Govinda Pageni and Aghalaya S Vatsala. Study of Three Systems of Non-Linear Caputo Fractional Differential Equations with Initial
Conditions and Applications
Neural, Parallel, and Scientific Computations 29 (2021) No.4, 211-229

https://doi.org/10.46719/npsc20212941

ABSTRACT.
We shall provide an analytical method for solving three linear coupled systems of Caputo fractional differential equations with fractional initial conditions. Because the Mittag-Leffler function doesn’t satisfy all the properties of the exponential function, we cannot use the integer order methods. Here we have used an efficient and convenient method, called the Laplace transform method, to solve the three systems of linear Caputo fractional differential equations with fractional initial conditions when the order of the fractional derivative is $q $ and $0 < q <1$. In addition, the Laplace-Adomian decomposition method allows us to obtain an approximation of the non-linear SIR epidemic model of fractional order $q$. All the methods we have adopted here yield integer results as a special case. Our method also works for scalar linear sequential Caputo fractional differential equations of order $nq$, since it can be reduced to $n$ systems of $qth$ order linear Caputo fractional differential equations with initial conditions.

AMS (MOS) Subject Classification. 34A12, 34A08.

Key Words: Mittag-Leffler Function, Caputo Fractional Derivative, Laplace-Adomian method.