Complex Transforms for Systems of Fractional Differential Equations

Abstract

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.

1. Introduction

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 [1], the Fourier transform [2], the traveling wave transform [3], the Bäcklund transformation [4], the integral transform [5], the fractional integral transforms [6], the fractional complex transform [7, 8], and Mellin transform [9].

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 [17], the fractional diffusion equations [18–23], and the generalized Chapman-Kolmogorov equation [24].

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 𝑓 of order 𝛼>0 is defined by 𝐼^𝛼_𝑎𝑓(𝑡)=^𝑡_𝑎(𝑡−𝜏)^𝛼−1Γ(𝛼)𝑓(𝜏)𝑑𝜏.(2.1) When =0, we write 𝐼^𝛼_𝑎𝑓(𝑡)=𝑓(𝑡)∗𝜙_𝛼(𝑡), where (∗) denoted the convolution product (see [16]), 𝜙_𝛼(𝑡)=𝑡^𝛼−1/Γ(𝛼),𝑡>0, 𝜙_𝛼(𝑡)=0,𝑡≤0 and 𝜙_𝛼→𝛿(𝑡) as 𝛼→0 where 𝛿(𝑡) is the delta function.

Definition 2.2. The fractional (arbitrary) order derivative of the function 𝑓 of order 0≤𝛼<1 is defined by 𝐷^𝛼_𝑎𝑓(𝑡)=𝑑𝑑𝑡^𝑡_𝑎(𝑡−𝜏)^−𝛼Γ()𝑓(𝜏)𝑑1−𝛼𝑑𝜏𝐼𝑑𝑡^𝑎1−𝛼𝑓(𝑡).(2.2)

Remark 2.3. From Definitions 2.1 and 2.2, 𝑎=0, we have 𝐷^𝛼𝑡^𝜇=Γ𝜇+1Γ𝑡𝜇−𝛼+1^𝜇−𝛼𝐼,𝜇>−1;0<𝛼<1,_𝛼𝑡^𝜇=Γ𝜇+1Γ𝑡𝜇+𝛼+1^𝜇+𝛼,𝜇>−1;𝛼>0.(2.3) The Leibniz rule is 𝐷^𝛼_𝑎𝑓(𝑡)𝑔(𝑡)=^∞_𝑘=0𝛼𝑘𝐷_𝑎^𝛼−𝑘𝑓(𝑡)𝐷^𝑘_𝑎𝑔(𝑡)=^∞_𝑘=0𝛼𝑘𝐷_𝑎^𝛼−𝑘𝑔(𝑡)𝐷^𝑘_𝑎𝑓(𝑡),(2.4) where 𝛼𝑘=Γ(𝛼+1)Γ(𝑘+1)Γ(𝛼+1−𝑘).(2.5)
In this paper we deal with the following fractional differential equation: 𝐷^𝛼𝑢(𝑡,𝑧)=𝑎(𝑡,𝑧)𝑢_𝑧𝑧+𝑏(𝑡,𝑧)𝑢_𝑧+𝑐(𝑡,𝑧)𝑢,(2.6) where 𝑎≠0,𝑏,𝑐,𝑢, are complex valued functions, analytic in the domain 𝒟∶=𝐽×𝑈;𝐽=[0,𝑇],𝑇∈(0,∞), and 𝑈∶={𝑧∈ℂ,|𝑧|≤1}.

Definition 2.4. The Caputo fractional derivative of order 𝜇>0 is defined, for a smooth function 𝑓(𝑡), by ^𝑐𝐷^𝜇𝑓(𝑡)1∶=Γ𝑛−𝜇^𝑡₀𝑓^(𝑛)(𝜁)(𝑡−𝜁)^{𝜇−𝑛+1}𝑑𝜁,(2.7) where 𝑛=[𝜇]+1 (the notation [𝜇] stands for the largest integer not greater than 𝜇).
Note that there is a relationship between Riemann-Liouville differential operator and the Caputo operator 𝐷^𝜇_𝑎𝑓(𝑡)=1Γ𝑓(𝑎)1−𝜇(𝑡−𝑎)^𝜇+^𝑐𝐷^𝜇_𝑎𝑓(𝑡),(2.8) and they are equivalent in a physical problem (i.e., a problem which specifies the initial conditions) [28].

