In this chapter, some real-life model problems that can be formulated as ordinary differential equations (ODEs) are introduced and numerically studied. These models are the variable-order fractional Hodgkin–Huxley model of neuronal excitation (VOFHHM) and other models with the variable-order fractional (VOF) time delay, such as the 4year life cycle of a population of lemmings model, the enzyme ...

In this paper, a fractional-order model for the spread of computer viruses with saturated treatment rate is presented. This model consists of three components: Susceptible, Infectious, and Recovered. The SIR epidemic model with saturated treatment rate is considered. The stability of the disease free and endemic equilibrium points are studied. The global stability of the disease free equilibriu...

Assuming that the population remains small, one can truncate at given values of n1 and n2 to yield a finite set of differential equations. For an initial condition, we take the population to be zero, that is, ρ1,1,1,1 = 1 with all other elements of the density matrix set to zero. Eq. A.6 can then be solved numerically (we used the Adams-Moulton-Bashforth procedure) until a steady state is reach...

This paper introduces techniques for the estimation of solutions to fractional order differential equations (FODEs) and the approximation of a function’s Caputo fractional derivative. These techniques are based on scattered data interpolation via reproducing kernel Hilbert spaces (RKHSs). Specifically, an RKHS is generated for the purpose of estimating fractional derivatives from the Mittag-Lef...

A recently proposed numerical scheme for solving nonlinear ordinary differential equations with integer and non-integer Liouville-Caputo derivative is applied to three systems chaotic solutions. The Adams-Bashforth involving Lagrange interpolation the fundamental theorem of fractional calculus. We provide existence uniqueness solutions, also convergence result state...

The solution of initial value problems of large systems of ordinary differential equations (ODEs) is computationally intensive and demands for efficient parallel solution techniques that take into account the complex architectures of modern parallel computer systems. This article discusses implementation techniques suitable for ODE systems with a special coupling structure, called limited acces...