Fractional exponential fitting backward differential formulas for solving differential equations of fractional order

In this paper, we develop an implicit type fractional exponential fitting backward differential formulas of second-order (FEBDF2) for solving fractional order differential equations of order α ∈ ( 0 , 1 ) in Caputo sense by constructing some new generating functions. The solutions belong to the space generated by the linear combinations of ⟨ 1 , e λ x , x e λ x ⟩ . The constraint on the parameter λ is discussed in detail. The novel introduced method joints the fractional BDF method when λ = 0 . Also, the stability of the method is discussed. We focus on linear and nonlinear equations and determine stability regions for the FEBDF methods. The stability regions of the new method are greater than the fractional BDF ones. We check the efficiency of the method by applying it to several examples and comparing the obtained results with the known ones.