Functions represented into fractional Taylor series

Abstract. Fractional Taylor series are studied. Then solutions of fractional linear ordinary differential equations (FODE), with respect to Caputo derivative, are approximated by fractional Taylor series. The Cauchy-Kowalevski theorem is proved to show the existence and uniqueness of local solutions for FODE with Cauchy initial data. Sufficient conditions for the global existence of the solution and the estimate of error are given for the method using fractional Taylor series. Two illustrative numerical examples are given to demonstrate the validity and applicability of this method.


Introduction
Taylor series method is a useful tool to approximate solutions of the ordinary differential equations (ODE) (see, for example, [1], [9], [12] and references therein) or solutions of the partial differential equations (PDE) (see, for example, [2], [4]). One advantage of the analytic methods is that the accuracy of solution can be evaluated directly. Thus the approximate solution can be replaced into the equation and the initial or boundary conditions. Fractional differential equations are useful tools for modeling many phenomena in fields of science and engineering (see [3], [5], [8], [11]). Several methods for approximation of solutions of ordinary differential equations were extended to fractional ordinary differential equations (FODE) (see, for example, [7]).
Fractional Taylor series method to approximate solutions for FODE, based on the corresponding Taylor's formula (see [10], [13]), can be found in [6], and references therein.
In this paper some properties of fractional Taylor series are given (see Theorems 1 and 2). We prove Cauchy-Kowalevski theorem in the case of linear FODE guaranteeing existence and uniqueness of local solutions with Cauchy initial data. Moreover, sufficient conditions for the existence of the global solution are given (see Theorem 3). By using the proof of the theorem we estimate the error in the method based on fractional Taylor series (see Remark 1). Finally, two numerical examples are presented to illustrate the results obtained.

Fractional Taylor series
Definition 1 A function f : (0, ∞) → R is said to be of class C µ (µ ∈ R) if there exists p > µ such that f (t) = t p g(t), ∀t > 0, where g : [0, ∞) → R is a continuous function. The function f is said to be of class C (n) µ (n ∈ N) if f (n) ∈ C µ . https://doi.org/10.1051/itmconf/20192901017 ICCMAE 2018 Definition 2 Let n = ⌈α⌉ (where ⌈x⌉ = min {z ∈ Z : z ≥ x} denotes the ceiling function). The Caputo fractional derivative of order α ≥ 0 of a function y ∈ C (n) −1 is defined as The Caputo fractional derivative has the following properties: A series of functions of the form � n≥0 a n t nα with α > 0 is said to be a fractional power series. Notice that it is sufficient to study the fractional power series with α ∈ (0, 1] because any fractional power series with α > 1 can be considered as a fractional power series of the form � n≥0 a ′ n t nα ′ with α ′ = α ⌈α⌉ ∈ (0, 1].
Because R = 1 lim sup n→∞ |a n | 1/n , we get (3). ii) Let g : [0, R) → R be the sum of the power series: g(x) = � n≥0 a n x n , so g(x) is continuous. Since f (t) = g(t α ) and the functions g(x) and t α are continuous, it follows that f (t) is continuous.
Hence we may choose a constant ∆ ≥ 1 such that, for every s, and S 1 (s) := Hence, g(s) is a decreasing function for s ≥ 2n − 1 and it follows that, for any s ≥ 2n − 1, {g(s − r)} r=0,n 1 is an increasing sequence. Since is a decreasing sequence, by Chebyshev's sum inequality we have and it follows that We can choose a positive integer s 2 = s 2 (δ) ≥ s 1 , s 2 ≥ 2n − 1 such that, for all s ≥ s 2 , We denote by S 2 (s) := and it follows that lim s→∞ S 2 (s) = 0. Hence, there exists a positive integer s 3 = s 3 (δ) ≥ s 2 such that, for all s ≥ s 3 , We consider C 1 ≥ 1 ε 3 such that, for all s ≤ s 3 , This begins an inductive proof. For s ≥ s 3 , assume (17) true for all s ′ ≤ s and we'll prove that By (9), (11), (12) and (17) we get Thus, by (14), (15) and (16), we get (18), and (17) holds for every s. Now, by (17), it follows that lim sup s→∞ |c s | 1/s ≤ 1 δ α . Since this is true for all δ ∈ (b, b ′ ) it follows that (10) holds. Finally, by Theorem 1 and (9), we get the lemma.
The following result is an extension of the Cauchy-Kowalevski theorem in the case of linear fractional ordinary differential equation.  (9), we can find c s , for all s ≤ m. As in the proof of Lemma 1, we get s i , i = 1, ..., 3 and a constant C 1 such that (17) holds. Then for any m ≥ s 3 , we approximate the solution by a fractional polynomial Thus y(t) = P m (t) + R m (t) and E m = sup |R m (t)| is the error of the approximation. By (17), Hence we get

Conclusion
In this paper an extension of the Cauchy-Kowalevski theorem for linear FODE and sufficient conditions for the global existence of the solution are presented. As a result, the solution of such an equation can be approximated by a fractional Taylor polynomial and an estimation of the error is given. The validity and applicability of the method is demonstrated by illustrative examples.