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- Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 814759, 15 pages
Complex Transforms for Systems of Fractional Differential Equations
Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia
Received 8 August 2012; Revised 11 October 2012; Accepted 11 October 2012
Academic Editor: Elena Litsyn
Copyright © 2012 Rabha W. Ibrahim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We provide a complex transform that maps the complex fractional differential equation into a system of fractional differential equations. The homogeneous and nonhomogeneous cases for equivalence equations are discussed and also nonequivalence equations are studied. Moreover, the existence and uniqueness of solutions are established and applications are illustrated.
Transform is a significant method to solve mathematical problems. Various practical transforms for solving various problems were materialized in open literature, such as the Laplace transform , the Fourier transform , the traveling wave transform , the Bäcklund transformation , the integral transform , the fractional integral transforms , the fractional complex transform [7, 8], and Mellin transform .
Fractional models have been studied by many researchers to sufficiently describe the operation of variety of computational, physical, and biological processes and systems. Accordingly, considerable attention has been paid to the solution of fractional differential equations, integral equations, and fractional partial differential equations of physical phenomena. Most of these fractional differential equations have analytic solutions, approximation, and numerical techniques [10–12]. Numerical and analytical methods have included finite difference method such as Adomian decomposition method, variational iteration method, homotopy perturbation method, and homotopy analysis method [13–16].
One of the most frequently used tools in the theory of fractional calculus is furnished by the Riemann-Liouville operators. It possesses advantages of fast convergence, higher stability, and higher accuracy to derive different types of numerical algorithms. In this paper, we will deal with scalar linear time-fractional differential equations. The time is taken in sense of the Riemann-Liouville fractional operators. Also, This type of differential equations arises in many interesting applications. For example the fractional Fokker-Planck differential equation which can be derived from the master equation , the fractional diffusion equations [18–23], and the generalized Chapman-Kolmogorov equation .
Here, we shall use the complex transform to reduce this fractional differential equation to a system of fractional order (homogeneous and nonhomogeneous) keeping the equivalency properties. Furthermore, we shall study the existence and uniqueness of solutions for the linear and nonlinear cases of these systems. Applications are imposed.
2. Fractional Calculus
The idea of the fractional calculus (i.e., calculus of integrals and derivatives of any arbitrary real or complex order) was planted over 300 years ago. Abel in 1823 investigated the generalized tautochrone problem and for the first time applied fractional calculus techniques in a physical problem. Later Liouville applied fractional calculus to problems in potential theory. Since that time the fractional calculus has haggard the attention of many researchers in all area of sciences (see [25–27]).
This section concerns with some preliminaries and notations regarding the fractional calculus.
Definition 2.1. The fractional (arbitrary) order integral of the function of order is defined by When , we write , where denoted the convolution product (see ), , and as where is the delta function.
Definition 2.2. The fractional (arbitrary) order derivative of the function of order is defined by
Remark 2.3. From Definitions 2.1 and 2.2, , we have
The Leibniz rule is
In this paper we deal with the following fractional differential equation: where , are complex valued functions, analytic in the domain , and .
Definition 2.4. The Caputo fractional derivative of order is defined, for a smooth function , by
where (the notation stands for the largest integer not greater than ).
Note that there is a relationship between Riemann-Liouville differential operator and the Caputo operator and they are equivalent in a physical problem (i.e., a problem which specifies the initial conditions) .
Equation (2.6) involves well known time fractional diffusion equations. Several researchers have studied fractional dynamic equations generalizing the diffusion or wave equations in terms of R-L or Caputo time fractional derivatives, and their fundamental solutions have been represented in terms of the Mittag-Leffler (M-L) functions and their generalizations. The mathematical study of fractional diffusion equations began with the work of Kochubeĭ [29, 30]. Later this study followed by the work of Metzler and Klafter  and Zaslavsky . Recently, Mainardi et al. obtained the time fractional diffusion equation from the standard diffusion equation [33, 34]. By using the Fourier-Laplace transform method, it was shown in  that the fundamental solution of the fractional diffusion equation with a generalized Riemann-Liouville time fractional derivative defined in the infinite domain can be expressed via Foxs H-function, when , where is a constant. Moreover, by employing the concept of majorant functions, it was proved in  the existence and uniqueness of holomorphic solutions to nonlinear fractional diffusion problems. Finally, by applying the fractional variational iteration method, it was provided in  the numerical solutions of time-fractional reaction-diffusion equation with modified Riemann-Liouville derivative.
