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We consider the numerical solution of (possibly nonlinear) fractional differential equations of the form y^(α)(t)=f(t,y(t),y^(β₁)(t),y^(β₂)(t),…,y^(β_n)(t)) with α>β_n>β_n−1>⋯>β₁ and α−β_n⩽1, β_j−β_j−1⩽1, 0<β₁⩽1, combined with suitable initial conditions. The derivatives are understood in the Caputo sense. We begin by discussing the analytical questions of existence and uniqueness of solutions, and we investigate how the solutions depend on the given data. Moreover we propose convergent and stable numerical methods for such initial value problems.