Geodesic
Cauchy Problem for Differential Equation with Caputo Derivative
Fractional calculus and applied analysis, Tome 7 (2004) no. 3, pp. 297-321.

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The paper is devoted to the study of the Cauchy problem for a nonlinear differential equation of complex order with the Caputo fractional derivative. The equivalence of this problem and a nonlinear Volterra integral equation in the space of continuously differentiable functions is established. On the basis of this result, the existence and uniqueness of the solution of the considered Cauchy problem is proved. The approximate-iterative method by Dzjadyk is used to obtain the approximate solution of this problem. Two numerical examples are given.
Keywords: Differential Equation of Fractional Order, Caputo Derivative, Existence and Uniqueness Theorem, Approximate-Iterative Method, 34A12, 34B15, 26A33, 65L10 
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Kilbas, Anatoly; Marzan, Sergei. Cauchy Problem for Differential Equation with Caputo Derivative. Fractional calculus and applied analysis, Tome 7 (2004) no. 3, pp. 297-321. http://geodesic.mathdoc.fr/item/FCAA_2004_7_3_a2/