordinary differential equations to calculus for random differential equations with delay, via Banach fixed-point theorem. We also relate solutions with sample-path solutions. Finally, we utilize the theoretical ideas to solve the random autonomous linear differential equation with discrete delay.This work has been supported by the Spanish Ministerio de Economía y Competitividad grant MTM2017 89664 PCalatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). Random differential equations with discrete delay. Stochastic Analysis and Applications. 37(5):699-707. https://doi.org/10.1080/07362994.2019.1608833S699707375Fridman, E., & Shaikhet, L. (2017). Stabilization by using artificial delays: An LMI approach. Automatica, 81, 429-437. doi:10.1016/j.automatica.2017.04.015Shaikhet, L., & Korobeinikov, A. (2015). Stability of a stochastic model for HIV-1 dynamics within a host. Applicable Analysis, 95(6), 1228-1238. doi:10.1080/00036811.2015.1058363Caraballo, T., Colucci, R., & Guerrini, L. (2018). On a predator prey model with nonlinear harvesting and distributed delay. Communications on Pure & Applied Analysis, 17(6), 2703-2727. doi:10.3934/cpaa.2018128Caraballo, T., J. Garrido-Atienza, M., Schmalfuss, B., & Valero, J. (2017). Attractors for a random evolution equation with infinite memory: Theoretical results. Discrete & Continuous Dynamical Systems - B, 22(5), 1779-1800. doi:10.3934/dcdsb.2017106Krapivsky, P. L., Luck, J. M., & Mallick, K. (2011). On stochastic differential equations with random delay. Journal of Statistical Mechanics: Theory and Experiment, 2011(10), P10008. doi:10.1088/1742-5468/2011/10/p10008Liu, S., Debbouche, A., & Wang, J. (2017). On the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths. Journal of Computational and Applied Mathematics, 312, 47-57. doi:10.1016/j.cam.2015.10.028Dorini, 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.2015.09.009Slama, 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/i2017-11763-6Nouri, K., Ranjbar, H., & Torkzadeh, L. (2019). Modified stochastic theta methods by ODEs solvers for stochastic differential equations. Communications in Nonlinear Science and Numerical Simulation, 68, 336-346. doi:10.1016/j.cnsns.2018.08.013Lupulescu, V., O’Regan, D., & ur Rahman, G. (2014). Existence results for random fractional differential equations. Opuscula Mathematica, 34(4), 813. doi:10.7494/opmath.2014.34.4.813Strand, J. . (1970). Random ordinary differential equations. Journal of Differential Equations, 7(3), 538-553. doi:10.1016/0022-0396(70)90100-2Villafuerte, 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.2009.08.061Granas, A., & Dugundji, J. (2003). Fixed Point Theory. Springer Monographs in Mathematics. doi:10.1007/978-0-387-21593-

Uncertainty quantification for random parabolic equations with non-homogeneous boundary conditions on a bounded domain via the approximation of the probability density function

[EN] This paper deals with the randomized heat equation defined on a general bounded interval [L-1, L-2] and with nonhomogeneous boundary conditions. The solution is a stochastic process that can be related, via changes of variable, with the solution stochastic process of the random heat equation defined on [0,1] with homogeneous boundary conditions. Results in the extant literature establish conditions under which the probability density function of the solution process to the random heat equation on [0,1] with homogeneous boundary conditions can be approximated. Via the changes of variable and the Random Variable Transformation technique, we set mild conditions under which the probability density function of the solution process to the random heat equation on a general bounded interval [L-1, L-2] and with nonhomogeneous boundary conditions can be approximated uniformly or pointwise. Furthermore, we provide sufficient conditions in order that the expectation and the variance of the solution stochastic process can be computed from the proposed approximations of the probability density function. Numerical examples are performed in the case that the initial condition process has a certain Karhunen-Loeve expansion, being Gaussian and non-Gaussian.This work has been supported by Spanish Ministerio de Economía y Competitividad grant MTM2017 89664 P. The author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). Uncertainty quantification for random parabolic equations with non-homogeneous boundary conditions on a bounded domain via the approximation of the probability density function. Mathematical Methods in the Applied Sciences. 42(17):5649-5667. https://doi.org/10.1002/mma.5333S564956674217Holden, H., Øksendal, B., Ubøe, J., & Zhang, T. (2010). Stochastic Partial Differential Equations. doi:10.1007/978-0-387-89488-1Casabán, 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.2014.04.063Xu, Z., Tipireddy, R., & Lin, G. (2016). Analytical approximation and numerical studies of one-dimensional elliptic equation with random coefficients. Applied Mathematical Modelling, 40(9-10), 5542-5559. doi:10.1016/j.apm.2015.12.041CalatayudJ CortésJC JornetM.On the approximation of the probability density function of the randomized heat equation.https://arxiv.org/pdf/1802.04190.pdfStrand, J. . (1970). Random ordinary differential equations. Journal of Differential Equations, 7(3), 538-553. doi:10.1016/0022-0396(70)90100-2Vaart, A. W. van der. (1998). Asymptotic Statistics. doi:10.1017/cbo9780511802256Villafuerte, 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.2009.08.061Pitman, J. (1993). Probability. doi:10.1007/978-1-4612-4374-8Williams, D. (1991). Probability with Martingales. doi:10.1017/cbo9780511813658LawlessJF.Truncated Distributions: Wiley StatsRef: Statistics Reference Online;2014

