Analysis of random non-autonomous logistic-type differential equations via the Karhunen–Loève expansion and the Random Variable Transformation technique

• Published in: Communications in nonlinear science & numerical simulation Vol. 72; pp. 121 - 138
• Main Authors: Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 30.06.2019; Elsevier Science Ltd
• Subjects: Differential equations; Diffusion; Diffusion coefficient; First probability density function; Karhunen–Loève expansion; Logistics; Mathematical analysis; Nonlinear stochastic processes; Probability; Probability density functions; Random logistic differential equation; Random Variable Transformation technique; Random variables; Statistical analysis; Stochastic models; Stochastic processes; Transformations (mathematics); Variables; Random Variable Transformation technique; First probability density function; Nonlinear stochastic processes; Random logistic differential equation; Karhunen–Loève expansion
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2018.12.013

•Random non-autonomous logistic-type differential equations are studied.•Random Variable Transformation method and Karhunen–Love expansion are combined.•First probability density function of the solution stochastic process is determined.•Numerical simulations for the mean, variance and PDF of the solution are performed.•A wide range of PDFs for input data are considered in numerical experiments. This paper deals with the study, from a probabilistic point of view, of logistic-type differential equations with uncertainties. We assume that the initial condition is a random variable and the diffusion coefficient is a stochastic process. The main objective is to obtain the first probability density function, f1(p, t), of the solution stochastic process, P(t, ω). To achieve this goal, first the diffusion coefficient is represented via a truncation of order N of the Karhunen–Loève expansion, and second, the Random Variable Transformation technique is applied. In this manner, approximations, say f1N(p,t), of f1(p, t) are constructed. Afterwards, we rigorously prove that f1N(p,t)⟶f1(p,t) as N → ∞ under mild conditions assumed on input data (initial condition and diffusion coefficient). Finally, three illustrative examples are shown.