Cracoviensis Studia

. In this paper, we have discussed the problem of existence and uniqueness of solutions under the self-similar form to the space-fractional diﬀusion equation. The space-fractional derivative which will be used is the generalized Riesz-Caputo fractional derivative. Based on the similarity variable η , we have introduced the equation satisﬁed by the self-similar solutions for the aforementioned problem. To study the existence and uniqueness of self-similar solutions for this problem, we have applied some known ﬁxed point theorems (i.e. Banach’s contraction principle, Schauder’s ﬁxed point theorem and the nonlinear alternative of Leray-Schauder type).


Introduction
Fractional calculus (FC) is a mathematical analysis subject which deals with different possible approaches of defining fractional order derivatives (FODs) and integrals (FOIs).The theory of classical (integer order) differential equations (IODEs) has been then generalized to the broader theory of fractional order differential equations (FDEs).For more details on the subject, the reader may refer to [7,26,31].
To define fractional integrals and derivatives, many approaches have been proposed in the literature, including Riemann-Liouville's (RL), Hadamard's, Caputo's, Riesz's, Erdelyi-Kober's approaches, etc.The development of each one of approches should go through a series of stages ranging from exponential functions to special functions.Later, in [23,24], a new fractional operator which generalized both the RL and Hadamard operators was introduced by Katugampola.Not long ago, in [3], a Caputo-type modification of this operator was proposed.This later is the Caputo type of generalized fractional derivative (CGFD).It represents a generalization of the Caputo and Caputo-Hadamard FDs.Aleem et al. in [1], presented a generalisation of the Riesz fractional operator, where this operator covers as particular cases the classical Riesz fractional derivative.In the same paper, the authors have also proposed a Caputo-type modification of this operator.This new fractional derivative (FD) was named as the generalized Riesz-Caputo fractional derivative (or the Riesz-Caputo generalized fractional derivative (R-CGFD)).In the same paper, some fundamental results have been introduced and proved.
Different fixed-point theorems have been used by researchers to develop solutions and their existence for non-linear initial value problems (IVPs) and boundary value problems (BVPs) of fractional differential equations (see [4,5,34,6,8,9]) Fractional partial DEs or simply FPDEs can be used for the modelling and study of many important phenomena in many different fields of science and engineering, such as diffusion processes, damping law, etc.One can find a variety of applications in [19,22,25,29,33].
The existence and uniqueness of solutions of non-linear FPDEs have been studied in many papers including [11,12,13,21,30].
Generally, for PDEs, we can search for special type solutions known as groupinvariant solutions.As in [14], by solving a reduced system of equations (which has fewer independent variables compared to the original problem), the groupinvariant solutions can be found.These solutions are also known as self-similar or scale-invariant solutions which are used to model many processes in mathematics and engineering's mechanics.The FPDEs which have self-similar solutions can easily be reduced to ordinary differential equations (ODEs).This latter process helps to simplify one's work on FPDEs.
The idea behind solutions' self-similarity along with Lie group analysis have also been applied in FDEs.For example, Luchko and Gorenflo in [27] and Buckwar in [14] have been discussed the application of Lie group analysis for the equation They have found a scale-invariant solutions for the fractional ordinary differential equation (FODE) with a new independent variable η = xt −α β .The left and right sided Erdélyi-Kober derivatives which depend on α, β of this equation and on the parameter γ of its scaled group are considered.They have derived a general solution in terms of the generalized Wright function.
Across the literature, one may easily be aware of the existence of plenty of research works on fractional (space, time and space-time) diffusion equations by using the similarity method.For more details, the reader may check [10,17,28].
In this paper, we discuss the existence, uniqueness and main properties of the solution of the following space-fractional diffusion equation, which is Existence results of self-similar solutions of FDE [51] under the following self-similar form where u(x, t) is a scalar function of space and time variables (x, t) ∂|x| α is the R-CGFD of order α with ρ > 0 and which is the main motivation of the present research, the "basic profile" f in ( 2) is not known in advance and is to be identified and β ∈ R is a constant chosen so that the solutions exist.
The rest of this paper is structured as follows.In the next section, we recall preliminaries related to some definitions of fractional integrals and derivatives, theorems and lemmas of FC.The main results are given in section 3. Finally, this paper is ended with a conclusion.

