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In this paper we study a number of nonlinear fractional equations, involving Caputo derivative in space or/and in time, admitting explicit solution in separating variable form. Some of these equations are particularly interesting because they admit completely periodic solutions. When time-fractional derivatives are introduced, this property is lost, but in the space-fractional case we can obtain new interesting equations admitting these solutions. This can be the starting point for a more general analysis about fractional isochronous partial differential equations.