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 [31] and Zaslavsky [32]. 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 [35] 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 𝑎(𝑡,𝑧)=𝐾, where 𝐾 is a constant. Moreover, by employing the concept of majorant functions, it was proved in [36] the existence and uniqueness of holomorphic solutions to nonlinear fractional diffusion problems. Finally, by applying the fractional variational iteration method, it was provided in [37] 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 𝑢(𝑡,𝑧)=𝜎(𝑧)𝑢(𝑡,𝑧),(3.1) where 𝜎≠0 is a complex valued function of complex variable 𝑧∈𝑈; therefor, we have 𝜎(𝑧)∶=𝜎₁(𝑧)+𝑖𝜎₂(𝑧)()()(),𝑎(),𝑢𝑡,𝑧=𝑣𝑡,𝑧+𝑖𝑤𝑡,𝑧𝑡,𝑧=𝑎₁(𝑡,𝑧)+𝑖𝑎₂(𝑡,𝑧),𝑏(𝑡,𝑧)=𝑏₁(𝑡,𝑧)+𝑖𝑏₂(),𝑐()𝑡,𝑧𝑡,𝑧=𝑐₁(𝑡,𝑧)+𝑖𝑐₂(𝑡,𝑧),𝑢(𝑡,𝑧)=𝑣(𝑡,𝑧)+𝑖𝑤(𝑡,𝑧).(3.2)

Substituting (3.1) in (2.6) yields 𝐷^𝛼𝑢(𝑡,𝑧)=𝑎(𝑡,𝑧)𝑢_𝑧𝑧+𝑏(𝑡,𝑧)𝑢_𝑧+𝑐(𝑡,𝑧)𝑢,(3.3) where 𝑎=𝑎,𝜎𝑏=𝑏+2𝑎^𝑧𝜎,𝜎𝑐=𝑐+𝑎^𝑧𝑧𝜎𝜎+𝑏^𝑧𝜎.(3.4)

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: 𝐷^𝛼𝑣=𝑎₁𝑣_𝑧𝑧−𝑎₂𝑤_𝑧𝑧+𝑏₁𝑣_𝑧−𝑏₂𝑤_𝑧+𝑐₁𝑣−𝑐₂𝐷𝑤,_𝛼𝑤=𝑎₁𝑤_𝑧𝑧+𝑎₂𝑣_𝑧𝑧+𝑏₁𝑤_𝑧+𝑏₂𝑣_𝑧+𝑐₁𝑤+𝑐₂𝑣.(3.5)

Since 𝑢=𝑣+𝑖𝑤=(𝜎₁(𝑧)+𝑖𝜎₂(𝑧))(𝑣(𝑡,𝑧)+𝑖𝑤(𝑡,𝑧))=𝜎₁𝑣−𝜎₂𝑤+𝑖(𝜎₁𝑤+𝜎₂𝑣), then we obtain the relations 𝑣=𝜎₁𝑣−𝜎₂𝑤,𝑤=𝜎₂𝑣+𝜎₁𝑤.(3.6) Consequently we have 𝑣_𝑧=𝜎₁𝑣_𝑧+𝜎_1𝑧𝑣−𝜎₂𝑤_𝑧−𝜎_2𝑧𝑣𝑤,_𝑧𝑧=𝜎₁𝑣_𝑧𝑧+2𝜎_1𝑧𝑣_𝑧+𝜎_1𝑧𝑧𝑣−𝜎₂𝑤_𝑧𝑧−2𝜎_2𝑧𝑤_𝑧−𝜎_2𝑧𝑧𝑤𝑤,_𝑧=𝜎₂𝑣_𝑧+𝜎_2𝑧𝑣+𝜎₁𝑤_𝑧+𝜎_1𝑧𝑣𝑤,_𝑧𝑧=𝜎₂𝑣_𝑧𝑧+2𝜎_2𝑧𝑣_𝑧+𝜎_2𝑧𝑧𝑣+𝜎₁𝑤_𝑧𝑧+2𝜎_1𝑧𝑤_𝑧+𝜎_1𝑧𝑧𝑤.(3.7)

By the linearity of the fractional differential operator 𝐷^𝛼, we impose that 𝐷^𝛼𝑣=𝜎₁𝐷^𝛼𝑣−𝜎₂𝐷^𝛼𝐷𝑤,_𝛼𝑤=𝜎₂𝐷^𝛼𝑣+𝜎₁𝐷^𝛼𝑤.(3.8)

