Λ-fractional Analysis. Basic Theory and Applications

Fractional Analysis is a mathematical method based on different principles from those governing the well-known mathematical principles of differential and integral calculus. The main difference from ordinary differential analysis lies in its property being a non-local analysis, not a local one. This analysis is essential in studying problems in physics, engineering, biology, biomechanics, and others that fall into the micro and nano areas. However, the main issue in fractional analysis is the mathematical imperfections presented by fractional derivatives. In fact, not all known fractional derivatives meet the differential topology requirements for mathematical derivatives. Hence, Λ-fractional differential geometry is invented and applied in various scientific areas, like physics, mechanics, biology, economy, and other fields. Apart from the basic mathematical theory concerning establishing the Λ-fractional derivative, the corresponding differential geometry, differential equations, variational methods, and fields theory are outlined. Proceeding to the applications, Λ-fractional continuum mechanics, Λ-fractional viscoelasticity, Λ-fractional physics, Λ-fractional beam and plate theory are discussed. It is pointed out that only globally stable states are allowed into the context of Λ-fractional analysis.