Series Solution Method Used to Solve Linear Caputo Fractional Initial Value Problems with Variable Coefficients

Computing the solution of the linear Caputo fractional differential equation with variable coefficients cannot be obtained in closed form as in the integer-order case. However, to use 'q', the order of the fractional derivative, as a parameter for our mathematical model, we need to compute the solution of the equation explicitly and/or numerically. The traditional methods, such as the integrating factor or variation of parameters methods used in the integer-order case, cannot be directly applied because the product rule of the integer derivative does not hold for the Caputo fractional derivative. In this work, we present a series solution method to compute the solution of the linear Caputo fractional differential equation with variable coefficients. This provides an opportunity to compare its solution with the corresponding integer solution, namely q=1. Additionally, we develop a series solution method using analytic functions in the space of Cq continuous functions. We also apply this series solution method to nonlinear Caputo fractional differential equations where the nonlinearity is in the form f(t,u)=u2. We have provided numerical examples to show the application of our series solution method.