Fractional calculus and time-fractional differential equations: revisit and construction of a theory

Masahiro Yamamoto

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arXiv · 2201.08769

Fractional calculus and time-fractional differential equations: revisit and construction of a theory

Masahiro Yamamoto

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Abstract

For fractional derivatives and time-fractional differential equations, we construct a framework on the basis of the operator theory in fractional Sobolev spaces. Our framework provides a feasible extension of the classical Caputo and the Riemann-Liouville derivatives within Sobolev spaces of fractional orders including negative ones. Our approach enables a unified treatment for fractional calculus and time-fractional differential equations. We formulate initial value problems for fractional ordinary differential equations and initial boundary value problems for fractional partial differential equations to prove the well-posedness and other properties.

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Masahiro Yamamoto. 2022-01-21. Fractional calculus and time-fractional differential equations: revisit and construction of a theory. https://arxiv.org/abs/2201.08769

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