On a class of semilinear stochastic systems in mechanics with quadratic- type nonlinearities: Time evolution of the probability density
Systems described by semilinear equations with random initial conditions or parameters are analyzed with the objective of defining existence and uniqueness conditions and providing a quantitative solution. In order to be specific, quadratic type nonlinearities are considered. The analysis obtains a complete solution for the time evolution of the state variable and for the probability density. The work is applicable to classical Lagrangian mechanics and a wide range of other physical problems and allows stochastic fluctuations in parameters without leading to problems of statistical separability. The work leads naturally and with easily computable terms to an excellent approximation. Though there are some superficial resemblances, the basic (decomposition) methodology is quite different than previously known methods and (as yet unpublished) recent Cyber 205 checks have shown a remarkable accuracy on the problems studied. The author's work helps to strengthen and clarify the mathematical framework of a method proving to be effective for wide classes of equations.
semilinear equations with random initial conditions
scientific article; zbMATH DE number 3994672
scientific article