By injecting (3.7) and (3.8) in (3.5) we receive the following system: 𝐷^𝛼𝑣=𝑎₁𝑣_𝑧𝑧−𝑎₂𝑤_𝑧𝑧+𝑏₁𝑣_𝑧−𝑏₂𝑤_𝑧+𝑐₁𝑣−𝑐₂𝐷𝑤,_𝛼𝑤=𝑎₁𝑤_𝑧𝑧+𝑎₂𝑣_𝑧𝑧+𝑏₁𝑤_𝑧+𝑏₂𝑣_𝑧+𝑐₁𝑤+𝑐₂𝑣,(3.9) where 𝑎₁=𝑎₁,𝑎₂=𝑎₂,𝑏₁=𝑏₁+2𝜎_1𝑧𝑎₁𝜎₁+𝑎₂𝜎₂+2𝜎_2𝑧𝑎₁𝜎₂−𝑎₂𝜎₁𝜎²₁+𝜎²₂,𝑏₂=𝑏₂−2𝜎_1𝑧𝑎₁𝜎₂−𝑎₂𝜎₁+2𝜎_2𝑧𝑎₂𝜎₂−𝑎₁𝜎₁𝜎²₁+𝜎²₂,𝑐₁=𝑐₁+𝜎_1𝑧𝑧𝑎₁𝜎₁+𝑎₂𝜎₂+𝜎_2𝑧𝑧𝑎₁𝜎₂−𝑎₂𝜎₁+𝜎_1𝑧𝑏₁𝜎₁+𝑏₂𝜎₂+𝜎_2𝑧𝑏₁𝜎₂−𝑏₂𝜎₁𝜎²₁+𝜎²₂,𝑐₂=𝑐₂−𝜎_1𝑧𝑧𝑎₁𝜎₂−𝑎₂𝜎₁+𝜎_2𝑧𝑧𝑎₁𝜎₁−𝑎₂𝜎₂+𝜎_1𝑧𝑏₁𝜎₂−𝑏₂𝜎₁−𝜎_2𝑧𝑏₁𝜎₁+𝑏₂𝜎₂𝜎²₁+𝜎²₂.(3.10)

Hence we conclude the following result.

