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The non-classical calculi such as q-calculus, fractional calculus and q-fractional calculus have been hot topics in both applied and pure sciences. Then some new linear and nonlinear models have appeared. This study mainly concentrates on the analytical aspects, and the variational iteration method is extended in a new way to solve an initial value problem.
Fractional differential equations have been caught much attention during the past decades. In this study, iteration formulae of a fractional differential equation with uncertainty are proposed and the approximate solutions for a simple case are derived via a fractional variational iteration method.
A new Adomian decomposition method is proposed to approximately solve fractional differential equations. The iteration procedure is based on a fractional Taylor series. An example is illustrated to show the presented method's efficiency and convenience.
The q-difference equations are a class of important models both in q-calculus and applied sciences. The variational iteration method is extended to approximately solve the initial value problems of q-difference equations. A q-analogue of the Lagrange multiplier is presented and three examples are illustrated to show the method's efficiency.