# Complex Transforms for Systems of Fractional Differential Equations

Authors: Rabha W. Ibrahim
Journal: Abstract and Applied Analysis
 (2012)

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## 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.


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## Body

## 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 functionf of order α>0 is defined by
(2.1)Iaαf(t)=∫at‍(t-τ)α-1Γ(α)f(τ)dτ.
When =0, we write Iaαf(t)=f(t)*ϕα(t), where (*) denoted the convolution product (see [16]), ϕα(t)=tα-1/Γ(α),t>0, ϕα(t)=0,t≤0 and ϕα→δ(t) as α→0 where δ(t) is the delta function.Definition 2.2.
The fractional (arbitrary) order derivative of the functionf of order 0≤α<1 is defined by
(2.2)Daαf(t)=ddt∫at‍(t-τ)-αΓ(1-α)f(τ)dτddtIa1-αf(t).Remark 2.3.
From Definitions2.1 and 2.2, a=0, we have
(2.3)Dαtμ=Γ(μ+1)Γ(μ-α+1)tμ-α,μ>-1;0<α<1,Iαtμ=Γ(μ+1)Γ(μ+α+1)tμ+α,μ>-1;α>0.
The Leibniz rule is
(2.4)Daα[f(t)g(t)]=∑k=0∞(αk)Daα-kf(t)Dakg(t)=∑k=0∞(αk)Daα-kg(t)Dakf(t),
where
(2.5)(αk)=Γ(α+1)Γ(k+1)Γ(α+1-k).
In this paper we deal with the following fractional differential equation:(2.6)Dαu(t,z)=a(t,z)uzz+b(t,z)uz+c(t,z)u,
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 orderμ>0 is defined, for a smooth function f(t), by
(2.7)cDμf(t):=1Γ(n-μ)∫0t‍f(n)(ζ)(t-ζ)μ-n+1dζ,
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(2.8)Daμf(t)=1Γ(1-μ)f(a)(t-a)μ+cDaμf(t),
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(3.1)u(t,z)=σ(z)u-(t,z),
where σ≠0 is a complex valued function of complex variable z∈U; therefor, we have
(3.2)σ(z)≔σ1(z)+iσ2(z),u(t,z)=v(t,z)+iw(t,z),a(t,z)=a1(t,z)+ia2(t,z),b(t,z)=b1(t,z)+ib2(t,z),c(t,z)=c1(t,z)+ic2(t,z),u¯(t,z)=v¯(t,z)+iw¯(t,z).Substituting (3.1) in (2.6) yields
(3.3)Dαu¯(t,z)=a¯(t,z)u¯zz+b¯(t,z)u¯z+c¯(t,z)u¯,
where
(3.4)a¯=a,b¯=b+2aσzσ,c¯=c+aσzzσ+bσzσ.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:
(3.5)Dαv=a1vzz-a2wzz+b1vz-b2wz+c1v-c2w,Dαw=a1wzz+a2vzz+b1wz+b2vz+c1w+c2v.Sinceu=v+iw=(σ1(z)+iσ2(z))(v¯(t,z)+iw¯(t,z))=σ1v¯-σ2w¯+i(σ1w¯+σ2v¯), then we obtain the relations
(3.6)v=σ1v¯-σ2w¯,w=σ2v¯+σ1w¯.
