Exponential quadrature rules for linear fractional differential equations
The authors investigate numerical solutions of linear fractional differential equations of the following form \[ D^{\alpha}_{t_{0}}y(t)+\lambda y(t)=f(t), \] where \( \alpha >0\) is a real number, \( \lambda \in \mathbb R \), \( y:[t_{0},T]\rightarrow \mathbb R \) and the forcing term \( f:[t_{0},T]\rightarrow \mathbb R \) is sufficiently smooth. The symbol \( D^{\alpha}_{t_{0}} \) denotes the derivative operator of non-integer order \( \alpha \), with respect to the origin \( t_{0}\), according to the Caputo definition \[ D^{\alpha}_{t_{0}}y(t)= \frac{1}{\Gamma(m-\alpha)}\int^{t}_{t_{0}}\frac{y^{(m)}(u)}{(t-u)^{\alpha+1-m}}du,\tag{\(*\)} \] with the gamma function \( \Gamma(.) \), \( m=\lceil\alpha\rceil \) the smallest integer such that \( \alpha < m \), \( y^{(m)} \) is the derivative of \( y \) of integer order \( m \) and the initial conditions \( y^{(k)}(t_{0})=y_{0,k}\), \(k=0,1,\dots,m-1 \). In order to obtain high accuracy under suitable and reasonable assumptions of smoothness on \( f(t)\), Eq.(*) is reformulated in terms of some special weighted integrals involving \( f(t) \) alone as the integrand and then specific quadrature rules are formulated for their numerical approximation. Numerical examples are given to illustrate the results obtained. The effectiveness of the present approach is demonstrated by means of comparisons with other standard methods.
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