Mahgoub Transform Method for Solving Linear Fractional Differential Equations

Transcription

1 Mahgoub Transform Mehod for Solving Linear Fraional Differenial Equaions A. Emimal Kanaga Puhpam 1,* and S. Karin Lydia 2 1* Assoiae Professor&Deparmen of Mahemais, Bishop Heber College Tiruhirappalli, Tamil Nadu, India Absra In his paper, Mahgoub Transform Mehod has been inrodued for solving linear Fraional Differenial Equaions (FDEs) wih onsan oeffiiens. The fraional derivaives are desribed in he Capuo sense. Some fundamenal properies of Mahgoub Transform neessary in solving FDE are derived. The effiieny of his mehod has been demonsraed using examples. Keywords - Mahgoub ransform, Fraional Differenial Equaions, Miag-Leffler. I. INTRODUCTION Fraional alulus is a generalizaion of ordinary differeniaion and inegraion o arbirary nonineger order. I has been used in various areas suh as signal proessing, image proessing, onrol engineering, bioengineering, polymer neworks, mehanis, and visoelasiiy [1]. Tradiional and new inegral ransform mehods have been applied o find he analyial soluion of FDE. Some of hem are Laplae, Mellin, Fourier, Sumudu, Naural, Elzaki and Kamal [2-8]. Many researhers have shown heir ineres in finding he numerial soluion o boh linear and nonlinear FDEs [9-11]. AbdelrahimMahgoub [12] inrodued Mahgoub ransform mehod o solve he ordinary differenial equaions. In his paper, we have inrodued he Mahgoub ransform mehod for finding he exa soluion of FDEs wih Capuo derivaives. This paper has been organized as follows: In Seion 2, basi definiions relaed o fraional alulus are given. In Seion 3, he Mahgoub ransform of fraional inegrals and derivaives have been disussed. In Seion 4, examples of FDE have been provided o illusrae he effiieny of his mehod. II. PRELIMINARIES AND NOTATIONS In his seion, fundamenal definiions and properies of fraional alulus referred in his paper are inrodued. Definiion 1: The Riemann Liouville fraional inegral I α f x of order α R, α > of funionf x C μ, μ 1 is defined as I α f = 1 Γα I f = f() (1) τ α 1 f τ dτ, > Definiion 2:The Capuo fraional derivaive of order α R, α > is given by D α f = I m α D m f x = 1 τ m α 1 f m τ dτ, > where m 1 α m, mεn + and Γ(. ) denoes he Gamma funion. Definiion 3: The Miag-Leffler funion of one parameer α is denoed by E α z and is defined as E α z = z k k=, α >, Re α >, zεc (3) Γ(αk ) The Miag-Leffler funion wih wo parameers αandβ is denoed by E α,β z and is defined as E α,β z = z k k=, α, β >, Re α, Re β > (4) Γ(αk +β ) where, C is he se of omplex numbers. For β = 1, we ge, E α,1 z = E α z whih is he dire generalizaion of exponenial series. (2) ISSN: hp:// Page 253

2 III. MAHGOUB TRANSFORM OF FRACTIONAL INTEGRALS AND DERIVATIVES Mahgoub ransform is defined on he se of oninuous funions and exponenial order. We onsider funions in he se A defined by A = f : f < Pe i if 1 i,, i = 1,2; i > (5) where 1, 2 may be finie or infinie and he onsan P mus be finie. Le f A, henmahgoub ransform is defined as M f = H u = u f e u d,, Mahgoub ransform of simple funions are given below: 1 u 2 (6) i M 1 = 1 (7) ii M = 1 u (8) iii M 2 = 2 u 2 (9) iv M n = n! Γ(n) un = (1) u n Mahgoub Transform for derivaives are: (i) M f = uh u uf() (11) ii M f = u 2 H u u 2 f uf (12) iii M f n = u n n 1 H u k= u n k f k () (13) A. Convoluion heorem Le F(u) and G(u) denoe he Mahgoub ransform of f() and g() respeively. Then M f() g() = 1 F u G(u) (14) u B. Inverse Mahgoub ransform If M f() = H u, henf() is alled he inverse Mahgoub ransform of H u. In symbol, f = M 1 H u where M 1 is alled he inverse Mahgoub ransform operaor. Some fundamenal properies of Mahgoub Transform neessary in solving FDE are given in he following heorems. Theorem 1: If H u is Mahgoubransformoff(), hen Mahgoub ransform ofriemann-liouville fraional inegral is M I α f() = u α H(u) (16) for m 1 < α m, m ε N. Consider he Riemann-Liouville fraional inegral I α f() = 1 Γ(α) = 1 Γ(α) τ α 1 f τ dτ α 1 f() Applying he Mahgoub ransform o Eqn. (17) and using (14), we ge M I α f() = u α H(u) This omplees he proof. ISSN: hp:// Page 254 (15) (17)

