Advances in Difference Equations This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Fractional complex transforms for fractional differential equations Advances in Difference Equations 1, 1:19 doi:1.1186/1687-1847-1-19 Rabha W Ibrahim (rabhaibrahim@yahoo.com) ISSN 1687-1847 Article type Research Submission date 1 July 1 Acceptance date 3 October 1 Publication date 6 November 1 Article URL http://www.advancesindifferenceequations.com/content/1/1/19 This peer-reviewed article can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Advances in Difference Equations go to http://www.advancesindifferenceequations.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com 1 Ibrahim This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/.), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ADE_98_edited [11/ 14:14] 1/1 Fractional complex transforms for fractional differential equations Rabha W Ibrahim Institute of Mathematical Sciences University Malaya, 563 Kuala Lumpur, Malaysia E-mail: rabhaibrahim@yahoo.com Abstract: The fractional complex transform is employed to convert fractional differential equations analytically in the sense of the Srivastava-Owa fractional operator and its generalization in the unit disk. Examples are illustrated to elucidate the solution procedure including the space-time fractional differential equation in complex domain, singular problems and Cauchy problems. Here, we consider analytic solutions in the complex domain. Keywords: fractional calculus; fractional differential equations; Srivastava-Owa fractional operators; unit disk; analytic function; fractional complex transform; Cauchy differential equation; Fox-Wright function AMS Mathematics Subject Classification: 3C45. 1 Introduction The theory of fractional calculus has been applied in the theory of analytic functions. The classical concepts of a fractional differential operator and a fractional integral operator and their generalizations have fruitfully been employed in finding, for example, the characterization properties, coefficients estimate [1], distortion inequalities [] and convolution properties for difference subclasses of analytic functions. Fractional differential equations are viewed as alternative models to nonlinear differential equations. Varieties of them play important roles and and serve as tools not only in mathematics, but also in physics, dynamical systems, control systems, and engineering to create the mathematical modeling of many physical phenomena. Furthermore, they are employed in social sciences such as food supplement, climate, and economics. Fractional differential equations concerning the Riemann-Liouville fractional operators or the Caputo derivative have been recommended by many authors (see [3-1]). Recently, the complex modelings of phenomena in nature and society have been the object of several investigations based on the methods originally developed in a physical context. These systems are the consequence of the ability of individuals to develop strategies. They occur in kinetic theory [11], complex dynamical systems [1], chaotic complex systems and hyperchaotic complex systems [13], and the complex Lorenz-like system which has been found in laser physics while analyzing baroclinic instability of the geophysical flows in the atmosphere (or in the ocean) [14,15]. Sainty [16] considered the complex heat equation using a complex valued Brownian. A model of complex fractional equations is introduced by Jumarie [17-] using different types of fractional derivatives. Baleanu et al. [1-3] imposed several applications of fractional calculus including complex modelings. The author studied various types of fractional differential equations in complex domain such as the Cauchy equation, the diffusion equation and telegraph equations [4-8]. 1