The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function

[EN] This paper deals with the damped pendulum random differential equation: (X) over double dot(t)+2 omega(0)xi(X) over dot(t) + omega X-2(0)(t) = Y(t), t is an element of [0, T], with initial conditions X(0) = X-0 and (X) over dot(0) = X-1. The forcing term Y(t) is a stochastic process and X-0 and X-1 are random variables in a common underlying complete probability space (Omega, F, P). The term X(t) is a stochastic process that solves the random differential equation in both the sample path and in the L-P senses. To understand the probabilistic behavior of X(t), we need its joint finite-dimensional distributions. We establish mild conditions under which X(t) is an absolutely continuous random variable, for each t, and we find its probability density function f(X(t))(x). Thus, we obtain the first finite-dimensional distributions. In practice, we deal with two types of forcing term: Y(t) is a Gaussian process, which occurs with the damped pendulum stochastic differential equation of Ito type; and Y(t) can be approximated by a sequence {Y-N(t)}(N-1)(infinity) in L-2([0, T] x Omega), which occurs with Karhunen-Loeve expansions and some random power series. Finally, we provide numerical examples in which we choose specific random variables X-0 and X-1 and a specific stochastic process Y(t), and then, we find the probability density function of X(t). (C) 2018 Elsevier B.V. All rights reserved.This work has been supported by the Spanish Ministerio de Economia y Competitividad grant MTM2017-89664-P. Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia. The authors are grateful for the valuable comments raised by the reviewers that have improved the final version of the paper.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2018). The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function. Physica A Statistical Mechanics and its Applications. 512:261-279. https://doi.org/10.1016/j.physa.2018.08.024S26127951

Improving the approximation of the first and second order statistics of the response stochastic process to the random Legendre differential equation