Preliminaries and definitions
In this section, we give the necessary definitions, notations, lemmas and theorems from FC theory which will be used through the whole of this work.Let J = [0, µ] be a finite interval of R with µ > 0. We denote by C(J, R) the Banach space of all continuous functions g : [0, µ] → R with the norm We denote also C n (J, R) with n ∈ N 0 the set of mappings having n times continuously differentiable on J.
As in [26], for 1 ≤ p ≤ ∞ and c ∈ R, consider the space X p c [a, b] as follows Definition 2.1 (Generalized fractional integrals.(see [2,23])) The left-sided and right-sided of the generalized fractional integrals of order α > 0 and parameter ρ > 0 of an integrable function g : [0, µ] → R with µ > 0 are defined respectively by and where Γ(.) is Euler's gamma function defined as [52] N. Ouagueni, Y. Arioua and N. Benhamidouche Definition 2.2 (CGFDs.(see [2])) Let µ > 0, ρ be a positive real number, α ∈ R + and n ∈ N be such that α ∈ (n − 1, n), and g : [0, µ] → R a function of class C n .The left-sided and right-sided of CGFDs of order α and parameter ρ are defined respectively by and The next two results justify the definition 2.2, since the Caputo-type of the generalized fractional derivative is an inverse operation of the generalized fractional integral.

Existence results of self-similar solutions of FDE [55]
Furthermore, if g(η) satisfies the inequality then the following inequality holds true where E α,1 (.) is a Mittag-Leffler function.
Theorem 2.14 (Banach's Fixed Point Theorem (see [15])) Let E be a Banach space and Theorem 2.15 (Schauder's Fixed Point Theorem (see [20])) Let E be a Banach space, and let P be a closed, convex and non-empty subset of E. Let T : P toP be a continuous mapping such that T (P ) is a relatively compact subset of E. Then T has at least one fixed point in P .
Theorem 2.16 (Nonlinear alternative of Leray-Schauder type (see [20])) Let E be a Banach space with P ⊂ E be a closed and convex.U be an open subset of P with 0 ∈ U .Assume that A : U → P is a continuous, compact (that is, A(U ) is a relatively compact subset of P ) map.Then either; (i) A has a fixed point in U ; or (ii) there is a point u ∈ ∂U and σ ∈ (0, 1) with u = σA(u).

Statement of the problem
In this subsection, we consider the following problem of the space-fractional diffusion equation We should first deduce the equation satisfied by the function f in (12). [56] N. Ouagueni, Y. Arioua and N. Benhamidouche , reduces the partial fractional differential equation (1) to the ordinary differential equation of fractional order of the form where . From (12), we obtain Furtheremore, for 1 < α ≤ 2, ρ > 0, by the definition 2.6 of the R-CGFD, equation (12) and by putting , we get Existence results of self-similar solutions of FDE By substituting ( 13) and ( 14) in (1), we get the following equation where µ = Xt −1 αρ 0 .