Theorem 3.1. Let 𝑎≠0,𝑏,𝑐,𝑢 be complex valued functions, analytic in the domain 𝒟=𝐽×𝑈, and let 𝜎≠0 be complex valued function in 𝑈. 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 𝑎₂𝑏=0,₂=2𝜎_1𝑧𝑎₁𝜎₂−2𝜎_2𝑧𝑎₁𝜎₁𝜎²₁+𝜎²₂,𝑐₂=𝜎_1𝑧𝑧𝑎₁𝜎₂+𝜎_2𝑧𝑧𝑎₁𝜎₁+𝜎_1𝑧𝑏₁𝜎₂−𝑏₂𝜎₁−𝜎_2𝑧𝑏₁𝜎₁+𝑏₂𝜎₂𝜎²₁+𝜎²₂.(3.11)
Then the fractional coupled system (3.5) is equivalent to the transformed fractional uncoupled system 𝐷^𝛼𝑣=𝑎₁𝑣_𝑧𝑧+𝑏₁𝑣_𝑧+𝑐₁𝐷𝑣,_𝛼𝑤=𝑎₁𝑤_𝑧𝑧+𝑏₁𝑤_𝑧+𝑐₁𝑤,(3.12) where 𝑎₁, 𝑏₁, and 𝑐₁ are defined in (3.10).
Now we proceed to consider the nonhomogeneous equation 𝐷^𝛼𝑢(𝑡,𝑧)=𝑎(𝑡,𝑧)𝑢_𝑧𝑧+𝑏(𝑡,𝑧)𝑢_𝑧+𝑐(𝑡,𝑧)𝑢+𝑓(𝑡,𝑧),(3.13) where 𝑎≠0,𝑏,𝑐,𝑢, and 𝑓 are complex valued functions, analytic in the domain 𝒟 such that 𝑓=𝑓₁+𝑖𝑓₂.
Assume the complex transform 𝑢(𝑡,𝑧)=𝜌(𝑡,𝑧)𝑢(𝑡,𝑧),(3.14) where 𝜌≠0 is a complex valued function of complex variable 𝑧∈𝑈, with 𝜌(𝑡,𝑧)∶=𝜌₁(𝑡,𝑧)+𝑖𝜌₂(𝑡,𝑧).(3.15)
Substituting (3.14) in (3.13) and using the Leibniz rule imply 𝐷^𝛼𝑢(𝑡,𝑧)=𝑎(𝑡,𝑧)𝑢_𝑧𝑧+𝑏(𝑡,𝑧)𝑢_𝑧+𝑐(𝑡,𝑧)𝑢+𝑓(𝑡,𝑧),(3.16) where 𝑎=𝑎,𝜌𝑏=𝑏+2𝑎^𝑧𝜌,𝜌𝑐=𝑐+𝑎^𝑧𝑧𝜌𝜌+𝑏^𝑧𝜌,𝑓𝑓=𝜌−𝛼𝜌_𝑡𝜌𝐼^1−𝛼𝑢.(3.17)
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: 𝐷^𝛼𝑣=𝑎₁𝑣_𝑧𝑧−𝑎₂𝑤_𝑧𝑧+𝑏₁𝑣_𝑧−𝑏₂𝑤_𝑧+𝑐₁𝑣−𝑐₂𝑤+𝑓₁,𝐷^𝛼𝑤=𝑎₁𝑤_𝑧𝑧+𝑎₂𝑣_𝑧𝑧+𝑏₁𝑤_𝑧+𝑏₂𝑣_𝑧+𝑐₁𝑤+𝑐₂𝑣+𝑓₂.(3.18)
Since 𝑣=𝜌₁𝑣−𝜌₂𝑤,𝑤=𝜌₂𝑣+𝜌₁𝑤,(3.19) then by taking the fractional derivative of the variables 𝑣 and 𝑤 in (3.19), we obtain 𝐷^𝛼𝑣=𝜌₁𝐷^𝛼𝑣−𝜌₂𝐷^𝛼𝑤+ℎ₁,𝐷^𝛼𝑤=𝜌₂𝐷^𝛼𝑣+𝜌₁𝐷^𝛼𝑤+ℎ₂,(3.20) where ℎ₁=𝜌_1𝑡𝐼^1−𝛼𝑣−𝜌_2𝑡𝐼^1−𝛼ℎ𝑤,₂=𝜌_2𝑡𝐼^1−𝛼𝑣+𝜌_1𝑡𝐼^1−𝛼𝑤.(3.21)
Consequently, we impose the following system: 𝐷^𝛼𝑣=𝑎₁𝑣_𝑧𝑧−𝑎₂𝑤_𝑧𝑧+𝑏₁𝑣_𝑧−𝑏₂𝑤_𝑧+𝑐₁𝑣−𝑐₂𝑤+𝑓₁,𝐷^𝛼𝑤=𝑎₁𝑤_𝑧𝑧+𝑎₂𝑣_𝑧𝑧+𝑏₁𝑤_𝑧+𝑏₂𝑣_𝑧+𝑐₁𝑤+𝑐₂𝑣+𝑓₂,(3.22) where 𝑎₁=𝑎₁,𝑎₂=𝑎₂,𝑏₁=𝑏₁+2𝜌_1𝑧𝑎₁𝜌₁+𝑎₂𝜌₂+2𝜌_2𝑧𝑎₁𝜌₂−𝑎₂𝜌₁𝜌²₁+𝜌²₂,𝑏₂=𝑏₂−2𝜌_1𝑧𝑎₁𝜌₂−𝑎₂𝜌₁+2𝜌_2𝑧𝑎₂𝜌₂−𝑎₁𝜌₁𝜌²₁+𝜌²₂,𝑐₁=𝑐₁+𝜌_1𝑧𝑧𝑎₁𝜌₁+𝑎₂𝜌₂+𝜌_2𝑧𝑧𝑎₁𝜌₂−𝑎₂𝜌₁+𝜌_1𝑧𝑏₁𝜌₁+𝑏₂𝜌₂+𝜌_2𝑧𝑏₁𝜌₂−𝑏₂𝜌₁𝜌²₁+𝜌²₂,𝑐₂=𝑐₂−𝜌_1𝑧𝑧𝑎₁𝜌₂−𝑎₂𝜌₁+𝜌_2𝑧𝑧𝑎₁𝜌₁−𝑎₂𝜌₂+𝜌_1𝑧𝑏₁𝜌₂−𝑏₂𝜌₁−𝜌_2𝑧𝑏₁𝜌₁+𝑏₂𝜌₂𝜌²₁+𝜌²₂,𝑓₁=𝜌₁f₁−ℎ₁+𝜌₂𝑓₂−ℎ₂𝜌²₁+𝜌²₂,𝑓₂=𝜌₂𝑓₁−ℎ₁+𝜌₁𝑓₂−ℎ₂𝜌²₁+𝜌²₂.(3.23)
Hence we conclude the following result.

