Generalized Monotone Method for Sequential Caputo Fractional Boundary Value Problems

Abstract

Generalized monotone method together with coupled lower and upper solutions yield monotone sequences which converge uniformly and monotonically to coupled minimal and maximal solutions of the nonlinear problem under consideration. In this work, we have developed generalized monotone method for sequential Caputo fractional boundary value problem with mixed boundary conditions which are in terms of Caputo fractional derivative. For that purpose, we have obtained a representation form for the corresponding linear Caputo sequential boundary value problem in terms of the Green's function. In addition, we have obtained a linear comparison result for the linear sequential differential inequalities with linear mixed boundary conditions. The comparison result is useful in proving the monotonicity of the iterates as well as the uniqueness of the solution of the nonlinear sequential boundary value problem. Our method yields the integer results as a special case. Some numerical examples for the linear sequential Caputo fractional boundary value problems have also been presented.

Numerical Results for Linear Sequential Caputo Fractional Boundary Value Problems

“…The linear comparison result has been developed in [31] in such a way that the linear sequential Caputo boundary value problem of Equation 7has a unique solution. This has been proved under the assumption that the left sequential Caputo fractional derivative of order nq will be the same as the right sequential Caputo fractional derivative of order nq for n = 1, 2 in [31]. We make a similar assumption throughout this paper also.…”

“…The main reason for this is the integer derivative is sequential, whereas the fractional derivative is not sequential. In the recent paper [31] the generalized monotone method combined with coupled lower and upper solutions for sequential boundary value problems of order 2q, which is sequential of order q, has been developed. In addition, the boundary conditions in [31] involve the left and right derivative of order q near the boundary.…”

“…In the recent paper [31] the generalized monotone method combined with coupled lower and upper solutions for sequential boundary value problems of order 2q, which is sequential of order q, has been developed. In addition, the boundary conditions in [31] involve the left and right derivative of order q near the boundary. In particular if q = 1, the sequential fractional boundary value problem will reduce to the integer boundary value problem including the mixed boundary conditions.…”

“…We denote the (left) sequential Caputo derivative of order q as ( sc D nq 0 + )u(x), where n ≥ 2 is an integer. From the paper[31], we consider the linear sequential Caputo fractional boundary value problem:…”