The present paper deles with a fractional differential equation

Fractional calculus is a rapidly growing subject of interest for physicists and mathematicians. The reason for this is that problems may be discussed in a much more stringent and elegant way than using traditional methods. Fractional differential equations have emerged as a new branch of applied mathematics which has been used for many mathematical models in science and engineering. In fact, fractional differential equations are considered as an alternative model to nonlinear differential equations [

The class of fractional differential equations of various types plays important roles and tools not only in mathematics but also in physics, control systems, dynamical systems, and engineering to create the mathematical modeling of many physical phenomena. Naturally, such equations required to be solved. Many studies on fractional calculus and fractional differential equations, involving different operators such as Riemann-Liouville operators, Erdlyi-Kober operators, Weyl-Riesz operators, Caputo operators, and Grünwald-Letnikov operators, have appeared during the past three decades with its applications in other field. Moreover, the existence and uniqueness of holomorphic solutions for nonlinear fractional differential equations such as Cauchy problems and diffusion problems in complex domain are established and posed [

The present article deals with a nonhomogeneous fractional differential equation. The nonhomogeneous fractional differential equations involving the Bessel differential equation which appears frequently in practical problems and applications. These equations have proved useful in many branches of physics and engineering. They have been used in problems of treating the boundary value problems exhibiting cylindrical symmetries.

In [

The fractional derivative of order

The fractional integral of order

From Definitions

In this section, we will express the solution for the nonhomogeneous problem

Consider the problem (

We assume that

That is, the power series for

Hence

A classical problem in the theory of functional equations is that if a function

In this section, we consider the Hyers-Ulam stability for fractional differential equation (

Let

We need the following results.

If

Let

In virtue of Theorem

Without loss of the generality, we consider

If

By setting

Since

From above, we conclude that fractional differential equations of Bessel type have holomorphic solutions in the unit disk. The uniqueness imposed by employing the Rouche’s theorem. Furthermore, this solution satisfied the generalized Ulam stability for infinite series of fractional power.