This report looks at the published literary sources on methods and approaches, which are based on fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) to solve the continuous and discrete mechanics problems. The problems solutions and comparative analysis results of fractional and classical models are presented. The report’s goal is to show an efficiency of using fractional calculus to describe the mechanical processes.

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Fractional analog of the reaction diffusion equation is used to model the subdiffusion process. Diffusion equation with fractional Riemann–Liouville operator is analyzed in this paper. We offer finite-difference methods that can be used to solve the initial-boundary value problems for some time-fractional order differential equations. Stability and convergence theorems are proved.

Fractional order diffusion equations are generalizations of classical diffusion equations, treating super‐diffusive flow processes. In this paper, we examine a splitting type numerical methods to solve a class of two‐dimensional initial‐boundary value fractional diffusive equations. Stability, consistency and convergence of the methods are investigated. It is shown that both schemes are unconditionally stable. A numerical example is presented.

In present work mechanics-mathematical models are constructed to define physical and mechanical properties of biological tissue. These models were adapted to the AFM experimental data so to be a theoretical basis for one. Researched theoretical results are compared with experimental data to confirm sufficiently high level adequacy of offered methodology for definition of physical and mechanical properties biotissue.