ADE_98_edited [11/ 14:14] /1 Transform is a significant technique to solve mathematical problems. Many useful transforms for solving various problems appeared in open literature such as wave transformation, the Laplace transform, the Fourier transform, the Bücklund transformation, the integral transform, the local fractional integral transforms and the fractional complex transform (see [9,3]). In this paper, we shall introduce two generalizations of the wave transformation in the complex domain. These generalizations depend on the fractional differential operators for complex variables. These transformations convert the fractional differential equations in complex domain into ordinary differential equations to obtain analytic solutions or exact solutions. Examples are illustrated. In [31], Srivastava and Owa provided the definitions for fractional operators (derivative and integral) in the complex z-plane C as follows. Definition 1.1 The fractional derivative of order α is defined, for a function f(z), by Dz α 1 d f(z) := Γ(1 α) dz f(ζ) dζ; α<1, (z ζ) α where the function f(z) is analytic in a simply-connected region of the complex z-plane C containing the origin, and the multiplicity of (z ζ) α is removed by requiring log(z ζ) to be real when(z ζ) >. Definition 1. The fractional integral of order α is defined, for a function f(z), by I α z f(z) := 1 Γ(α) f(ζ)(z ζ) α 1 dζ; α>, where the function f(z) is analytic in a simply-connected region of the complex z-plane (C) containing the origin, and the multiplicity of (z ζ) α 1 is removed by requiring log(z ζ) toberealwhen (z ζ) >. Remark 1.1 From Definitions 1.1 and 1., we have D α z z β = Γ(β +1) Γ(β α +1) zβ α, β > 1; α<1 and Iz α z β Γ(β +1) = Γ(β + α +1) zβ+α, β > 1; α>. In [3] the author derived a formula for the generalized fractional integral. Consider for natural n N = {1,,...} and real μ, the n-fold integral of the form which yields I α,μ z f(z) = ζ μ 1 dζ 1 ζ1 ζn 1 ζ μ dζ... ζnf(ζ μ n )dζ n, (1) ζ μ 1 dζ 1 ζ1 ζ μ f(ζ)dζ = ζ μ f(ζ)dζ Repeating the above step n 1times,weobtain ζ μ 1 dζ 1 ζ1 ζ μ dζ... ζn 1 which implies the fractional operator type ζ ζ μ nf(ζ n )dζ n = Iz α,μ (μ +1)1 α f(z) = Γ(α) ζ μ 1 dζ 1 = 1 μ +1 (μ +1)1 n (n 1)! (z ζ )ζ μ f(ζ)dζ. (z ζ ) n 1 ζ μ f(ζ)dζ, (z ζ ) α 1 ζ μ f(ζ)dζ, ()
ADE_98_edited [11/ 14:14] 3/1 where a and μ 1 are real numbers and the function f(z) is analytic in a simply-connected region of the complex z-plane C containing the origin, and the multiplicity of (z ζ ) α is removed by requiring log(z ζ )toberealwhen(z ζ ) >. When μ =, we get the standard Srivastava-Owa fractional integral, which is applied to define the Srivastava-Owa fractional derivatives. The computation implies [3] Iz α,μ z ν = zα()+ν ν+ Γ( ) (μ +1) α Γ(α + ν+ (3) ). When μ =, we obtain Iz α z ν = Γ(ν+1) Γ(α+ν+1) zα+ν (see Remark 1.1). Corresponding to the generalized fractional integrals (), we imposed the generalized differential operator. Definition 1.3 The generalized fractional derivative of order α is defined, for a function f(z), by Dz α,μ (μ +1)α d f(z) := Γ(1 α) dz ζ μ f(ζ) (z ζ dζ; α<1, (4) ) α where the function f(z) is analytic in a simply-connected region of the complex z-plane