# Using Homo-Separation of Variables for Solving Systems of Nonlinear Fractional Partial Differential Equations

Authors: Abdolamir Karbalaie; Hamed Hamid Muhammed; Bjorn-Erik Erlandsson
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## Abstract

A new method proposed and coined by the authors as the homo-separation of variables method is utilized to solve systems of linear and nonlinear fractional partial differential equations (FPDEs). The new method is a combination of two well-established mathematical methods, namely, the homotopy perturbation method (HPM) and the separation of variables method. When compared to existing analytical and numerical methods, the method resulting from our approach shows that it is capable of simplifying the target problem at hand and reducing the computational load that is required to solve it, considerably. The efficiency and usefulness of this new general-purpose method is verified by several examples, where different systems of linear and nonlinear FPDEs are solved.


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## Body

## 1. Introduction

In the recent years, it has turned out that many phenomena in various technical and scientific fields can be described very successfully by using fractional calculus. In particular, fractional calculus can be employed to solve many problems within the biomedical research field and get better results. Such a practical application of fractional order models is to use these models to improve the behavior and efficiency of bioelectrodes. The importance of this application is based on the fact that bioelectrodes are usually needed to be used for all types of biopotential recording and signal measurement purposes such as electroencephalography (EEG), electrocardiography (ECG), and electromyography (EMG) [1–3]. Another promising biomedical application field is proposed by Arafa et al. [4] where a fractional-order model of HIV-1 infection of CD4+ T cells is introduced. Other examples of applications of fractional calculus in life sciences and technology can be found in botanics [5], biology [6], rheology [7], and elastography [8].Numerous analytical methods have been presented in the literature to solve FPDEs, such as the fractional Greens function method [9], the Fourier transform method [2], the Sumudu transform method [10], the Laplace transform method, and the Mellin transform method [11]. Some numerical methods have also widely been used to solve systems of FPDEs, such as the variational iteration method [12], the Adomian decomposition method [2], the homotopy perturbation method [13] and the homotopy analysis method [14]. Some of these methods use specific transformations and others give the solution as a series which converges to the exact solution.In addition, some numerical methods use a combination of utilizing specific transformations and obtaining series which converge to the exact solutions. An example of such a method is the iterative Laplace transform method which is a combination of the Laplace transform method and an iterative method [15]. Another such a combination is the homotopy perturbation transformation method, which is constructed by combining two powerful methods, namely, the Laplace transform method and the homotopy perturbation method [16]. A third example is the Sumudu decomposition method, which is a combination of the Sumudu transform method and Adomian decomposition method [17]. A fourth such an approach is combining the Sumudu transformation method with the homotopy perturbation method, which gives a new method called the homotopy perturbation Sumudu transform method [18].Recently, the homotopy perturbation method and the Adomian decomposition method are frequently used for solving nonlinear FPDE problems.According to the combinational methods mentioned above, it is possible to notice that HPM has a strong potential to be combined with another method to produce a more efficient approach. The main reason is that HPM is an efficient method for solving PDEs and ODEs (ordinary differential equations) with integer or fractional order. Recently, Karbalaie et al. [19] found the exact solution of one-dimensional FPDEs by using truncated versions of modified HPM. Therefore, by getting inspiration of the ideas, methods, and tools of previous works, the novel approach called the homo-separation of variables method is developed and utilized to find the exact solutions of systems of FPDEs. The methods presented in Karbalaie et al. [19] are extended and developed further to achieve methods that can solve n-dimensional systems of FPDEs. This new approach is constructed by a smart combination of HPM and the separation of variables method. By using this method, the system of FPDEs to be solved is changed into a system of FODEs. Consequently, the original problem is simplified considerably which makes it more straightforward and easier to solve.The structure of the current paper is as follows. For easy reader-friendly reference, some basic definitions and properties of fractional differential equations are presented in Section2. In Section 3, the homo-separation of variables method is described. After that, in Section 4, three examples are presented to show the simplicity and efficiency (at the same time) of the homo-separation of variables method. Finally, in Section 5, relevant conclusions are drawn.


