Splitting Schemes for Non-Stationary Problems with a Rational\n Approximation for Fractional Powers of the Operator

2020/12/05 by Petr N. Vabishchevich, Vabishchevich, Petr N.
Mathematics · #26A33 #35R11 #65F60 #65M06 #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2012.03059

openalex publication_date 2020/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Problems of the numerical solution of the Cauchy problem for a first-order\ndifferential-operator equation are discussed. A fundamental feature of the\nproblem under study is that the equation includes a fractional power of the\nself-adjoint positive operator. In computational practice, rational\napproximations of the fractional power operator are widely used in various\nversions. The purpose of this work is to construct special approximations in\ntime when the transition to a new level in time provided a set of standard\nproblems for the operator and not for the fractional power operator. Stable\nsplitting schemes with weights parameters are proposed for the additive\nrepresentation of rational approximation for a fractional power operator.\nPossibilities of using similar time approximations for other problems are\nnoted. The numerical solution of a two-dimensional non-stationary problem with\na fractional power of the Laplace operator is also presented.\n

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