In this paper, we propose the numerical approximation of fractional initial and boundary value problems using Haar wavelets. In contrast to the Haar wavelet methods available in literature, where the fractional derivative of the function is approximated using the Haar basis, we approximate the function and its classical derivatives using Haar basis functions. Moreover, error bounds in the approximation of fractional integrals and the fractional derivatives are derived, which depend on the index J of the approximation space V_J and the fractional order α. A neural network problem modeled by a system of nonlinear fractional differential equations is also solved using the proposed method. The numerical results show that the proposed numerical approach is efficient.

中文翻译：

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基于Haar小波的分数阶微积分近似方法及其在初值和边值问题中的应用

在本文中，我们提出了使用Haar小波的分数初值和边值问题的数值逼近。与文献中可用的Haar小波方法相反，在该方法中，使用Haar基来逼近函数的分数导数，而使用Haar基函数来逼近函数及其经典导数。此外，在误差积分分数和分数衍生物衍生的，这取决于该索引的近似界定Ĵ近似空间的V _Ĵ和分数阶α。使用提出的方法还解决了由非线性分数阶微分方程系统建模的神经网络问题。数值结果表明，所提出的数值方法是有效的。