Some applications of Wright functions in fractional differential equations
aa r X i v : . [ m a t h . C A ] J u l SOME APPLICATIONS OF WRIGHT FUNCTIONS INFRACTIONAL DIFFERENTIAL EQUATIONS.
R.GARRA AND F. MAINARDI Abstract.
In this note we prove some new results about the application ofWright functions of the first kind to solve fractional differential equations withvariable coefficients. Then, we consider some applications of these results in orderto obtain some new particular solutions for nonlinear fractional partial differentialequations.
Keywords : Wright functions, linear and nonlinear fractional equations
MSC 2010 : 34A08 Introduction
The problem to find explicit solutions for fractional ordinary differential equationswith variable coefficients is a topic of interest, also in the context of the studies aboutspecial functions. For example the so called Saigo-Kilbas function emerges in thestudy of fractional ODE with variable coefficient (see [2] and the references therein).Recently different authors are working on systematic methods to solve fractionalODEs with variable coefficients, we refer in particular to the recent paper [1]. Inother cases, it is possible to prove that some interesting classes of special functionssolve fractional equations with variable coefficients. For example, in [7] the authorsproved that a generalized Le Roy function solves a particular integro-differentialequation with variable coefficients involving Hadamard fractional operators.In the recent paper [6], a new interesting result has been pointed out about thesolution of a new class of fractional ODEs by means of the classical Wright functionsof the first kind.In particular, it was proved, by direct calculations, that the following fractionalODE(1.1) ddt t λ d λ udt λ − λut − λ = 0 , t ≥ , λ ∈ (0 , λ is solved by the function(1.2) u ( t ) = W λ, ( t λ ) = ∞ X k =0 t λk k !Γ( kλ + 1) , λ ∈ (0 , , under the initial condition that u ( t = 0) = 1. We note that W λ, is a particular caseof special transcendental functions known as Wright functions that we will briefly AND F. MAINARDI discuss in the next section, distinguishing them in two kinds according to the valuesof the first parameter λ .Observe that Eq. (1.1) can be viewed as a sort of fractional generalization ofthe Bessel-type differential equations. Indeed, for λ = 1 we obtain the followingequation(1.3) ddt t dudt = u, whose solution is the so-called Tricomi function (see [3], [4] and the referencestherein)(1.4) C ( t ) = ∞ X k =0 t k k ! . that turns out to be related to modified Bessel function of the first kind and orderzero I by(1.5) C ( t ) = I (2 t / ) . In the next section we will also discuss in some detail the relations between theWright functions of the first kind with functions of the Bessel type.The operator ddt t ddt appearing in (1.3) is also named Laguerre derivative in theliterature.The aim of this short note is twofold. First of all we provide a more general resultconnecting Wright functions of the first kind with fractional ODE with variable co-efficients. We underline the role of these special functions in the theory of fractionaldifferential equations. Then, we discuss some simple applications of these resultsto solve nonlinear fractional PDEs admitting solutions by generalized separatingvariable solution.2.
Preliminaries about Wright functions
The classical
Wright function that we denote by W λ,µ ( z ), is defined by the seriesrepresentation convergent in the whole complex plane,(2.1) W λ,µ ( z ) := ∞ X n =0 z n n !Γ( λn + µ ) , λ > − , µ ∈ C , The integral representation reads as:(2.2) W λ,µ ( z ) = 12 πi Z Ha − e σ + zσ − λ dσσ µ , λ > − , µ ∈ C , where Ha − denotes the Hankel path: this one is a loop which starts from −∞ alongthe lower side of negative real axis, encircling it with a small circle the axes originand ends at −∞ along the upper side of the negative real axis. OME APPLICATIONS OF WRIGHT FUNCTIONS 3 W λ,µ ( z ) is then an entire function for all λ ∈ ( − , + ∞ ). Originally, Wrightassumed λ ≥ − < λ <
0, [24].In view of the asymptotic representation in the complex domain and of the Laplacetransform for positive argument z = r > r can be the time variable t or the spacevariable x ), the Wright functions are distinguished in first kind ( λ ≥
0) and secondkind ( − < λ <
