Fractional Laplace Transform and Fractional Calculus

Transcription

1 Inernaional Mahemaical Forum, Vol. 12, 217, no. 2, HIKARI Ld, hps://doi.org/ /imf Fracional Laplace Transform and Fracional Calculus Gusavo D. Medina 1, Nelson R. Ojeda 2, José H. Pereira 3 and Luis G. Romero 4 1,2,3,4 Faculy of Humaniies 1 Faculy of Naural Resources Naional Formosa Universiy. Av. Gunisky, 32 Formosa 36, Argenina Copyrigh c 217 Gusavo D. Medina, Nelson R. Ojeda, José H. Pereira and Luis G. Romero. This aricle is disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Absrac In his work we sudy he acion of he Fracional Laplace Transform inroduced in [6] on he Fracional Derivaive of Riemann-Liouville. The properies of he ransformaion in he convoluion produc defined as Miana were also presened. As an example we calculae he soluion of a differenial equaion. Mahemaics Subjec Classificaion: 26A33, 44A1 Keywords: Inegral Laplace ransform. Convoluion producs. Fracional Derivaive 1 Inroducion and Preliminaries We sar by recalling some elemenary definiions of page. 13 of [7]. Definiion 1. Le f f( be a funcion of R + f(s is given by he inegral.the Laplace ransform f(s L[f(] (s e s f(d (1.1

2 992 Gusavo D. Medina, Nelson R. Ojeda, José H. Pereira and Luis G. Romero for s R Definiion 2. Le A(R + a funcion of he space: i f is piecewise coninuous in he inerval T for any T R +. ii f i is of exponenial order, f( Ke a for M where M, K y a are real posiive consans. The parameer a is called he abscissa of convergence of he Laplace ransform.therefore we have he nex classic Definiion 3. Le f f( a funcion defined in R + The incomplee Laplace ransform f(s is given by he inegral L[f(, b] (s b e s f(d (1.2 for b, s R Medina, Ojeda, Pereira and Romero (cf.[6] has inroduced he following Definiion 4. Le f f( by a funcion of R +.The Inegral Laplace Transform f (s of order R + is given he inegral f (s L [f(](s e s1/ f(d (1.3 for s R The Inegral Laplace Transform i is a generalizaion of he Laplace ransform so ha when 1. Tha is o say L 1 [f(](s L[f(](s (1.4 Then we can generalize Theorem 2. If f( A(R +, hen here f (s L [f(](s for s > a Noe ha i is naural o enunciae he following Lemma 2. Le f be a sufficienly well-behaved funcion and le be a real number, < < 1. The fraccional Laplace ransform of he f funcion is given by L [f](s L[f](µ, µ s 1 Proof Follow from he definiion (1.3

3 Fracional Laplace ransform and fracional calculus 993 Properies If f (k ( A(R + con k 1, 2,...n y n N hen [( n ] df( L (s s n L [f(](s d Proof: Recall [( n ] df( L (µ µ n L [f(](s d and how we obained hen L [( df( d n k1 s n k f k 1 ( (1.5 n µ n k f k 1 ( (1.6 k1 L [f](s L[f](µ, µ s 1 n ] (s s n L [f(](s n k1 Now, we are able o find he inversion formula for he k-tl. L [f](s L[f](µ g 1 (µ, µ s 1 f( L 1 [L [f](s] L 1 (g 1 (µ( applying he Laplace inverse ransform gives L 1 (g 1 (µ( 1 2πi a+i a i e µ g 1 (µdµ 1 2πi s n k f k 1 ( (1.7 a+i a i e µ L[f](µdµ(1.8 and making he change of variable µ s 1, where dµ 1 s 1 1 ds L 1 (g 1 (µ( 1 2πi a +i a i e s L [f](s 1 s 1 1 ds (1.9 From his expression we have he following Definiion 5. Le f be a sufficienly well-behaved funcion and le be a real number, < < 1. The inverse Inegral Laplace Transform is given by L 1 [ f (s]( 1 2πi a +i a i e s f (ss 1 ds (1.1 Remark. Making he change of variable µ s 1, and aking ino accoun he formula ha esablish he relaionship beween he convenional and he fracional Laplace ransform, easily we can prove ha