[EN] In this paper, we deal with uncertainty quantification for the random Legendre differential equation, with input coefficient A and initial conditions X-0 and X-1. In a previous study (Calbo et al. in Comput Math Appl 61(9):2782-2792, 2011), a mean square convergent power series solution on (-1/e, 1/e) was constructed, under the assumptions of mean fourth integrability of X-0 and X-1, independence, and at most exponential growth of the absolute moments of A. In this paper, we relax these conditions to construct an L-p solution (1 <= p <= infinity) to the random Legendre differential equation on the whole domain (-1, 1), as in its deterministic counterpart. Our hypotheses assume no independence and less integrability of X-0 and X-1. Moreover, the growth condition on the moments of A is characterized by the boundedness of A, which simplifies the proofs significantly. We also provide approximations of the expectation and variance of the response process. The numerical experiments show the wide applicability of our findings. A comparison with Monte Carlo simulations and gPC expansions is performed.This work has been supported by the Spanish Ministerio de Economia y Competitividad Grant MTM2017-89664-P. Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). Improving the approximation of the first and second order statistics of the response stochastic process to the random Legendre differential equation. Mediterranean Journal of Mathematics. 16(3):1-14. https://doi.org/10.1007/s00009-019-1338-6S114163Soong, T.T.: Random Differential Equations in Science and Engineering. Academic Press, New York (1973)Strand, J.L.: Random ordinary differential equations. J. Differ. Equ. 7(3), 538–553 (1970)Smith, R.C.: Uncertainty quantification. Theory, implementation, and application. SIAM Comput. Sci. Eng. New York (2013) ISBN 9781611973211Fishman, G.: Monte Carlo: Concepts, Algorithms, and Applications. Springer, Berlin (2013)Cortés, J.-C., Romero, J.-V., Roselló, M.-D., Santonja, F.-J., Villanueva, R.-J.: Solving continuous models with dependent uncertainty: a computational approach. Abstr. Appl. Anal. 2013, 983839 (2013). https://doi.org/10.1155/2013/983839Xiu, D.: Numerical Methods for Stochastic Computations. A Spectral Method Approach. Cambridge Texts in Applied Mathematics. Princeton University Press, New York (2010)El-Tawil, M.A.: The approximate solutions of some stochastic differential equations using transformations. Appl. Math. Comput. 164(1), 167–178 (2005)Cortés, J.-C., Sevilla-Peris, P., Jódar, L.: Constructing approximate diffusion processes with uncertain data. Math. Comput. Simul. 73(1–4), 125–132 (2006)Cortés, J.-C., Jódar, L., Villafuerte, L., Villanueva, R.-J.: Computing mean square approximations of random diffusion models with source term. Math. Comput. Simul. 76(1–3), 44–48 (2007)Khodabin, M., Maleknejad, K., Rostami, M., Nouri, M.: Numerical solution of stochastic differential equations by second order Runge–Kutta methods. Math. Comput. Model. 53(9–10), 1910–1920 (2011)Nouri, K., Ranjbar, H.: Mean square convergence of the numerical solution of random differential equations. Mediterran. J. Math. 12(3), 1123–1140 (2015)Nouri, N.: Study on stochastic differential equations via modified Adomian decomposition method. U.P.B. Sci. Bull. Ser. A 78(1), 81–90 (2016)Khodabin, M., Rostami, M.: Mean square numerical solution of stochastic differential equations by fourth order Runge–Kutta method and its application in the electric circuits with noise. Adv. Differ. Equ. 623, 1–19 (2015)Díaz-Infante, S., Jerez, S.: Convergence and asymptotic stability of the explicit Steklov method for stochastic differential equations. J. Comput. Appl. Math. 291(1), 36–47 (2016)Soheili, Ali R, Toutounian, F., Soleymani, F.: A fast convergent numerical method for matrix sign function with application in SDEs (Stochastic Differential Equations). J. Comput. Appl. Math. 282, 167–178 (2015)Øksendal, B.: Stochastic Differential Equations. Springer, Berlin (2003)Villafuerte, L., Braumann, C.A., Cortés, J.-C., Jódar, L.: Random differential operational calculus: theory and applications. Comput. Math. Appl. 59(1), 115–125 (2010)Licea, J., Villafuerte, L., Chen-Charpentier, B.M.: Analytic and numerical solutions of a Riccati differential equation with random coefficients. J. Comput. Appl. Math. 309(1), 208–219 (2013)Cortés, J.-C., Jódar, L., Camacho, J., Villafuerte, L.: Random Airy type differential equations: mean square exact and numerical solutions. Comput. Math. Appl. 60(5), 1237–1244 (2010)Calbo, G., Cortés, J.-C., Jódar, L.: Random Hermite differential equations: mean square power series solutions and statistical properties. Appl. Math. Comput. 218(7), 3654–3666 (2011)Calbo, G., Cortés, J.-C., Jódar, L., Villafuerte, L.: Solving the random Legendre differential equation: mean square power series solution and its statistical functions. Comput. Math. Appl. 61(9), 2782–2792 (2011)Cortés, J.C., Jódar, L., Villafuerte, L.: Mean square solution of Bessel differential equation with uncertainties. J. Comput. Appl. Math. 309(1), 383–395 (2017)Golmankhaneh, A.K., Porghoveh, N.A., Baleanu, D.: Mean square solutions of second-order random differential equations by using homotopy analysis method. Roman. Rep. Phys. 65(2), 350–362 (2013)Khudair, A.K., Ameen, A.A., Khalaf, S.L.: Mean square solutions of second-order random differential equations by using Adomian decomposition method. Appl. Math. Sci. 51(5), 2521–2535 (2011)Khudair, A.K., Haddad, S.A.M., Khalaf, S.L.: Mean square solutions of second-order random differential equations by using the differential transformation method. Open J. Appl. Sci. 6, 287–297 (2016)Norman, L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions, vol. 1. Wiley, Oxford (1994)Ernst, O.G., Mugler, A., Starkloff, H.-J., Ullmann, E.: On the convergence of generalized polynomial chaos expansions. ESAIM Math. Modell. Num. Anal. 46(2), 317–339 (2012)Shi, W., Zhang, C.: Error analysis of generalized polynomial chaos for nonlinear random ordinary differential equations. Appl. Num. Math. 62(12), 1954–1964 (2012)Calatayud, J., Cortés, J.-C., Jornet, M.: On the convergence of adaptive gPC for non-linear random difference equations: theoretical analysis and some practical recommendations. J. Nonlinear Sci. Appl. 11(9), 1077–1084 (2018

Solving a class of random non-autonomous linear fractional differential equations by means of a generalized mean square convergent power series

[EN] The aim of this paper is to solve a class of non-autonomous linear fractional differential equations with random inputs. A mean square convergent series solution is constructed in the case that the fractional order a of that Caputo derivative lies in ]0,1] using a random Frobenius approach. The analysis is conducted by using the so-called mean square random calculus. The mean square convergence of the series solution is established assuming mild conditions on random inputs (diffusion coefficient and initial condition). We show that these conditions are satisfied for a variety of unbounded random variables. In addition, explicit expressions to approximate the mean, the variance and the covariance functions of the random series solution are given. Two full illustrative examples are shown. (C) 2017 Elsevier Ltd. All rights reserved.Authors gratefully acknowledge the comments made by reviewers, which have greatly enriched the manuscript. This work has been partially supported by Ministerio de Economia y Competitividad grant MTM2013-41765-P.Burgos-Simon, C.; Calatayud-Gregori, J.; Cortés, J.; Villafuerte, L. (2018). Solving a class of random non-autonomous linear fractional differential equations by means of a generalized mean square convergent power series. Applied Mathematics Letters. 78:95-104. https://doi.org/10.1016/j.aml.2017.11.009S951047