On a nonlocal extension of differentiation
On a nonlocal extension of differentiation
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关于微分的非局部扩展
DOI：
10.1016/j.jmaa.2016.03.054
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
R. Shankar
中科院分区：
文献类型：
--
作者：
R. Shankar
研究了一个推广了反微分问题的积分方程。我们用一个已知的非局部算子来代替经典导数，类似于断裂力学和非局部扩散中的算子。我们证明当非定域参数消失时，该算子弱收敛于经典导数。利用傅里叶变换，我们求出了积分方程的通解。我们证明了非局部不定积分除了包含一个任意常数外，还包含一个无限维的函数集。然而，当非定域性参数消失时，这些函数弱收敛于零。对于特殊类型的积分核，我们证明了当非定域参数消失时，非定域不定积分弱收敛于其经典对应物。
We study an integral equation that extends the problem of anti-differentiation. We formulate this equation by replacing the classical derivative with a known nonlocal operator similar to those applied in fracture mechanics and nonlocal diffusion. We show that this operator converges weakly to the classical derivative as a nonlocality parameter vanishes. Using Fourier transforms, we find the general solution to the integral equation. We show that the nonlocal antiderivative involves an infinite dimensional set of functions in addition to an arbitrary constant. However, these functions converge weakly to zero as the nonlocality parameter vanishes. For special types of integral kernels, we show that the nonlocal antiderivative weakly converges to its classical counterpart as the nonlocality parameter vanishes.