Fractional Kinetics Compartmental Models

Abstract

Dynamic models of many processes in the physical and biological sciences give rise to systems of differential equations called compartmental systems. These assume that state variables are continuous and describe the movement of material from compartment to compartment as continuous flows. Together with the mass balance requirements of compartmental systems, these assumptions lead to highly constrained systems of ordinary differential equations, which satisfy certain physical and/or physiological constraints. In this chapter we deal with equivalent structures represented using systems of differential equations of fractional order, that is fractional compartmental systems. The calculus of fractional integrals and derivatives is almost as old as calculus itself going back as early as 1695, to a correspondence between Gottfried von Leibnitz and Guillaume de l’Hopital. Until a few decades ago, however, expressions involving fractional derivatives, integrals and differential equations were mostly restricted to the realm of mathematics. The first modern examples of applications can be found in the classic papers by Caputo (Caputo) and Caputo and Mainardi (Caputo and Mainardi) (dealing with the modeling of viscoelastic materials), but it is only in recent years that it has turned out that many phenomena can be described successfully by models using fractional calculus. In physics fractional derivatives and integrals have been applied to fractional modifications of the commonly used diffusion and Fokker–Planck equations, to describe sub-diffusive (slower relaxation) processes as well as super-diffusion (Sokolov, Klafter et al.). Other examples are of applications are in diffusion processes (Oldham and Spanier), signal processing (Marks and Hall), diffusion problems (Olmstead and Handelsman). More recent applications are in mainly in physics: finite element implementation of viscoelastic models (Chern), mechanical systems subject to damping (Gaul, Klein et al.), relaxation and reaction kinetics of polymers (Glockle and Nonnenmacher), so-called ultraslow processes (Gorenflo and Rutman), relaxation in filled polymer networks (Metzler, Schick et al.), viscoelastic materials (Bagley and Torvik), although there are recent applications in splines and wavelets (Unser and Blu ; Forster, Blu et al.), control theory (Podlubny ; Xin and Fawang), and biology (El-Sayed, Rida et al.) (bacterial chemotaxis), pharmacokinetics (Dokoumetzidis and Macheras ; Popovic, Atanackovic et al. ; Verotta), and pharmacodynamics (Verotta). Surveys with collections of applications can also be found in Matignon and Montseny , Nonnenmacher and Metzler (Nonnenmacher and Metzler), and Podlubny (Podlubny). A brief history of the development of fractional calculus can be found in Miller and Ross (Miller and Ross).