4 994 Gusavo D. Medina, Nelson R. Ojeda, José H. Pereira and Luis G. Romero L [L 1 ] Id where Id denoe he ideniy operaor. Definiion 6. Le f and g funcions belonging o L 1 (R +, he ussual or classic convoluion produc is given by (f ( f(τg( τdτ, > (1.11 Definiion 7. Le f and g funcions belonging o L 1 (R +, Miana in [2] inroduce he convoluion produc as he inegral (f g( f(τ g(τdτ, > (1.12 Theorem 5. If f(, g( A(R + such ha f (s L [f(](s and g (s L [g(](s, hen L [f( g(](s f (s. g (s ( Main Resuls Properies Le λ R +, f and g funcions belonging o L 1 (R + and he exponenial funcion e λ 1/ : e λ1/ hen: i f e λ 1/ L [f](λ.e λ 1/ ii e λ 1/ f L [f](λ e λ 1/ (e λ f( iii L (f g(s L (gl (f,.( s 1/ (s Proof i From definiion 7 we have (f e λ 1/( if u τ, hen du dτ (f e λ 1/( [ f(τ e λ1/τ dτ f(ue λ1/ (u+ du ] f(ue λ1/u du.e λ 1/ L [f](λ.e λ 1/

5 Fracional Laplace ransform and fracional calculus 995 ii From definiion 7 we have (e λ 1/ f( e λ1/ (τ f(τdτ (2.1 as f y e λ 1/ are funcions belonging o L 1 (R +, hen e λ 1/ f L 1 (R + we obain ( (e λ 1/ f( e λ1/ (τ f(τdτ (e λ f( ( e λ1/τ f(τdτ e λ 1/ (e λ f( L [f](λe λ 1/ (e λ 1/ f( iii Le f and g funcions belonging o L 1 (R +, from definiion 7 we have (f g( applying definiion 4 we obain L [(f g(](s f(τ g(τdτ, > e s1/ (f g(d ( f(τ g(τdτ e s1/ Applying Fubini s Theorem we have ( e s1/ f(τ g(τdτ d d ( τ g(τ e s1/ f(τ d dτ If τ < <, < τ < and we consider changing he variable u τ, hen τ u +, < u < and he differenial d du ( τ L [(f g(](s g(τ e s1/ (τ u f(udu dτ ( τ e s1/τ g(τ e s1/u f(udu dτ L (gl(f,.( s 1/ (s 3 -Laplace Transform of Fracional Riemann- Liouville Operaor In his las secion we consider he Riemann-Liouville fracional operaors and we show he resuls of applies our -Laplace Transform o hem.

6 996 Gusavo D. Medina, Nelson R. Ojeda, José H. Pereira and Luis G. Romero Previously we need some elemenary definiions and resuls. Definiion 8. Le f be a locally inegrable funcion on (a, +. The Riemann-Liouville inegral of order, of he funcion f is given by I x f(. 1 Γ( x here Γ( denoes he Gamma funcion of Euler Γ(z a (x 1 f(d (3.1 e z 1 d (3.2 For > 1, and >, le j ( 1, be he singular kernel of Riemann- Γ( Liouville. I can be proved ha he Riemann-Liouville fracional inegral may be expressed as he convoluion ( Ix 1 f( Γ( f (x (3.3 The Riemann-Liouville fracional derivaive of order, is defined inverse D x I x id Anoher way o defined his fracional derivaive is as follows. Definiion 9. Le be a real number, and le m be an ineger. Then he Riemann-Liouville fracional derivaive of order is given by D x f( ( d dx m I m x f( (3.4 Lemma 1. Le f be a sufficienly well-behaved funcion and le be a real number, < < 1. The Laplace ransform of he Riemann-Liouville fracional inegral of he f funcion is given by L[I f](s (s L[f](s (3.5 Lemma 2. Le f be a sufficienly well-behaved funcion and le be a real number, < < 1. The Laplace ransform of he Riemann-Liouville fracional derivaive of he f funcion is given by L[D f(](s s L[f(](s I f( (3.6 Lemma 3. Le f be a sufficienly well-behaved funcion and le be a real number, < < 1. The Laplace ransform of he Riemann-Liouville fracional inegral of he f funcion is given by

