Computing probabilistic solutions of the Bernoulli random differential equation

The random variable transformation technique is a powerful method to determine the probabilistic solution for random differential equations represented by the first probability density function of the solution stochastic process. In this paper, that technique is applied to construct a closed form expression of the solution for the Bernoulli random differential equation. In order to account for the general scenario, all the input parameters (coefficients and initial condition) are assumed to be absolutely continuous random variables with an arbitrary joint probability density function. The analysis is split into two cases for which an illustrative example is provided. Finally, a fish weight growth model is considered to illustrate the usefulness of the theoretical results previously established using real data.

[1]  J.-V. Romero,et al.  Probabilistic solution of random homogeneous linear second-order difference equations , 2014, Appl. Math. Lett..

[2]  M. M. Selim,et al.  Solution of the stochastic radiative transfer equation with Rayleigh scattering using RVT technique , 2012, Appl. Math. Comput..

[3]  Juan Carlos Cortés López,et al.  Determining the first probability density function of linear random initial value problems by the Random Variable Transformation (RVT) technique: A comprehensive study , 2014 .

[4]  S. Zacks,et al.  Introduction to stochastic differential equations , 1988 .

[5]  J. Pitchford,et al.  Stochastic von Bertalanffy models, with applications to fish recruitment. , 2007, Journal of theoretical biology.

[6]  N. Bershad,et al.  Random differential equations in science and engineering , 1975, Proceedings of the IEEE.

[7]  R Webster West,et al.  Modeling the simple epidemic with deterministic differential equations and random initial conditions. , 2005, Mathematical biosciences.

[8]  S. Shapiro,et al.  An Analysis of Variance Test for Normality (Complete Samples) , 1965 .

[9]  S. Aachen Stochastic Differential Equations An Introduction With Applications , 2016 .

[10]  Juan Carlos Cortés,et al.  Probabilistic solution of the homogeneous Riccati differential equation: A case-study by using linearization and transformation techniques , 2016, J. Comput. Appl. Math..

[11]  Ralph C. Smith,et al.  Uncertainty Quantification: Theory, Implementation, and Applications , 2013 .

[12]  L. Bertalanffy,et al.  A quantitative theory of organic growth , 1938 .

[13]  Md. Asaduzzaman Shah,et al.  Stochastic Logistic Model for Fish Growth , 2014 .

[14]  L. Bertalanffy Quantitative Laws in Metabolism and Growth , 1957 .

[15]  Milani Chaloupka,et al.  GREEN TURTLE SOMATIC GROWTH MODEL: EVIDENCE FOR DENSITY DEPENDENCE , 2000 .

[16]  Clifford H. Thurber,et al.  Parameter estimation and inverse problems , 2005 .

[17]  Misael Morales,et al.  Edad y crecimiento del pez Haemulon steindachneri (Perciformis: Haemulidae) en el suroeste de la isla de Margarita, Venezuela , 2009 .

[18]  G. Casella,et al.  Statistical Inference , 2003, Encyclopedia of Social Network Analysis and Mining.

[19]  María Dolores Roselló,et al.  Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique , 2015, Commun. Nonlinear Sci. Numer. Simul..

[20]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[21]  Yongjin Wang,et al.  Skew Ornstein-Uhlenbeck processes and their financial applications , 2015, J. Comput. Appl. Math..

[22]  Seifedine Kadry On the generalization of probabilistic transformation method , 2007, Appl. Math. Comput..

[23]  M. M. Selim,et al.  A developed solution of the stochastic Milne problem using probabilistic transformations , 2010, Appl. Math. Comput..

[24]  E. Allen Modeling with Itô Stochastic Differential Equations , 2007 .

[25]  Albert Tarantola,et al.  Inverse problem theory - and methods for model parameter estimation , 2004 .

[26]  M. Carletti Numerical solution of stochastic differential problems in the biosciences , 2006 .

[27]  M. M. Selim,et al.  Solution of the stochastic transport equation of neutral particles with anisotropic scattering using RVT technique , 2009, Appl. Math. Comput..

[28]  Magdy A. El-Tawil,et al.  The approximate solutions of some stochastic differential equations using transformations , 2005, Appl. Math. Comput..

[29]  Magdy A. El-Tawil,et al.  Toward a solution of a class of non-linear stochastic perturbed PDEs using automated WHEP algorithm , 2013 .