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On the Approximation of Riemann-Liouville Integral by Fractional Nabla H-Sum and Applications L Khitri-Kazi-Tani, H Dib

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On the Approximation of Riemann-Liouville Integral by Fractional Nabla H-Sum and Applications L Khitri-Kazi-Tani, H Dib On the approximation of Riemann-Liouville integral by fractional nabla h-sum and applications L Khitri-Kazi-Tani, H Dib To cite this version: L Khitri-Kazi-Tani, H Dib. On the approximation of Riemann-Liouville integral by fractional nabla h-sum and applications. 2016. hal-01315314 HAL Id: hal-01315314 https://hal.archives-ouvertes.fr/hal-01315314 Preprint submitted on 12 May 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On the approximation of Riemann-Liouville integral by fractional nabla h-sum and applications L. Khitri-Kazi-Tani∗ H. Dib y Abstract First of all, in this paper, we prove the convergence of the nabla h- sum to the Riemann-Liouville integral in the space of continuous functions and in some weighted spaces of continuous functions. The connection with time scales convergence is discussed. Secondly, the efficiency of this approximation is shown through some Cauchy fractional problems with singularity at the initial value. The fractional Brusselator system is solved as a practical case. Keywords. Riemann-Liouville integral, nabla h-sum, fractional derivatives, fractional differential equation, time scales convergence MSC (2010). 26A33, 34A08, 34K28, 34N05 1 Introduction The numerous applications of fractional calculus in the mathematical modeling of physical, engineering, economic and chemical phenomena have contributed to the rapidly growing of this theory [12, 19, 21, 18], despite the discrepancy on definitions of the fractional operators leading to nonequivalent results [6]. Each definition has its own advantages for applications. Tacking into account some developments of fractional calculus theory, we can refer to [22, 17, 14, 20] and the references therein. On the other side, the discrete fractional calculus is considered as new attempt. The fractional operators are obtained using the forward operators delta [16, 8] or backward operators nabla [11, 2] or combined delta and nabla fractional op- erators [3]. Besides that, since Hilger's original concept of measure chain which provides a unifying framework to investigate both continuous and discrete time systems [13], many definitions of fractional integral/derivative in time scale arise to ex- tend and unify results on the real line. Invoke that any time scale is a closed non-empty subset of R (for more details see [1, 24, 4, 5]). ∗corresponding author: l [email protected] yh [email protected] 1 In this context, the h-fractional operators, when the time scale is chosen to be the subset (hZ) of real numbers, was our source of inspiration to consider an ap- proximation of Riemann-Liouville fractional integral. Especially, the fractional summation obtained by the backward differences. We prove the convergence of this h-sum to the Riemann-Liouville integral firstly in the space of continuous functions, and then in more general weighted spaces. Moreover, we propose a numerical solution to the weighted fractional differential Cauchy problem. The paper is organized as the following. Section 2 deals with some preliminary notions and results in the purpose of making this paper self-contained. Sec- tion 3 is about the convergence results. Section 4 treats the connection with convergence over time scales. Section 5 is extremely devoted to the approxi- mation application to some fractional differential equations. First we show the accuracy of this approach for the Riemann-Liouville initial value problem in weighted spaces of continuous functions. Then the numerical solution for the non-linear fractional Brusselator system is presented as a practical case. Some new research directions are suggested in section 6. 