JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 678174 10.1155/2012/678174 678174 Research Article Numerical Solution for Complex Systems of Fractional Order Ibrahim Rabha W. Öziş Turgut Institute of Mathematical Sciences University of Malaya 50603 Kuala Lumpur Malaysia um.edu.my 2012 20 12 2012 2012 18 10 2012 04 12 2012 2012 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.

By using a complex transform, we impose a system of fractional order in the sense of Riemann-Liouville fractional operators. The analytic solution for this system is discussed. Here, we introduce a method of homotopy perturbation to obtain the approximate solutions. Moreover, applications are illustrated.

1. Introduction

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 . Numerical and analytical methods have included finite difference methods such as Adomian decomposition method, variational iteration method, homotopy perturbation method, and homotopy analysis method .

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 drawn the attention of many researchers in all areas of sciences (see ).

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-space fractional differential equations. The time is taken in sense of the Riemann-Liouville fractional operators. Also, This type of differential equation arises in many interesting applications. For example, the Fokker-Planck partial differential equation, bond pricing equations, and the Black-Scholes equations are in this class of differential equations (partial and fractional).

In , the author used complex transform to obtain a system of fractional order (nonhomogeneous) keeping the equivalency properties. By employing the homotopy perturbation method, the analytic solution is presented for coupled system of fractional order. Furthermore, applications are imposed such as wave equations of fractional order.

2. Fractional Calculus

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 (2.1)Iaαf(t)=at(t-τ)α-1Γ(α)f(τ)dτ. When a=0, we write Iaαf(t)=f(t)*ϕα(t), where (*) denoted the convolution product (see ), ϕα(t)=tα-1/Γ(α),t>0 and ϕα(t)=0,t0 and ϕαδ(t) as α0 where δ(t) is the delta function.

Definition 2.2.

The fractional (arbitrary) order derivative of the function f of order 0α<1 is defined by (2.2)Daαf(t)=ddtat(t-τ)-αΓ(1-α)f(τ)dτ=ddtIa1-αf(t).

Remark 2.3.

From Definitions 2.1 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).

Definition 2.4.

The Caputo fractional derivative of order μ>0 is defined, for a smooth function f(t) by (2.6)  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 (2.7)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).

In this paper, we consider the following fractional differential equation: (2.8)Dαu(t,z)=a(t,z)uzz+b(t,z)uz+c(t,z)u+f(t,z), where a0,b,c,u,f are complex valued functions, analytic in the domain 𝒟:=J×U;J=[0,T],T(0,) and U:={z,|z|1}.

The above equation involves well-known time fractional diffusion equations.

3. Complex Transforms

In this section, we will transform the fractional differential equation (2.8) into a coupled nonlinear system of fractional order has similar form. It was shown in  that the complex transform (3.1)u(t,z)=σ(z)u¯(t,z), where σ0 is a complex valued function of complex variable zU, reduces (2.8) into the system (3.2)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 (3.3)a¯1=a1,a¯2=a2b¯1=b1+2σ1z(a1σ1+a2σ2)+2σ2z(a1σ2-a2σ1)σ12+σ22b¯2=b2-2σ1z(a1σ2-a2σ1)+2σ2z(a2σ2-a1σ1)σ12+σ22c¯1=c1+σ1zz(a1σ1+a2σ2)+σ2zz(a1σ2-a2σ1)+σ1z(b1σ1+b2σ2)+σ2z(b1σ2-b2σ1)σ12+σ22c¯2=c2-σ1zz(a1σ2-a2σ1)+σ2zz(a1σ1-a2σ2)+σ1z(b1σ2-b2σ1)-σ2z(b1σ1+b2σ2)σ12+σ22.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). Also, it was shown that the complex transform (3.4)u(t,z)=ρ(t,z)u¯(t,z), reduces the nonhomogenous equation (3.5)Dαu(t,z)=a(t,z)uzz+b(t,z)uz+c(t,z)u+f(t,z), into the system (3.6)Dαv¯=a¯1v¯zz-a¯2w¯zz+b¯1v¯z-b¯2w¯z+c¯1v¯-c¯2w¯+f¯1Dαw¯=a¯1w¯zz+a¯2v¯zz+b¯1w¯z+b¯2v¯z+c¯1w¯+c¯2v¯+f¯2, where (3.7)a¯1=a1,a¯2=a2b¯1=b1+2ρ1z(a1ρ1+a2ρ2)+2ρ2z(a1ρ2-a2ρ1)ρ12+ρ22b¯2=b2-2ρ1z(a1ρ2-a2ρ1)+2ρ2z(a2ρ2-a1ρ1)ρ12+ρ22c¯1=c1+ρ1zz(a1ρ1+a2ρ2)+ρ2zz(a1ρ2-a2ρ1)+ρ1z(b1ρ1+b2ρ2)+ρ2z(b1ρ2-b2ρ1)ρ12+ρ22c¯2=c2-ρ1zz(a1ρ2-a2ρ1)+ρ2zz(a1ρ1-a2ρ2)+ρ1z(b1ρ2-b2ρ1)-ρ2z(b1ρ1+b2ρ2)ρ12+ρ22f¯1=ρ1(f1-h1)+ρ2(f2-h2)ρ12+ρ22f¯2=ρ2(f1-h1)+ρ1(f2-h2)ρ12+ρ22,f¯=fρ-αρtρI1-αu¯=f¯1+if¯2h1=ρ1tI1-αv¯-ρ2tI1-αw¯h2=ρ2tI1-αv¯+ρ1tI1-αw¯,ρ(t,z):=ρ1(t,z)+iρ2(t,z)0.

