On nonlinear equations related to the distributions of composition of processes
Анотація
The relationship between non-trivial linear partial differential equations and the probability densities of compositions of relevant processes has been pointed out in the recent literature.
In this note, we construct exact solutions for nonlinear partial differential equations starting from these linear equations by using simple transformations. In this way, we have an interesting bridge between the fundamental solutions of linear equations with a clear probabilistic meaning and the construction of exact interesting solutions for the corresponding nonlinear equations.
This method can be obviously generalized to many other cases.
Посилання
L. Beghin, E. Orsingher, L. Sakhno, Equations of mathematical physics and compositions of Brownian and Cauchy processes, Stochastic analysis and applications, 29 (2011) №4, 551–569.
M. D’Ovidio, E. Orsingher, Composition of processes and related partial differential equations, Journal of Theoretical Probability, 24 (2011), №2, 342–375.
E. Orsingher, B. Toaldo, Shooting randomly against a line in Euclidean and non-Euclidean spaces, Stochastics An International Journal of Probability and Stochastic Processes, 86 (2014), №1, 16–45.
A.D. Polyanin, V.F. Zaitsev, Handbook of nonlinear partial differential equations. Chapman and Hall/CRC, 2003.
N.K. Vitanov, On the method of transformations: obtaining solutions of nonlinear differential equations by means of the solutions of simpler linear or nonlinear differential equations, Axioms, 12 (2023), №12, 1106.
Авторське право (c) 2025 C. Cesarano, R. Garra, E. Orsingher
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