Probabilistic approach to the Lame equations of linear elastostatics

A probabilistic approach to systems of partial differential equations is developed on the basis of the well-known Feynman–Kac and Bismut formulas providing explicit probabilistic representations of the solutions and of their derivatives of scalar differential equations. Some numerical examples are also included. In particular the Lamé equations of elastostatics are solved and the results are compared with some known exact analytic solutions to demonstrate the efficiency of the approach.