4. Numerical Solution

Let us put (4.1)F1(t,z,v¯,w¯)=ϕ1(t,z)-L1(v¯,w¯)-N1(v¯,w¯)F2(t,z,v¯,w¯)=ϕ2(t,z)-L2(v¯,w¯)-N2(v¯,w¯), where ϕ1(t,z) and ϕ2(t,z) are arbitrary functions; (4.2)L1(v¯,w¯)=-1(v¯)+1(w¯)=-(a¯1v¯zz+b¯1v¯z+c¯1v¯)+(a¯2w¯zz+b¯2w¯z+c¯2w¯),L2(v¯,w¯)=-(2(v¯)+2(w¯))=-(a¯2v¯zz+b¯2v¯z+c¯2v¯+a¯1w¯zz+b¯1w¯z+c¯1w¯) are the linear parts of F1 and F2, respectively. While N1 and N2 are the nonlinear parts of F1 and F2, respectively. Moreover, let us set the homotopy system (4.3)(1-p)Dαv¯(t,z)+pDαv¯(t,z)-ϕ1(t,z)+L1(v¯,w¯)+N1(v¯,w¯)=0,p[0,1](1-p)Dαw¯(t,z)+pDαw¯(t,z)-ϕ2(t,z)+L2(v¯,w¯)+N2(v¯,w¯)=0, where (4.4)v¯(t,z)=n=0vn(t,z)pn,w¯(t,z)=n=0wn(t,z)pn,N1(v¯,w¯)=k=0Nkpk,N2(v¯,w¯)=k=0N~kpk. Hence we obtain the following system: (4.5)Dα(v¯0(t,z)v¯1(t,z)v¯2(t,z)v¯n(t,z))=1(0v¯0(t,z)v¯1(t,z)v¯n-1(t,z))-1(0w¯0(t,z)w¯1(t,z)w¯n-1(t,z))-(0N0(v¯0(t,z))N1(v¯0(t,z),v¯1(t,z))Nn-1(v¯0(t,z),v¯1(t,z),,v¯n-1(t,z)))+(0ϕ1(t,z)00),Dα(w¯0(t,z)w¯1(t,z)w¯2(t,z)w¯n(t,z))=2(0v¯0(t,z)v¯1(t,z)v¯n-1(t,z))+2(0w¯0(t,z)w¯1(t,z)w¯n-1(t,z))-(0N~0(w¯0(t,z))N~1(w¯0(t,z),w¯1(t,z))N~n-1(w¯0(t,z),w¯1(t,z),,w¯n-1(t,z)))+(0ϕ2(t,z)00), where (4.6)v¯0(t,z)=j=0k-1ν(z)v0(j)tjj!,α(k-1,k),ν(z)=n=0νnzn,v¯1(t,z)=-Iα(L1v¯0(t,z))-IαN0(v¯0(t,z))+Iαϕ1(t,z),v¯n(t,z)=-Iα(L1v¯n-1(t,z))-IαNn-1(v¯0(t,z),,v¯n-1(t,z)),w¯0(t,z)=j=0k-1ϖ(z)w0(j)tjj!,  α(k-1,k),ϖ(z)=n=0ϖnzn,w¯1(t,z)=-Iα(L2w¯0(t,z))-IαN~0(w¯0(t,z))+Iαϕ2(t,z),w¯n(t,z)=-Iα(L2w¯n-1(t,z))-IαN~n-1(w¯0(t,z),,w¯n-1(t,z)). Consequently, we have the approximate solution (4.7)v¯(t,z)=j=0ν(z)v0(j)tjj!-Iα(j=0L1v¯j(t,z)+j=0Nj-ϕ1(t,z))w¯(t,z)=j=0ϖ(z)w0(j)tjj!-Iα(j=0L2w¯j(t,z)+j=0N~j-ϕ2(t,z)). Thus, we impose a nonlinear integral equation in the following formula: (4.8)v¯(t,z)=j=0ν(z)v0(j)tjj!+0t(t-τ)α-1Γ(α)F1(τ,ζ,u)dτw¯(t,z)=j=0ϖ(z)w0(j)tjj!+0t(t-τ)α-1Γ(α)F2(τ,ζ,u)dτ. Now we can sake the main result of this section.

