Abstract
This contribution provides a practical view to the computation of the first probability density function of the solution stochastic process to ordinary and partial differential equations with randomness using the Random Variable Transformation technique. The analysis is performed via a set of simple examples, belonging to different areas like Physics, Biology and Engineering, with the aim of illustrating key ideas from a practical standpoint.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
The Whishart distribution is a probability distribution for random matrices. A random matrix is a matrix whose entries are random variables. We say that a k × k random matrix A is absolutely continuous (respectively discrete) if its vectorization, vec(A), is an absolutely continuous (respectively discrete) k 2 × 1 random vector. In this case, the density or mass function of A is defined as the density or mass function of vec(A). We say that a k × k random matrix A follows a Wishart distribution, A ∼Wishart(H, n), H being k × k symmetric and positive definite matrix and n ≥ k, if it is symmetric, positive definite and absolutely continuous, with density
$$\displaystyle \begin{aligned} P_A(a)=c[\det(a)]^{\frac{n}{2}-\frac{k+1}{n}}e^{-\frac{n}{2}\text{tr}(H^{-1}a)}, \end{aligned}$$being c a constant depending on H and n.
References
Soong, T.T.: Random Differential Equations in Science and Engineering. Academic, New York (1973)
Smith, R.C.: Uncertainty Quantification. Theory, Implementation and Applications. SIAM Computational Science & Engineering. SIAM, Philadelphia (2014)
Øksendal, B.: Stochastic Differential Equations. An Introduction with Applications. Stochastic Modelling and Applied Probability, vol. 23. Springer, Heidelberg (2003)
Kloeden, P., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (2011)
Allen, E.: Modeling with Itô Stochastic Differential Equations. Mathematical Modelling: Theory and Applications. Springer, Amsterdam (2007)
Papoulis, A., Pillai, U.: Probability, Random Variables and Stochastic Processes, 4th edn. McGraw-Hill, New York (2002)
Casella, G., Berger, R.: Statistical Inference. Cambridge Texts in Applied Mathematics. Brooks/Cole, New York (2002)
Kadry, S.: On the generalization of probabilistic transformation method. Appl. Math. Comput. 190(15), 1284–1289 (2007). https://doi.org/10.1016/j.amc.2011.12.088
Lord, G.J., Powell, C.E., Shardlow, T.: An Introduction to Computational Stochastic PDEs. Cambridge Texts in Applied Mathematics. Cambridge University Press, New York (2014)
Casabán, M.-C., Cortés, J.-C., Romero, J.-V., Roselló, M.-D.: Determining the first probability density function of linear random initial value problems by the random variable transformation (R.V.T.) technique: a comprehensive study. Abstr. Appl. Anal. 2014, 1–25, 248512 (2014). https://doi.org/10.1155/2013/248512
Casabán, M.-C., Cortés, J.-C., Romero, J.-V., Roselló, M.-D.: Solving random homogeneous linear second-order differential equations: a full probabilistic description. Mediterr. J. Math. 13(6), 3817–3836 (2016). https://doi.org/10.1007/s00009-016-0716-6
Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., Villanueva, R.-J.: Computing probabilistic solutions of the Bernoulli random differential equation. J. Comput. Appl. Math. 309, 396–407 (2017). https://doi.org/10.1016/jcam.2016.02.034
Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., Villanueva, R.-J.: A comprehensive probabilistic solution of random SIS-type epidemiological models using the Random Variable Transformation technique. Commun. Nonlinear Sci. Numer. Simul. 32, 199–210 (2016). https://doi.org/10.1016/j.cnsns.2015.08.009
Dorini, F.A., Cecconello, M.S., Dorini, L.B.: On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density. Commun. Nonlinear Sci. Numer. Simul. 33, 160–173 (2016). https://doi.org/10.1016/j.cnsns.2015.09.009
Seber, G.A.F., Wild, C.J.: Nonlinear Regression. Wiley, Hoboken (2003)
Lesaffre, E., Lawson, A.B.: Bayesian Biostatistics. Statistics in Practice. Wiley, Hoboken (2012)
Hussein, A., Selim, M.M.: Solution of the stochastic generalized shallow-water wave equation using RVT technique. Eur. Phys. J. Plus 139, 249 (2015). https://doi.org/10.1140/epjp/i2015-15249-3
Hussein, A., Selim, M.M.: Solution of the stochastic radiative transfer equation with Rayleigh scattering using RVT technique. Appl. Math. Comput. 218(13), 7193–7203 (2012). https://doi.org/10.1016/j.amc.2011.12.088
Angulo Pava, J.: Nonlinear Dispersive Equations. Existence of Stability of Solitary and Periodic Travelling Wave Solutions. Mathematical Surveys and Monographs, vol. 156. American Mathematical Society, Providence (2009)
Hussein, A., Selim, M.M.: New soliton solutions for some important nonlinear partial differential equations using a generalized Bernoulli method. Int. J. Math. Anal. Appl. 1(1), 1–8 (2014)
Yang, X.-F., Deng, Z.-C., Wei, Y.: A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application. Adv. Difference Equ. 2015, 117 (2015). https://doi.org/10.1186/s13662-015-0452-4
Loève, M.: Probability Theory, vol. 1. Springer, New York (1977)
Braumann, C.A., Villafuerte, L., Cortés, J.-C., Jódar, L.: Random differential operational calculus: theory and applications. Comput. Math. Appl. 59, 115–125 (2010). https://doi.org/10.1016/j.camwa.2009.08.061
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Calatayud, J., Cortés, JC., Jornet, M., Navarro-Quiles, A. (2019). Solving Random Ordinary and Partial Differential Equations Through the Probability Density Function: Theory and Computing with Applications. In: Sadovnichiy, V., Zgurovsky, M. (eds) Modern Mathematics and Mechanics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-96755-4_15
Download citation
DOI: https://doi.org/10.1007/978-3-319-96755-4_15
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-96754-7
Online ISBN: 978-3-319-96755-4
eBook Packages: EngineeringEngineering (R0)