Cauchy problem for differential equation with Caputo derivative. (English) Zbl 1114.34005

Summary: 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.


34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
26A33 Fractional derivatives and integrals
65L10 Numerical solution of boundary value problems involving ordinary differential equations
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