The Dodson mass diffusion equation with exponentially diffusivity is analyzed through approximate integral solutions. Integral-balance solutions were developed to integer-order versions as well as to formally fractionalized models. The formal fractionalization considers replacement of the time derivative with a fractional version with either singular (Riemann-Liouville or Caputo) or non-singular fading memory. The solutions developed allow seeing a new side of the Dodson equation and to separate the formal fractional model with Caputo-Fabrizio time derivative with an integral-balance allowing relating the fractional order to the physical relaxation time as adequate to the phenomena behind.

Based upon the Mittag-Leffler function, new derivatives with fractional order were constructed. With the same line of idea, improper derivatives based on the Weyl approach are constructed in this work. To further model some complex physical problems that cannot be modeled with existing derivatives with fractional order, we propose, a new derivative based on the more generalized Mittag-Leffler function known as Prabhakar function. Some new results are presented together with some applications.

The present paper is devoted to the study of the fractional calculus operators to obtain a number of key results for the generalized multiindex Bessel function involving Saigo hypergeometric fractional integral and differential operators in terms of generalized Wright function. Various particular cases and consequences of our main fractional-calculus results as classical Riemann-Liouville and Erde´lyi-Kober fractional integral and differential formulas are deduced.

In this study, a uniqueness theorem is given for Sturm-Liouville problem with special singular potential. We prove that singular potential function can be uniquely determined by the spectral set \( \left\{ \lambda _{n}\left( q_{0},h_{m}\right) \right\} _{m=1}^{+\infty }.\)

In this paper, we propose the random \(\mathcal{Z}\)-contraction, prove a stochastic fixed point theorem for this contraction, and show that a solution for a non-linear stochastic integral equations exists in Banach spaces.