Cauchy formula for repeated integration is \[ f^{(-n)}(x) = \frac{1}{(n-1)!}\int_a^x (x-t)^{-1}f(t) dt. \] where \[ f^{(-n)}(x) = \int_a^x \int_a^{\sigma_1} \cdots \int_a^{\sigma_{n-1}} f(\sigma_n) d\sigma_n \dots d\sigma_2 d \sigma_1. \] Remarkably, the formula holds for any real $n$, leading to the concepts of fractional integrals and derivatives. There is a lot of research interest in the numerical solution of fractional order differential equations. In this project we'll investigate how the Matlab toolbox, Chebfun, can be used to solve these problems. A student who undertakes this project would investigate the mathematics that underpins the system, while developing new programmes and examples to solve problems with chebfuns. An excellent reference for this project is Approximation Theory and Approximation Practice by Nick Trefethen, available from the library at 511.4 TRE. |