Approximate Solutions for Fractional Differential Equation in the Unit Disk

In this paper we establish the existence solution approximately for differential equation of fractional order takes the form z α D[D α u(z)] + (b − z)u ′ (z) − au(z) = 0, a = 0, |b| < 1, 0 < α < 1, subject to the initial conditions u(0) = u 0 and u ′ (0) = u 1 , in the unit disk U := {z ∈ C : |z| < 1}. The uniqueness of this solution is discussed. The general analytic solutions are posed. Moreover, the Hyers-Ulam stability is studied. An example is illustrated. 1 Introduction. Recently, fractional differential equations and inclusions have been of great interest. It is caused both by the intensive development of the theory of fractional calculus [9] itself and by the applications of such constructions in various sciences and topics such Moreover, fractional differential equations in complex domain have been studied and established [5-7]. In this paper, we deal with the existence of Kummer differential equation of fractional order in the complex plane. The Kummer differential equation which is also called the confluent hypergeometric differential equation, appears frequently in practical problems and applications. These equations have proved useful in many branches of physics. They have been used in problems involving diffusion, for example, in isotope separation and protein molecular weight determinations in the ultracentrifuge.

equation for the velocity distribution of electrons in high frequency gas discharges may frequently be expressed in terms of these functions.The high frequency breakdown electric field may then be predicted theoretically for gases by the use of such solutions together with kinetic theory [3,8,17,19].Furthermore, an immensely useful class of special functions (namely, the generalized hypergeometric function) played a rather crucial role in the theory of analytic and univalent functions.These latter developments in an area other than the so-called traditional areas of applications of generalized hypergeometric functions have naturally provided a new impetus for the study of such an important class of special functions.Finally, hypergeometric function and its generalizations employed to define certain families of linear operators which reduced in terms of (for example) fractional derivatives and fractional integrals, Hadamard product or convolution [13,14].
In [15], Srivastava and Owa, gave definitions for fractional operators (derivative and integral) in the complex z-plane C as follows: Definition 1.1.The fractional derivative of order α is defined, for a function f (z) by where the function f (z) is analytic in simply-connected region of the complex z-plane C containing the origin and the multiplicity of (z −ζ) −α is removed by requiring log(z −ζ) to be real when(z − ζ) > 0.
Definition 1.2.The fractional integral of order α is defined, for a function f (z), by where the function f (z) is analytic in simply-connected region of the complex z-plane (C) containing the origin and the multiplicity of (z and In this paper we study the existence of holomorphic solution at the origin of the fractional pantograph equation in the unit disk takes the form EJQTDE, 2011 No. 64, p. 2 subject to the initial conditions in the unit disk U such that u : U → C.

Existence and Uniqueness
We have the following main results: Theorem 2.1.Consider the initial-value problem (1).If one of the following cases is hold: • a = −j and α → 0, • a = −j and α → 1, for some non negative integer m then (1) has no holomorphic solution U.
Proof.Let u be a solution to the initial-value problem (1) such that u is a holomorphic at z = 0 then u can be represented as a power series of the form Substituting this expression into the equation (1), we obtain For u 0 = 0 and u 1 = 0 we have b 0 = 0.If a = −j for non negative integer j, then it is clear that the solution is a polynomial which is converge in U. Now let a = −j then we have the ratio For a = −j and α → 0, we obtain that this poses that there is a converge solution for the problem (1) in the unit disk.Also for a = −j and α → 1, we have thus the problem (1) has a solution.Finally, if b satisfies the relation (2), this implies that b n is undefined and hence the problem (1) has no solution in the unit disk. 1) has a unique solution.
Proof.Let u be a solution to the initial-value problem (1) such that u is a holomorphic at z = 0.The differential equation ( 1) can be converted into the equation: where We establish the existence of a local solution by showing that the integral equation is a contraction mapping.Let H(U ) denote the Banach space of functions in equipped with the norm ., u = sup z∈U |u|, and let P : H(U ) → H(U ) be the operator defined by EJQTDE, 2011 No. 64, p. 4 Let Hence there is a constant 0 < ρ < 1 such that Therefore the initial-value problem (1) has a unique solution in H(U ).

General Solution
In this section, we will determine the general solution of the inhomogeneous fractional differential equation where neither a nor b is a non-positive integer and the coefficients a n of the power series are given such that the radius of convergence is σ > 0. A power series solution of is given by where (a) n is the Pochhammer symbol defined by Here we assume that b 0 = 1.For some values of a and b, the power series w(z) converges for all values z ∈ U.
Theorem 3.1.Let a, b be real constants such that a / ∈ Z − and ab γ 0 = 1.Assume that the radius of convergence of the power series ∞ n=0 a n z n , z ∈ U is 0 < σ < 1 and that there exists a positive real number µ satisfies 1+µ |1+b| ≤ 1 and EJQTDE, 2011 No. 64, p. 5 for all sufficiently large integer n.Then every solution u of the non-homogenous equation ( 5) can be represented by where u h is a solution for (6).
Proof.Let u p (z) = u(z) − u h (z).We show that u p (z) satisfies the equation ( 5).Since we obtain Next we prove that the power expression of u p (z) converges for all z ∈ U.By applying the ratio test to the power series expression of u p (z) and using the assumption (7) we have and for α → 0 we pose Therefore the power series of u p (z) converges for all z ∈ U and consequently those of u ′ p (z) and D α u p (z). Hence the power series expression for z α D[D α u(z)] + (b − z)u ′ (z) − au(z) has the same convergence region (the maximum open disk where the relevant power series converges) as that of u p (z).This completes the proof.EJQTDE, 2011 No. 64, p. 6

Hyers-Ulam Stability
Assume that X and Y are a topological vector space and a normed space, respectively, and that I is an open subset of X.If for any function f : I → Y satisfying the differential inequality for all x ∈ I and for some ε ≥ 0, there exists a solution f 0 : I → Y of the differential equation then we say that the above differential equation satisfies the Hyers-Ulam stability ( or the local Hyers-Ulam stability if the domain I is not the whole space X) (see [2]).In the following theorem, we shall prove a local Hyers-Ulam stability of the equation (1) under additional conditions.Theorem 4.1.Let a, b be real constants such that a / ∈ Z − .Suppose a function u : U → C is represented by the power series ∞ n=0 b n z n , z ∈ U. Assume that there exist nonnegative constants µ = 0 and ν satisfying the condition and there is a constant L ≥ 0 such that where If u satisfies the fractional differential inequality for all z ∈ U and for some ε ≥ 0 then there exists a solution u h : U → C of the equation ( 1) such that where n 0 = max{0, ⌈−a⌉}.
EJQTDE, 2011 No. 64, p. 7 Proof.By the definition of a n , we have Consequently yields In view of Theorem 3.1, we have Hence by using Abel ′ s formula (see [16]), we can estimate EJQTDE, 2011 No. 64, p. 8 Thus we have for all z ∈ U.This completes the proof.