Abstract
An operator-based approach for the construction of closed-form solutions to fractional differential equations is presented in this paper. The technique is based on the analysis of Caputo and Riemann-Liouville algebras of fractional power series. Explicit solutions to a class of linear fractional differential equations are obtained in terms of Mittag-Leffler and fractionally-integrated exponential functions in order to demonstrate the viability of the proposed technique.
Highlights
Fractional differential equations (FDEs) have become one of the cornerstones in the modeling of various real-world systems in recent years
An operator-based approach for the construction of closed-form solutions to fractional differential equations is presented in this paper
Caputo and Riemann-Liouville fractional differentiation and integration operators are defined for respective sets of fractional power series
Summary
Fractional differential equations (FDEs) have become one of the cornerstones in the modeling of various real-world systems in recent years. An analytical technique based on power series is exploited in [5] to predict and represent the multiplicity of solutions to nonlinear boundary value problems of fractional order. This approach is further generalized in [12] and [4] in which the authors propose schemes for the construction of exact analytical solutions to linear and nonlinear equations based on the generalized Taylor series formula. The presented technique is based on Caputo and Riemann-Liouville algebras of fractional power series. The viability of our approach is demonstrated using the fractional damped harmonic oscillator – it is shown that as the fractional derivative order approaches the standard integer value, the exact solution of the FDE converges to the exponential-function exact solution of the respective ODE
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