An Efficient Fractional Secant-Type Method and Its Application to Boundary Value Problems

An emerging field of study is the application of fractional calculus to iteratively solve the nonlinear equations. Recently, several Newton-type techniques have been proposed that make use of the notion of fractional order derivatives. However, the existence of at least first order derivative is essentially required for the convergence of these methods. On the contrary, we propose a new secant-type method, which is inherently derivative-free, although its construction is based on the idea of conformable fractional derivative of order $$\alpha \in (0,1]$$ . The primary objective for the development is to analyze how fractional derivatives have the effect of enlarging the convergence domain. In this regard, the proposed scheme is examined for its convergence characteristics and dynamical features for different values of $$\alpha $$ in the specified range. Furthermore, the efficacy of method is demonstrated through solving various applied nonlinear problems including the fractional order Burgers’ equation.