A Cubic Spline Projection Method for Computing Stationary Density Functions of Frobenius-Perron Operator

Abstract

Stationary density functions of Frobenius-Perron operators have critical applications in many fields of science and engineering. Accordingly, approximating stationary density functions f* is important and the focus of this dissertation. Among the computational methods of approximating the smooth f*, the linear spline and quadratic spline projection methods have been proven effective. However, we intend to improve the convergence rate of the previous methods. We will fulfill this goal by using cubic spline functions since cubic spline functions are twice continuously differentiable on the whole domain. Theoretically, we prove the existence of a nonzero sequence of cubic spline functions {fₙ} that converges to the stationary density function f* of the Frobenius-Perron operator in L¹-norm. The numerical experimental results assure that the cubic spline projection method gives the fastest convergence rate so far. In addition, when the stationary density function f* lies in the cubic spline space, the cubic spline projection method computes f* exactly no matter what n may be.