3 Theorem 2: If H u is Mahgoub ransform of f, hen Mahgoub ransform of Capuo fraional derivaive is M D α f() = u α m 1 H u k= u α k f k () (18) form 1 < α m, m ε N. Consider he Capuo fraional derivaive = 1 D α f = 1 m α 1 f (m ) τ m α 1 f m τ dτ, > Applying Mahgoub ransform o Eqn. (19) and using (14), we ge M D α 1 f() = Γ(m α) M(m α 1 )M(f m ) = u α m 1 H u k= u α k f k () afer simplifiaion. This omplees he proof. C. Mahgoub ransform of Miage-Leffler funion Theorem 3: The Mahgoub ransform of he Miage-Leffler funion is given by he following heorem. 1 uα β M = β 1 E α,β a α, u α a < 1 (2) Using he definiion of Mahgoub ransform, we have Then, M β 1 E α,β a α = u β 1 E α,β a α e u d 1 uα β M = = k= a k u Γ(αk +β) αk +β 1 e u d a k k= M αk +β 1 = Γ αk +β = β 1 E α,β a α. uα β (19) Example 1: Consider he inhomogeneous Bagley-Torvik equaion IV. ILLUSTRATIVE EXAMPLES D 2 y + D 3 2y + y = 1 + (21) Using Mahgoub Transform o Eqn. (21), we ge Using he properies (13) and (18), we ge y = y = 1 M(D 2 y ) + M( D 3 2y ) + M(y ) = M(1 + ) H u = u By aking he Inverse Mahgoub Transform, we ge he exa soluion of his problem as ISSN: hp:// Page 255

4 y = 1 +. Example 2: Consider he inhomogeneous linear equaion D α y + y = 2 2 α 1 α Γ(3 α) y =, were α 1 Applying Mahgoub Transform o Eqn. (22), we ge + Γ(2 α) 2, (22) M( D α y ) + M(y ) = M Using he properies (13) and (18), we ge H u = 2 1 u 2 u 2 2 α Γ 3 α M 1 α Γ 2 α + M( 2 ) By aking he Inverse Mahgoub Transform, we ge he exa soluion of his problem as Example 3: y = 2. Consider he linear iniial value problem D α y + y =, (23) y = 1, y =,were α 2 ApplyingMahgoub Transform o Eqn. (23), we ge M( D α y ) + M(y ) =. Using he properies (13) and (18), we ge H u = uα (u α ) By aking he Inverse Mahgoub Transform, we ge he exa soluion of his problem as Example 4: y = E α α. Consider he linear iniial value problem D α y = y + 1, were α 1 y =, (24) Using Mahgoub Transform o Eqn. (24), we ge M( D α y ) = M y + M 1. Using he properies (13) and (18), we ge H u = 1 u α 1 By aking he Inverse Mahgoub Transform, we ge he exa soluion of his problem as y = α E α,α α V. CONCLUSION In his paper, he Mahgoub Transformaion mehod has been suessfully applied o obain an exa soluion of linear fraional ordinary differenial equaions wih onsan oeffiiens.some fundamenal properies of Mahgoub Transform neessary in solving FDE are derived.by solving he illusraive examples, i is onluded ha he Mahgoub Transform is effiien, reliable and powerful for finding analyi soluion of linear fraional differenial equaions wih onsan oeffiiens. ISSN: hp:// Page 256

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