Picard and Adomian decomposition methods for a quadratic integral equation of fractional order

We study two analytical methods: the classical method of successive approximations (Picard method) (Curtain and Pritchard in Functional analysis in modern applied mathematics, Academic press, London, 1977) and Adomian method which gives the solution as a series (see Adomian in Stochastic system, Academic press, New York, 1983; Adomian in Nonlinear stochastic operator equations. Academic press, San Diego, 1986; Adomian in Nonlinear stochastic systems: theory and applications to physics. Kluwer, Dordrecht, 1989; Adomian et al. in J Math Comput 23:17–23, 1992; Abbaoui and Cherruault in Comput Math Appl 28:103–109, 1994; Adomian in Solving frontier problems of physics: the decomposition method. Kluwer, Dordrecht, 1995; Cherruault in Kybernetes 18:31–38, 1989; Cherruault et al. in Int J Biomed Comput 38:89–93, 1995). The existence and uniqueness of the solution and the convergence will be discussed for each method and some examples will be studied.