Theorem 4.1.

Consider the fractional differential system (3.6) subject to the initial conditions (4.9)(v¯(m)(0,z)=v¯0(m)(z),w¯(m)(0,z)=w¯0(m)(z),m=0,1,2,,k-1).

The homotopy perturbation technique implies that the initial value problem ((3.6)–(4.9)) can be expressed as a nonlinear integral equation of the form (4.8).

We proceed to prove the analytical convergence of our solution.

Theorem 4.2.

Suppose the sequence un(t,z)=(v¯n(t,z)w¯n(t,z)) of the homotopy series v¯(t,z)=n=0v¯n(t,z)pn and w¯(t,z)=n=0w¯n(t,z)pn is defined for p[0,1]. Assume the initial approximation u0(t,z)=(v¯0(t,z)w¯0(t,z)) inside the domain of the solution u(t,z)=(v¯(t,z)w¯(t,z)). If un+1ρun for all n, where 0<ρ<1, then the solution is absolutely convergent when p=1.

Proof.

Let Cn(t,z) be the sequence of partial sum of the homotopy series. Our aim is to show that Cn(t,z) is a Cauchy sequence. Consider (4.10)Cn+1(t,z)-Cn(t,z)=un+1(t,z)ρun(t,z)ρ2un-1(t,z)ρn+1u0(t,z). For nm,n, we have (4.11)Cn(t,z)-Cm(t,z)=Cn(t,z)-Cn-1(t,z)+Cn-1(t,z)-Cn-2(t,z)++Cm+1(t,z)-Cm(t,z)Cn(t,z)-Cn-1(t,z)+Cn-1(t,z)-Cn-2(t,z)++Cm+1(t,z)-Cm(t,z)1-ρn-m1-ρρm+1u0(t,z). Hence (4.12)limn,mCn(t,z)-Cm(t,z)=0; therefore, Cn(t,z) is a Cauchy sequence in the complex Banach space and consequently yields that the series solution is convergent. This completes the proof.

Recently the homotopy methods are used to obtain approximate analytic solutions of the time-fractional nonlinear equation and time-space-fractional nonlinear equation (see ).

5. Applications

In this section, we will consider the pump wave equations along the fiber (Schrödinger equations). These types of equations are the fundamental equations for describing non-relativistic quantum mechanical behavior taking the form (5.1)iDαu(t,z)=-12uzz(t,z)-|u|2u(t,z).

Under the transform u=u¯=v¯+iw¯ such that either |u|2=|v¯|2 or |u|2=|w¯|2, we have the uncoupled system (5.2)iDαv¯(t,z)=-12v¯zz(t,z)-|v¯|2v¯(t,z)iDαw¯(t,z)=-12w¯zz(t,z)-|w¯|2w¯(t,z), where 0<α1. Subject to the initial conditions (5.3)v¯0(0,z)=eiz,w¯0(0,z)=1. Operating (5.2) by Iα, we have (5.4)iv¯(t,z)=v¯0(0,z)+Iα[-12v¯zz(t,z)-|v¯|2v¯(t,z)]iw¯(t,z)=w¯0(0,z)+Iα[-12w¯zz(t,z)-|w¯|2w¯(t,z)]. By the same computation as in Section 5, we receive (5.5)v¯0=eiz,w¯0=1v¯1=itα2Γ(α+1)eiz,w¯1=itαΓ(α+1)v¯2=(itα)222Γ(2α+1)eiz,w¯2=(itα)2Γ(2α+1)v¯n=(itα)n2nΓ(nα+1)eiz,  w¯n=(itα)nΓ(nα+1). Thus the solution u¯ is given by (5.6)u¯(t,z)=(n=0(itα)n2nΓ(nα+1)eiz,n=0(itα)nΓ(nα+1))T. Moreover, under the same transform, (5.1) reduces to coupled system (5.7)iDαv¯(t,z)=-12v¯zz(t,z)-(|v¯|2+|w¯|2)v¯(t,z)iDαw¯(t,z)=-12w¯zz(t,z)-(|v¯|2+|w¯|2)w¯(t,z), Operating (5.7) by Iα, we have (5.8)iv¯(t,z)=v¯0(0,z)+Iα[-12v¯zz(t,z)-(|v¯|2+|w¯|2)v¯(t,z)]iw¯(t,z)=w¯0(0,z)+Iα[-12w¯zz(t,z)-(|v¯|2+|w¯|2)w¯(t,z)]. Therefore, (5.9)u¯(t,z)=(n=0(itα)n2nΓ(nα+1)eiz,n=0(itα)nΓ(nα+1))T,|eiz|=1.

