Numerical solution for complex systems of fractional order

Transcription

1 Shiraz University of Technology From the SelectedWorks of Habibolla Latifizadeh 213 Numerical solution for complex systems of fractional order Habibolla Latifizadeh, Shiraz University of Technology Available at:

2 Numerical solution for complex systems of fractional order Rabha W Ibrahim Institute of Mathematical Sciences University Malaya, 563 Kuala Lumpur, Malaysia rabhaibrahim@yahoocom Abstract: By using a complex transform, we impose a system of fractional order in 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 Keywords: Fractional calculus; fractional differential equations; unit disk; analytic function; complex transform AMS Mathematics Subject Classification: 34A12 1 Introduction Fractional models have been studied by many researchers to sufficiently describe the operation of variety of computational, physical biological processes systems Accordingly, considerable attention has been paid to the solution of fractional differential equations, integral equations fractional partial differential equations of physical phenomena Most of these fractional differential equations have analytic solutions, approximation numerical techniques [1-3] Numerical analytical methods have included finite difference method such as Adomian decomposition method, variational iteration method, homotopy perturbation method, homotopy analysis method [4-7] The idea of the fractional calculus that is, calculus of integrals derivatives of any arbitrary real or complex order was planted over 3 years ago Abel in 1823 investigated the generalized tautochrone problem 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 areas of sciences see [8-1] 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 higher accuracy to derive different types of numerical algorithms In this note, we shall deal with scalar linear timespace 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 examples the Fokker-Planck partial differential equation, bond pricing equations the Black-Scholes equations are in this class of differential equations partial fractional In [11] the author used complex transform to obtain a system of fractional order non-homogeneous 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 1

3 2 Fractional calculus This section concerns with some preliminaries notations regarding the fractional calculus Definition 21 The fractional arbitrary order integral of the function f of order α > is defined by I α a ft = t a t τ α 1 fτdτ Γα When a =, we write I α a ft = ft φ α t, where denoted the convolution product see [16], φ α t = tα 1 Γα, t > φ αt =, t φ α δt as α where δt is the delta function Definition 22 The fractional arbitrary order derivative of the function f of order α < 1 is defined by Da α ft = d t t τ α dt Γ1 α fτdτ = d dt I1 α a ft Remark 21 From Definition 21 Definition 22, a =, we have The Leibniz rule is where D α a [ftgt] = D α t µ = I α t µ = α k k= a Γµ + 1 Γµ α + 1 tµ α, µ > 1; < α < 1 Γµ + 1 Γµ + α + 1 tµ+α, µ > 1; α > α k D α k a ftd k agt = = α k k= Γα + 1 Γk + 1Γα + 1 k Da α k gtdaft, k Definition 23 The Caputo fractional derivative of order µ > is defined, for a smooth function ft by c D µ 1 t f n ζ ft := dζ, Γn µ t ζ µ n+1 where n = [µ] + 1, the notation [µ] sts for the largest integer not greater than µ Note that there is a relationship between Riemann-Liouville differential operator the Caputo operator D a µ 1 fa ft = Γ1 µ t a µ + c D a µ ft; they are equivalent in a physical problem ie, a problem which specifies the initial conditions In this paper we consider the following fractional differential equation: D α ut, z = at, zu zz + bt, zu z + ct, zu + ft, z, 1 where a, b, c, u, f are complex valued functions, analytic in the domain D := J U; J = [, T ], T, U := {z C, z 1} The above equation involves well known time fractional diffusion equations 2

4 3 Complex transforms In this section, we shall transform the fractional differential equation 1 into a coupled nonlinear system of fractional order has the smiler form It was shown in [11], that the complex transform ut, z = σzut, z, 2 where σ is a complex valued function of complex variable z U; reduces the equation 1 into the system D α v = a 1 v zz a 2 w zz + b 1 v z b 2 w z + c 1 v c 2 w D α w = a 1 w zz + a 2 v zz + b 1 w z + b 2 v z + c 1 w + c 2 v, 3 where a 1 = a 1, a 2 = a 2 b 1 = b 1 + 2σ 1za 1 σ 1 + a 2 σ 2 + 2σ 2z a 1 σ 2 a 2 σ 1 σ σ2 2 b 2 = b 2 2σ 1za 1 σ 2 a 2 σ 1 + 2σ 2z a 2 σ 2 a 1 σ 1 σ σ2 2 c 1 = c 1 + σ 1zza 1 σ 1 + a 2 σ 2 + σ 2zz a 1 σ 2 a 2 σ 1 + σ 1z b 1 σ 1 + b 2 σ 2 + σ 2z b 1 σ 2 b 2 σ 1 σ1 2 + σ2 2 c 2 = c 2 σ 1zza 1 σ 2 a 2 σ 1 + σ 2zz a 1 σ 1 a 2 σ 2 + σ 1z b 1 σ 2 b 2 σ 1 σ 2z b 1 σ 1 + b 2 σ 2 σ1 2 + σ2 2 σz := σ 1 z + iσ 2 z, ut, z = vt, z + iwt, z at, z = a 1 t, z + ia 2 t, z, bt, z = b 1 t, z + ib 2 t, z ct, z = c 1 t, z + ic 2 t, z, ut, z = vt, z + iwt, z Also, it was shown that the complex transform ut, z = ρt, zut, z, 4 reduces the non-homogenous equation D α ut, z = at, zu zz + bt, zu z + ct, zu + ft, z, 5 into the system D α v = a 1 v zz a 2 w zz + b 1 v z b 2 w z + c 1 v c 2 w + f 1 D α w = a 1 w zz + a 2 v zz + b 1 w z + b 2 v z + c 1 w + c 2 v + f 2, 6 where 3