Consequently we have
(3.7)vz=σ1v¯z+σ1zv¯-σ2w¯z-σ2zw¯,vzz=σ1v¯zz+2σ1zv¯z+σ1zzv¯-σ2w¯zz-2σ2zw¯z-σ2zzw¯,wz=σ2v¯z+σ2zv¯+σ1w¯z+σ1zw¯,vzz=σ2v¯zz+2σ2zv¯z+σ2zzv¯+σ1w¯zz+2σ1zw¯z+σ1zzw¯.By the linearity of the fractional differential operatorDα, we impose that
(3.8)Dαv=σ1Dαv¯-σ2Dαw¯,Dαw=σ2Dαv¯+σ1Dαw¯.By injecting (3.7) and (3.8) in (3.5) we receive the following system:
(3.9)Dαv¯=a¯1v¯zz-a¯2w¯zz+b¯1v¯z-b¯2w¯z+c¯1v¯-c¯2w¯,Dαw¯=a¯1w¯zz+a¯2v¯zz+b¯1w¯z+b¯2v¯z+c¯1w¯+c¯2v¯,
where
(3.10)a¯1=a1,a¯2=a2,b¯1=b1+2σ1z(a1σ1+a2σ2)+2σ2z(a1σ2-a2σ1)σ12+σ22,b¯2=b2-2σ1z(a1σ2-a2σ1)+2σ2z(a2σ2-a1σ1)σ12+σ22,c¯1=c1+σ1zz(a1σ1+a2σ2)+σ2zz(a1σ2-a2σ1)+σ1z(b1σ1+b2σ2)+σ2z(b1σ2-b2σ1)σ12+σ22,c¯2=c2-σ1zz(a1σ2-a2σ1)+σ2zz(a1σ1-a2σ2)+σ1z(b1σ2-b2σ1)-σ2z(b1σ1+b2σ2)σ12+σ22.Hence we conclude the following result.Theorem 3.1.
Leta≠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(3.11)a2=0,b2=2σ1za1σ2-2σ2za1σ1σ12+σ22,c2=σ1zza1σ2+σ2zza1σ1+σ1z(b1σ2-b2σ1)-σ2z(b1σ1+b2σ2)σ12+σ22.
Then the fractional coupled system (3.5) is equivalent to the transformed fractional uncoupled system
(3.12)Dαv¯=a¯1v¯zz+b¯1v¯z+c¯1v¯,Dαw¯=a¯1w¯zz+b¯1w¯z+c¯1w¯,
where a¯1, b¯1, and c¯1 are defined in (3.10).
Now we proceed to consider the nonhomogeneous equation(3.13)Dαu(t,z)=a(t,z)uzz+b(t,z)uz+c(t,z)u+f(t,z),
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(3.14)u(t,z)=ρ(t,z)u¯(t,z),
where ρ≠0 is a complex valued function of complex variable z∈U, with
(3.15)ρ(t,z):=ρ1(t,z)+iρ2(t,z).
Substituting (3.14) in (3.13) and using the Leibniz rule imply
(3.16)Dαu¯(t,z)=a¯(t,z)u¯zz+b¯(t,z)u¯z+c¯(t,z)u¯+f¯(t,z),
where
(3.17)a¯=a,b¯=b+2aρzρ,c¯=c+aρzzρ+bρzρ,f¯=fρ-αρtρI1-αu¯.
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:
(3.18)Dαv=a1vzz-a2wzz+b1vz-b2wz+c1v-c2w+f1,Dαw=a1wzz+a2vzz+b1wz+b2vz+c1w+c2v+f2.
Since(3.19)v=ρ1v¯-ρ2w¯,w=ρ2v¯+ρ1w¯,
then by taking the fractional derivative of the variables v and w in (3.19), we obtain
(3.20)Dαv=ρ1Dαv¯-ρ2Dαw¯+h1,Dαw=ρ2Dαv¯+ρ1Dαw¯+h2,
where
(3.21)h1=ρ1tI1-αv¯-ρ2tI1-αw¯,h2=ρ2tI1-αv¯+ρ1tI1-αw¯.
