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# Fractional complex transforms for fractional differential equations

- Rabha W Ibrahim
^{1}Email author

**2012**:192

https://doi.org/10.1186/1687-1847-2012-192

© Ibrahim; licensee Springer 2012

**Received:**10 July 2012**Accepted:**23 October 2012**Published:**6 November 2012

## Abstract

The fractional complex transform is employed to convert fractional differential equations analytically in the sense of the Srivastava-Owa fractional operator and its generalization in the unit disk. Examples are illustrated to elucidate the solution procedure including the space-time fractional differential equation in complex domain, singular problems and Cauchy problems. Here, we consider analytic solutions in the complex domain.

**MSC:**30C45.

## Keywords

- fractional calculus
- fractional differential equations
- Srivastava-Owa fractional operators
- unit disk
- analytic function
- fractional complex transform
- Cauchy differential equation
- Fox-Wright function

## 1 Introduction

The theory of fractional calculus has been applied in the theory of analytic functions. The classical concepts of a fractional differential operator and a fractional integral operator and their generalizations have fruitfully been employed in finding, for example, the characterization properties, coefficients estimate [1], distortion inequalities [2] and convolution properties for difference subclasses of analytic functions.

Fractional differential equations are viewed as alternative models to nonlinear differential equations. Varieties of them play important roles and serve as tools not only in mathematics, but also in physics, dynamical systems, control systems, and engineering to create the mathematical modeling of many physical phenomena. Furthermore, they are employed in social sciences such as food supplement, climate, and economics. Fractional differential equations concerning the Riemann-Liouville fractional operators or the Caputo derivative have been recommended by many authors (see [3–10]).

Recently, the complex modelings of phenomena in nature and society have been the object of several investigations based on the methods originally developed in a physical context. These systems are the consequence of the ability of individuals to develop strategies. They occur in kinetic theory [11], complex dynamical systems [12], chaotic complex systems and hyperchaotic complex systems [13], and the complex Lorenz-like system which has been found in laser physics while analyzing baroclinic instability of the geophysical flows in the atmosphere (or in the ocean) [14, 15]. Sainty [16] considered the complex heat equation using a complex valued Brownian. A model of complex fractional equations is introduced by Jumarie [17–20] using different types of fractional derivatives. Baleanu *et al.* [21–23] imposed several applications of fractional calculus including complex modelings. The author studied various types of fractional differential equations in complex domain such as the Cauchy equation, the diffusion equation and telegraph equations [24–28].

Transform is a significant technique to solve mathematical problems. Many useful transforms for solving various problems appeared in open literature such as wave transformation, the Laplace transform, the Fourier transform, the Bücklund transformation, the integral transform, the local fractional integral transforms and the fractional complex transform (see [29, 30]).

In this paper, we shall introduce two generalizations of the wave transformation in the complex domain. These generalizations depend on the fractional differential operators for complex variables. These transformations convert the fractional differential equations in complex domain into ordinary differential equations to obtain analytic solutions or exact solutions. Examples are illustrated.

In [31], Srivastava and Owa provided the 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 a simply-connected region of the complex *z*-plane ℂ containing the origin, and the multiplicity of ${(z-\zeta )}^{-\alpha}$ is removed by requiring $log(z-\zeta )$ to be real when $(z-\zeta )>0$.

**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 a simply-connected region of the complex *z*-plane $(\mathbb{C})$ containing the origin, and the multiplicity of ${(z-\zeta )}^{\alpha -1}$ is removed by requiring $log(z-\zeta )$ to be real when $(z-\zeta )>0$.

