mahemaics Aricle A Noe on he Equivalence of Fracional Relaxaion Equaions o Differenial Equaions wih Varying Coefficiens Francesco Mainardi Deparmen of Physics and Asronomy, Universiy of Bologna, and he Naional Insiue of Nuclear Physics (INFN), Via Irnerio, 46, I-426 Bologna, Ialy; francesco.mainardi@bo.infn.i; Tel.: +39-52968 Received: 4 December 27; Acceped: 5 January 28; Published: 9 January 28 Absrac: In his noe, we show how an iniial value problem for a relaxaion process governed by a differenial equaion of a non-ineger order wih a consan coefficien may be equivalen o ha of a differenial equaion of he firs order wih a varying coefficien. This equivalence is shown for he simple fracional relaxaion equaion ha poins ou he relevance of he Miag Leffler funcion in fracional calculus. This simple argumen may lead o he equivalence of more general processes governed by evoluion equaions of fracional order wih consan coefficiens o processes governed by differenial equaions of ineger order bu wih varying coefficiens. Our main moivaion is o solici he researchers o exend his approach o oher areas of applied science in order o have a deeper knowledge of cerain phenomena, boh deerminisic and sochasic ones, invesigaed nowadays wih he echniques of he fracional calculus. Keywords: Capuo fracional derivaives; Miag Leffler funcions; anomalous relaxaion MSC: 26A33; 33E2; 34A8; 34C26. Inroducion Le us consider he following relaxaion equaion dψ d = r() Ψ(),, () subjeced o he iniial condiion, for he sake of simpliciy, Ψ( + ) =, (2) where Ψ() and r() are posiive funcions, sufficienly well-behaved for. In Equaion (), Ψ() denoes a non-dimensional field variable and r() he varying relaxaion coefficien. The soluion of he above iniial value problem reads Ψ() = exp[ R()], R() = I is easy o recognize from re-arranging Equaion () ha, for, r( ) d >. (3) r() = Φ() Ψ(), Φ() = Ψ() () = dψ (). (4) d Mahemaics 28, 6, 8; doi:.339/mah68 www.mdpi.com/journal/mahemaics
Mahemaics 28, 6, 8 2 of 5 The soluion (3) can be derived by solving he iniial value problem by separaion of variables Ψ() >From Equaion (3), we also noe ha dψ( ) Ψ( ) = Φ( ) Ψ( ) d = r( ) d = R(). (5) R() = log[ψ()]. (6) As a maer of fac, we have shown well-known resuls ha will be relevan for he nex secions. 2. Miag Leffler Funcion as a Soluion of he Fracional Relaxaion Process Le us now consider he following iniial value problem for he so-called fracional relaxaion process { D αψ α(() = Ψ α (),, Ψ α ( + (7) ) =, wih α (, ]. Above, we have labeled he field variable wih Ψ α o poin ou is dependence on α and considered he Capuo fracional derivaive, defined as: Γ( α) D α Ψ α () = Ψ () α ( ) ( ) α d, < α <, d d Ψ α(), α =. As found in many reaises of fracional calculus, and, in paricular, in he 27 survey paper by Mainardi and Gorenflo [] o which he ineresed reader is referred for deails and addiional references, he soluion of he fracional relaxaion problem (7) can be obained by using he echnique of he Laplace ransform in erms of he Miag Leffler funcion. Indeed, we ge in an obvious noaion by applying he Laplace ransform o Equaion (7) so ha s α Ψ α (s) s α = Ψ α (s), hence Ψ α (s) = sα s α +, (9) Ψ α () = E α ( α ) = n= αn (8) ( ) n Γ(αn + ). () For more deails on he Miag Leffler funcion, we refer o he recen reaise by Gorenflo e al. [2]. In Figure, for readers convenience, we repor he plos of he soluion () for some values of he parameer α (, ]. I can be noiced ha, for α, he soluion of he iniial value problem reduces o he exponenial funcion exp( ) wih a singular limi for because of he asympoic represenaion for α (, ), E α ( α ) α Γ( α + ),. () Now, i is ime o carry ou he comparison beween he wo iniial value problems described by Equaions () and (7) wih heir corresponding soluions (3), (). I is clear ha we mus consider he derivaive of he Miag Leffler funcion in (), namely Φ α () = d d Ψ α() = d d E α( α ) = α E α,α ( α ). (2)
