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In this article, a new reproducing kernel approach is developed for obtainingnumerical solution of nonlinear three-point boundary value problems withfractional order. This approach is based on reproducing kernel which isconstructed by shifted Legendre polynomials. In considered problem, fractionalderivatives with respect to $\alpha$ and $\beta$ are defined in Caputo sense.This method has been applied to some examples which have exact solutions. Inorder to shows the robustness of the proposed method, some numerical resultsare given in tabulated forms.

A new reproducing kernel approach for nonlinear fractional three-point boundary value problems