Application of Differential Transformation Method to Fractional Differential Equations With Delay

ABSTRACT Delay fractional differential equations (DFDEs) are an important class of models describing natural and technical phenomena that involve both memory effects and delayed system response. Examples include biological population models with maturation time, anomalous diffusion with material memory, and delayed feedback control systems. Solving such equations is highly nonlinear and analytically challenging, especially when the derivative is in the Caputo sense and the delay does not allow for simple function decomposition. In this paper, we propose a modified differential transformation method (DTM), which allows for the construction of a solution using a power series expansion and a delayed term approximation. This method allows for the systematic determination of the solution's recursive coefficients and can be effectively applied even in the absence of an explicit analytical solution. We demonstrate its effectiveness in, among other things, the following applications: example of a logistic growth model, population growth with delayed maturation and fractional memory, indicating the convergence, accuracy and ease of implementation of the method.