0) as outlined in the Appendix F of the book by Mainardi [15], seealso the recent survey article [16]. In particular, for the asymptotic behavior, werefer the interested reader to the surveys by Luchko and by Paris in the Handbookof Fractional Calculus and Applications, see, respectively, [14, 18], and referencestherein.We note that the Wright functions are an entire of order 1 / (1 + λ ); hence, onlythe first kind functions ( λ ≥
0) are of exponential order, whereas the second kindfunctions ( − < λ <
0) are not of exponential order. The case λ = 0 is trivialsince W ,µ ( z ) = e z / Γ( µ ) . As a consequence of the difference in the orders, we mustpoint out the different Laplace transforms proved e.g., in [10, 15], see also the recentsurvey on Wright functions by Luchko [14]. We have: • for the first kind, when λ ≥ W λ,µ ( ± r ) ÷ s E λ,µ (cid:18) ± s (cid:19) ; • for the second kind, when − < λ < ν = − λ so 0 < ν < W − ν,µ ( − r ) ÷ E ν,µ + ν ( − s ) , whre E λ,µ ( z ) denotes the Mittag-Leffler function, see for details [9].In the present paper we need to restrict our attention to the Wright function ofthe first kind in order to point out their relations with functions of the Bessel type.Indeed, it is easy to recognize that the Wright functions of the first kind turn outto be related to the well-known Bessel functions J ν and I ν for λ = 1 and µ = ν + 1.In fact, by using the well known series definitions for the Bessel functions and theseries definitions for the Wright functions, we get the identities:(2.5) J ν ( z ) := (cid:16) z (cid:17) ν ∞ X n =0 ( − n ( z/ n n ! Γ( n + + ν + 1) = (cid:16) z (cid:17) ν W ,ν +1 (cid:18) − z (cid:19) ,W ,ν +1 ( − z ) := ∞ X n =0 ( − n z n n ! Γ( n + ν + 1) = z − ν/ J ν (2 z / ) . R.GARRA AND F. MAINARDI and(2.6) I ν ( z ) := (cid:16) z (cid:17) ν ∞ X n =0 ( z/ n n ! Γ( n + + ν + 1) = (cid:16) z (cid:17) ν W ,ν +1 (cid:18) z (cid:19) ,W ,ν +1 ( z ) := ∞ X n =0 z n n ! Γ( n + ν + 1) = z − ν/ I ν (2 z / ) . As far as the standard Bessel functions J ν are concerned, the following observationsare worth noting. We first note that the Wright functions W ,ν +1 ( − z ) are related tothe entire functions J Cν ( z ) known as Bessel-Clifford functions herewith defined(2.7) J Cν ( z ) := z − ν/ J ν (2 z / ) = ∞ X k =0 ( − k z k k ! Γ( k + ν + 1)We note that different variants of the Bessel functions (that is without the singularfactor) were adopted independently by Tricomi to get entire functions as in the caseof Bessel-Clifford in his treatise on Special Functions published in the late 1950’s[21], later revisited and enlarged by Gatteschi [8],(2.8) J Tν ( z ) := ( z/ − ν J ν ( z ) = ∞ X k =0 ( − k k !Γ( k + ν + 1) (cid:16) z (cid:17) k . Then, in view of the first equation in (2.5), some authors refer to the Wright function(of the first kind) as the
Wright generalized Bessel function (misnamed also as the
Bessel-Maitland function ) and introduce the notation for λ ≥
0, see e.g. [13], p. 336,and [17](2.9) J ( λ ) ν ( z ) := (cid:16) z (cid:17) ν ∞ X n =0 ( − n ( z/ n n !Γ( λn + ν + 1) = (cid:16) z (cid:17) ν W λ,ν +1 (cid:18) − z (cid:19) . Similar remarks can be extended to the modified Bessel functions I ν . Even if forBessel functions the parameter ν can take aribitrary real and/or complex values,from now on we restrict our analysis to ν ≥ Fractional ordinary differential equations with variablecoefficients
We here prove a new connection between Wright functions of first kind and frac-tional ODE.
Theorem 3.1.
The fractional equation (3.1) d β dt β (cid:18) t ν dfdt (cid:19) = βt ν − f ( t ) , OME APPLICATIONS OF WRIGHT FUNCTIONS 5 involving a fractional derivative in the sense of Caputo of order β ∈ (0 , , admits asolution of the form (3.2) f ( t ) = W β,ν ( t β ) . Proof.
By direct calculations we have that d β dt β (cid:18) t ν ddt (cid:19) W β,ν ( t β ) = β d β dt β ∞ X k =1 t kβ + ν − ( k − kβ + ν )= β ∞ X k =1 t kβ + ν − − β ( k − kβ + ν − β ) = βt ν − W β,ν ( t β ) . where we used the fact that(3.3) d β dt β t s = Γ( s + 1)Γ( s + 1 − β ) t s − β , where s > (cid:3) This result is clearly more general than the one proved in [6], but also in thiscase we have a direct connection with Bessel-type equations involving Laguerrederivatives. Indeed, for ν = β = 1 we obtain again the equation(3.4) ddt t dudt = u, studied in [4].In a similar way, we can prove that the function(3.5) u ( t ) = ∞ X k =0 t kβ k ! Γ( kβ + ν ) , solves the equation(3.6) (cid:18) d β dt β t ν ddt t ddt (cid:19) f ( t ) = βt ν − f ( t ) . This can be simply generalized to higher order equations.4.