## 2. Basic Definitions and Properties

In this section, to make it easier to introduce the new approach of this research work, a number of good-to-know concepts and properties within the field of fractional differential equations are defined and presented briefly.Definition 1.
A real functionf(t),t>0, is said to be in the space Cσ,σ∈ℝ, if there exists a real numberp>σ, such that f(t)=tpf1(t), where f1(t)∈C[0,∞), and it is said to be in the space Cσm if fm∈Cσ,m∈ℕ.Definition 2.
The left sided Riemann-Liouville fractional integral of orderα≥0, of a function f∈Cσ,σ≥-1, is defined as
(1)Jtαf(t)=def1Γ(α)∫0t‍(t-τ)α-1f(τ)dτ,
where α>0,t>0, and Γ(α)=∫0∞‍rα-1e-rdr is the gamma function.
Properties of the operatorJtα (for f∈Cμ,μ≥-1,α,β≥0,γ≥-1) can be found in [20] and are defined as follows:(1)
Jt0f(t)=f(t),(2)
JtαJtβf(t)=Jtα+βf(t),(3)
Jtαtγ=(Γ(γ+1)/Γ(α+γ+1))tα+γ.Definition 3.
The left sided Caputo fractional derivative off,(f∈Cμn,n∈ℕ∪{0}), in Caputo sense is defined by [11] as follows:
(2)Dtαf(t)=def{1Γ(n-α)∫0t‍(t-τ)n-α-1f(n)(τ)dτ,n-1<α≤n,Dtnf(t),α=n.
Note that, according to [21], (1) and (2) become
(3)Jtαf(x,t)=1Γ(α)∫0t‍(t-τ)α-1f(x,τ)dτ,α>0,t>0,Dtαf(x,t)=1Γ(n-α)∫0t‍(t-τ)n-α-1f(n)(x,τ)dτ,n-1<α≤n.Definition 4.
The single-parameter and the two-parameter variants of the Mittag-leffler function are denoted byEα(t) and Eα,β(t), respectively. These two variants are relevant to be considered here because of their connection with fractional calculus and are defined as
(4)Eα(t)=∑j=0∞‍tjΓ(αj+1),α>0,t∈ℂ,Eα,β(t)=∑j=0∞‍tjΓ(αj+β),α,β>0,t∈ℂ.
Thekth derivatives of these two variants are
(5)Eα(k)(t)=dkdtkEα(t)=∑j=0∞‍(k+j)!tjj!Γ(αj+αk+1),k=0,1,2,…,Eα,β(k)(t)=dkdtkEα,β(t)=∑j=0∞‍(k+j)!tjj!Γ(αj+αk+β),k=0,1,2,….
Some special cases and properties of the Mittag-Leffler functions are the following:(1)
Eα,1(t)=Eα(t),(2)
E1,1(t)=E1(t)=et,(3)
E2,1(t2)=cosh(t),(4)
E2,2(t2)=sinh(t)/t,(5)
Eα,β(t)+Eα,β(-t)=2E2α,β(t2),(6)
Eα,β(t)-Eα,β(-t)=2tE2α,α+β(t2),(7)
