The time-fractional kinetic equation for the non-equilibrium processes

In this study, we consider the non-Markovian dynamics of the generic non-equilibrium kinetic process. We summarize the generalized master equation, the continuous and discrete forms of the time-fractional diffusion equation. Using path integral formulation, we generalized the solutions of the Markovian system to the non-Markovian for the non-equilibrium kinetic processes. Then, we obtain the time-fractional kinetic equation for the non-equilibrium systems in terms of free energy. Finally, we introduce a time-fractional equation to analyse time evolution of the open probability for the deformed voltage-gated ion-channel system as an example.

On Weighted (k, s)-Riemann-Liouville Fractional Operators and Solution of Fractional Kinetic Equation

In this article, we establish the weighted (k,s)-Riemann-Liouville fractional integral and differential operators. Some certain properties of the operators and the weighted generalized Laplace transform of the new operators are part of the paper. The article consists of Chebyshev-type inequalities involving a weighted fractional integral. We propose an integro-differential kinetic equation using the novel fractional operators and find its solution by applying weighted generalized Laplace transforms.

FRACTIONAL KINETIC EQUATIONS INVOLVING GENERALIZED V-FUNCTION VIA LAPLACE TRANSFORM

In this study, a new and further generalized form of the fractional kinetic equation involving the generalized V$-$function has been developed. We have discussed the manifold generality of the generalized V$-$function in terms of the solution of the fractional kinetic equation. Also, the graphical interpretation of the solutions by employing MATLAB is given. The results are very general in nature, and they can be used to generate a large number of known and novel results.

On a class of stochastic fractional kinetic equation with fractional noise

In this article we study a class of stochastic fractional kinetic equations with fractional noise which are spatially homogeneous and are fractional in time with $H>1/2$ H > 1 / 2 . The diffusion operator involved in the equation is the composition of the Bessel and Riesz potentials with any fractional parameters. We prove the existence of the solution under some mild conditions which generalized some results obtained by Dalang (Electron. J. Probab. 4(6):1–29, 1999) and Balan and Tudor (Stoch. Process. Appl. 120:2468–2494 , 2010). We study also its Hölder continuity with respect to space and time variables with $b=0$ b = 0 . Moreover, we prove the existence for the density of the solution and establish the Gaussian-type lower and upper bounds for the density by the techniques of Malliavin calculus.

Solution of Fractional Kinetic Equations Associated with the p,q-Mathieu-Type Series

In this paper, our aim is to finding the solutions of the fractional kinetic equation related with the p,q-Mathieu-type series through the procedure of Sumudu and Laplace transforms. The outcomes of fractional kinetic equations in terms of the Mittag-Leffler function are presented.

Fractional Calculus involving (p, q)-Mathieu Type Series

Aim of the present paper is to establish fractional integral formulas by using fractional calculus operators involving the generalized (p, q)-Mathieu type series. Then, their composition formulas by using the integral transforms are introduced. Further, a new generalized form of the fractional kinetic equation involving the series is also developed. The solutions of fractional kinetic equations are presented in terms of the Mittag-Leffler function. The results established here are quite general in nature and capable of yielding both known and new results.

Fractional Kinetic Equations Associated with Incomplete I-Functions

In this paper, we investigate the solution of fractional kinetic equation (FKE) associated with the incomplete I-function (IIF) by using the well-known integral transform (Laplace transform). The FKE plays a great role in solving astrophysical problems. The solutions are represented in terms of IIF. Next, we present some interesting corollaries by specializing the parameters of IIF in the form of simpler special functions and also mention a few known results, which are very useful in solving physical or real-life problems. Finally, some graphical results are presented to demonstrate the influence of the order of the fractional integral operator on the reaction rate.