Other q-Fractional Calculi
In this chapter we investigate q-analogues of some known fractional operators. This chapter includes as well the fractional generalization of the Askey–Wilson operator introduced in (Ismail and Rahman, J. Approx. Theor. 114(2), 269–307, 2002). At the end
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Other q-Fractional Calculi
Abstract In this chapter we investigate q-analogues of some known fractional operators. This chapter includes as well the fractional generalization of the Askey– Wilson operator introduced in (Ismail and Rahman, J. Approx. Theor. 114(2), 269–307, 2002). At the end of this chapter we introduce a fractional generalization of the q-difference operator introduced in (Rubin, J. Math. Anal. Appl. 212(2), 571– 582, 1997).
¨ 5.1 Basic Grunwald–Letnikov Fractional Derivative In this section we define a q-analogue of the Gr¨unwald–Letnikov fractional derivative. If f is a function defined on Œ0; a (a > 0) then from (1.25), Dqk f .x/, k 2 N0 , x 2 .0; a is given by Dqk f .x/
k X f .q j x/ 1 j k D k .1/ : x .1 q/k j D0 j q q j.j 1/=2Cj.kj /
(5.1)
h i Since
k j q
D 0 for all j > k, then
Dqk f .x/ D
1 X f .q j x/ 1 j k .1/ : x k .1 q/k j D0 j q q j.j 1/=2Cj.kj /
(5.2)
The right hand side of (5.2) has a meaning when we replace k by any real number ˛. Therefore, we define the q-fractional operator D˛q f .x/
1 X 1 f .q j x/ j ˛ D ˛ .1/ : x .1 q/˛ j D0 j q q j.j 1/=2Cj.˛j /
M.H. Annaby and Z.S. Mansour, q-Fractional Calculus and Equations, Lecture Notes in Mathematics 2056, DOI 10.1007/978-3-642-30898-7 5, © Springer-Verlag Berlin Heidelberg 2012
(5.3)
147
5 Other q-Fractional Calculi
148
The operator (5.3) is a q-analogue of the Gr¨unwald–Letnikov fractional operator defined above. From (1.5) .1/j
.q ˛ I q/j ˛ q j.j C1/=2j˛ D q j .qI q/j j q
.j 2 N0 / :
Then D˛q f .x/ D x ˛ .1 q/˛ D x ˛ .1 q/˛
1 X ˛ .1/j q j.j C1/=2q ˛j f .q j x/ j q j D0 1 X j D0
qj
.q ˛ I q/j f .q j x/: .qI q/j
From (4.26), the q-fractional operator D˛q , ˛ 2 R, coincides with the Riemann– Liouville fractional operator Dq˛ .
5.2 Caputo Fractional q-Derivative Definition 5.2.1. For ˛ > 0, the Caputo fractional q-derivative of order ˛ is defined by c ˛ Dq f .x/ WD Iqn˛ Dqn f .x/ .n WD d˛e/ : (5.4) It is obvious form the definition that if ˛ is a nonnegative integer, then c
Dqn f .x/ D Iq0 Dqn f .x/ D Dqn f .x/: .n/
Theorem 5.1. Let ˛ > 0 and n D d˛e. If f 2 A Cq Œ0; a then c Dq˛ f .x/ 2 Lq1 Œ0; a. .n/
Proof. The proof is straightforward since if f 2 A Cq Œ0; a then Dqn f Lq1 Œ0; a. Hence, applying Lemma 4.9 yields c
2
Dq˛ f .x/ D Iqn˛ Dqn f .x/ 2 Lq1 Œ0; a: t u
Example 5.2.1. Let f .x/ D x ˇ1 , ˇ > 0 . Then 8 0; if ˇ 2 f1; 2; 3; : : : ; ng ; < c ˛ q .ˇ/ Dq f .x/ D x ˇ˛1 ; ˇ > n: : q .ˇ ˛/
(5.5)
5.2 Caputo Fractional q-Derivative
149
A few additional examples of Caputo-type derivatives of certain important functions are collected for the readers convenience in Table A.5. In the following theorem, we discuss the results of combining the Riemann– Liouville fractional q-derivative and Caputo fractional q-derivative of not necessarily equal orders. Theorem 5.2. Let ˛ and ˇ be positive numbers and let n D d˛e and m D dˇe. Then the following relations hold: .m/
1. If f 2 A Cq Œ0; a, then 8 j < Iqˇ˛ f .x/ Pm1 Dq f .0C / x ˇ˛Cj ; ˇ ˛; j D
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