Consequently, we impose the following system:(3.22)Dαv¯=a¯1v¯zz-a¯2w¯zz+b¯1v¯z-b¯2w¯z+c¯1v¯-c¯2w¯+f¯1,Dαw¯=a¯1w¯zz+a¯2v¯zz+b¯1w¯z+b¯2v¯z+c¯1w¯+c¯2v¯+f¯2,
where
(3.23)a¯1=a1,a¯2=a2,b¯1=b1+2ρ1z(a1ρ1+a2ρ2)+2ρ2z(a1ρ2-a2ρ1)ρ12+ρ22,b¯2=b2-2ρ1z(a1ρ2-a2ρ1)+2ρ2z(a2ρ2-a1ρ1)ρ12+ρ22,c¯1=c1+ρ1zz(a1ρ1+a2ρ2)+ρ2zz(a1ρ2-a2ρ1)+ρ1z(b1ρ1+b2ρ2)+ρ2z(b1ρ2-b2ρ1)ρ12+ρ22,c¯2=c2-ρ1zz(a1ρ2-a2ρ1)+ρ2zz(a1ρ1-a2ρ2)+ρ1z(b1ρ2-b2ρ1)-ρ2z(b1ρ1+b2ρ2)ρ12+ρ22,f¯1=ρ1(f1-h1)+ρ2(f2-h2)ρ12+ρ22,f¯2=ρ2(f1-h1)+ρ1(f2-h2)ρ12+ρ22.
Hence we conclude the following result.Theorem 3.3.
Leta≠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(3.24)a2=0,b2=2ρ1za1ρ2-2ρ2za1ρ1ρ12+ρ22,c2=ρ1zza1ρ2+ρ2zza1ρ1+ρ1z(b1ρ2-b2ρ1)-ρ2z(b1ρ1+b2ρ2)ρ12+ρ22.
Then the fractional coupled system (3.18) is equivalent to the transformed fractional uncoupled system
(3.25)Dαv¯=a¯1v¯zz+b¯1v¯z+c¯1v¯+f¯1,Dαw¯=a¯1w¯zz+b¯1w¯z+c¯1w¯+f¯2,
where a¯1, b¯1, and c¯1 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
(3.26)Dαu¯(t,z)=a¯(t,z)u¯zz+b¯(t,z)u¯z+c¯(t,z)u¯+g(t,z),
where
(3.27)a¯=a,b¯=b+2aρzρ,c¯=c+aρzzρ+bρzρ,g(t,z)=αρtρI1-αu¯:=g1(t,z)+ig2(t,z).A computation implies the following system:(3.28)Dαv=a1vzz-a2wzz+b1vz-b2wz+c1v-c2w+g1,Dαw=a1wzz+a2vzz+b1wz+b2vz+c1w+c2v+g2.Now in view of (3.19)–(3.21), we have the system
(3.29)Dαv¯=a¯1v¯zz-a¯2w¯zz+b¯1v¯z-b¯2w¯z+c¯1v¯-c¯2w¯+g¯1,Dαw¯=a¯1w¯zz+a¯2v¯zz+b¯1w¯z+b¯2v¯z+c¯1w¯+c¯2v¯+g¯2,
where
(3.30)a¯1=a1,a¯2=a2,b¯1=b1+2ρ1z(a1ρ1+a2ρ2)+2ρ2z(a1ρ2-a2ρ1)ρ12+ρ22,b¯2=b2-2ρ1z(a1ρ2-a2ρ1)+2ρ2z(a2ρ2-a1ρ1)ρ12+ρ22,c¯1=c1+ρ1zz(a1ρ1+a2ρ2)+ρ2zz(a1ρ2-a2ρ1)+ρ1z(b1ρ1+b2ρ2)+ρ2z(b1ρ2-b2ρ1)ρ12+ρ22,c¯2=c2-ρ1zz(a1ρ2-a2ρ1)+ρ2zz(a1ρ1-a2ρ2)+ρ1z(b1ρ2-b2ρ1)-ρ2z(b1ρ1+b2ρ2)ρ12+ρ22,g¯1=ρ1(g1-h1)+ρ2(g2-h2)ρ12+ρ22,g¯2=ρ2(g1-h1)+ρ1(g2-h2)ρ12+ρ22.Thus we receive the following result.Theorem 3.5.
Leta≠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(3.31)a2=0,b2=2ρ1za1ρ2-2ρ2za1ρ1ρ12+ρ22,c2=ρ1zza1ρ2+ρ2zza1ρ1+ρ1z(b1ρ2-b2ρ1)-ρ2z(b1ρ1+b2ρ2)ρ12+ρ22.
