A solution to the more than 300-years old problem of geometric and physical interpretation of fractional integration and differentiation (i.e., integration and differentiation of an arbitrary real order) is suggested for the Riemann-Liouville fractional integration and differentiation, the Caputo fractional differentiation, the Riesz potential, and the Feller potential. It is also generalized f...

In this paper, we discuss existence and uniqueness of solutions to nonlinear fractional order ordinary differential equations with boundary conditions in an ordered Banach space. We use the Caputo fractional differential operator and the nonlinearity depends on the fractional derivative of an unknown function. Schauder’s fixed point Theorem is the main tool used here to establish the existence ...

1. Mohney BG, Huffaker RK. Common forms of childhood exotropia. Ophthalmology 2003;110:20936. 2. Duane TD, Schatz NJ, Caputo AR. Pseudo-Duane's retraction syndrome. Trans Am Ophthalmol Soc 1976;74:122-32. 3. Gittinger JW Jr, Hughes JP, Suran EL. Medial orbital wall blow-out fracture producing an acquired retraction syndrome. J Clin Neuroophthalmol 1986;6:153-6. 4. Gao Z, Guo H, Chen W. Initial ...

In the paper, we extend the differential transform method to solve nonlinear fractional partial differential equations. The timeand space-fractional Burgers equations with initial conditions are chosen to illustrate our method. As a result, we successfully obtain some available approximate solutions of them. The results reveal that the proposed method is very effective and simple for obtaining ...

Jean-Guy Caputo,1,2 Yuri Gaididei,3 Volodymyr P. Kravchuk,4 Franz G. Mertens,5 and Denis D. Sheka4,* 1Laboratoire de Mathématiques, INSA de Rouen, Boîte Postale 8, 76131 Mont-Saint-Aignan Cedex, France 2Laboratoire de Physique Theorique et Modelisation, Université de Cergy-Pontoise and CNRS, Cergy-Pontoise 95011, France 3Institute for Theoretical Physics, 03143 Kiev, Ukraine 4National Taras She...

In this paper, we study periodic boundary value problems for a class of linear fractional evolution equations involving the Caputo fractional derivative. Utilizing compactness of the constructed evolution operators and Fredholm alternative theorem, some interesting alternative results for the mild solutions are presented. Periodic motion controllers that are robust to parameter drift are also d...

*Correspondence: [email protected] 1Institute of Mathematics and Computer Science, Jan Długosz University, Czȩstochowa, 42-200, Poland Full list of author information is available at the end of the article Abstract The one-dimensional time-fractional advection-diffusion equation with the Caputo time derivative is considered in a line segment. The fundamental solution to the Dirichlet pro...

We prove a necessary optimality condition for isoperimetric problems under nabla-differentiable curves. As a consequence, the recent results of [M.R. Caputo, A unified view of ostensibly disparate isoperimetric variational problems, Appl. Math. Lett. (2008), doi:10.1016/j.aml.2008.04.004], that put together seemingly dissimilar optimal control problems in economics and physics, are extended to ...

Based on the theory of fractional calculus, the contraction mapping principle, Krasnoselskii fixed point theorem and the inequality technique, a class of Caputo fractional-order BAM neural networks with delays in the leakage terms is investigated in this paper. Some new sufficient conditions are established to guarantee the existence and uniqueness of the nontrivial solution. Moreover, uniform ...

We develop and extend earlier results related to mathematical modelling of the liver formation zone by the adoption of noninteger order derivative. The hidden uncertainties in the model are captured and controlled thanks to the Caputo derivative. The stationary states are investigated and the time-dependent solution is approximated using two recent iteration methods. In particular, we discuss t...