Solving a well-posed fractional initial value problem by a complex approach

Abstract

Nonlinear fractional differential equations have been intensely studied using fixed point theorems on various different function spaces. Here we combine fixed point theory with complex analysis, considering spaces of analytic functions and the behaviour of complex powers. It is necessary to study carefully the initial value properties of Riemann–Liouville fractional derivatives in order to set up an appropriate initial value problem, since some such problems considered in the literature are not well-posed due to their initial conditions. The problem that emerges turns out to be dimensionally consistent in an unexpected way, and therefore suitable for applications too.

Examining of a tumor system with Caputo derivative

“…Definition 2.1 [6] Let , ( 1) nn  −   is the order for the derivative and ( ) gt be a function, then the definition of Caputo derivative is defined as:…”

“…Fractional calculus is more important real-life problems [5][6][7][8][9][10][11][12][13][14]. Recently, work on this subject has increased about physic, engineering, disease etc.…”

“… (6). and get results are illustrated about helper, cytotoxic, myeloid (tumor) and dendritic cells, IL-2 in Fig.1-5 especially fractional order 0.7, 0.65, 0.6  =…”