Approximate solutions of randomized non-autonomous complete linear differential equations via probability density functions
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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.
CitationCalatayud, J., Cortés, J. C., & Jornet, M. (2019). Approximate solutions of randomized non-autonomous complete linear differential equations via probability density functions. Electronic Journal of Differential Equations, 2019(85), pp. 1-40.
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