Dtα(tα-1Eα,α(atα))=tα-1Eα,α(atα).
Further properties of the Mittag-Leffler functions can be found in [21]. These functions are generalizations of the exponential function; therefore, most linear differential equations of fractional order (FPDEs) have solutions that are expressed in terms of these functions.Important Note. According to [22], Eα(atα) cannot satisfy the following semigroup property:
(6)Eα(a(t+s)α)=Eα(atα)Eα(asα),t,s≥0,a∈ℝ
for any noninteger α>0. The above property is not true unless in the special cases where α=1 or =0. Unfortunately, this misunderstanding can be found in a number of publications, which can be misleading and confusing.Definition 5.
Consider the following relaxation-oscillation equation, which is discussed in [23]:
(7)Dtαu(t)+Au(t)=f(t),t>0,
subject to the initial condition
(8)∂i∂tiu(t)|t=0=ui(0)=bii=1,2,…,n-1,
where bi(i=1,2,…,n-1) are real constants and n-1<α≤n. The solution of (7) is obtained according to [23] as follows:
(9)u(t)=∫0t‍(t-τ)α-1Eα,α(-A(t-τ)α)f(τ)dτ+∑j=0n-1‍bjDtα-j-1(tα-1Eα,α(-Atα)),
where Eα,α is the Mittag-Leffler function. It is easy to realize that if 0<α≤1, then the solution of (7) becomes as follows:
(10)u(t)=∫0t‍(t-τ)α-1Eα,α(-A(t-τ)α)f(τ)dτ+Dtα-1(tα-1Eα,α(-Atα)),
which will be used to solve the systems of FPDEs in the examples discussed in this paper.Definition 6.
According to [24], consider the following linear system of fractional differential equations with its initial condition:
(11)Dtαx(t)=Ax(t),0<t≤a,x(0)=x0,
where x∈ℝn,a>0, the matrix A∈ℝn×ℝn and Dtα is the Caputo fractional derivative of orderα, where 0<α≤1. The general analytical solution for (11) is given by
(12)x(t)=c1u(1)Eα(λ1tα)+c2u(2)Eα(λ2tα)+⋯+cnu(n)Eα(λntα),
where c1,c2,…,cn are arbitrary constants, Eα is the Mittag-Leffler function, and λ1,λ2,…,λn and u(1),u(2),…,u(n)are the eigenvalues and the corresponding eigenvectors of the characteristic equation (A-λiI)u(i)=0, where i=1,2,…,n and I is an n×n identity matrix.