Then the fractional coupled system (3.28) is equivalent to the transformed fractional uncoupled system
(3.32)Dαv¯=a¯1v¯zz+b¯1v¯z+c¯1v¯+g¯1,Dαw¯=a¯1w¯zz+b¯1w¯z+c¯1w¯+g¯2,
where a¯1,b¯1, and c¯1 are defined in (3.30).


## 3.1. Equivalence Transforms

Assume the complex transform(3.1)u(t,z)=σ(z)u-(t,z),
where σ≠0 is a complex valued function of complex variable z∈U; therefor, we have
(3.2)σ(z)≔σ1(z)+iσ2(z),u(t,z)=v(t,z)+iw(t,z),a(t,z)=a1(t,z)+ia2(t,z),b(t,z)=b1(t,z)+ib2(t,z),c(t,z)=c1(t,z)+ic2(t,z),u¯(t,z)=v¯(t,z)+iw¯(t,z).Substituting (3.1) in (2.6) yields
(3.3)Dαu¯(t,z)=a¯(t,z)u¯zz+b¯(t,z)u¯z+c¯(t,z)u¯,
where
(3.4)a¯=a,b¯=b+2aσzσ,c¯=c+aσzzσ+bσzσ.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:
(3.5)Dαv=a1vzz-a2wzz+b1vz-b2wz+c1v-c2w,Dαw=a1wzz+a2vzz+b1wz+b2vz+c1w+c2v.Sinceu=v+iw=(σ1(z)+iσ2(z))(v¯(t,z)+iw¯(t,z))=σ1v¯-σ2w¯+i(σ1w¯+σ2v¯), then we obtain the relations
(3.6)v=σ1v¯-σ2w¯,w=σ2v¯+σ1w¯.
Consequently we have
(3.7)vz=σ1v¯z+σ1zv¯-σ2w¯z-σ2zw¯,vzz=σ1v¯zz+2σ1zv¯z+σ1zzv¯-σ2w¯zz-2σ2zw¯z-σ2zzw¯,wz=σ2v¯z+σ2zv¯+σ1w¯z+σ1zw¯,vzz=σ2v¯zz+2σ2zv¯z+σ2zzv¯+σ1w¯zz+2σ1zw¯z+σ1zzw¯.By the linearity of the fractional differential operatorDα, we impose that
(3.8)Dαv=σ1Dαv¯-σ2Dαw¯,Dαw=σ2Dαv¯+σ1Dαw¯.By injecting (3.7) and (3.8) in (3.5) we receive the following system:
(3.9)Dαv¯=a¯1v¯zz-a¯2w¯zz+b¯1v¯z-b¯2w¯z+c¯1v¯-c¯2w¯,Dαw¯=a¯1w¯zz+a¯2v¯zz+b¯1w¯z+b¯2v¯z+c¯1w¯+c¯2v¯,
where
(3.10)a¯1=a1,a¯2=a2,b¯1=b1+2σ1z(a1σ1+a2σ2)+2σ2z(a1σ2-a2σ1)σ12+σ22,b¯2=b2-2σ1z(a1σ2-a2σ1)+2σ2z(a2σ2-a1σ1)σ12+σ22,c¯1=c1+σ1zz(a1σ1+a2σ2)+σ2zz(a1σ2-a2σ1)+σ1z(b1σ1+b2σ2)+σ2z(b1σ2-b2σ1)σ12+σ22,c¯2=c2-σ1zz(a1σ2-a2σ1)+σ2zz(a1σ1-a2σ2)+σ1z(b1σ2-b2σ1)-σ2z(b1σ1+b2σ2)σ12+σ22.Hence we conclude the following result.Theorem 3.1.
Leta≠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(3.11)a2=0,b2=2σ1za1σ2-2σ2za1σ1σ12+σ22,c2=σ1zza1σ2+σ2zza1σ1+σ1z(b1σ2-b2σ1)-σ2z(b1σ1+b2σ2)σ12+σ22.