C containing the origin, and the multiplicity of (z ζ ) α is removed by requiring log(z ζ )tobereal when (z ζ ) >. The calculation yields D α,μ z z ν = (μ +1)α 1 Γ( ν +1) Γ( ν +1 α) z (1 α)()+ν 1. (5) When μ =, we obtain Dz α z ν = Γ(ν+1) Γ(ν+1 α) zν α (see Remark 1.1). For analytic functions of the form f(z) = a n z n, z U, (6) we have the following property: D α,μ z I α,μ z n=1 f(z) =I α,μ z D α,μ Fractional complex transform z f(z) =f(z), z U. In recent times, one of the most important and useful methods for fractional calculus called fractional complex transform has appeared [33-36]. Fractional complex transform is to renovate the fractional differential equations into ordinary differential equations, yielding a tremendously simple solution procedure. In this section, we illustrate some fractional complex transform using properties of the Srivastava-Owa fractional operator and its generalization. Analogous to wave transformation where a, b, andc are constants, the fractional complex transform is η = az + bw + cu +..., (7) η = az α + bw β + cu γ +... (8) for the fractional differential equations in the sense of the Srivastava-Owa fractional operators. While the fractional complex transform of the form η = Az α() + Bw β() + Cu γ() +... (9) 3
ADE_98_edited [11/ 14:14] 4/1 is applied to fractional differential equations in the sense of the generalized operators () and (4). It is obvious that when μ =, (9) reduces to (8). Furthermore, in a real case, (9) implies the fractional complex transform defined in [34]. We impose the fractional complex transform D α z f(z) = f Z Dα z Z, Z := z α (1) if we denote Dz α f(z) := α f z, it yields α α f z α = f α Z Z z α := f (11) Z θ α, where θ α is the fractal index, which is usually determined in terms of gamma functions. Similarly, we can receive Dz α,μ f(z) = f Z Dα,μ z Z, Z := z α() (1) if we let Dz α,μ f(z) := α() f, which implies z α() α() f f α() Z = zα() Z z α() := f Z Θ α,μ. (13) Example.1 Let Z = z α and f = Z n,n then in view of Remark 1.1, we obtain α f z α = f α Z Z z α Γ(1 + nα)znα α = Γ(1 + nα α) := f Z θ α = nθ α z nα α. We hence can receive that θ α = Γ(1 + nα) nγ(1 + nα α). Example. Let Z = z α() and f = Z n,n then in virtue of (5), we have We, therefore, have α() f f α() Z n = zα() Z z α() = (μ +1)α 1 Γ( nα +1) Γ( nα +1 α) z (1 α)()+nα 1 := f Z Θ α,μ = nθ α,μ μ +1 z(1 α)()+nα 1. Θ α,μ = (μ +1)α Γ( nα +1) nγ( nα. +1 α) 4
ADE_98_edited [11/ 14:14] 5/1 3 Applications Example 3.1 Consider the following equation: ρt 1/ 1/ u(t,z) 1.18 + D β t 1/ z u(t, z) =, t J =[, 1] u(,z)=, in a neighborhood of z =, (14) where u(t, z) is the unknown function ρ (, 1) and β (, 1]. We propose to show that (14) has a unique analytic solution by using the Banach fixed point theorem. By assuming u(t, z) =μ(z)t + v(t, z) (v(t, z) =O(t )) as a formal solution, where μ(z) =O(z β ), calculations imply t 1/ 1/ u(t, z) t 1/ =1.18μ(z)t + t 1/ v α (t, z), α =1/, and Therefore, μ(z) satisfies which is equivalent to where D β z u(t, z) =D β z ( ) μ(z)t + v(t, z) = tdz β μ(z)+v β (t, z). ρμ(z)+d β z μ(z) =, D β z μ(z) =g(z,μ(z)), (15) g(z,μ(z)) = ρμ(z). Now g(z,μ(z)) is a contraction mapping whenever ρ (, 1); therefore, in view of the Banach fixed point theorem, Eq.(15) has a unique analytic solution in the unit disk and consequently the problem (14). To calculate the fractal index for the equation D β z μ(z)+ρμ(z) =, μ() = 1, (16) we assume the transform Z = z β and the solution can be expressed in a series in the form μ(z) = μ m Z m, (17) m= where μ m are constants. Substituting (17) into Eq. (16) yields θ βm μ m Z m + ρ μ m Z m =. (18) Z Since m= θ βm = then the computation imposes the relation m= Γ(1 + mβ) mγ(1 + mβ β), Γ(1 + mβ) Γ(1 + mβ β) μ m + ρμ m 1 = 5
