Transformational and Numerical Techniques for Solving Non-Integer Order Dynamical Systems

Abstract

In this research study, two types of integral transform techniques called the Natural and the Aboodh transform have been employed to get exact solutions of initial value problems in non integer order ordinary differential equations, that is., continuous dynamical systems having non-integer order derivatives. Theorems for each integral transform have been proved which ultimately help to find the exact solution of the problems under consideration. Upon compari son with other existing transform techniques such as the Laplace, Sumudu, Elzaki and Mellin, the obtained results would match. However, such transform techniques are only applicable for continuous linear dynamical systems. Having known the fact that most of the dynamical systems particularly with non-linearity do not have closed form solutions, two different numer ical techniques while considering a Volterra operator with a weakly singular convolution kernel have been used under Caputo’s type differential operator to find approximate solutions of such problems. The numerical techniques have been analyzed for their stability characteristics, con vergence and error bounds (local and global). Absolute errors at the final nodal point on the integration interval [0, T], T > 0 as well as over the entire time interval have been computed and the numerical techniques are compared with some existing techniques in literature. Upon comparison, the techniques are observed to be better in terms of errors and the run time com puted in (seconds.milliseconds). This comparison is made using continuous dynamical systems of varying nature. For example, some of them are autonomous and non-autonomous whereas others are linear and nonlinear. Apart from these, the dynamical systems from fields like chem istry, physics and life sciences are also tested by replacing their first order derivatives with non-integer order ones. Various orders for non-integer order derivatives have been used while simulating the dynamical systems. MATLAB software having version 9.3.0.713579 (R2017b), using Intel(R) Core(TM) i3-4500U processor running on 1.70 GHz has been used for all the necessary computations.