3. Complex Transforms
In this section, we shall illustrate two types of complex transform. The first type is equivalence transforms for the homogeneous and nonhomogeneous equations. The second is non equivalence transform.
3.1. Equivalence Transforms
Assume the complex transform where is a complex valued function of complex variable ; therefor, we have
Thus under the map (3.1), (2.6) remains homogeneous with new coefficients. Our aim is to translate (3.3) into a system of fractional order using the relations in (3.2). Again substituting (3.2) in (2.6) implies the following system:
Since , then we obtain the relations Consequently we have
By the linearity of the fractional differential operator , we impose that
Hence we conclude the following result.
Theorem 3.1. Let be complex valued functions, analytic in the domain , and let be complex valued function in . Then the fractional coupled system (3.5) is equivalent to the transformed fractional coupled system (3.9) if and only if (3.10) holds.
Corollary 3.2. Let
Then the fractional coupled system (3.5) is equivalent to the transformed fractional uncoupled system where , , and are defined in (3.10).
Now we proceed to consider the nonhomogeneous equation where , and are complex valued functions, analytic in the domain such that .
Assume the complex transform where is a complex valued function of complex variable , with
Substituting (3.14) in (3.13) and using the Leibniz rule imply where
Obviously, (3.16) remains nonhomogeneous with new coefficients. Our goal is to reduce (3.16) to a system of fractional order using the relations (3.2) and (3.15). A calculation yields the following system:
Since then by taking the fractional derivative of the variables and in (3.19), we obtain where
Consequently, we impose the following system: where
Hence we conclude the following result.
Theorem 3.3. Let and be complex valued functions, analytic in the domain . Then the fractional coupled system (3.18) is equivalent to the transformed fractional coupled system (3.22) if and only if (3.23) holds.
3.2. Nonequivalence Transforms
A computation implies the following system:
Thus we receive the following result.
Theorem 3.5. Let , and be complex valued functions, analytic in the domain . Then the fractional coupled system (3.28) is equivalent to the transformed fractional coupled system (3.29) if and only if (3.30) holds.
4. Existence and Uniqueness
In this section we shall discuss the existence and uniqueness of solution for the linear and nonlinear systems. For this purpose, we need the following concept.
Definition 4.1 (see ). A vector-valued function is said to satisfy a Lipschitz condition in a region in -space if, for some constant (called the Lipschitz constant), we have
A convex domain means that for any two points and in , for .
Lemma 4.2 (see ). If has continuous partial derivatives on a bounded closed convex domain , then it satisfies a Lipschitz condition.
Proof. System (3.9) can be assumed as in a matrix form where . Since the domain is bounded closed convex and all the functions are analytic in then in view of Lemma 4.2, satisfies a Lipschitz condition. Hence by the Banach fixed point theorem, (4.2) has a unique solution and has a unique solution as well as system (3.9). This completes the proof.
In this section, we shall illustrate some examples.
Example 5.1. Consider the equation
where and are real constants, and the complex transform
Equation (5.1) reduces to the homogeneous type where
Hence (5.3) can be translated to the uncoupled system where . Furthermore, it can be represented as coupled system
Example 5.2. Consider the equation
where and are real constants, and the complex transform
Thus (5.7) becomes where
Hence (5.7) can be formulated to the uncoupled system where . Furthermore, it can be represented as coupled system
Next example is a time fractional diffusion equation, which describes the reconstruction of a source. Recently, for real case, this equation is studied by Wei and Zhang .
Example 5.3. Consider the fractional diffusion equation where is the solute concentration, is the source term, and . Equation (5.13) is a special case of (3.13) with , ; thus it can be reduced to the coupled system where ,
We suggested two types of complex transforms for systems of fractional differential equations. We concluded that the complex fractional differential equations can be transformed into coupled and uncoupled system of homogeneous and nonhomogeneous types. Moreover, the existence and uniqueness of these systems are studied by using the Banach fixed point theorem. Some examples are illustrated.
The author is thankful to the referees for helpful suggestions for the improvement of this paper.
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