Mahemaics 28, 6, 8 3 of 5..8 Ψ().6.4 α=.25 α=.5 α=.75.2. 2 3 4 5 Figure. Plos of he Miag Leffler funcion Ψ α () for α =.25,.5,.75, versus [, 5]. In Figure 2, we show he plos of posiive funcion Φ α () for some values of α (, ]...8 Φ().6.4 α=.25 α=.5 α=.75.2. 2 3 4 5 Figure 2. Plos of he posiive funcion Φ α () for α =.25,.5,.75, versus [, 5]. The above discussion leads o he varying relaxaion coefficien of he equivalen ordinary relaxaion process: r α () = Φ α() Ψ α () = r α() = α E α,α ( α ) E α ( α ) α Γ(α), +, α, +. (3)
Mahemaics 28, 6, 8 4 of 5 Figure 3 depics he plos of r α () for some raional values of α, including he sandard case α =, in which he raio reduces o he consan...8 r().6.4 α=.25 α=.5 α=.75.2. 2 3 4 5 Figure 3. Plos of he raio r α () for α =.25,.5,.75, versus [, 5]. We conclude by ploing in Figure 4 he funcion R α () = log[ψ α ()] for some values of he parameer α (, ]. 5 4 R() 3 2 α=.25 α=.5 α=.75 2 3 4 5 Figure 4. Plos of R α () = log[ψ α ()] for α =.25,.5,.75, versus [, 5]. 3. Conclusions In his noe, we have shown how he fracional relaxaion process governed by a fracional differenial equaion wih a consan coefficien is equivalen o a relaxaion process governed by an ordinary differenial equaion wih a varying coefficien. These consideraions provide a differen look
Mahemaics 28, 6, 8 5 of 5 a his fracional process over all for experimenaliss who can measure he varying relaxaion coefficien versus ime. Indeed, if his coefficien is found o fi he analyical or asympoical expressions in Label (3), he researcher canno disinguish if he governing equaion is fracional or simply ordinary. To make he difference, we hus need oher experimenal resuls. We are convinced ha i is possible o adap he above reasoning o oher fracional processes, including anomalous relaxaion in viscoelasic and dielecric media and anomalous diffusion in complex sysems. This exension is lef o percepive readers who can explore hese possibiliies. Las bu no leas, we do no claim o be original in using he above analogy in view of he grea simpliciy of he argumen: for example, a similar procedure has recenly been used by Sandev e al. [3] in dealing wih he fracional Schrödinger equaion. Acknowledgmens: The auhor is very graeful o Leonardo Benini, a Maser s suden in Physics (Universiy of Bologna), for his valuable help in ploing. As a maer of fac, he has used he MATLAB rouine for he Miag Leffler funcion appoined by Robero Garrappa (see hps://i.mahworks.com/malabcenral/ fileexchange/4854-he-miag-leffler-funcion). The auhor would like o devoe his noe o he memory of he lae Rudolf Gorenflo (93 27), wih whom for 2 years he had published join papers. The auhor presumes ha his noe is wrien in he spiri of Gorenflo, being based on he simpler consideraions. This work has been carried ou in he framework of he aciviies of he Naional Group of Mahemaical Physics (GNFM-INdAM). Furhermore, he auhor has appreciaed consrucive remarks and suggesions of he anonymous referees ha helped o improve his noe. Conflics of Ineres: The auhor declares no conflic of ineres. References. Mainardi, F.; Gorenflo, R. Time-fracional derivaives in relaxaion processes: A uorial survey. Frac. Calc. Appl. Anal. 27,, 269 38. 2. Gorenflo, R.; Kilbas, A.A.; Mainardi, F.; Rogosin, S. Miag Leffler Funcions, Relaed Topics and Applicaions; Springer: Berlin/Heidelberg, Germany, 24. 3. Sandev, T.; Pereska, I.; Lenzi, E.K. Effecive poenial from he generalized ime-dependen Schrödinger equaion. Mahemaics 26, 4, 59 68. c 28 by he auhor. Licensee MDPI, Basel, Swizerland. This aricle is an open access aricle disribued under he erms and condiions of he Creaive Commons Aribuion (CC BY) license (hp://creaivecommons.org/licenses/by/4./).