Theorem 3.3. Let 𝑎≠0,𝑏,𝑐,𝑢,𝑓 and 𝜌 be complex valued functions, analytic in the domain 𝒟=𝐽×𝑈. Then the fractional coupled system (3.18) is equivalent to the transformed fractional coupled system (3.22) if and only if (3.23) holds.

Corollary 3.4. Let 𝑎₂𝑏=0,₂=2𝜌_1𝑧𝑎₁𝜌₂−2𝜌_2𝑧𝑎₁𝜌₁𝜌²₁+𝜌²₂,𝑐₂=𝜌_1𝑧𝑧𝑎₁𝜌₂+𝜌_2𝑧𝑧𝑎₁𝜌₁+𝜌_1𝑧𝑏₁𝜌₂−𝑏₂𝜌₁−𝜌_2𝑧𝑏₁𝜌₁+𝑏₂𝜌₂𝜌²₁+𝜌²₂.(3.24)
Then the fractional coupled system (3.18) is equivalent to the transformed fractional uncoupled system 𝐷^𝛼𝑣=𝑎₁𝑣_𝑧𝑧+𝑏₁𝑣_𝑧+𝑐₁𝑣+𝑓₁,𝐷^𝛼𝑤=𝑎₁𝑤_𝑧𝑧+𝑏₁𝑤_𝑧+𝑐₁𝑤+𝑓₂,(3.25) where 𝑎₁, 𝑏₁, and 𝑐₁ are defined in (3.23).

3.2. Nonequivalence Transforms

Here, we shall show that the transform (3.14) reduces (2.6) into nonhomogeneous equation. And Hence we obtain nonhomogeneous system. Substituting (3.14) in (2.6), we have 𝐷^𝛼𝑢(𝑡,𝑧)=𝑎(𝑡,𝑧)𝑢_𝑧𝑧+𝑏(𝑡,𝑧)𝑢_𝑧+𝑐(𝑡,𝑧)𝑢+𝑔(𝑡,𝑧),(3.26) where 𝑎=𝑎,𝜌𝑏=𝑏+2𝑎^𝑧𝜌,𝜌𝑐=𝑐+𝑎^𝑧𝑧𝜌𝜌+𝑏^𝑧𝜌,𝑔(𝑡,𝑧)=𝛼𝜌_𝑡𝜌𝐼^1−𝛼𝑢∶=𝑔₁(𝑡,𝑧)+𝑖𝑔₂(𝑡,𝑧).(3.27)

A computation implies the following system: 𝐷^𝛼𝑣=𝑎₁𝑣_𝑧𝑧−𝑎₂𝑤_𝑧𝑧+𝑏₁𝑣_𝑧−𝑏₂𝑤_𝑧+𝑐₁𝑣−𝑐₂𝑤+𝑔₁,𝐷^𝛼𝑤=𝑎₁𝑤_𝑧𝑧+𝑎₂𝑣_𝑧𝑧+𝑏₁𝑤_𝑧+𝑏₂𝑣_𝑧+𝑐₁𝑤+𝑐₂𝑣+𝑔₂.(3.28)