## 3. The Homo-Separation of Variables Method

This approach is achieved by a novel combination of HPM and separation of variables. In this section, the algorithm of this new method is presented briefly by considering the following system of FPDEs:(13)Dtαiui=fi(x¯,t)+ℒi(u1,u2,…,un)+𝒩i(u1,u2,…,un),i=1,2,…,n,
subject to the initial conditions
(14)ui(x¯,0)=gi(x¯),
where x¯∈ℝn-1, and for i=1,2,…,n, the terms ℒi represent linear operators, the terms 𝒩i are nonlinear operators, while the terms fi represent known analytical functions. In addition, Dtαi=∂αi/∂tαi is the Caputo fractional derivative of order αi, where 0<αi≤1 for i=1,2,…,n.According to the homotopy perturbation technique (HPM), we can construct the following homotopies:(15)(p-1)(DtαiVi-Dtαiui0)+p(DtαiVi-ℒi~-𝒩i~-fi(x¯,t))=0,i=1,2,…,n,
where ℒ~i=ℒi(V1,V2,…,Vn), 𝒩~i=𝒩i(V1,V2,…,Vn), and the homotopy parameter, denoted by p, is considered as small (p∈[0,1]). In addition, fori=1,2,…,n, ui0=ui0(x¯,t) is an initial approximation of the solution of (13) which satisfies the initial condition in (14).We can assume that the solution of (15) can be expressed as a power series in p, as given below in (16). 						(16)V1=∑j=0∞‍pjV1j=V10+pV11+p2V12+p3V13+⋯,V2=∑j=0∞‍pjV2j=V20+pV21+p2V22+p3V23+⋯,⋮Vn=∑j=0∞‍pjVnj=Vn0+pVn1+p2Vn2+p3Vn3+⋯.By substituting (16) into (15) and equating the terms with identical powers of p, a set of equations is obtained as follows:
(17)p0:DtαiVi0-Dtαiui0=0,(18)p1:DtαiVi1+Dtαiu10-ℒi(V10,V20,…,Vn0)-ℳi0-fi=0,(19)⋮pj:DtαiVij-ℒi(V1j-1,V2j-1,…,Vnj-1)-ℳij-1=0,
where the terms ℳij(i=1,2,…,n and j=0,1,…) are obtained from (17)–(19) by equating the coefficients of the nonlinear operators 𝒩ij(i=1,2,…,n  andj=0,1,…) with the identical powers j of p, that is, the terms containing pj.In case thep-parameter is considered as small, the best approximate solution of (13) can be readily obtained as follows:
(20)ui(x¯,t)=limp→1∑j=0∞‍pjVij(x¯,t)=Vi0(x¯,t)+Vi1(x¯,t)+Vi2(x¯,t)+⋯,i=1,2,…,n,
where x¯=(x1,x2,…,xn-1).If in (20) there exists some ViN(x¯,t)=0, where N≥1 and i=1,2,…,n, then the exact solution can be written in the following form:
(21)ui(x¯,t)=Vi0(x¯,t)+pVi1(x¯,t)+⋯+pN-1ViN-1(x¯,t),i=1,2,…,n.For simplicity, we assume thatVi1(x¯,t)≡0 in (21), which means that the exact solution in (13) is simplified as follows:
(22)ui(x¯,t)=Vi0(x¯,t),i=1,2,…,n.Now when solving (17), we obtain the following result:
(23)Vi0(x¯,t)=ui0(x¯,t),i=1,2,…,n.By using (22) and (23) we have
(24)ui(x¯,t)=Vi0(x¯,t)=ui0(x¯,t),i=1,2,…,n.The core of the new method proposed in this research work is to formulate the initial approximation of (13) in the form of separation of variables, as follows:
(25)ui(x¯,t)=ui0(x¯,t)=ui(x¯,0)ci1(t)+(∑j=1n‍Dxjui(x¯,0))ci2(t)=gi(x¯)ci1(t)+(∑j=1n‍Dxjgi(x¯))ci2(t),i=1,2,…,n.The task now is to find the termsci1(t) and ci2(t) to obtain the exact solution of the system of FPDEs in (13). Since (25) satisfies the initial conditions in (14), we get
(26)ci1(0)=1,ci2(0)=0,i=1,2,…,n.By substituting (25) into (18), we get
(27)DtαiVi1=-Dtαi(gi(x¯)ci1(t)+(∑j=1n‍Dxjgi(x¯))ci2(t))-Li-Mi0+fi(x¯,t)≡0,
wherei=1,2,…,n and
(28)Li=ℒi(g1(x¯)c11(t)+(∑j=1n‍Dxjg1(x¯))c12(t),…,ℒi(gn(x¯)cn1(t)+(∑j=1n‍Dxjgn(x¯))cn2(t))),Mi0=ℳi0(g1(x¯)c11(t)+(∑j=1n‍Dxjg1(x¯))c12(t),…,ℳi0(gn(x¯)cn1(t)+(∑j=1n‍Dxjgn(x¯))cn2(t))).Obviously, our goal is finally achieved and the system of FPDEs is changed into a system of FODEs. Consequently, the problem at hand is simplified considerably as it is well known that solving a system of FODEs is generally more straightforward and much easier than solving the corresponding system of FPDEs.Finally, the target unknownsci1(t) and ci2(t) can be obtained by utilizing and solving the system of FODEs and the initial conditions in (26).


