ABSTRACT
The development of a single numerical method that is able to solve different forms of fractional-order differential equations and fractional-order differential-algebraic equations is the prime objective of this chapter. Before construction of the numerical method, it is shown that the general form of system of fractional-order differential equations encompassing the aforesaid different forms has a unique solution in the given interval. Convergence analysis is carried out to show that the approximate solution obtained by the proposed method can approach the original solution as the step size decreases to zero. The proposed method is applied to physical process models such as the Bagley-Torvik equation, the two-point Bagley-Torvik equational, the plant-herbivore model, the computer virus model, the chemical Akzo Nobel problem, Robertson’s system describing the kinetics of autocatalytic reaction, and the high irradiance response of photo morphogenesis. In addition to the proposed method, the most popular semianalytical techniques such as the Adomian decomposition method (ADM), the homotopy analysis method (HAM), and the fractional differential transform method with Adomian polynomials (FDTM) are implemented as well on physical process models involving a stiff system of differential equations and stiff differential-algebraic equations. It is astonishing to note that ADM, HAM, and FDTM fail to simulate those process models even in the neighborhood of the initial time point 0, although they have successfully simulated many other physical process models. By contrast, the proposed method is able to produce valid approximate solution not only in the vicinity of the initial time point 0 but also in the desired time interval, which can be quite a bit larger than [0, 1].