ADE_98_edited [11/ 14:14] 6/1 with μ() = 1, and consequently we obtain Thus, we have the following solution: which is equivalent to μ(z) = μ m = μ(z) = m= m= ( ρ)m Γ(1 + mβ). ( ρ) m Γ(1 + mβ) Zm ( ρ) m Γ(1 + mβ) zmβ = E β ( ρz β ), where E β is a Mittag-Leffler function. The last assertion is the exact solution for the problem (16) and consequently for (14). Example 3. Consider the following equation: t 1/ / 1/ u(t,z) 1.18 +4z u(t,z) t 1/ z + Dz β u(t, z) =z β t, t J =[, 1] u(,z)=, in a neighborhood of z =, (19) where u(t, z) is the unknown function and β (, 1]. In the same manner of Example 3.1, we let as a formal solution, where μ(z) =O(z β )and u(t, z) =μ(z)t + v(t, z) (v(t, z) =O(t )) μ (z) ν (z) <λ μ(z) ν(z), λ< 1 8. Estimations imply and Therefore, μ(z) satisfies which is equivalent to where t 1/ 1/ u(t, z) =1.18μ(z)t + t 1/ v t 1/ α (t, z), α =1/, u(t, z) z = ztμ (z)+zv z (t, z) z ( ) Dz β u(t, z) =Dz β μ(z)t + v(t, z) = tdz β μ(z)+v β (t, z). μ(z) +4zμ (z)+d β z μ(z) z β =, D β z μ(z) =G(z,μ(z),zμ (z)), () G(z,μ(z),zμ (z)) = z β 1/μ(z) 4zμ (z). Now, to show that G(z,μ(z),zμ (z)) is a contraction mapping, G(z,μ(z),zμ (z)) G(z,ν(z),zν (z)) = z β 1/μ(z) 4zμ (z) (z β 1/ν(z) 4zν (z)) 1 μ(z) ν(z) +4λ μ(z) ν(z) =( 1 +4λ) μ(z) ν(z). 6
ADE_98_edited [11/ 14:14] 7/1 Thus, in view of the Banach fixed point theorem, Eq.() has a unique analytic solution in the unit disk and consequently the problem (19). To evaluate the fractal index for the equation Dz β μ(z)+ μ(z) +4zμ (z) z β =, μ() = 1, (1) we assume the transform Z = z β and the solution can be articulated as in (17). Substituting (17) into Eq. (1), we have θ βm μ m Z m + 1 μ m Z m +4 mμ m Z m Z =, Z () m= m= m=1 where Γ(1 + mβ) θ βm = mγ(1 + mβ β). Hence, the computation imposes the relation ( Γ(1 + mβ) ) Γ(1 + mβ β) +4m μ m + 1 μ m 1 = with μ() = 1, and consequently we obtain μ m := (B m) m Γ(1 + mβ), where B m in terms of a gamma function. If we let B := max m {B m }, then the solution approximates to (B) m μ(z) Γ(1 + mβ) Zm, m= which is equivalent to (B) m μ(z) = Γ(1 + mβ) zmβ = E β (Bz β ). m= The last assertion is the exact solution for the problem (1) and consequently for (19). Next, we consider the Cauchy problem by employing the generalized fractional differential operator (4). We shall show that the solution of such a problem can be determined in terms of the Fox-Wright function [37]: (a 1,A 1 ),..., (a p,a p ); pψ q w = p Ψ q [(a j,a j ) 1,p ;(b j,b j ) 1,q ; w] (b 1,B 1 ),..., (b q,b q ); Γ(a 1 + na 1 )...Γ(a p + na p ) w n := Γ(b 1 + nb 1 )...Γ(b q + nb q ) n! = n= q j=1 Γ(a j + na j ) p j=1 Γ(b j + nb j ) where A j > for all j =1,..., p, B j > for all j =1,..., q, and1+ q j=1 B j p j=1 A j for suitable values w < 1anda i,b j are complex parameters. n= w n n!, Example 3.3 Consider the Cauchy problem in terms of the differential operator (4) D α,μ z u(z) =F (z,u(z)), (3) 7
