# Fractional Derivatives, Fractional Integrals, And Fractional ....56

We generalize the fractional Caputo derivative to the fractional derivative C D γα,β, which is a convex combination of the left Caputo fractional derivative of order α and the right Caputo fractional derivative of order β. The fractional variational problems under our consideration are formulated in terms of C D γα,β. The Euler-Lagrange equations for the basic and isoperimetric problems, as well as transversality conditions, are proved.

## Fractional Derivatives, Fractional Integrals, And Fractional ....56

We study a coupled system of multi-term Hilfer fractional differential equations of different orders involving non-integral and autonomous type Riemann-Liouville mixed integral nonlinearities supplemented with nonlocal coupled multi-point and Riemann-Liouville integral boundary conditions. The uniqueness result for the given problem is based on the contraction mapping principle, while the existence results are derived with the aid of Krasnosel'ski$\rm\mathord\buildrel\lower3pt\hbox$\scriptscriptstyle\smile$ \over i $'s fixed point theorem and Leray-Schauder nonlinear alternative. Examples illustrating the main results are presented.

In the human retina, rod and cone cells detect incoming light with a molecule called rhodopsin. After rhodopsin molecules are activated (by photon impact), these molecules activate the rest of the signalling process for a brief period of time until they are deactivated by a multistage process. First, active rhodopsin is phosphorylated multiple times. Following this, they are further inhibited by the binding of molecules called arrestins. Finally, they decay into opsins. The time required for each of these stages becomes progressively longer, and each stage further reduces the activity of rhodopsin. However, while this deactivation process itself is well researched, the roles of the above stages in signal (and image) processing are poorly understood. In this paper, we will show that the activity of rhodopsin molecules during the deactivation process can be described as the fractional integration of an incoming signal. Furthermore, we show how this affects an image; specifically, the effect of fractional integration in video and signal processing and how it reduces noise and the improves adaptability under different lighting conditions. Our experimental results provide a better understanding of vertebrate and human vision, and why the rods and cones of the retina differ from the light detectors in cameras.

In this paper, we investigate whether the multi-stage deactivation process of rhodopsin and related residual activities offer any signal processing benefits, and how it affects signals in general. In our previous work, we used the model presented in [15] to show that this process has the potential to approximate fractional integral-like behaviour. To investigate the effect of the deactivation process, we have expanded this model with the activity of arrestin bound rhodopsin, as it was not previously included.

In addition, we show that the activity of rhodopsin still approximates fractional integration after the addition of the arrestin binding process to the cone model. Furthermore, the addition of the arrestin binding process model expands the frequency range of the approximation. Our main purpose for including these results is to demonstrate that residual activities can accumulate in signalling processes; therefore, they should not be neglected. Finally, as the activity itself can be described as fractional integration, its effects can be predicted without explicitly modelling the process.

In our case, fractional integrals add power law dynamics to the model, as their impulse response follows power law dynamics. Thus, the response to sustained inputs can reach higher levels than would be possible with only an exponential decay (see Fig 5). In addition, as pupils contract in response to light, the possible magnitudes are restricted (under normal lighting conditions). Therefore, power law dynamics provided by fractional integrals can allow the process to reach higher levels of overall activity in response to sustained inputs that would otherwise be impossible to achieve. This would allow the processes inside and outside cones to adapt to (and differentiate between) lighting conditions and temporarily high input levels.

We consider an inverse problem for a time fractional advection-dispersion equation in a 1-D semi-infinite setting. The fractional derivative is interpreted in the sense of Caputo and advection and dispersion coefficients are constant. The inverse problem consists on the recovery of the boundary distribution of solute concentration and dispersion flux from measured (noisy) data known at an interior location. This inverse problem is ill-posed and thus the numerical solution must include some regularization technique. Our approach is a finite difference space marching scheme enhanced by adaptive discrete mollification. Error estimates and illustrative numerical examples are provided.

From a standard advection-dispersion equation, new problems arise by considering fractional time and/or space derivatives. Furthermore, for the new problems it makes sense to consider inverse problems of different types. These mathematical problems have applications in physics, chemistry, biology and several branches of engineering, among others (see [4,14,15,16,17]). Fractional advection-dispersion equations and the inverse problems based on them are specially valuable in groundwater hydrology and other instances of transport in porous media [3,5].

Several inverse problems based on time fractional differential equations have been successfully solved by discrete mollification [12,13] but to the best of our knowledge, this is the first time that this regularization method is implemented for the stable solution of a time fractional inverse advection-dispersion equation (TFIADP).

where u is the solute concentration, u x is the dispersion flux, the constants a > 0, d > 0 represent the average fluid velocity and the dispersion coefficient, respectively, and denotes the Caputo fractional derivative of order α,

The presence of a derivative inside the integral in (2) is an indicator of the ill-conditioning of Caputo fractional derivatives when applied to noisy data. More precisely, if D (α) g is sought but we only have an approximation of g(t) denoted by and satisfying instead of recovering D (α) g we look for a mollified approximation . This approximation satisfies the following error estimate [13]:

Let be a positive real number and let Δx = h and Δt = k be the parameters of the finite difference discretization. For each with and each with we denote by and the computed approximations of the mollified solute concentration v(mh, nk), mollified flux of solute v x (mh, nk), and time fractional partial derivative of time partial derivative of mollified solute concentration respectively. Following Murio [12], the space marching finite difference scheme proposed is given by the system

The next algorithm uses the finite differences scheme (19) to approximate the solution of the mollified problem (18). Notice that the evaluation of the time fractional derivative has to be performed in the indicated increasing order of values of time at each space grid position.

w 1(t) = 0, and time fractional orders α = 0.10,0.50,0.90, grid sizes h = 0.01, k = 1/128 and noise level ε = 0.05. The numerical results for mollified solute concentration v and dispersion flux v x , after applying the mollified space marching algorithm (1) are shown in Figures (1) and (2).

Direct problems (28) and (29) are solved using scheme (27) with parameters h = 1/100 and k = 1 /128, time fractional orders α = 0.10, 0. 50, 0. 90 and noise level ε = 0. 05. Numerical results are presented in Figures (3) and (4).

This paper is devoted to the study of Caputo modification of the Hadamard fractional derivatives. From here and after, by Caputo-Hadamard derivative, we refer to this modified fractional derivative (Jarad et al. in Adv. Differ. Equ. 2012:142, 2012, p.7). We present the generalization of the fundamental theorem of fractional calculus (FTFC) in the Caputo-Hadamard setting. Also, several new related results are presented.

The presence of the δ-differential operator (δ=x d d x ) in the definition of Hadamard fractional derivatives could make their study uninteresting and less applicable than Riemann-Liouville and Caputo fractional derivatives. More so, this operator appears outside the integral in the definition of the Hadamard derivatives just like the usual derivative D= d d x is located outside the integral in the case of Riemann-Liouville, which makes the fractional derivative of a constant of these two types not equal to zero in general. The authors in [11] studied and modified the Hadamard derivatives into a more useful type using Caputo definitions.