Last modified: 2023-05-15

#### Abstract

Qualitative and quantitative study of differential equations with initial and boundary conditions plays an important role in applications, since the dynamic equations arise from modeling problems in science and engineering. Here we develop a methodology to find the solution of nonlinear Caputo fractional differential equations by generalized iterative methods, when the non-linear function is the sum of convex, concave, increasing and decreasing functions, using coupled lower and upper solution. We use generalized iterative methods such as generalized monotone method and/ or generalized quasilinearization method coupled with lower and upper solutions or coupled lower and upper solutions. The generalized iterative methods we employ here are both theoretical as well as computational. To start with we will use the natural lower and upper solution to start with since they are relatively easy to compute. We will use these natural lower and upper solutions to compute the coupled lower and upper solutions. The first approximations which we obtain in general are on some interval [0, t], and not necessarily on the whole interval [0, T]. We develop a method to construct coupled lower and upper solution on the interval [0, T]. Further, using these coupled lower and upper solution, we will construct monotone sequences on the desired interval, which converge uniformly and monotonically to coupled minimal and maximal solutions. These minimal and maximal solutions will converge to a unique solution if the nonlinear term satisfies the Lipshitz condition. We will develop a numerical code using MatLab to obtain the iterates.