ADE_98_edited [11/ 14:14] 8/1 where F (z,u(z)) is analytic in u and u(z) is analytic in the unit disk. Thus, F can be expressed by F (z,u) =φu(z). Let Z = z α(). Then the solution can be formulated as follows: u(z) = u m Z m, (4) m= where u m are constants. Substituting (4) into Eq. (3) implies Since Z Θ α,μ,m u m Z m φ u m Z m =. (5) m= then the calculation yields the relation consequently, we obtain Thus, we have the following solution: which is equivalent to u(z) = m= Θ α,μ,m = (μ +1)α Γ( mα +1) mγ( nα, +1 α) (μ +1) α Γ( mα +1) Γ( mα +1 α) u m φu m 1 =; (m 1)α φ Γ( +1 α)γ( mα +1 α) u m =[ ]m (μ +1) α Γ( (m 1)α +1)Γ( mα +1). u(z) = φ [ (μ +1) m= φ [ (μ +1) m= Since φ is an arbitrary constant, we assume that Thus, for a suitable α, we present or where z < 1. u(z) = m= = 3 Ψ u(z) = 3 Ψ (m 1)α Γ( +1 α)γ( mα +1 α) ]m α Γ( (m 1)α +1)Γ( mα +1) Z m (m 1)α Γ(m +1)Γ( +1 α)γ( mα +1 α) Zm ]m α +1) Γ( (m 1)α +1)Γ( mα φ := (μ +1) α. Γ(m +1)Γ( (m 1)α +1 α)γ( mα +1 α) Zm Γ( (m 1)α +1)Γ( mα +1) m! (1, 1), (1 α α, α ), (1 α, α ); Z (1 α, α ), (1, α ); (1, 1), (1 α α, α ), (1 α, α ); z α() (1 α, α ), (1, α ); m!. 8
ADE_98_edited [11/ 14:14] 9/1 4 Conclusion A generalized fractional complex transform is suggested in this paper to find exact solutions of fractional differential equations in the unit disk. We have converted some classes of fractional differential equations in the sense of the Srivastava-Owa fractional operator and its generalization into ordinary differential equations. Hence, the exact solutions are imposed. The solution procedure is simple and might find wide applications in image processing and signal processing by using fractional filter mask. Examples 3.1 and 3. showed the conversion of time-space fractional differential equations into a normal case. The exact solutions are introduced in the expression of a Mittag-Leffler function. While Example 3.3, the Cauchy problem of fractional order in the unit disk, proposed the exact solution in terms of the Fox-Wright function. 5 Acknowledgements The author is thankful to the referees for helpful suggestions for the improvement of this article. 6 Competing interests The author declares that they have no competing interests. References [1] M. Darus, R.W. Ibrahim, Radius estimates of a subclass of univalent functions, Math Vesnik (63)(11) 55-58. [] H.M. Srivastava, Y. Ling, G. Bao, Some distortion inequalities associated with the fractional derivatives of analytic and univalent functions, JIneqPureApplMath()(1) 1-6. [3] K.S. Miller and B. Ross, An Introduction to The Fractional Calculus and Fractional Differential Equations, John-Wiley and Sons, Inc., 1993. [4] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. [5] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, River Edge, NJ, USA,. [6] B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Institute for Nonlinear Science, Springer, New York, NY, USA, 3. [7] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 4 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 6. [8] J. Sabatier, O. P. Agrawal, and J. A. Machado, Advance in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 7. [9] D. Baleanu, B. Guvenc and J. A. Tenreiro, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, NY, USA, 1. [1] V. Lakshmikantham, S. Leela, J. Vasundhara, Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge 9. [11] C. Bianca, On set of nonlinearity in thermostatted active particles models for complex systems, Nonlinear Analysis: Real World Applications, 13(1) 593-68. [1] G. Mahmoud, E. Mahmoud, Modified projective Lag synchronization of two nonidentical hyperchaotic complex nonlinear systems, International Journal of Bifurcation and Chaos, (8) (11) 369-379. [13] G. Mahmoud, E. Mahmoud, M. Ahmed, On the hyperchaotic complex Lü system, Nonlinear Dyn, 58(9) 75-738. 9
