*Correspondence: [email protected] School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, PR China Abstract In this paper, we study boundary-value problems for the following nonlinear fractional differential equations involving the Caputo fractional derivative: D0+x(t) = f (t, x(t), Dβ0+x(t)), t ∈ [0, 1], x(0) + x′(0) = y(x), ∫ 1 0 x(t)dt =m, x′′(0) = x′′′(0) = · · · = x(n–...

We establish four necessary and sufficient conditions for the existence of the Averch−Johnson effect in a generalized version of their famous model of the rate−of−return regulated firm. The four necessary and sufficient conditions are then compared to the two stronger sufficient conditions for the Averch−Johnson effect found in the literature. Our analysis also permits us to put to rest a somew...

Discrete maps with long-term memory are obtained from nonlinear differential equations with Riemann–Liouville and Caputo fractional derivatives. These maps are generalizations of the well-known universal map. The memory means that their present state is determined by all past states with special forms of weights. To obtain discrete maps from fractional differential equations, we use the equival...

A universal approach by Laplace transform to the variational iteration method for fractional derivatives with the nonsingular kernel is presented; in particular, the Caputo-Fabrizio fractional derivative and the Atangana-Baleanu fractional derivative with the non-singular kernel is considered. The analysis elaborated for both non-singular kernel derivatives is shown the necessity of considering...

In this paper, we apply the concept of Caputo’s H-differentiability, constructed based on the generalized Hukuhara difference, to solve the fuzzy fractional differential equation (FFDE) with uncertainty. This is in contrast to conventional solutions that either require a quantity of fractional derivatives of unknown solution at the initial point (Riemann–Liouville) or a solution with increasing...

Numerical evaluations of Caputo fractional derivatives for scattered noisy data is an important problem in scientific research and practical applications. Fractional derivatives have been applied recently to the numerical solution of problems in fluid and continuum mechanics. The Caputo fractional derivative of order α is given as follows f (t) = 1 Γ(1− α) ∫ t 0 f (s) (t− s)α ds, 0 < α < 1 The ...

In this work, we consider boundary value problems involving Caputo and Riemann-Liouville fractional derivatives of order α ∈ (1, 2) on the unit interval (0, 1). These fractional derivatives lead to non-symmetric boundary value problems, which are investigated from a variational point of view. The variational problem for the Riemann-Liouville case is coercive on the space H α/2 0 (0, 1) but the ...

From the available literature, the allometric scaling laws generally exist in biology, ecology, etc. These scaling laws obey power law distributions. A possibly better approach to characterize the power law is to utilize fractional derivatives. In this paper, we establish a fractional differential equation model for this allometry by using the Caputo fractional derivatives.

In this article, the analytical method, Homotopy perturbation method (HPM) has been successfully implemented for solving nonlinear logistic model of fractional order. The fractional derivatives are described in the Caputo sense. Using initial value, the explicit solutions of population size for different particular cases have been derived. Numerical results show that the method is extremely eff...

*Correspondence: [email protected] Department of Mathematics, Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, Białystok, 15-351, Poland Abstract In this paper viability results for nonlinear fractional differential equations with the Caputo derivative are proved. We give a necessary condition for fractional viability of a locally closed set with respect to a nonli...