论文标题
贝塞尔操作员的分数及其数值计算
Fractional powers of Bessel operator and its numerical calculation
论文作者
论文摘要
本文讨论了贝塞尔操作员的分数及其数值实现。广泛的文献致力于研究拉普拉斯操作员及其应用的分数力量。 Such degrees are used in the construction of functional spaces, in the natural generalization of the Schrödinger equation in quantum theory, in the construction of the models of acoustic wave propagation in complex media (for example, biological tissues) and space-time models of anomalous (very slow or very fast) diffusion, in spectral theory etc. If we assume the radiality of the function on which the Laplace operator acts, then we receive the problem of constructing the贝塞尔操作员的分数力量。我们建议使用一种组成方法来构造前面提到的运算符，这导致其性质的构造与Riesz衍生物相似。 Hankel变换被认为是基本的积分变换。从V.V.提出的组成方法的基础上Katrakhov和S.M. Sitnik构建了贝塞尔操作员的负功率。最终的操作员在内核中包含高斯超几何函数。为了进一步研究，文章中考虑了广义翻译操作员，并证明了其性能。为了构建贝塞尔操作员的积极分数，积分的正则化方法已知。然后，提出了贝塞尔操作员分数幂的数值计算方案。该方案基于B.M.获得的Taylor-Delsarte公式。利维坦。
The article discusses the fractional powers of the Bessel operator and their numerical implementation. An extensive literature is devoted to the study of fractional powers of the Laplace operator and their applications. Such degrees are used in the construction of functional spaces, in the natural generalization of the Schrödinger equation in quantum theory, in the construction of the models of acoustic wave propagation in complex media (for example, biological tissues) and space-time models of anomalous (very slow or very fast) diffusion, in spectral theory etc. If we assume the radiality of the function on which the Laplace operator acts, then we receive the problem of constructing the fractional power of the Bessel operator. We propose to use a compositional method for constructing the operators mentioned earlier, which leads to constructions similar in their properties to the Riesz derivatives. The Hankel transform is considered as a basic integral transformation. On its basis, the compositional method proposed by V.V. Katrakhov and S.M. Sitnik, negative powers of the Bessel operator are constructed. The resulting operator contains the Gaussian hypergeometric function in the kernel. For further study, the generalized translation operator is considered in the article, and its properties are proved. For constructing a positive fractional power of the Bessel operator known methods of regularization of the integral are considered. Then, a scheme for the numerical calculation of fractional powers of the Bessel operator is proposed. This scheme is based on the Taylor--Delsarte formula obtained by B.M. Levitan.
