分数阶导数的一种新的扩散表示及其应用
A New Diffusive Representation for Fractional Derivatives and its Application
摘要：
分数阶导数的扩散表示已被证明是构造求解分数阶微分方程的快速高效数值方法的有用工具。在这种方法的许多已知变体中的共同挑战是，它们需要在无界积分上的一些积分的数值逼近，该无界积分的被积函数衰减相当缓慢，这意味着它们的数值处理是困难且昂贵的。我们提出了一个新的变种，这样的扩散表示。这种形式也需要对无界区域上的积分进行数值逼近，但被积函数衰减得更快。这允许使用具有更好的收敛特性的良好建立的求积规则。
英文摘要：
Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known variants of this approach is that they require the numerical approximation of some integrals over an unbounded integral whose integrand decays rather slowly which implies that their numerical handling is difficult and costly. We present a novel variant of such a diffusive representation. This form also requires the numerical approximation of an integral over an unbounded domain, but the integrand decays much faster. This allows to use well established quadrature rules with much better convergence properties.
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