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We make use of the theory of strongly continuous solution operators for fractional models together with the subordination principle for fractional evolution equations (Bazhlekova (2000) and Prüss (1993)) to analyze and show existence results for a fractional fragmentation model with growth characterized by its growth rate

Despite its three centuries of age, fractional calculus remains lightly unpopular amongst science and engineering community. However, there is a growing interest in extending analysis involving normal calculus with integer orders to noninteger orders (real or complex order) [

Classical models of clusters’ fission with normal derivative

As said above,

Most of works on fragmentation with normal derivative

These reasons are the sources of the increasing volition to try new approaches and extend classical models to models with fractional derivative and investigate them with various and different techniques in order to establish broader outlooks on the real phenomena they describe. For instance, we have differential equations with fractional orders increasingly used to model many problems in applied sciences including engineering, applied mathematics, physics, biology, chemistry, economic, and other domains of applications. In the process, differential equations with fractional derivative have become a useful tool for describing nonlinear phenomena of science and engineering models. In the same way, the process of fragmentation of clusters occurs in many branches of natural sciences ranging from physics to chemistry, engineering, biology, ecology, and numerous domains of applied sciences, such as the depolymerization, the rock fractures, and of break of droplets.

With this in mind, we obtain the following fractional model of fragmentation process with growth:

Recall that fission models of type (

As said earlier, in the model with growth, one of the major problems in the analysis occurs when

To describe the dynamics here, we use the cluster (or group) density function

Clusters grow as a result of divisions of phytoplankton cells. The growth rate is denoted by

During a small time interval

The boundary condition considered here is the McKendrick-von Foerster renewal condition:

The analysis is performed in the space

A vital role in the analysis of the model is played by the integrability of

We are interested in investigating the fractional differential model given by

Since

It is obvious to see that

To proceed we need the following definition.

Consider an operator

It is well known [

Note that if we have a contraction solution operator, we can use Definition

An operator

This theorem is a particular version of [

In our analysis we will need some interesting properties of the Mittag-Leffler relaxation function

Now, we consider the full operator in (

Consider

We exploit the Marchaud type representation (

There is an extension of the operator

We make use of the subordination principle developed in [

Under the McKendrick-von Foerster renewal condition (

The author declares that there is no conflict of interests regarding the publication of this paper.