Existence of Time-Scale Class of Three Dimensional Fractional Differential Equations

The holomorphic results for fractional differential operator formals have been established. The analytic continuation of these outcomes has been studied for the fractional differential formal where U is the open unit disk. The benefit of such a problem is that a generalization of two significant problems: the Cauchy problem and the diffusion problem. Moreover, the analytic solution is given inside the open unit disk, this leads to discuss the solution geometrically. The upper bound of outcomes is determined by suggesting a majorant analytic function in U (for two functions characterized by a power series, a majorant is the summation of a power series with positive coefficients which are not less than the absolute values of the conforming coefficients of the assumed series). This technique is very useful in approximation theory.


Introduction
Time scales calculus [1] cards us to teaching the dynamic equations, which contains both differences and differential equations, both of which are substantial in understanding applications. The dynamical behavior of different classes of fractional operating formals on time scales is presently experiencing active studies.
Several authors considered the existence and uniqueness solutions for problems involving classical fractional derivative (see [2]- [10]). Holomorphic solution for some complex fractional classes is given in [5]- [7]. In this work, we use a majorant technique of analytic functions to prove the convergent of outcomes. We generalize some properties by applying the concept of classic fractional derivative formal operator.
Our construction is furnished by the Riemann-Liouville fractional operators. Definition 1. The Riemann-Liouville fractional integral formal of the function φ of arbitrary order α > 0 is given by Definition 2. The Riemann-Liouville fractional differential formal of the function φ of arbitrary order

Definition 3. [8]
The majorant formula is given by : Define the family of majorant functions: for each k ∈ N, we set (1.1) Clearly that for every k ∈ N, ν ≥ 1, the functional Ξ (k) ν converges for all values |z| < 1. Further, this functional has some significant majorant correlations as follows: The following inequalities hold.
In the same manner of Proposition 1, we have the following result: for any 0 < r < εr 0 .

Fractional operator formal
Assume that H(℘, z, υ, v, w), ℘ ∈ J = [a, T ] is a holomorphic function in a proximity of the four dim.
point (a, b, c, d, e) ∈ J ×C 4 , and suppose that ψ(z) is a holomorphic function in a proximity of z = b achieving Consider the initial value problem and diffusion problem of fractional order when Theorem 3. Consider the initial value problem (2.1), then it has a unique holomorphic outcome (υ(℘, z)), in a proximity of (a, b) ∈ J × C.
Proof. Move the point (a, b) into the origin (0, 0) and change the variable as follows: where ϕ(℘, z) is the new variable, then we get Here, the functional Θ(℘, z, ϕ, ∂ϕ ∂z , ∂ 2 ϕ ∂z 2 ) is holomorphic in a proximity of the origin in I × C 4 , ℘ ∈ I = [0, 1]. Therefore, it is sufficient to consider (2.2). Let the above equation has a unique outcome: We show that ϕ(℘, z) converges.
Assume that Θ is bounded by M in this domain. Since Θ is holomorphic, then it has the following construction: In virtue of the Cauchy's inequality and the certainty that the coefficient a p,q,s (z) is holomorphic in a proximity of {z ∈ C; |z| ≤ r 0 }, implies that a p,q,s,l (z) M τ p ρ q+s+l In this case, the problem turns to evaluate a function ϑ(℘, z) satisfying the majorant inequalities then the function ϑ(℘, z) majorizes the formal solution ϕ(℘, z). Assume 0 < r < r 0 and define Operating by the fractional differential formal with respect to ℘ we get Then by Proposition 1 (vi) we get where C ν := (2ν) 12 . For a constant K 0 > 0 again in virtue of Proposition 1 (ii) and (iii) we find p,q,s,l M ρ q+s+l Comparing (2.7) and (2.8) with the inequality then we obtain the majorant inequalities in (2.4) are achieved.

Continuation outcomes
Suppose that Ω is a proximate of the origin (0, 0) and H(℘, z, υ, v, w), ℘ ∈ I, is a holomorphic function in Ω × C υ × C v × C w . Consider the following equation: Then we have Define the following two sets: Clearly, H is linear if and only if S = ∅; and it is nonlinear otherwise.
Next, we show that υ(℘, z) = O(℘ κ ) is analytically continued up to a proximate of the origin. Now we suggest the following initial value problem in We aim to show that the formal χ(℘, z) converges in a domain including the origin. This leads to υ(℘, z) can be continued analytically by χ(℘, z) up to some neighborhood of the origin.
Therefore, the function W (℘, z) achieves the majorant conclusion is one majorant function for the formal solution χ(℘, z). In the similar manner of the proof of Theorem 3 by choosing suitable values for ρ > 0, c > 0, and letting ε = cτ 2 , we have achieves the majorant formal given in (3.11). Thus, W (℘, z) is holomorphic in a domain involving (0,0); consequently must be true for χ(℘, z).