**Remark 1.1**From Definitions 1.1 and 1.2, we have

*μ*, the

*n*-fold integral of the form

*a*and $\mu \ne -1$ are real numbers and the function $f(z)$ is analytic in a simply-connected region of the complex

*z*-plane ℂ containing the origin, and the multiplicity of ${({z}^{\mu +1}-{\zeta}^{\mu +1})}^{-\alpha}$ is removed by requiring $log({z}^{\mu +1}-{\zeta}^{\mu +1})$ to be real when $({z}^{\mu +1}-{\zeta}^{\mu +1})>0$. When $\mu =0$, we get the standard Srivastava-Owa fractional integral, which is applied to define the Srivastava-Owa fractional derivatives. The computation implies [32]

When $\mu =0$, we obtain ${I}_{z}^{\alpha}{z}^{\nu}=\frac{\mathrm{\Gamma}(\nu +1)}{\mathrm{\Gamma}(\alpha +\nu +1)}{z}^{\alpha +\nu}$ (see Remark 1.1).

Corresponding to the generalized fractional integrals (2), we imposed the generalized differential operator.

**Definition 1.3**The generalized fractional derivative of order

*α*is defined, for a function $f(z)$, by

*z*-plane ℂ containing the origin, and the multiplicity of ${({z}^{\mu +1}-{\zeta}^{\mu +1})}^{-\alpha}$ is removed by requiring $log({z}^{\mu +1}-{\zeta}^{\mu +1})$ to be real when $({z}^{\mu +1}-{\zeta}^{\mu +1})>0$. The calculation yields

When $\mu =0$, we obtain ${D}_{z}^{\alpha}{z}^{\nu}=\frac{\mathrm{\Gamma}(\nu +1)}{\mathrm{\Gamma}(\nu +1-\alpha )}{z}^{\nu -\alpha}$ (see Remark 1.1).

## 2 Fractional complex transform

*a*,

*b*, and

*c*are constants, the fractional complex transform is

**Example 2.1**Let $Z={z}^{\alpha}$ and $f={Z}^{n}$, $n\ne 0$ then in view of Remark 1.1, we obtain

**Example 2.2**Let $Z={z}^{\alpha (\mu +1)}$ and $f={Z}^{\frac{n}{\mu +1}}$, $n\ne 0$ then in virtue of (5), we have

## 3 Applications

**Example 3.1**Consider the following equation:

where $u(t,z)$ is the unknown function $\rho \in (0,1)$ and $\beta \in (0,1]$.

Now $g(z,\mu (z))$ is a contraction mapping whenever $\rho \in (0,1)$; therefore, in view of the Banach fixed point theorem, Eq. (15) has a unique analytic solution in the unit disk and consequently the problem (14).

where ${E}_{\beta}$ is a Mittag-Leffler function. The last assertion is the exact solution for the problem (16) and consequently for (14).

**Example 3.2**Consider the following equation:

Thus, in view of the Banach fixed point theorem, Eq. (20) has a unique analytic solution in the unit disk and consequently the problem (19).

The last assertion is the exact solution for the problem (21) and consequently for (19).

where ${A}_{j}>0$ for all $j=1,\dots ,p$, ${B}_{j}>0$ for all $j=1,\dots ,q$, and $1+{\sum}_{j=1}^{q}{B}_{j}-{\sum}_{j=1}^{p}{A}_{j}\ge 0$ for suitable values $|w|<1$ and ${a}_{i}$, ${b}_{j}$ are complex parameters.

**Example 3.3**Consider the Cauchy problem in terms of the differential operator (4)

*u*and $u(z)$ is analytic in the unit disk. Thus,

*F*can be expressed by

*ϕ*is an arbitrary constant, we assume that

where $|z|<1$.

## 4 Conclusion

A generalized fractional complex transform is suggested in this paper to find exact solutions of fractional differential equations in the unit disk. We have converted some classes of fractional differential equations in the sense of the Srivastava-Owa fractional operator and its generalization into ordinary differential equations. Hence, the exact solutions are imposed. The solution procedure is simple and might find wide applications in image processing and signal processing by using fractional filter mask. Examples 3.1 and 3.2 showed the conversion of time-space fractional differential equations into a normal case. The exact solutions are introduced in the expression of a Mittag-Leffler function. While Example 3.3, the Cauchy problem of fractional order in the unit disk, proposed the exact solution in terms of the Fox-Wright function.

## Declarations

### Acknowledgements

The author is thankful to the referees for helpful suggestions for the improvement of this article.

## Authors’ Affiliations

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