Existence and uniqueness results of the basic profile
In this subsection, to study the following problem, we will need the results in subsection 3.1 along with Theorem 3.1, with the conditions where β ∈ R and ρ, µ > 0.
In what follows, to derive the principal theorems, we will need the following lemmas.
Proof.Let µ be a positive parameter.It is obvious that the space E with the norm .∞ is a subspace of the Banach space C[0, µ].So, to show that E is a Banach space, it is enough to demonstrate that this later is closed in C[0, µ].
Since f n is continuous, we get and This implies that Then, for η = 0 and υ = µ, we have also Existence results of self-similar solutions of FDE [59] In the next lemma, we will give the solution of problem ( 15)-( 16).
Then the problem (15)-( 16) is equivalent to the following integral equation where Proof.First, by applying the Riesz-generalized fractional integral RG 0 I α,ρ µ defined in (7) to both sides of equation ( 15), we obtain From Lemma 2.7 and Remark 2.8, we get Then the fractional integral equation (19), can be re-written as follows Applying ( 16) to (20) yields Then, according to (18), the problem ( 15)-( 16) is equivalent to [60] N. Ouagueni, Y. Arioua and N. Benhamidouche Lemma 3.4 Let T be an integral operator defined by provided that the supremum norm is Then, T maps E into itself (T : E → E).
where E is the Banach space defined by (17).Then, from ( 21), we have It follows from ( 9) and (10) in Remark 2.9 that Next, we will deal with the existence and uniqueness of solution for ( 15)-( 16).Firstly, using Banach's Fixed Point Theorem, we will derive the conditions of the solutions' existence.
Existence results of self-similar solutions of FDE [61] Proof.First, we will transform the problem ( 15)-( 16) into a fixed point problem.By Lemma 3.3, we define the operator T : E → E as follows Since the problem ( 15)-( 16) can be written in the form of the fractional integral equation ( 23), the fixed point of T is to be considered as a solution for ( 15)-( 16).
Secondly, using the fixed point theorem of Schauder, we will derive the conditions of the solutions' existence.
Proof.Let the operator T be defined in (23).We have already transformed the problem ( 15)-( 16) into a fixed point problem We shall show that T satisfies the assumption of Schauder's Fixed Point Theorem 2.15.The proof will be given in three claims.
Claim 2: According to (25), let and define a subset Thus, E R is a closed, bounded and convex subset of E.
Let f ∈ E R and T be the integral operator defined in (23).Then, we prove that T (E R ) ⊂ E R .In fact, by (11) in Remark 2.9, we have This implies that Existence results of self-similar solutions of FDE (27), we get N. Ouagueni, Y. Arioua and N. Benhamidouche We have and and we have also and Existence results of self-similar solutions of FDE [67] By substituting ( 29), ( 30), ( 31) and ( 32) in (28), we obtain So, the right-hand side of the above inequality tends to zero as η 2 → η 1 .Hence, we obtain that T (E R ) is equicontinuous.Therefore, combining claims 1 to 3 and by the means of the Ascoli-Arzela Theorem 2.12, we get that T : E R → E R is continuous and relatively compact.As a consequence, Schauder's Fixed Point Theorem assures the existence of at least one fixed point of operator (23) which is the solution of the problem ( 15)-( 16).
Finally, using the fixed point theorem of Leray-Schauder, we will derive the conditions of the solutions' existence.
Proof.Consider the operator T defined in (23).Then we shall show that all assumption of Leray-Schauder Fixed Point Theorem 2.16 are satisfied by the operator T .The proof will be divided to four claims.
Claim 1: It is clear that T is continuous.Claim 2: T maps bounded sets into bounded sets in E. Actually, it is enough to show that for any θ > 0, there exists N > 0 such that for each As a similar way as in ( 27), we have Claim 3: It is clear that T maps bounded sets into equicontinuous sets of E. From Claim1-Claim3, we conclude that T : E → E is continuous and completely continuous.
By choosing of H, there is no f ∈ ∂H, such that f = λT (f ) for some λ ∈ (0, 1).Consequently, by the nonlinear alternative of Leray-Schauder's Fixed Point Theorem 2.16, the operator T has a fixed point f in H, which is a solution to the problem ( 15)-( 16) on [0, µ].Now, we prove the principal theorems.

Existence results to the original problem
In this subsection, we demonstrate the existence and uniqueness of solutions of the following space-fractional diffusion equation  Then, for f ∈ E, (36) has a unique solution in the self-similar form (37).
Proof.The transformation (37) reduces the space-fractional diffusion equation (36) to the ordinary fractional differential equation of the following form , Y. Arioua and N. Benhamidouche , Y. Arioua and N. Benhamidouche