Some applications of wright functions in fractional differential equations

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In this note we prove some new results about the application of Wright functions of the first kind to solve fractional differential equations with variable coefficients. Then, we consider some applications of these results in order to obtain some new particular solutions for nonlinear fractional partial differential equations.

Introduction

The problem to find explicit solutions for fractional ordinary differential equations with variable coefficients is a topic of interest, also in the context of the studies about special functions. For example the so-called Kilbas–Saigo function emerges in the study of fractional ODE with variable coefficient (see [3] and the references therein). Recently different authors are working on systematic methods to solve fractional ODEs with variable coefficients, we refer in particular to the recent paper [1]. In other cases, it is possible to prove that some interesting classes of special functions solve fractional equations with variable coefficients. For example, in [8] the authors proved that a generalized Le Roy function solves a particular integro-differential equation with variable coefficients involving Hadamard fractional operators.

In the recent paper [7], a new interesting result has been pointed out about the solution of a new class of fractional ODEs by means of the classical Wright functions of the first kind.

In particular, it was proved, by direct calculations, that the following fractional ODE

ddttλdλudtλλut1λ=0,t0,λ(0,1),

involving the Caputo fractional derivative of order λ is solved by the function

u(t)=Wλ,1(tλ)=k=0ttkk!Γ(kλ+1),λ(0,1),

under the initial condition that u(t = 0) = 1. We note that Wλ, 1 is a particular case of special transcendental functions known as Wright functions that we will briefly discuss in the next section, distinguishing them in two kinds according to the values of the first parameter λ. We recall that the fractional derivative in the sense of Caputo of order ν > 0 is defined as

(dνdtνf)(t):={1Γ(mν)0tf(m)(τ)(tτ)ν+1mdτ,form1<ν<m,dmdtmf(t),forν=m.

Therefore, Eq. (1.1) can be viewed as a sort of fractional generalization of the Bessel-type differential equations. Indeed, for λ = 1 we obtain the following equation

ddttdudt=u,

whose solution is the so-called Tricomi function (see [4, 5] and the references therein)

C0(t)=k=0tkk!2,

that turns out to be related to modified Bessel function of the first kind and order zero I0 by

C0(t)=I0(2t1/2).

In the next section we will also discuss in some detail the relations between the Wright functions of the first kind with functions of the Bessel type.

The operator ddttdudt appearing in (1.4) is also named Laguerre derivative in the literature. Laguerre derivatives have been recently studied by different authors in the framework of the so-called monomiality principle pointed out for example in [5]. Applications of Laguerre derivatives in population dynamics have been considered in [2]. More recently, in [26], mathematical models of heat propagation based on Laguerre derivatives in space have been studied.

The aim of this short note is twofold. First of all we provide a more general result connecting Wright functions of the first kind with fractional ODE with variable coefficients. We underline the role of these special functions in the theory of fractional differential equations. Then, we discuss some simple applications of these results to solve nonlinear fractional PDEs admitting solutions by generalized separating variable solution.

Section snippets

Preliminaries about Wright functions

The classical Wright function that we denote by Wλ, μ(z), is defined by the series representation convergent in the whole complex plane,

Wλ,μ(Z):=n=0Znn!Γ(λn+μ),λ>1,μC.

The integral representation reads as

Wλ,μ(z)=12πiHaeσ+zσλdσσμ,λ>1μC,

where Ha denotes the Hankel path: this one is a loop which starts from –∞ along the lower side of the negative real axis, encircling it with a small circle the axes origin and ends at –∞ along the upper side of the negative real axis.

Wλ, μ(z) is then an

Fractional ordinary differential equations with variable coefficients

We here prove a new connection between Wright functions of first kind and fractional ODE.

New results about nonlinear fractional diffusion equations

A simple but useful method for constructing exact solutions for nonlinear PDEs is given by the generalized separation of variables. This method permits to find particular classes of exact solutions mainly based on the reduction to nonlinear ODEs that can be exactly solved. It is possible to prove that wide classes of nonlinear PDEs admit such solutions in separating variable form by using for example the invariant subspace method (see the relevant monograph [6]). Many papers have been devoted

Conclusions

The main aim of this paper is to underline a new interesting application of Wright functions of the first kind to solve fractional ordinary differential equations with variable coefficients that generalize Bessel-type equations. In practice the solutions are formal because obtained by the method of substitution, showing the direct connection between fractional Bessel-type equations and Wright functions. Then, we discuss the applications of this new result to solve linear and nonlinear

Acknowledgements

The work of the authors has been carried out in the framework of the activities of the National Group for Mathematical Physics (GNFM).

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