Application of the Fractional Complex Transform to Fractional Differential Equations

Transcription

1 From the SelectedWks of Ji-Huan He 2011 Application of the Fractional Comple Transfm to Fractional Differential Equations Zheng-Biao Li Ji-Huan He Available at:

2 Z.B. Li, J.H. He, Nonlinear Sci. Lett.A, Vol.2, No.3, , 2011 Application of the Fractional Comple Transfm to Fractional Differential Equations Zheng-Biao Li 1, Ji-Huan He *2 1. College of Mathematics and Infmation Science, Qujing Nmal University, Qujing, Yunnan , China; 2. National Engineering Labaty f Modern Silk, College of Tetile and Engineering, Soochow University, 199 Ren-ai Road, Suzhou , China Abstract The fractional comple transfm is used to analytically deal with fractional differential equations. Two eamples are given to elucidate the solution procedure, showing it is etremely accessible to nonmathematicians. Keywds: Fractional Comple Transfm, Modified Riemann-Liouville Derivative, Fractional Differential Equation 1. Introduction Transfm is an imptant method to solve mathematical problems. Many useful transfms f solving various problems were appeared in open literature, such as the travelling wave transfm [1], the Laplace transfm [2], the Fourier transfm [3], the Bäcklund transfmation [4], the integral transfm [5], and the local fractional integral transfms[6]. Very recently the fractional comple transfm [7-10] was suggested to convert fractional der differential equations with modified Riemann-Liouville derivatives [11-13] into integer der differential equations, and the resultant equations can be solved by advanced calculus. 2. Jumarie s Fractional Derivative Jumarie's fractional derivative is a modified Riemann-Liouville derivative defined as[11-13] 1 1 ( ξ) ( f( ξ) f(0)) dξ, < 0, Γ ( ) 0 1 d D f( ) = ( ξ) ( f( ξ) f(0)) dξ, 0< < 1, Γ (1 ) d (1) 0 ( n) ( n) [ f ( )], n < n+ 1, n 1, * Cresponding auth. hejihuan@suda.edu.cn (J.H. He) Copyright 2011 Asian Academic Publisher Ltd. Journal Homepage:

3 122 ISSN : Nonlinear Science Letters A- Mathematics, Physics and Mechanics where f : R R, f( ) denotes a continuous (but not necessarily differentiable) function. Some useful fmulas and results of Jumarie s modified Riemann Liouville derivative were summarized in Refs. [11-13]: D c= 0, > 0, c=constant. (2) D [ cf( )] = cd f( ), > 0, c=constant. (3) D Γ (1 + ) =, > > 0. Γ (1 + ) D [ f( ) g( )] = [ D f( )] g( ) + f( )[ D g( )]. (5) D f t = f t. (6) ' ( ) ( ( )) ( ) ( ) (4) 3. The Fractional Comple Transfm The fractional comple transfm was first proposed in Refs. [7]. Consider a fractional differential equation 3 γ λ u u u u u u 3 γ λ t y z = 0. (7) where u/ t is Jumarie's fractional derivative of der with respect to t. The fractional comple transfm requires qt T =, Γ (1 + ) p X =, Γ (1 + ) γ ky Y =, Γ (1 + γ ) λ lz Z =. Γ (1 + λ) (8) where p, q, k, and l are unknown constants, 0< 1, 0< 1, 0< γ 1, 0< λ 1. Using the basic properties of the fractional derivative listed in Eqs. (2)~(6), we can convert the fractional derivatives into classical derivatives:

4 Z.B. Li, J.H. He, Nonlinear Sci. Lett.A, Vol.2, No.3, , = q, t T = p, X γ = k, γ y Y λ = l. λ z Z (9) Eq. (7) becomes 3 u u 3 u u u q + 2pu + 3p + 4γ + 5λ = 0, (10) 3 T X X Y Z which can be solved by, f eample, the ep-function method [14-15]. 4. Applications Eample 1 As the first eample, we consider a space-time fractional equation [16] f the transpt equation in pous media: ut (, ) ut (, ) + c = 0, 0<, 1, (11) t where c is a constant. Using the following fractional comple transfm t T =, Γ (1 + ) (12) X =, Γ (1 + ) We obtain ut (, ) u T = = 1 =, (13) t T t T T Eq. (11) becomes ut (, ) u X = = 1 =. X X X (14) u T u + c = 0, X (15) Solving Eq. (15) results in 1 u1( X, T) = f( T X), (16) c

