From fractional order equations to integer order equations

Abstract

From fractional order equations to integer order equationsThis document is a preprint of a currently submitted article. There might be significant differences between this manuscript and the future final version (adding, deleting or improving some contents). The authors do strongly recommend to the reader to take this document as a sketch (with proofs) of the most important results of a future final version. In this sense, the authors request the reader to consult the final document, when published in a scientific journal, for a more detailed and corrected version. AbstractThe main goal of this article is to show a new method to solve some Fractional Order Integral Equations (FOIE), more precisely the ones which are linear, have constant coefficients and all the integration orders involved are rational. The method essentially turns a FOIE into an Ordinary Integral Equation (OIE) by applying a suitable fractional integral operator.After discussing the state of the art, we present the idea of our construction in a particular case (Abel integral equation). After that, we propose our method in a general case, showing that it does work when dealing with a family of "additive" operators over a vector space. Later, we show that our construction is always possible when dealing with any FOIE under the above-mentioned hypotheses. Furthermore, it is shown that our construction is "optimal" in the sense that the OIE that we obtain has the least possible order.

Constant Sign Solutions to Linear Fractional Integral Problems and Their Applications to the Monotone Method

“…We also summarize the results in [3,4] about the existence and uniqueness of solutions to linear fractional integral problems, in the following theorem.…”

“…Finally, (IV) holds trivially, in virtue of Remark 6: The integral orders associated to the fractional operator are greater or equal to one, the upper and lower solutions are bounded, and the function f (t, a 1 , a 2 ) = 1 − Γ 5 2 a 1 • 1 − Γ 9 4 a 2 is continuously differentiable. Thus, the previous problem is under the hypotheses of Theorem 7 when b = 3 5 . In particular, we know that the problem has at least one solution defined in [0, 3 5 ], whose image lies in the interval [0, 1].…”

“…The study of fractional integral equations is relevant by itself and also for the study of the properties of the solutions to fractional differential equations. Linear problems for fractional equations can be addressed by passing to integer order equations [3,4]. On the other hand, for nonlinear problems, one interesting approach is the development of iterative techniques based on the use of upper and lower solutions [5].…”

Backpropagation algorithm with fractional derivatives

“…In the presented model (differently than in the classic approach), the transition function of a single neuron is taken as a Grünwald-Letnikov fractional derivative (GL) [16][17][18][19][20] of log base function. Based on the definition of the integer derivative and the fractional derivative, the GL derivative is described by the formula:…”

The Solutions of Some Riemann–Liouville Fractional Integral Equations

“…Riemann developed an approach to noninteger order derivatives in terms of convergent series, conversely to the Riemann-Liouville approach, that was given as an integral. Many researchers focused on developing the theoretical aspects, methods of solution, and applications of fractional integral equations, see [37][38][39][40][41][42][43][44][45].…”