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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.
MSC:
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
34B15Nonlinear boundary value problems for ODE
26A33Fractional derivatives and integrals (real functions)
65L10Boundary value problems for ODE (numerical methods)