5 a 1 = a 1, a 2 = a 2 b 1 = b 1 + 2ρ 1za 1 ρ 1 + a 2 ρ 2 + 2ρ 2z a 1 ρ 2 a 2 ρ 1 ρ ρ2 2 b 2 = b 2 2ρ 1za 1 ρ 2 a 2 ρ 1 + 2ρ 2z a 2 ρ 2 a 1 ρ 1 ρ ρ2 2 c 1 = c 1 + ρ 1zza 1 ρ 1 + a 2 ρ 2 + ρ 2zz a 1 ρ 2 a 2 ρ 1 + ρ 1z b 1 ρ 1 + b 2 ρ 2 + ρ 2z b 1 ρ 2 b 2 ρ 1 ρ ρ2 2 c 2 = c 2 ρ 1zza 1 ρ 2 a 2 ρ 1 + ρ 2zz a 1 ρ 1 a 2 ρ 2 + ρ 1z b 1 ρ 2 b 2 ρ 1 ρ 2z b 1 ρ 1 + b 2 ρ 2 ρ ρ2 2 f 1 = ρ 1f 1 h 1 + ρ 2 f 2 h 2 ρ ρ2 2 f 2 = ρ 2f 1 h 1 + ρ 1 f 2 h 2 ρ 2 1 +, ρ2 2 f = f ρ αρ t ρ I1 α u = f 1 + if 2 h 1 = ρ 1t I 1 α v ρ 2t I 1 α w h 2 = ρ 2t I 1 α v + ρ 1t I 1 α w, ρt, z := ρ 1 t, z + iρ 2 t, z 4 Numerical solution Let us put F 1 t, z, v, w = φ 1 t, z L 1 v, w N 1 v, w F 2 t, z, v, w = φ 2 t, z L 2 v, w N 2 v, w where φ 1 t, z φ 2 t, z are arbitrary functions; L 1 v, w = l 1 v + l 1 w = a 1 v zz + b 1 v z + c 1 v + a 2 w zz + b 2 w z + c 2 w L 2 v, w = l 2 v + l 2 w = a 2 v zz + b 2 v z + c 2 v + a 1 w zz + b 1 w z + c 1 w are the linear parts of F 1 F 1 respectively While N 1 N 2 are the nonlinear parts of F 1 F 1 respectively Moreover, let us set the homotopy system where 1 pd α vt, z + pd α vt, z φ 1 t, z + L 1 v, w + N 1 v, w =, p [, 1] 1 pd α wt, z + pd α wt, z φ 2 t, z + L 2 v, w + N 2 v, w =, vt, z = v n t, zp n, wt, z = w n t, zp n Hence we obtain the following system: k= N 1 v, w = N k p k, N 2 v, w = Ñ k p k k= 4

6 where D α D α v t, z v 1 t, z v 2 t, z v n t, z w t, z w 1 t, z w 2 t, z w n t, z = l 1 = l 2 v t, z v 1 t, z v n 1 t, z l 1 N v t, z N 1 v t, z, v 1 t, z w t, z w 1 t, z w n 1 t, z N n 1 v t, z, v 1 t, z,, v n 1 t, z v t, z v 1 t, z v n 1 t, z + l 2 w t, z w 1 t, z w n 1 t, z Ñ w t, z Ñ 1 w t, z, w 1 t, z Ñ n 1 w t, z, w 1 t, z,, w n 1 t, z + + φ 1 t, z φ 2 t, z,, k 1 v t, z = νzv j t j j!, α k 1, k, νz = ν n z n j= v 1 t, z = I α L 1 v t, z I α N v t, z + I α φ 1 t, z v n t, z = I α L 1 v n 1 t, z I α N n 1 v t, z,, v n 1 t, z k 1 w t, z = ϖzw j t j j!, α k 1, k, ϖz = ϖ n z n j= w 1 t, z = I α L 2 w t, z I α Ñ w t, z + I α φ 2 t, z w n t, z = I α L 2 w n 1 t, z I α Ñ n 1 w t, z,, w n 1 t, z Consequently, we have the approximate solution vt, z = νzv j t j j! Iα L 1 v j t, z + N j φ 1 t, z j= j= j= wt, z = ϖzw j t j j! Iα L 2 w j t, z + Ñ j φ 2 t, z j= j= j= 7 5

