Direct and inverse source problems of a fractional diffusion equation with regularized Caputo-like counterpart hyper-Bessel operator are considered. Solutions to these problems are constructed based on appropriate eigenfunction expansion and results on existence and uniqueness are established. To solve the resultant equations, a solution to a non-homogeneous fractional differential equation with regularized Caputo-like counterpart hyper-Bessel operator is also presented.

Inverse initial and inverse source problems of a time-fractional differential equation with Bessel operator are considered. Results on existence and uniqueness of solutions to these problems are presented. The solution method is based on series expansions using a set of Bessel functions of order zero. Convergence of the obtained series solutions is also discussed.

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The contact rate is defined as the average number of contacts adequate for disease transmission by an individual per unit time and it is usually assumed to be constant in time. However, in reality, the contact rate is not always constant throughout the year due to different factors such as population behavior, environmental factors and many others. In the case of serious diseases with a high level of infection, the population tends to reduce their contacts in the hope of reducing the risk of infection. Therefore, it is more realistic to consider it to be a function of time. In particular, the study of models with contact rates decreasing in time is well worth exploring. In this paper, an SIR model with a time-varying contact rate is considered. A new form of a contact rate that decreases in time from its initial value till it reaches a certain level and then remains constant is proposed. The proposed form includes two important parameters, which represent how far and how fast the contact rate is reduced. These two parameters are found to play important roles in disease dynamics. The existence and local stability of the equilibria of the model are analyzed. Results on the global stability of disease-free equilibrium and transcritical bifurcation are proved. Numerical simulations are presented to illustrate the theoretical results and to demonstrate the effect of the model parameters related to the behavior of the contact rate on the model dynamics. Finally, comparisons between the constant, variable contact rate and variable contact rate with delay in response cases are presented.

In this paper, we have considered two different sub-diffusion equations involving Hilfer, hyper-Bessel and Erdelyi-Kober fractional derivatives. Using a special transformation, we equivalently reduce the considered boundary value problems for fractional partial differential equation to the corresponding problem for ordinary differential equation. An essential role is played by certain properties of Erd\'elyi-Kober integral and differential operators. We have applied also successive iteration method to obtain self-similar solutions in an explicit form. The obtained self-similar solutions are represented by generalized Wright type function. We have to note that the usage of imposed conditions is important to present self-similar solutions via given data.