ADE_98_edited [11/ 14:14] 1/1 [14] A. C. Fowler, J. D. Gibbon, M. J. McGuinness, The Complex Lorenz Equations, Physica D 4 (198) 139-163. [15] A. Rauth, L. Hannibal, N. B. Abraham, Global Stability properties of the Complex Lorenz Model, Physica D 99 (1996) 45-58. [16] P. Sainty, Contraction of a complex-valued fractional Brownian of order n, J. Math. Phys. 33(9)(199) 318-3149. [17] G. Jumarie, Fractional Brownian motion with complex variance via random walk in the complex plane and applications, Chaos, Sollitions and Fractals 11() 197-111. [18] G. Jumarie, Schrodinger equation for quantum fractal space time of order n via the complex-valued fractional Brownian motion. Int J Mod Phys A, 16(1)561-584. [19] G. Jumarie, Fractional Brownian motions via random walk in the complex plane and via fractional derivative. Comparison and further results. Chaos, Solitons and Fractals, (4)97-95. [] G. Jumarie, Fractionalization of the complex-valued Brownian motion of order n using Riemann Liouville derivative. Applications to mathematical finance and stochastic mechanics, Chaos, Solitons and Fractals 8 (6) 185-135. [1] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus Models and Numerical Methods (Series on Complexity, Nonlinearity and Chaos). World Scientific (1). [] D. Baleanu and J. J. Trujillo, On exact solutions of a class of fractional Euler-Lagrange equations. Nonlinear Dyn. 5 (8) 331-335. [3] R. Magin, X. Feng X and D. Baleanu, Solving the fractional order Bloch equation. Concepts in Magnetic Resonance Part A, 34A(9) 16-3. [4] R. W. Ibrahim, M. Darus, Subordination and superordination for univalent solutions for fractional differential equations, J. Math. Anal. Appl., 345(8) 871-879. [5] R. W. Ibrahim, On solutions for fractional diffusion problems, Electronic Journal of Differential Equations, 147 (1) 1-11. [6] R. W. Ibrahim, Existence and uniqueness of holomorphic solutions for fractional Cauchy problem, J. Math. Anal. Appl., 38 (11) 3-4 [7] R. W. Ibrahim, Ulam stability for fractional differential equation in complex domain, Abstract and Applied Analysis, (Volume 1, Article ID 649517, 8 pages doi:1.1155/1/649517) [8] R. W. Ibrahim, On holomorphic solution for space and time fractional telegraph equations in complex domain, Journal of Function Spaces and Applications, (Volume 1, Article ID 73681, 1 pages. doi:1.1155/1/73681). [9] P. R. Gordoa, A. Pickering, Z. N. Zhu, Bücklund transformations for a matrix second Painlev equation, Physics Letters A, 374 (34) (1) 34-344. [3] R. Molliq, B. Batiha, Approximate analytic solutions of fractional Zakharov-Kuznetsov equations by fractional complex transform, International Journal of Engineering and Technology, 1 (1) (1) 1-13. [31] H.M. Srivastava, S. Owa, Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press, John Wiley and Sons, New York, Chichester, Brisban, and Toronto, 1989. [3] R.W. Ibrahim, On generalized Srivastava-Owa fractional operators in the unit disk, Advances in Difference Equations (55)( 11) 1-1. [33] Z.B. Li, J.H. He, Fractional complex transform for fractional differential equations, Math Comput Appl 15 (1) 97-973. [34] Z.B. Li An extended fractional complex transform, J Non Sci Num Simu, 11 (1) 335-337. [35] Z.B. Li, J.H. He, Application of the fractional complex transform to fractional differential equations, Nonlinear Sci Lett A, (11) 11-16. [36] Ji-Huan He, S.K. Elagan, Z.B. Li, Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Physics Letters A 376 (1) 57-59. [37] H.M. Srivastava, P.W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985. 1