Now in view of (3.19)–(3.21), we have the system 𝐷^𝛼𝑣=𝑎₁𝑣_𝑧𝑧−𝑎₂𝑤_𝑧𝑧+𝑏₁𝑣_𝑧−𝑏₂𝑤_𝑧+𝑐₁𝑣−𝑐₂𝑤+𝑔₁,𝐷^𝛼𝑤=𝑎₁𝑤_𝑧𝑧+𝑎₂𝑣_𝑧𝑧+𝑏₁𝑤_𝑧+𝑏₂𝑣_𝑧+𝑐₁𝑤+𝑐₂𝑣+𝑔₂,(3.29) where 𝑎₁=𝑎₁,𝑎₂=𝑎₂,𝑏₁=𝑏₁+2𝜌_1𝑧𝑎₁𝜌₁+𝑎₂𝜌₂+2𝜌_2𝑧𝑎₁𝜌₂−𝑎₂𝜌₁𝜌²₁+𝜌²₂,𝑏₂=𝑏₂−2𝜌_1𝑧𝑎₁𝜌₂−𝑎₂𝜌₁+2𝜌_2𝑧𝑎₂𝜌₂−𝑎₁𝜌₁𝜌²₁+𝜌²₂,𝑐₁=𝑐₁+𝜌_1𝑧𝑧𝑎₁𝜌₁+𝑎₂𝜌₂+𝜌_2𝑧𝑧𝑎₁𝜌₂−𝑎₂𝜌₁+𝜌_1𝑧𝑏₁𝜌₁+𝑏₂𝜌₂+𝜌_2𝑧𝑏₁𝜌₂−𝑏₂𝜌₁𝜌²₁+𝜌²₂,𝑐₂=𝑐₂−𝜌_1𝑧𝑧𝑎₁𝜌₂−𝑎₂𝜌₁+𝜌_2𝑧𝑧𝑎₁𝜌₁−𝑎₂𝜌₂+𝜌_1𝑧𝑏₁𝜌₂−𝑏₂𝜌₁−𝜌_2𝑧𝑏₁𝜌₁+𝑏₂𝜌₂𝜌²₁+𝜌²₂,𝑔₁=𝜌₁𝑔₁−ℎ₁+𝜌₂𝑔₂−ℎ₂𝜌²₁+𝜌²₂,𝑔₂=𝜌₂𝑔₁−ℎ₁+𝜌₁𝑔₂−ℎ₂𝜌²₁+𝜌²₂.(3.30)

Thus we receive the following result.

Theorem 3.5. Let 𝑎≠0,𝑏,𝑐,𝑢, and 𝜌 be complex valued functions, analytic in the domain 𝒟=𝐽×𝑈. Then the fractional coupled system (3.28) is equivalent to the transformed fractional coupled system (3.29) if and only if (3.30) holds.

Corollary 3.6. Let 𝑎₂𝑏=0,₂=2𝜌_1𝑧𝑎₁𝜌₂−2𝜌_2𝑧𝑎₁𝜌₁𝜌²₁+𝜌²₂,𝑐₂=𝜌_1𝑧𝑧𝑎₁𝜌₂+𝜌_2𝑧𝑧𝑎₁𝜌₁+𝜌_1𝑧𝑏₁𝜌₂−𝑏₂𝜌₁−𝜌_2𝑧𝑏₁𝜌₁+𝑏₂𝜌₂𝜌²₁+𝜌²₂.(3.31)
Then the fractional coupled system (3.28) is equivalent to the transformed fractional uncoupled system 𝐷^𝛼𝑣=𝑎₁𝑣_𝑧𝑧+𝑏₁𝑣_𝑧+𝑐₁𝑣+𝑔₁,𝐷^𝛼𝑤=𝑎₁𝑤_𝑧𝑧+𝑏₁𝑤_𝑧+𝑐₁𝑤+𝑔₂,(3.32) where 𝑎₁,𝑏₁, and 𝑐₁ are defined in (3.30).

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 [38]). A vector-valued function 𝑋(𝑡,𝑥) is said to satisfy a Lipschitz condition in a region ℛ in (𝑡,𝑥)-space if, for some constant 𝐿 (called the Lipschitz constant), we have 𝑋(𝑡,𝑥)−𝑋𝑡,𝑦≤𝐿𝑥−𝑦,(4.1) whenever (𝑡,𝑥),(𝑡,𝑦)∈ℛ.
A convex domain 𝐷 means that for any two points 𝑥 and 𝑦 in 𝐷, (1−𝑡)𝑥+𝑡𝑦∈𝐷 for 0≤𝑡≤1.

Lemma 4.2 (see [38]). If 𝑋(𝑡,𝑥) has continuous partial derivatives on a bounded closed convex domain 𝐷, then it satisfies a Lipschitz condition.

Theorem 4.3. Assume that all the functions in the system (3.9) are analytic in the domain 𝒟. Then the system (3.9)-(3.10) has a unique solution in 𝒟.