Then the fractional coupled system (3.5) is equivalent to the transformed fractional uncoupled system
(3.12)Dαv¯=a¯1v¯zz+b¯1v¯z+c¯1v¯,Dαw¯=a¯1w¯zz+b¯1w¯z+c¯1w¯,
where a¯1, b¯1, and c¯1 are defined in (3.10).
Now we proceed to consider the nonhomogeneous equation(3.13)Dαu(t,z)=a(t,z)uzz+b(t,z)uz+c(t,z)u+f(t,z),
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(3.14)u(t,z)=ρ(t,z)u¯(t,z),
where ρ≠0 is a complex valued function of complex variable z∈U, with
(3.15)ρ(t,z):=ρ1(t,z)+iρ2(t,z).
Substituting (3.14) in (3.13) and using the Leibniz rule imply
(3.16)Dαu¯(t,z)=a¯(t,z)u¯zz+b¯(t,z)u¯z+c¯(t,z)u¯+f¯(t,z),
where
(3.17)a¯=a,b¯=b+2aρzρ,c¯=c+aρzzρ+bρzρ,f¯=fρ-αρtρI1-αu¯.
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:
(3.18)Dαv=a1vzz-a2wzz+b1vz-b2wz+c1v-c2w+f1,Dαw=a1wzz+a2vzz+b1wz+b2vz+c1w+c2v+f2.
Since(3.19)v=ρ1v¯-ρ2w¯,w=ρ2v¯+ρ1w¯,
then by taking the fractional derivative of the variables v and w in (3.19), we obtain
(3.20)Dαv=ρ1Dαv¯-ρ2Dαw¯+h1,Dαw=ρ2Dαv¯+ρ1Dαw¯+h2,
where
(3.21)h1=ρ1tI1-αv¯-ρ2tI1-αw¯,h2=ρ2tI1-αv¯+ρ1tI1-αw¯.
Consequently, we impose the following system:(3.22)Dαv¯=a¯1v¯zz-a¯2w¯zz+b¯1v¯z-b¯2w¯z+c¯1v¯-c¯2w¯+f¯1,Dαw¯=a¯1w¯zz+a¯2v¯zz+b¯1w¯z+b¯2v¯z+c¯1w¯+c¯2v¯+f¯2,
where
(3.23)a¯1=a1,a¯2=a2,b¯1=b1+2ρ1z(a1ρ1+a2ρ2)+2ρ2z(a1ρ2-a2ρ1)ρ12+ρ22,b¯2=b2-2ρ1z(a1ρ2-a2ρ1)+2ρ2z(a2ρ2-a1ρ1)ρ12+ρ22,c¯1=c1+ρ1zz(a1ρ1+a2ρ2)+ρ2zz(a1ρ2-a2ρ1)+ρ1z(b1ρ1+b2ρ2)+ρ2z(b1ρ2-b2ρ1)ρ12+ρ22,c¯2=c2-ρ1zz(a1ρ2-a2ρ1)+ρ2zz(a1ρ1-a2ρ2)+ρ1z(b1ρ2-b2ρ1)-ρ2z(b1ρ1+b2ρ2)ρ12+ρ22,f¯1=ρ1(f1-h1)+ρ2(f2-h2)ρ12+ρ22,f¯2=ρ2(f1-h1)+ρ1(f2-h2)ρ12+ρ22.
Hence we conclude the following result.Theorem 3.3.
Leta≠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(3.24)a2=0,b2=2ρ1za1ρ2-2ρ2za1ρ1ρ12+ρ22,c2=ρ1zza1ρ2+ρ2zza1ρ1+ρ1z(b1ρ2-b2ρ1)-ρ2z(b1ρ1+b2ρ2)ρ12+ρ22.
