After a brief historical view of this problem, we will demonstrate the derivation of first order linear differential equations with random perturbations. Students in their semester research project under a course “special topics” or “independent study” will learn a past century's attempts to solve such differential equations. In addition, the concept of differentiability and integrability will be reviewed.
We will use the concept of “noise” to study the random perturbation on a differential equation as a nowhere differentiable function. The noise in historical Langevin stochastic differential equations will be treated as a model with Brownian motion. A short introduction of Wiener process leading to Ito's calculus is used in derivation of the mean and variance of the solutions to the Langevin Equations. A computational algorithm is developed and applied to study linear stochastic differential equations. Symbolic computation and simulation of a computer algebra system will be used to demonstrate the behavior of the solution to the Langevin Stochastic Differential Equation.
Cite this work
Researchers should cite this work as follows: