Numerical methods for the solution of fractional differential equations
The fractional calculus is a generalisation of the calculus of Newton and
Leibniz. The substitution of fractional differential operators in ordinary differential
equations substantially increases their modelling power.
Fractional differential operators set exciting new challenges to the computational
mathematician because the computational cost of approximating
fractional differential operators is of a much higher order than that necessary
for approximating the operators of classical calculus.
1. We present a new formulation of the fractional integral.
2. We use this to develop a new method for reducing the computational
cost of approximating the solution of a fractional differential equation.
3. This method can be implemented with two levels of sophistication.
We compare their rates of convergence, their algorithmic complexity,
and their weight set sizes so that an optimal choice, for a particular
application, can be made.
4. We show how linear multiterm fractional differential equations can be
approximated as systems of fractional differential equations of order at