Then the fractional coupled system (3.18) is equivalent to the transformed fractional uncoupled system
(3.25)Dαv¯=a¯1v¯zz+b¯1v¯z+c¯1v¯+f¯1,Dαw¯=a¯1w¯zz+b¯1w¯z+c¯1w¯+f¯2,
where a¯1, b¯1, and c¯1 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
(3.26)Dαu¯(t,z)=a¯(t,z)u¯zz+b¯(t,z)u¯z+c¯(t,z)u¯+g(t,z),
where
(3.27)a¯=a,b¯=b+2aρzρ,c¯=c+aρzzρ+bρzρ,g(t,z)=αρtρI1-αu¯:=g1(t,z)+ig2(t,z).A computation implies the following system:(3.28)Dαv=a1vzz-a2wzz+b1vz-b2wz+c1v-c2w+g1,Dαw=a1wzz+a2vzz+b1wz+b2vz+c1w+c2v+g2.Now in view of (3.19)–(3.21), we have the system
(3.29)Dαv¯=a¯1v¯zz-a¯2w¯zz+b¯1v¯z-b¯2w¯z+c¯1v¯-c¯2w¯+g¯1,Dαw¯=a¯1w¯zz+a¯2v¯zz+b¯1w¯z+b¯2v¯z+c¯1w¯+c¯2v¯+g¯2,
where
(3.30)a¯1=a1,a¯2=a2,b¯1=b1+2ρ1z(a1ρ1+a2ρ2)+2ρ2z(a1ρ2-a2ρ1)ρ12+ρ22,b¯2=b2-2ρ1z(a1ρ2-a2ρ1)+2ρ2z(a2ρ2-a1ρ1)ρ12+ρ22,c¯1=c1+ρ1zz(a1ρ1+a2ρ2)+ρ2zz(a1ρ2-a2ρ1)+ρ1z(b1ρ1+b2ρ2)+ρ2z(b1ρ2-b2ρ1)ρ12+ρ22,c¯2=c2-ρ1zz(a1ρ2-a2ρ1)+ρ2zz(a1ρ1-a2ρ2)+ρ1z(b1ρ2-b2ρ1)-ρ2z(b1ρ1+b2ρ2)ρ12+ρ22,g¯1=ρ1(g1-h1)+ρ2(g2-h2)ρ12+ρ22,g¯2=ρ2(g1-h1)+ρ1(g2-h2)ρ12+ρ22.Thus we receive the following result.Theorem 3.5.
Leta≠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(3.31)a2=0,b2=2ρ1za1ρ2-2ρ2za1ρ1ρ12+ρ22,c2=ρ1zza1ρ2+ρ2zza1ρ1+ρ1z(b1ρ2-b2ρ1)-ρ2z(b1ρ1+b2ρ2)ρ12+ρ22.
Then the fractional coupled system (3.28) is equivalent to the transformed fractional uncoupled system
(3.32)Dαv¯=a¯1v¯zz+b¯1v¯z+c¯1v¯+g¯1,Dαw¯=a¯1w¯zz+b¯1w¯z+c¯1w¯+g¯2,
where a¯1,b¯1, and c¯1 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 functionX(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
(4.1)∥X(t,x)-X(t,y)∥≤L∥x-y∥,
whenever (t,x),(t,y)∈ℛ.
A convex domainD 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]).
IfX(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
(4.2)(Dαv¯Dαw¯)=(a-1-a-2b-1-b-2c-1-c-2a-2a-1b-2b-1c-2c-1)(v-zzw-zzv-zw-zv-w-)≔AZ,
where Z=(Z1(t,z),…,Z6(t,z))T. Since the domain 𝒟=J×U is bounded closed convex and all the functions Zi,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
(4.3)(Dαv¯Dαw¯)=(a-1-a-2b-1-b-2c-1-c-2a-2a-1b-2b-1c-2c-1)(v-zzw-zzv-zw-zv-w-)+(f-1f-2)≔AZ+F.
