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### Convergence of Riemann-Liouvelli and Caputo Derivative Definitions for Practical Solution of Fractional Order Differential Equation

Shantanu Das

#### Abstract

The classical Fractional Differential Equations with Riemann-Liouvelli require initial states of fractional order-which sometimes is physically difficult to visualize. Though one can solve the Fractional Differential Equation, assuming those fractional states exists and gives mathematically pure linearly independent solutions, yet engineers and applied scientists would prefer practical answers and would like to utilize what is easily realizable; and that is integer order initial states. Well, the Caputo’s derivative was thus developed and defined in 1967, to have integer order initial states. This dichotomy has restricted the use of fractional calculus and is still not very popular amongst application scientists and engineers. Can these two definitions, of Fractional Derivatives of opposite nature, merge and give transparent Fractional Differential Equation, which can be practically and easily solved-at least for fundamental solution? Yes, perhaps it can be, as these two definitions of Fractional Calculus seem to merge when we decompose the Fractional Differential Equation System; giving same results with both these formulations. This gives a powerful tool to engineer and applied scientists to describe the system in Riemann-Liouvelli scheme and yet obtain solution with integer order initial states as for integer order classical calculus. This observation is not recorder earlier, have been provided as lemma with proof and example.

#### Keywords

Caputo and Riemann-Liouvelli (RL) fractional derivative, Fractional initial states, Decomposition, Fractional Differential Equation (FDE)

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