Abstract: The solution for sequential Caputo linear fractional differential equations with variable coefficients of order q, 0 < q < 1 can be obtained from symbolic representation form. Since the iterative method developed in Chapter 2 is time-consuming even for the simple linear fractional differential equations with variable coefficients, the direct numerical approximation developed in Chapter 3 is very useful tool when computing the linear and non-linear fractional differential equations of a specific type. This direct numerical method is useful in developing the monotone method and the quasilinearization method for non-linear problems. As an application of this result, we have obtained the numerical solution for a special Ricatti, type of differential equation which blows up in finite time. The generalized monotone iterative method with coupled lower and upper solutions yields monotone natural sequence which converges uniformly and monotonically to coupled minimal and maximal solutions of Caputo fractional boundary value problem. We obtain the existence and uniqueness of sequential Caputo fractional boundary value problems with mixed boundary conditions with the Green\'s function representation.
Keywords: Coupled lower and upper solutions,Fractional boundary value problems,Green's function,Mittag-Leffler function, Sequential caputo fractional derivative