Hence Lemma4.2 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(5.1)Dαu(t,z)=Az2uzz+Bzuz+Cu,
where A≠0,B and C are real constants, and the complex transform
(5.2)u(t,z)=z2u¯(t,z):=(σ1+iσ2)(v¯+iw¯).
Equation (5.1) reduces to the homogeneous type
(5.3)Dαu¯(t,z)=A¯z2uzz+B¯zuz+C¯u,
where
(5.4)A¯=A,B¯=(4A+B),C¯=2A+2B+C.
Hence (5.3) can be translated to the uncoupled system
(5.5)Dαv¯=A¯σ1v¯zz+B¯(±r+σ12)v¯z+C¯v¯,Dαw¯=A¯σ2w¯zz+B¯(±r-σ12)w¯z+C¯w¯,
where r=σ12+σ22. Furthermore, it can be represented as coupled system
(5.6)Dαv¯=A¯σ1v¯zz-A¯σ2w¯zz+B¯(±r+σ12)vz-B¯(±r-σ12)w¯z+C¯v¯-C¯w¯,Dαw¯=A¯σ1w¯zz+A¯σ2v¯zz+B¯(±r+σ12)w¯z+B¯(±r-σ12)v¯z+C¯w¯+C¯v¯.Example 5.2.
Consider the equation(5.7)Dαu(t,z)=Atz2uzz+Btzuz+Ctu+eit,t∈[a,1],a≠0,
where A≠0,B and C are real constants, and the complex transform
(5.8)u(t,z)=tz2u¯(t,z):=t(ρ1+iρ2)(v¯+iw¯).
Thus (5.7) becomes
(5.9)Dαu¯(t,z)=A¯tz2uzz+B¯tzuz+C¯tu+f¯,
where
(5.10)A¯=A,B¯=(4A+B),C¯=2A+2B+C,f¯=eit-αI1-αu¯t:=(cost-ϕ1)+i(sint-ϕ2):=f¯1+if¯2.
Hence (5.7) can be formulated to the uncoupled system
(5.11)Dαv¯=A¯ρ1v¯zz+B¯t(±r+ρ12)v¯z+C¯tv¯+f¯1,Dαw¯=A¯ρ2w¯zz+B¯t(±r-ρ12)w¯z+C¯tw¯+f¯2,
where r=ρ12+ρ22. Furthermore, it can be represented as coupled system
(5.12)Dαv¯=A¯ρ1v¯zz-A¯ρ2w¯zz+B¯t(±r-ρ12)vz-B¯t(±r-ρ12)w¯z+C¯tv¯-C¯tw¯+f¯1,Dαw¯=A¯ρ1w¯zz+A¯ρ2v¯zz+B¯t(±r+ρ12)w¯z+B¯t(±r-ρ12)v¯z+C¯tw¯+C¯tv¯+f¯2.
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(5.13)Dαu(t,z)=uzz+f(z)p(t),
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
(5.14)Dαv¯=v¯zz-b¯1v¯z-b¯2w¯z+c¯1v¯-c¯2w¯+f¯1,Dαw¯=w¯zz+b¯1w¯z+b¯2v¯z+c¯1w¯+c¯2v¯+f¯2,
where f¯=(fp(t)/ρ)-(αρt/ρ)I1-αu¯:=f¯1+if¯2,
(5.15)a¯1=1,a¯2=0,b¯1=2ρ1z(a1ρ1)+2ρ2z(a1ρ2)ρ12+ρ22,b¯2=-2ρ1z(a1ρ2)+2ρ2z(-a1ρ1)ρ12+ρ22,c¯1=ρ1zz(a1ρ1)+ρ2zz(a1ρ2)ρ12+ρ22,c¯2=-ρ1zz(a1ρ2)+ρ2zz(a1ρ1)ρ12+ρ22,f¯1=ρ1(f1-h1)+ρ2(f2-h2)ρ12+ρ22,f¯2=ρ2(f1-h1)+ρ1(f2-h2)ρ12+ρ22.


## 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.



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*Source: 814759-2012-11-26.xml*