## 4. Applications

In this section, we illustrate the applicability, simplicity, and efficiency of the homo-separation of variables method for solving systems of linear and nonlinear FPDEs.Example 7.
Consider the following system of linear FPDEs:(29)Dtαu=vx-v-u,Dtαv=ux-v-u,
where 0<α≤1, subject to the initial conditions
(30)u(x,0)=sinh(x),v(x,0)=cosh(x).
To solve (29) and (30) by using the proposed homo-separation of variables method, we choose the initial approximations for (29) as follows:
(31)u0(x,t)=u(x,0)b1(t)+Dxu(x,0)b2(t),v0(x,t)=v(x,0)c1(t)+Dxv(x,0)c2(t).
By substituting the two initial conditions given by (30) into these two initial approximations, we get the following initial approximations:
(32)u0(x,t)=b1(t)sinh(x)+b2(t)cosh(x),v0(x,t)=c1(t)cosh(x)+c2(t)sinh(x).
By considering and following the approach which resulted in (27) and substituting the results presented by (32) into the system of linear FPDEs in (29), we get
(33)Dtu1=sinh(x)(Dtαb1(t)+b1(t)-c1(t)+c2(t))+cosh(x)(Dtαb2(t)+b2(t)+c1(t)-c2(t))≡0,Dtv1=cosh(x)(Dtαc1(t)+c1(t)-b1(t)+b2(t))+sinh(x)(Dtαc2(t)+c2(t)+b1(t)-b2(t))≡0.
By solving the latter system of linear FPDEs presented in (33), we obtain the following system of linear FODEs:
(34)Dtαb1(t)+b1(t)-c1(t)+c2(t)=0,Dtαb2(t)+b2(t)+c1(t)-c2(t)=0,Dtαc1(t)+c1(t)-b1(t)+b2(t)=0,Dtαc2(t)+c2(t)+b1(t)-b2(t)=0
together with the initial conditions
(35)b2(0)=1,b2(0)=0,c1(0)=1,c1(0)=0.
By solving (34) and (35) by considering and applying Definition 6 which is discussed in Section 2, we obtain the following results:
(36)b1(t)=12(Eα(tα)+Eα(-tα)),b2(t)=12(-Eα(tα)+Eα(-tα)),c1(t)=12(Eα(tα)+Eα(-tα)),c2(t)=12(-Eα(tα)+Eα(-tα)).
By substituting (36) into (32) and applying Definition 4, in particular properties (2), (5) and (6) which are discussed in Section 2, we obtain the exact solution for the system of linear FPDEs in (29), as follows:
(37)u(x,t)=E2α,1(t2α)sinh(x)-tαE2α,α+1(t2α)cosh(x),v(x,t)=E2α,1(t2α)cosh(x)-tαE2α,α+1(t2α)sinh(x).
If we now putα→1 in (37) or solve (29) and (30) for α=1, we obtain the following exact solution for the corresponding system of linear PDEs:
(38)u(x,t)=sinh(x-t),v(x,t)=cosh(x-t).Example 8.
Consider the following system of inhomogeneous FPDEs:(39)Dtαu+uxv+u=1,Dtβv-vxu-v=1,
where 0<α and β≤1, subject to the initial conditions
(40)u(x,0)=ex,v(x,0)=e-x.
To solve (39) and (40) by using the new homo-separation of variables method, we choose the initial approximations for (39) as follows:
(41)u0(x,t)=u(x,0)c1(t)+Dxu(x,0)c2(t),v0(x,t)=v(x,0)b1(t)+Dxv(x,0)b2(t).
By substituting the two initial conditions given by (40) into these two initial approximations, we get the following initial approximations:
(42)u0(x,t)=exc1(t)+exc2(t),v0(x,t)=e-xb1(t)-e-xb2(t).
By considering and following the approach which resulted in (27) and substituting the results presented by (42) into the system of inhomogeneous FPDEs that we would like to solve, which is presented by (39), we get
(43)Dtαu1=ex(Dtα(c1(t)+c2(t))+(c1(t)+c2(t)))+(c1(t)+c2(t))(b1(t)-b2(t))-1≡0,Dtβv1=ex(Dtβ(b1(t)-b2(t))-(b1(t)-b2(t)))+(c1(t)+c2(t))(b1(t)-b2(t))-1≡0.
By solving the latter system of FPDEs presented in (43), we obtain the following system of FODEs:
(44)Dtα(c1(t)+c2(t))+(c1(t)+c2(t))=0,Dtβ(b1(t)-b2(t))-(b1(t)-b2(t))=0,(c1(t)+c2(t))(b1(t)-b2(t))-1=0
together with the initial conditions
(45)b1(0)=1,b2(0)=0,c1(0)=1,c2(t)=0.
By solving (44) and (45) by considering and applying Definition 5 which is discussed in Section 2, we obtain the following results:
(46)c1(t)+c2(t)=Eα(-tα),b1(t)-b2(t)=Eβ(tβ),Eα(-tα)Eβ(tβ)=1.
By substituting (46) into (42), we obtain the exact solution for the system of inhomogeneous FPDEs presented in (39), as follows:
(47)u(x,t)=exEα(-tα),v(x,t)=e-xEβ(tβ),Eα(-tα)Eβ(tβ)=1.
Thus, this means thatu(x,t)=1/v(x,t). If we now put α,β→1 in (47), or solve (39) and (40) for α=β=1, we obtain the following exact solution for the corresponding system of inhomogeneous PDEs:
(48)u(x,t)=ex-t,v(x,t)=e-x+t.Example 9.
Consider the following system of nonlinear FPDEs:(49)Dtαu=-u-vxwy-vywx,Dtβv=v-uxwy-uywx,Dtγw=w-uxvy-uyvx,
where 0<α  and  β,γ≤1,  subject to the initial conditions
(50)u(x,y,0)=ex+y,v(x,y,0)=ex-y,w(x,y,0)=e-x+y.
To solve (49) and (50) by using the new homo-separation of variables method, we choose the initial approximations for (49) as follows:
(51)u0(x,y,t)=u(x,y,0)b1(t)+(Dxu(x,y,0)+Dyu(x,y,0))b2(t),v0(x,y,t)=v(x,y,0)c1(t)+(Dxv(x,y,0)+Dyv(x,y,0))c2(t),w0(x,y,t)=w(x,y,0)d1(t)+(Dxw(x,y,0)+Dyw(x,y,0))d2(t).
By substituting the initial conditions given by (50) into these initial approximations, we get the following initial approximations:
(52)u0(x,y,t)=ex+yb1(t)+2ex+yb2(t),v0(x,y,t)=ex-yc1(t),w0(x,y,t)=e-x+yd1(t).
By considering and following the approach which resulted in (27) and substituting the results presented by (52) into the system of nonlinear FPDEs that we would like to solve, which is presented by (49), we get
(53)Dtαu1=e-x+y(Dtαb1(t)+2Dtαb2(t)+b1(t)+2b2(t))≡0,DtβDtv1=ex-y(Dtβc1(t)-c1(t))≡0,Dtγw1=e-x+y(Dtγd1(t)-d1(t))≡0.
By solving the latter system of FPDEs presented in (53), we obtain the following system of FODEs:
(54)Dtα(b1(t)+2b2(t))+(b1(t)+2b2(t))=0,Dtβc1(t)-c1(t)=0,Dtγd1(t)-d1(t)=0,
together with the initial conditions
(55)b1(0)=1,b2(0)=0,c1(0)=1,c2(t)=0,d1(0)=1,d2(t)=0.
By solving (54) and (55) by considering and applying Definition 5 which is discussed in Section 2 of this paper, we obtain the following results:
(56)b1(t)-2b2(t)=Eα(-tα),c1(t)=Eβ(tβ),d1(0)=Eγ(tγ).
By substituting (56) into (52), we obtain the exact solution for the system of nonlinear FPDEs in (49), as follows:
(57)u(x,y,t)=ex+yEα(-tα),v(x,y,t)=ex-yEβ(tβ),w(x,y,t)=e-x+yEγ(tγ).
If we now putα,β,γ→1 in (57), or solve (49) and (50) for α=β=γ=1, we obtain the following exact solution for the corresponding system of nonlinear PDEs:
(58)u(x,y,t)=ex+y-t,v(x,y,t)=ex-y+t,w(x,y,t)=e-x+y+t.


## 5. Conclusions

In this paper, a novel approach was introduced and utilized to solve linear and nonlinear systems of FPDEs. The new technique was coined by the authors of this paper as the homo-separation of variables method. In this research work, it was demonstrated through different examples how this new method can be used for solving various systems of FPDEs.When compared with the existing published methods, it is easy to notice that the new method has many advantages. It is straightforward, easy to understand, and fast, requiring much less computations to perform a limited number of steps of the simple procedure that can be applied to find the exact solution of a wide range of types of FPDE equations’ systems. Furthermore, there is no need for using linearization or restrictive assumptions when employing this new method.The basic principle of the current method is to perform an elegant combination (which is simple and smart) of two existing techniques. These two techniques are the popular homotopy perturbation method (HPM) that was originally proposed by He [25, 26] and another popular approach called separation of variables. Finally, the resulting homo-separation of variables method, which is analytical, can be used to solve various systems of PDEs with integer and fractional order, for example, with respect to time.



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