7 Fracional Laplace ransform and fracional calculus 997 L [I β x f](s (s β/ L [f](s (3.7 Proof Remember ha is > y β R for [able of 6, page 61] L [ β ] From definiion 4 and (3.8 we have Γ(β + 1 s β+1 (3.8 L [j β (](s s β/ (3.9 recall (3.3 applying definiion 4 o (3.1 and (3.8 properie I x f(x j β ( f( (3.1 L (I β f(x L [j β ( f(](s L [j β (](s.l [f](s s β/.l [f](s Lemma 4. Le f be a sufficienly well-behaved funcion and le be a real number, < < 1. The Laplace ransform of he Riemann-Liouville fracional derivaive of he f funcion is given by L [D f(](s s β/ L [f(](s I 1 f( (3.11 Proof by definiion 9 we have ha if < β 1, m 1 y by Lemma 2 we have we ge he hesis L [ d dx I1 β L [D β xf(](s L [ d dx I1 β x f(](s (3.12 x f(](s s β L [Ix 1 β f] I 1 β x s 1/ s (1 β/ L [f] I 1 β x s 1/ s (1 β/ L [f] I 1 β x s β/ L [f] I 1 β x

8 998 Gusavo D. Medina, Nelson R. Ojeda, José H. Pereira and Luis G. Romero 4 Miag-Leffler The called funcions of he Miag-Leffler ype, play an imporan role in he heory of fracional differenial equaions (FDEs. Firs we inroduce a woparameer Miag-Leffler funcion defined by formula (4.1 E,β (λ k (λ k Γ(k + β (4.1 As we will see laer, classical derivaives of he Miag-Leffler funcion appear in so- luion of FDEs. Since he series (4.1 is uniformly convergen we may differeniae erm by erm and obain E (m,β (λ k (k + m! k! (λ k Γ(k + m + β Theorem 6. Le γ, β C, R(γ >, R(β >, λ R. Then hold ( L γm+β 1 E (m γ,β (λγ k (4.2 s γ β (4.3 (s γ/ λ m+1 Proof Remember he nex series convergence (k + m! x k m! (4.4 k! (1 x m+1 Then ( L γm+β 1 E (m γ,β (λγ (k + m!λ k L [ γk+γm+β 1 ] k! Γ(γk + γm + β (s k (k + m!λ k Γ(γk + γm + β k k! (k + m! k! k s γm β s γm β s γm β k Γ(γk + γm + βs γk+γm+β 1+1 λ k s γk+γm+β (k + m! (λs γ/ k k! m! (1 λs γ/ m+1 m! s (m+1γ/ (s γ/ λ m+1 s γ β (s γ/ λ m+1

9 Fracional Laplace ransform and fracional calculus Example A sligh generalizaion of an equaion solved in [4, page 157] D 1 2 f( + af( ; I 1 2 f( C (5.1 applying he The Inegral Laplace Transform, wih 1, we obained 2 (D 1 2 f( + af( (5.2 L 1 2 sl 1 [f(](s I 1 2 f( + al L 1 [f(](s 2 and applying definiion (1.1 gives he soluion of (5.1 ( ( C L 1 1 L 1 [f(](s L s + a is idenical o soluion obained in [8,page 139] (5.3 C s + a (5.4 (5.5 (5.6 f( C 1 2 E 1 2, 1 2 ( a 1 2 (5.7 References [1] A.A. Kilbas, H.M. Srivasava, J.J. Trujillo, Theory and Applicaions of Fracional Differenial Equaions, Norh-Holland Mahemaics Sudies, Vol. 24, Elsevier Science, 26. hps://doi.org/1.116/s34-28(6x81-5 [2] P.J. Miana, Convoluion producs in L 1 (R +, inegral ransforms and fracional calculus, Fracional Calculus and Applied Analysis, 8 (25, no. 4, [3] K.S. Miller, B. Ross, An Inroducion o he Fracional Calculus and Fracional Differenial Equaions, John Willey, [4] K. Oldham, J. Spanier, The Fracional Calculus, Academic Press, [5] L.G. Romero, R.A. Cerui, L.L. Luque, A New Fracional Fourier Transform and Convoluion Producs, Inernaional Journal of Pure and Applied Mahemaics, 66 (211, no. 4,

10 1 Gusavo D. Medina, Nelson R. Ojeda, José H. Pereira and Luis G. Romero [6] L.G. Romero, G.D. Medina, N.R. Ojeda, J.H. Pereira, A new alfa- Inegral Laplace Transform, Asian Journal of Curren Engineering and Mahs., 5 (216, [7] I. Podlubny, Fracional Differenial Equaions, Academic Press, Unied Saes, [8] S. Samko, A. Kilbas, O. Marichev, Fracional Inegrals and Derivaives, Gordon and Breach, Received: December 9, 217; Published: December 29, 217