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, we employed the homotopy perturbation scheme for solving the nonlinear complex fractional differential systems. The convergence of the method is discussed in a domain that contains the initial solution. The Schrödinger equation is illustrated as an application. This type of equation is used in the quantum mechanics, which describes how the quantum state of a physical system changes with time. In the standard quantum mechanics, the wave function is the most complete explanation that can be specified to a physical system. Solutions of the Schrdinger’s equation describe not only molecular, atomic, and subatomic systems, but also macroscopic systems (see Figure 1).

((a)–(d)) The solution v¯ when α=0.5,α=0.75,α=0.9, and α=1, respectively. ((e),(f)) the solution (u¯,v¯) when α=0.5 and α=1.

Ibrahim R. W. Existence and uniqueness of holomorphic solutions for fractional Cauchy problem Journal of Mathematical Analysis and Applications 2011 380 1 232 240 10.1016/j.jmaa.2011.03.001 2786198 ZBL1214.30027 Ibrahim R. W. On holomorphic solution for space- and time-fractional telegraph equations in complex domain Journal of Function Spaces and Applications 2012 2012 10 703681 2944701 ZBL1250.35176 Ibrahim R. W. Approximate solutions for fractional differential equation in the unit disk Electronic Journal of Qualitative Theory of Differential Equations 2011 64 11 2832770 Momani S. Odibat Z. Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations Computers & Mathematics with Applications 2007 54 7-8 910 919 10.1016/j.camwa.2006.12.037 2395628 ZBL1141.65398 Molliq R. Y. Noorani M. S. M. Hashim I. Variational iteration method for fractional heat- and wave-like equations Nonlinear Analysis. Real World Applications 2009 10 3 1854 1869 10.1016/j.nonrwa.2008.02.026 2502991 ZBL1172.35302 Sayevand K. Golbabai A. Yildirim A. Analysis of differential equations of fractional order Applied Mathematical Modelling 2012 36 9 4356 4364 10.1016/j.apm.2011.11.061 2929835 Jafari H. Firoozjaee M. A. Homotopy analysis method for solving KdV equations Surveys in Mathematics and Its Applications 2010 5 89 98 2652568 Podlubny I. Fractional Differential Equations 1999 London, UK Academic Press 1658022 Kilbas A. A. Srivastava H. M. Trujillo J. J. Theory and applications of fractional differential equations 2006 Elsevier North-Holland Mathematics Studies 2218073 Srivastava H. M. Owa S. Univalent Functions, Fractional Calculus, and Their Applications 1989 New York, NY, USA Halsted Press 1199135 Ibrahim R. W. Complex transforms for systems of fractional differential equations Abstract and Applied Analysis 2012 2012 11 814759 10.1155/2012/814759 Hesameddini E. Latifizadeh H. Homotopy analysis method to obtain numerical solutions of the Painlevé equations Mathematical Methods in the Applied Sciences 2012 35 12 1423 1433 10.1002/mma.2521 2957527 Ibrahim R. W. Jalab H. Analytic solution for fractional differential equation in the unit disk Wulfenia Journal 2012 19 8 105 114 Sakar M. G. Erdogan F. Yıldırım A. Variational iteration method for the time-fractional Fornberg-Whitham equation Computers & Mathematics with Applications 2012 63 9 1382 1388 10.1016/j.camwa.2012.01.031 2912063 ZBL1247.65138 Salah A. Khan M. Gondal M. A novel solution procedure for fuzzy fractional heat equationsby homotopy analysis transform method Neural Computing & Applications 2012 2012 3 10.1007/s00521-012-0855-z El-Ajou A. Abu Arqub O. Momani S. Homotopy analysis method for second-order boundary value problems of integrodifferential equations Discrete Dynamics in Nature and Society 2012 2012 18 365792 Vishal K. Das S. Solution of the nonlinear fractional diffusion equation with absorbent term and external force using optimal homotopy-analysis method Zeitschrift fr NAturforschung A 2012 67 203 209