5 124 ISSN : Nonlinear Science Letters A- Mathematics, Physics and Mechanics u ( X, T) = f( X ct), (17) 2 where the function f ( ξ ) is an arbitrary and first der function which is differentiable with respect to ξ. We, therefe, obtain the solution, which reads (, ) ( t u1 t = f Γ (1 + ) cγ (1 + ) ), (18) (, ) t u2 t = c f( ), Γ (1 + ) Γ (1 + ) (19) Eample 2 We investigate a fractional differential equation in the fm [16]: t u(, t) u(, t) + = 0, 0<, 1, Γ (1 + ) t Γ (1 + ) (20) Using the fractional comple transfm, Eq. (12), Eq. (20) is converted into the following partial differential equation Tu where u = u( X, T). Its general solution can be epressed as T + Xu = 0, (21) X uxt (, ) = ft ( / X), (22) uxt (, ) = f( X/ T), (23) where the function f ( ξ ) is an arbitrary and first der function differentiable with respect to ξ. We obtain the needed solution, which reads Introducing ξ defined as t ut (, ) = f( / ), Γ (1 + ) Γ (1 + ) (24) t ut (, ) = f( / ), Γ (1 + ) Γ (1 + ) (25) t ξ = /. Γ (1 + ) Γ(1 + ) (26) and using Jumarie's chain rule, we have

6 Z.B. Li, J.H. He, Nonlinear Sci. Lett.A, Vol.2, No.3, , ut (, ) ( ) = fξξt = fξ/, (27) t Γ (1 + ) and ut 2 (, ) t ( ) / = fξξ = fξ. (28) 2 Γ (1 + ) ( Γ (1 + )) It is easy to prove that Eq. (24) is the solution of Eq. (20). Substituting the results into the left side of Eq. (20) yields t ut (,) ut (,) t / t / + = fξ fξ = 0. Γ (1 + ) t Γ (1 + ) Γ (1 + ) Γ (1 + ) Γ (1 + ) Γ (1 + ) (29) 4. Conclusions The fractional comple transfm is etremely simple but effective f solving fractional differential equations. The method is accessible to all with basic knowledge of Advanced Calculus and with little Fractional Calculus. Acknowledgement The wk is suppted by PAPD (A Project Funded by the Priity Academic Program Development of Jiangsu Higher Education Institutions), National Natural Science Foundation of China under Grant No , Natural Science Foundation of Yunnan Province under Grant No. 2010CD086 and the Project (A study on eact solutions of partial differential equation with fraction der) of the Science and Technology Department of Yunnan Province of China, References [1] Bekir A. New eact travelling wave solutions of some comple nonlinear equations, Communications in Nonlinear Science and Numerical Simulation, 14(4) (2009) [2] Cassol M, Wtmann S, Rizza U. Analytic modeling of two-dimensional transient atmospheric pollutant dispersion by double GITT and Laplace Transfm techniques, Environmental Modelling & Software, 24(1) (2009) [3] Sejdić E, Djurović I, et al. Fractional Fourier transfm as a signal processing tool: An overview of recent developments, Signal Processing, 91(6) (2011) [4] Gdoa PR, Pickering A, Zhu ZN. Bäcklund transfmations f a matri second Painlevé equation, Physics Letters A, 374 (34) (2010) [5] Cotta RM, Mikhailov MD, Integral transfm method, Applied Mathematical Modelling, 17 (3) (1993) [6] Yang X, Local Fractional Integral Transfms, Progress in Nonlinear Science, 4(2011) [7] Li ZB, He JH. Fractional Comple Transfm f Fractional Differential Equations, Mathematical and Computational Applications, 15 (5) (2010) [8] Li ZB. An Etended Fractional Comple Transfm, Journal of Nonlinear Science and Numerical Simulation, 11 (2010) [9] He JH, Li ZB. Converting Fractional differential equations into partial differential equations, Thermal Science, DOI REFERENCE: /TSCI H

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