On the solutions of stochastic initial-value problems in continuum mechanics
The authors study the stochastic operator equation \[ (1)\quad L^{(2)}_{t,x}u+R^{(2)}_{t,x}u=f(t,\underline x,u,L^{(1)}_{t,x}u,r(\underline x,t;\omega),\eta) \] \[ u(\underline x,t=0)=\psi_ 1(\underline x);\quad u_ t(\underline x;t=0)=\psi_ 2(\underline x), \] which is a mathematical model for a number of physical systems. Under some assumptions, too complex to be presented here, the equation (1) is investigated using a perturbation procedure leading to a sequence of stochastic linear problems solved by application of the known Adomian method in both cases where \(\eta\) is a small parameter and \(\eta\) is not small. The calculation of moments of the solutions is included.
scientific article; zbMATH DE number 3932137