Fractional powers of Bessel operator and its numerical calculation

Abstract

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 Schr\"odinger 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.