2 Preliminaries In this section we present some definitions and results which will be used in the sequel. For β > 0 the weighted space of continuous function is defined by β Cβ [a; b] := f 2 C (a; b] : lim (t − a) f(t) exists : t!a+ endowed with the norm β kfkβ := max (t − a) jf(t)j t2[a;b] Cβ[a; b] is a Banach space. Remark that C0 [a; b] = C [a; b], the Banach space of all real continuous functions. We recall the expression of the norm of a continuous linear operator T from Cβ1 [a; b] to Cβ2 [a; b]: kT (f)k kT k = sup β2 β1,β2 kfk f6=0 β1 The following chain of inclusions holds C [a; b] ⊂ Cβ1 [a; b] ⊂ Cβ2 [a; b] if 0 < β1 < β2. Definition 1 (Riemann-liouville operators) Let f be a continuous function on [a; b], 0 < α < 1 and a ≤ x ≤ b: The fractional integral of order α is defined by 1 Z x Iαf(x) = (x − t)α−1f(t)dt Γ(α) a The Riemann-Liouville fractional derivative of order α is defined by 1 d Z x Dαf(x) = (x − t)α−1f(t)dt Γ(α) dx a 2 Definition 2 (Caputo fractional derivative) Let f 2 C1 [a; b], 0 < α < 1 and a ≤ x ≤ b: The Caputo fractional derivative of order α is defined by 1 Z x C Dαf(x) = (x − t)α−1f 0(t)dt Γ(α) a In the sequel, we will take [a; b] = [0; 1]. Some useful properties of the Riemann- Liouville operator are listed bellow. Proposition 2.1 Let β 2 ]0; 1[ , α − β ≥ 0 and f 2 Cβ[0; 1] then 1. Iαf is continuous on [0; 1] : Γ(1 − β) 2. kIαk ≤ : β;0 Γ(α + 1 − β) Proof 1. Let 0 ≤ x < y ≤ 1 and " > 0 1 Z 1 Iαf(y) − Iαf(x) = (1 − t)α−1 (yαf(yt) − xαf(xt)) dt Γ(α) 0 1 Z 1 = (1 − t)α−1 yα−β − xα−β)yβf(yt) + xα−β(yβf(yt) − xβf(xt) dt Γ(α) 0 Therefore, Z 1 α α 1 α−1 −β α−β α−β β β jI f(y) − I f(x)j ≤ (1 − t) t (y − x )y t f(yt) dt Γ(α) 0 α−β Z 1 x α−1 −β β β β β + (1 − t) t (y t f(yt) − x t f(xt) dt Γ(α) 0 As f 2 Cβ [0; 1] ; there exists δ1 > 0; such that α α jx − yj < δ1 ) jtx − tyj < δ1; 8t 2 [0; 1] ) j(yt) f(yt) − (xt) f(xt)j < " α−β The continuity of the function x , leads to the existence of δ2 > 0; such that α−β α−β jx − yj < δ2 ) y − x < ": Thus, jx − yj < min(δ1; δ2) implies 1 " kfkβ + 1 Z " kfkβ + 1 Γ(1 − β) jIαf(y) − Iαf(x)j ≤ (1−t)α−1t−βdt ≤ Γ(α) 0 Γ(1 + α − β) 2. 8f 2 Cβ[0; 1]; 8x 2 [0; 1] 1 Z x jIαf(x)j ≤ (x − t)α−1t−βtβ jf(t)j dt Γ(α) 0 Z x 1 α−1 −β Γ(1 − β) ≤ kfkβ (x − t) t dt ≤ kfkβ Γ(α) 0 Γ(α + 1 − β) and hence the result. 3 In order to use de nabla h-sum of order α; we start by defining a uniform discretization of [0; 1] : Let 0 = t0 < t1 < ··· < tN = 1 be a mesh with step size h > 0: First we recall the definition of the fractional h-sum of order α at the point tn = nh by (see [15, formula (7)] ) n hα X Γ(n − i + α) r−αf(t ) = f(ih) h n Γ(α) Γ(n − i + 1) i=1 −α with, by convention, rh f(t0) = 0. This definition suggest to introduce the fractional h-sum operator by Definition 3 (Nabla h-sum operator) Let f be a continuous function, 0 ≤ x ≤ 1 and 1 ≤ n ≤ N; such that tn−1 < x ≤ tn: The fractional h-sum of order α is given by: 1 r−αf(x) = (t − x) r−αf(t ) + (x − t )r−αf(t ) h h n h n−1 n−1 h n −α Note that rh f(:) is nothing but the broken continuous line joining the −α −α points (tn; rh f(tn)): Hence rh f(:) is a continuous function. The first prop- −α erty of the linear operator rh is given by the following −α Proposition 2.2 rh : C [0; 1] ! C [0; 1] is a continuous linear operator with −α 1 r ≤ : h 0;0 Γ(α + 1) For the proof the following lemma is needed Lemma 2.3 (See [10, Theorem 5] ) Let n ≥ 1 be an integer, and a; b > 0 be two real numbers. then n XΓ(k + a) 1 Γ(n + 1 + a) Γ(1 + a) = − : Γ(k + b) a − b + 1 Γ(n + b) Γ(b) k=1 Proof (of the prop 2.2) Let f 2 C [0; 1] and tn = nh; n 2 f0; 1;:::;Ng ; then α n α n−1 −α h XΓ(n − i + α) h X Γ(i + α) r f(tn) ≤ jf(ih)j ≤ kfk (1) h Γ(α) Γ(n − i + 1) Γ(α) 0 Γ(i + 1) i=1 i=0 From lemma 2.3, we get n−1 X Γ(i + α) 1 Γ(n + α) = Γ(i + 1) α Γ(n) i=0 Inequality (1) can be written as α −α h Γ(n + α) r f(tn) ≤ kfk h Γ(α + 1) 0 Γ(n) from Wendel's inequality (see [23]) Γ(x + β) ≤ xβ; 0 < β < 1; x > 0 Γ(x) 4 we get α −α h α kfk0 r f(tn) ≤ kfk n ≤ 8n 2 ; 0 ≤ n ≤ N h Γ(α + 1) 0 Γ(α + 1) N Furthermore, 8x 2 [0; 1] −α 1 −α −α kfk0 r f(x) = (tn − x)r f(tn−1) + (x − tn−1)r f(tn) ≤ h h h h Γ(α + 1) −α Thus, rh f is a bounded operator on [0; 1] : The following technical lemmas play a key role in the proofs of ours main results.
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