7 Thus, we impose a nonlinear integral equation in the following formula: vt, z = wt, z = νzv j t j t j! + t τ α 1 F 1 τ, ζ, udτ Γα ϖzw j t j t j! + t τ α 1 F 2 τ, ζ, udτ Γα j= j= 8 Now we can sake the main result of this section Theorem 51 Consider the fractional differential system 6 subject to the initial conditions v m, z = v m z, w m, z = w m z, m =, 1, 2,, k 1 9 The homotopy perturbation technique implies that the initial value problem 6-9 can be expressed as a nonlinear integral equation of the form 8 We proceed to prove the analytical convergence of our solution vn t, z Theorem 52 Suppose the sequence u n t, z = of the homotopy series vt, z = w n t, z v nt, zp n wt, z = w nt, zp n is defined for p [, 1] Assume the initial approximation u t, z = inside the domain of the solution ut, z = If u v t, z vt, z w t, z wt, z n+1 ρ u n for all n, where < ρ < 1, then the solution is absolutely convergent when p = 1 Proof Let C n t, z be the sequence of partial sum of the homotopy series Our aim is to show that C n t, z is a Cauchy sequence Consider C n+1 t, z C n t, z = u n+1 t, z ρ u n t, z ρ 2 u n 1 t, z ρ n+1 u t, z For n m, n N we have C n t, z C m t, z = C n t, z C n 1 t, z + C n 1 t, z C n 2 t, z + + C m+1 t, z C m t, z Hence C n t, z C n 1 t, z + C n 1 t, z C n 2 t, z + + C m+1 t, z C m t, z 1 ρn m ρ m+1 u t, z 1 ρ lim C nt, z C m t, z = ; n,m therefore, C n t, z is a Cauchy sequence in the complex Banach space 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 time-space-fractional nonlinear equation see [12-17] 6

8 5 Applications In this section, we shall consider the pump wave equations along the fiber Schrödinger equations These type of equations are the fundamental equations for describing non-relativistic quantum mechanical behavior take the form id α ut, z = 1 2 u xxt, z u 2 ut, z 1 Under the transform u = u = v + iw such that either u 2 = v 2 or u 2 = w 2, we have the uncoupled system id α vt, z = 1 2 v xxt, z v 2 vt, z id α wt, z = 1 2 w xxt, z w 2 wt, z, 11 where < α 1 Subject to the initial conditions Operating 11 by I α, we have v, z = e iz, w, z = 1 ivt, z = v, z + I α [ 1 2 v xxt, z v 2 vt, z] iwt, z = w, z + I α [ 1 2 w xxt, z w 2 wt, z] 12 By the same computation as in Section 5, we receive v = e iz, w = 1 it α it α v 1 = 2Γα + 1 eiz, w 1 = Γα + 1 v 2 = it α Γ2α + 1 eiz, w 2 = itα 2 Γ2α Thus the solution u is given by v n = ut, z = it α n 2 n Γnα + 1 eiz, w n = itα n Γnα + 1 it α n 2 n Γnα + 1 eiz, it α n T Γnα + 1 Moreover, under the same transform, the equation 1 reduces to coupled system id α vt, z = 1 2 v xxt, z v 2 + w 2 vt, z id α wt, z = 1 2 w xxt, z v 2 + w 2 wt, z, 14 Operating 14 by I α, we have 7

9 Figure 1: a-d The solution v when α = 5, α = 75, α = 9, α = 1 respectively e,f The solution u, v when α = 5 α = 1 ivt, z = v, z + I α [ 1 2 v xxt, z v 2 + w 2 vt, z] iwt, z = w, z + I α [ 1 2 w xxt, z v 2 + w 2 wt, z] 15 Therefore, ut, z = it α n 2 n Γnα + 1 eiz, it α n T, e iz = 1 Γnα 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 uncoupled system of homogeneous non-homogeneous 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 containing 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 stard 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, subatomic systems, but also macroscopic systems References [1] R W Ibrahim, Existence uniqueness of holomorphic solutions for fractional Cauchy problem, J Math Anal Appl

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