Proof. System (3.9) can be assumed as in a matrix form 𝐷^𝛼𝑣𝐷^𝛼𝑤=𝑎₁−𝑎₂𝑏₁−𝑏₂𝑐₁−𝑐₂𝑎₂𝑎₁𝑏₂𝑏₁𝑐₂𝑐₁𝑣_𝑧𝑧𝑤_𝑧𝑧𝑣_𝑧𝑤_𝑧𝑣𝑤∶=𝐴𝑍,(4.2) where 𝑍=(𝑍₁(𝑡,𝑧),…,𝑍₆(𝑡,𝑧))^𝑇. Since the domain 𝒟=𝐽×𝑈 is bounded closed convex and all the functions 𝑍_𝑖,𝑖=1,…,6 are analytic in 𝒟 then in view of Lemma 4.2, 𝑍 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.

Theorem 4.4. Let the functions in the system (3.22) be analytic in the domain 𝒟. Moreover let 𝑓 be a Lipschitz function. Then the system (3.22)-(3.23) has a unique solution in 𝒟.

Proof. System (3.22) can be written in a matrix form 𝐷^𝛼𝑣𝐷^𝛼𝑤=𝑎₁−𝑎₂𝑏₁−𝑏₂𝑐₁−𝑐₂𝑎₂𝑎₁𝑏₂𝑏₁𝑐₂𝑐₁𝑣_𝑧𝑧𝑤_𝑧𝑧𝑣_𝑧𝑤_𝑧𝑣𝑤+𝑓₁𝑓₂∶=𝐴𝑍+𝐹.(4.3)
Hence Lemma 4.2 and Banach fixed point theorem imply the uniqueness of (4.3) and consequently (3.22). This completes the proof.

5. Applications

In this section, we shall illustrate some examples.

Example 5.1. Consider the equation 𝐷^𝛼𝑢(𝑡,𝑧)=𝐴𝑧²𝑢_𝑧𝑧+𝐵𝑧𝑢_𝑧+𝐶𝑢,(5.1) where 𝐴≠0,𝐵 and 𝐶 are real constants, and the complex transform 𝑢(𝑡,𝑧)=𝑧²𝑢(𝑡,𝑧)∶=𝜎₁+𝑖𝜎₂(𝑣+𝑖𝑤).(5.2)
Equation (5.1) reduces to the homogeneous type 𝐷^𝛼𝑢(𝑡,𝑧)=𝐴𝑧²𝑢_𝑧𝑧+𝐵𝑧𝑢_𝑧+𝐶𝑢,(5.3) where 𝐴=𝐴,𝐵=(4𝐴+𝐵),𝐶=2𝐴+2𝐵+𝐶.(5.4)
Hence (5.3) can be translated to the uncoupled system 𝐷^𝛼𝑣=𝐴𝜎₁𝑣_𝑧𝑧+𝐵±𝑟+𝜎₁2𝑣_𝑧+𝐶𝐷𝑣,_𝛼𝑤=𝐴𝜎₂𝑤_𝑧𝑧+𝐵±𝑟−𝜎₁2𝑤_𝑧+𝐶𝑤,(5.5) where 𝑟=𝜎²₁+𝜎²₂. Furthermore, it can be represented as coupled system 𝐷^𝛼𝑣=𝐴𝜎₁𝑣_𝑧𝑧−𝐴𝜎₂𝑤_𝑧𝑧+𝐵±𝑟+𝜎₁2𝑣_𝑧−𝐵±𝑟−𝜎₁2𝑤_𝑧+𝐶𝑣−𝐶𝐷𝑤,_𝛼𝑤=𝐴𝜎₁𝑤_𝑧𝑧+𝐴𝜎₂𝑣_𝑧𝑧+𝐵±𝑟+𝜎₁2𝑤_𝑧+𝐵±𝑟−𝜎₁2𝑣_𝑧+𝐶𝑤+𝐶𝑣.(5.6)