New results about nonlinear fractional diffusion equations
A simple but useful method for constructing exact solutions for nonlinear PDEs isgiven by the generalized separation of variables. This method permits to find partic-ular classes of exact solutions mainly based on the reduction to nonlinear ODEs thatcan be exactly solved. It is possible to prove that wide classes of nonlinear PDEsadmit such solutions in separating variable form by using for example the invariantsubspace method (see the relevant monograph [5]). Many papers have been devotedto show the utility of this method in order to obtain exact results for nonlineardiffusive equations, we refer for example to [19] and [22] and the references therein.
R.GARRA AND F. MAINARDI In the recent literature the construction of exact solutions for nonlinear fractionalPDEs based on this method have gained some interest, see for example [20]. Indeed,there are few exact results for nonlinear PDEs involving space or time-fractionalderivatives. On the other hand, the applications of Lie group methods play a centralrole in this framework for a complete mathematical analysis. We refer in particularto the review chapters [11] and [12] published on the recent
Handbook of FractionalCalculus with Applications .We here prove some new interesting results that can be obtained as a directconsequence of Theorem 3.1.
Proposition 4.1.
The nonlinear fractional equation (4.1) t − ν ∂ β ∂t β t ν ∂u∂t + u ∂u∂x = − u − βu, β ∈ (0 , admits the solution (4.2) u ( x, t ) = e − x · W β,ν ( t β ) . Proof.
We search a particular solution by separating variable form u ( x, t ) = f ( t ) e − x .By substitution, we have that(4.3) e − x t − ν ∂ β ∂t β (cid:18) t ν ∂f∂t (cid:19) − f e − x = − f e − x − βf e − x and therefore to the following fractional ODE on f(t)(4.4) d β dt β t ν dfdt = − βt ν − f ( t ) . Under the condition that f ( t = 0) = 1 the solution for (4.4) is given by f ( t ) = W β,ν ( − t β )and we obtain the claimed result. (cid:3) Remark 4.2.
Observe that, if we consider the case β = ν = 1 , we obtain an explicitsolution for the equation (4.5) ∂∂t t ∂u∂t + u ∂u∂x = − u − u, that is a sort of nonlinear hyperbolic equation with an advective term and a nonlinearreaction term, admitting the following particular solution (4.6) u ( x, t ) = C ( − t ) · e − x . Proposition 4.3.
The equation (4.7) t − λ ∂∂t t λ ∂ λ u∂t λ = ∂ u m ∂x − u, m > , OME APPLICATIONS OF WRIGHT FUNCTIONS 7 admits a solution of the form (4.8) u ( x, t ) = W λ, ( − t λ ) x /m Proof.
Let us assume that the equation admits a solution, via separation of variables,of the form x /m f ( t ), then if we plug in this ansatz we get(4.9) (cid:18) t − λ ∂∂t t λ ∂ λ ∂t λ (cid:19) x /m f ( t ) = − x /m f ( t ) , whose solution is given by f ( t ) = W λ, ( − t λ ) , as claimed. (cid:3) We have previously discussed applications to equations involving fractional Bessel-type operator in time, another interesting case is the applications to linear space-fractional diffusive-type equations involving derivatives in the sense of Caputo. Wehave the following result.
Proposition 4.4.
The fractional equation (4.10) ∂u∂t = 1 β x ν − ∂ β ∂x β (cid:18) x ν ∂u∂x (cid:19) , x ≥ , β ∈ (0 , , admits a solution of the form (4.11) u ( x, t ) = e − t · W β,ν ( − x β ) . We neglect the complete proof that can be directly obtained by substitution.5.
Conclusions
The main aim of this paper is to underline a new interesting application of Wrightfunctions of the first kind to solve fractional ordinary differential equations with vari-able coefficients that generalize Bessel-type equations. In practice the solutions areformal becuse obtained by the method of subtition, showing the direct connectionbetween fractional Bessel-type equations and Wright functions. Then, we discussthe applications of this new result to solve linear and nonlinear fractional partialdifferential equations admitting solutions obtained by means of the generalized sepa-ration of variable method. As a consequence we are able to find new exact solutionsfor time or space-fractional linear and nonlinear diffusive-type equations with vari-able coefficients.A more general classification of linear or nonlinear fractional equations admittingsolutions that can be obtained by using the results here discussed should be object offurther research. Moreover, the possible applications of these fractional Bessel-typeequations must be deepened.
Acknowledgments:
The work of the authors has been carried out in the frame-work of the activities of the National Group for Mathematical Physics (GNFM).
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