The aim of this tutorial survey is to revisit the basic theory of relaxation processes governed by linear differential equations of fractional order. The fractional derivatives are intended both in the Rieamann-Liouville sense and in the Caputo sense. After giving a necessary outline of the classical theory of linear viscoelasticity, we contrast these two types of fractional derivatives in thei...

This survey paper is concerned with some of the most recent results on Lyapunov-type inequalities for fractional boundary value problems involving a variety derivative operators and conditions. Our work deals Caputo, Riemann-Liouville, ?-Caputo, ?-Hilfer, hybrid, Caputo-Fabrizio, Hadamard, Katugampola, Hilfer-Katugampola, p-Laplacian, proportional operators.

This study introduces some remarks on generalized fractional integral and differential operators, which generalize familiar derivative with respect to a given function. We briefly explain how formulate representations of operators. Then, mainly, we propose predictor–corrector algorithm for the numerical simulation initial value problems involving Caputo-type derivatives another Numerical soluti...

In this paper, we introduce and study fractional versions of the Bell–Touchard process, Poisson-logarithmic process generalized Pólya–Aeppli process. The state probabilities these compound Poisson processes solve a system differential equations that involves Caputo derivative order 0<β<1. It is shown are limiting cases recently introduced namely, counting We obtain mean, variance, covaria...

In this paper, a hybrid variable-order mathematical model for multi-vaccination COVID-19 is analyzed. The derivative defined as linear combination of the integral Riemann–Liouville and Caputo derivative. A symmetry parameter ? presented in order to be consistent with physical problem. existence, uniqueness, boundedness positivity proposed are given. Moreover, stability discussed. theta finite d...

Many dynamic systems can be modeled by fractional differential equations in which some external parameters occur under uncertainty. Although these increase the complexity, they present more acceptable solutions. With aid of Atangana-Baleanu-Caputo (ABC) operator, an advanced numerical-analysis approach is considered and applied this work to deal with different classes fuzzy integrodifferential ...

This paper introduces techniques for the estimation of solutions to fractional order differential equations (FODEs) and the approximation of a function’s Caputo fractional derivative. These techniques are based on scattered data interpolation via reproducing kernel Hilbert spaces (RKHSs). Specifically, an RKHS is generated for the purpose of estimating fractional derivatives from the Mittag-Lef...

In this study, fractional differential transform method (FDTM), which is a semi analytical-numerical technique, is used for computing the eigenelements of the Sturm-Liouville problems of fractional order. The fractional derivatives are described in the Caputo sense. Three problems are solved by the present method. The calculated results are compared closely with the results obtained by some exi...

The present paper is devoted to constructing L2 type difference analog of the Caputo fractional derivative. fundamental features this operator are studied and it used construct schemes generating approximations second fourth order in space (3−α)th-order time for diffusion equation with variable coefficients. Difference were also constructed variable-order generalized fractional-order Sobolev ty...

The first-order differential equation of exponential relaxation can be generalized by using either the fractional derivative in the Riemann–Liouville (R-L) sense and in the Caputo (C) sense, both of a single order less than 1. The two forms turn out to be equivalent. When, however, we use fractional derivatives of distributed order (between zero and 1), the equivalence is lost, in particular on...