Example 5.2. Consider the equation 𝐷^𝛼𝑢(𝑡,𝑧)=𝐴𝑡𝑧²𝑢_𝑧𝑧+𝐵𝑡𝑧𝑢_𝑧+𝐶𝑡𝑢+𝑒^𝑖𝑡,𝑡∈[𝑎,1],𝑎≠0,(5.7) where 𝐴≠0,𝐵 and 𝐶 are real constants, and the complex transform 𝑢(𝑡,𝑧)=𝑡𝑧²𝑢(𝑡,𝑧)∶=𝑡𝜌₁+𝑖𝜌₂(𝑣+𝑖𝑤).(5.8)
Thus (5.7) becomes 𝐷^𝛼𝑢(𝑡,𝑧)=𝐴𝑡𝑧²𝑢_𝑧𝑧+𝐵𝑡𝑧𝑢_𝑧+𝐶𝑡𝑢+𝑓,(5.9) where 𝐴=𝐴,𝐵=(4𝐴+𝐵),𝐶=2𝐴+2𝐵+𝐶,𝑓=𝑒^𝑖𝑡−𝛼𝐼^1−𝛼𝑢𝑡∶=cos𝑡−𝜙₁+𝑖sin𝑡−𝜙₂∶=𝑓₁+𝑖𝑓₂.(5.10)
Hence (5.7) can be formulated to the uncoupled system 𝐷^𝛼𝑣=𝐴𝜌₁𝑣_𝑧𝑧+𝐵𝑡±𝑟+𝜌₁2𝑣_𝑧+𝐶𝑡𝑣+𝑓₁,𝐷^𝛼𝑤=𝐴𝜌₂𝑤_𝑧𝑧+𝐵𝑡±𝑟−𝜌₁2𝑤_𝑧+𝐶𝑡𝑤+𝑓₂,(5.11) where 𝑟=𝜌²₁+𝜌²₂. Furthermore, it can be represented as coupled system 𝐷^𝛼𝑣=𝐴𝜌₁𝑣_𝑧𝑧−𝐴𝜌₂𝑤_𝑧𝑧+𝐵𝑡±𝑟−𝜌₁2𝑣_𝑧−𝐵𝑡±𝑟−𝜌₁2𝑤_𝑧+𝐶𝑡𝑣−𝐶𝑡𝑤+𝑓₁,𝐷^𝛼𝑤=𝐴𝜌₁𝑤_𝑧𝑧+𝐴𝜌₂𝑣_𝑧𝑧+𝐵𝑡±𝑟+𝜌₁2𝑤_𝑧+𝐵𝑡±𝑟−𝜌₁2𝑣_𝑧+𝐶𝑡𝑤+𝐶𝑡𝑣+𝑓₂.(5.12)
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 [39].

Example 5.3. Consider the fractional diffusion equation 𝐷^𝛼𝑢(𝑡,𝑧)=𝑢_𝑧𝑧+𝑓(𝑧)𝑝(𝑡),(5.13) where 𝑢(𝑡,𝑧) is the solute concentration, 𝑝(𝑡) is the source term, and 𝑡∈(0,𝑇),𝑧∈𝑈. Equation (5.13) is a special case of (3.13) with 𝑎(𝑡,𝑧)=1, 𝑏(𝑡,𝑧)=𝑐(𝑡,𝑧)=0; thus it can be reduced to the coupled system 𝐷^𝛼𝑣=𝑣_𝑧𝑧−𝑏₁𝑣_𝑧−𝑏₂𝑤_𝑧+𝑐₁𝑣−𝑐₂𝑤+𝑓₁,𝐷^𝛼𝑤=𝑤_𝑧𝑧+𝑏₁𝑤_𝑧+𝑏₂𝑣_𝑧+𝑐₁𝑤+𝑐₂𝑣+𝑓₂,(5.14) where 𝑓=(𝑓_𝑝(𝑡)/𝜌)−(𝛼𝜌_𝑡/𝜌)𝐼^1−𝛼𝑢∶=𝑓₁+𝑖𝑓₂, 𝑎₁=1,𝑎₂=0,𝑏₁=2𝜌_1𝑧𝑎₁𝜌₁+2𝜌_2𝑧𝑎₁𝜌₂𝜌²₁+𝜌²₂,𝑏₂=−2𝜌_1𝑧𝑎₁𝜌₂+2𝜌_2𝑧−𝑎₁𝜌₁𝜌²₁+𝜌²₂,𝑐₁=𝜌_1𝑧𝑧𝑎₁𝜌₁+𝜌_2𝑧𝑧𝑎₁𝜌₂𝜌²₁+𝜌²₂,𝑐₂𝜌=−^1𝑧𝑧𝑎₁𝜌₂+𝜌_2𝑧𝑧𝑎₁𝜌₁𝜌²₁+𝜌²₂,𝑓₁=𝜌₁𝑓₁−ℎ₁+𝜌₂𝑓₂−ℎ₂𝜌²₁+𝜌²₂,𝑓₂=𝜌₂𝑓₁−ℎ₁+𝜌₁𝑓₂−ℎ₂𝜌²₁+𝜌²₂.(5.15)

6. Conclusion

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.

Acknowledgments

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