q-Integral Transforms for Solving Fractional q-Difference Equations
Integral transforms like Laplace, Mellin, and Fourier transforms are used in finding explicit solutions for linear differential equations, linear fractional differential equations, and diffusion equations. See for example (Kilbas et al., Theory and Applic
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q-Integral Transforms for Solving Fractional q-Difference Equations
Abstract Integral transforms like Laplace, Mellin, and Fourier transforms are used in finding explicit solutions for linear differential equations, linear fractional differential equations, and diffusion equations. See for example (Kilbas et al., Theory and Applications of Fractional Differential Equations, Elsevier, London, first edition, 2006; Mainardi, Appl. Math. Lett. 9(6), 23–28, 1996; Nikolova and Boyadjiev, Fract. Calc. Appl. Anal. 13(1), 57–67, 2010; Wyss, J. Math. Phys. 27(11), 2782– 2785, 1986). This chapter is devoted to the use of the q-Laplace, q-Mellin, and q 2 -Fourier transforms to find explicit solutions of certain linear q-difference equations, linear fractional q-difference equations, and certain fractional q-diffusion equations.
9.1 q-Laplace Transform of Fractional q-Integrals and q-Derivatives In the following theorem we compute the q Ls transform of the Riemann–Liouville fractional q-integral and q-derivatives. Theorem 9.1. If F 2 Lq1 Œ0; a and ˚.s/ WD q Ls F .x/ then q
Ls Iq˛ F .x/ D
.1 q/˛ ˚.s/ s˛
for any ˛ > 0:
(9.1)
.n/
If n 1 < ˛ n and Iqn˛ F .x/ 2 A Cq Œ0; a then ˛ q Ls Dq F .x/ D
n X s˛ s m1 ˚.s/ Dq˛m F .0C / : ˛ .1 q/ .1 q/m mD1
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 9, © Springer-Verlag Berlin Heidelberg 2012
(9.2)
271
9 Applications of q-Integral Transforms
272
Proof. Since Iq˛ F .x/ D .1 q/
x ˛1 q F .x/ D .1 q/ ˛1 .x/ q F .x/ ; q .˛/
we obtain (9.1) from (1.88), (1.91), and (1.92). Now we prove (9.2). From (4.49), (1.93), and (9.1) we get ˛ n n˛ F .x/ q Ls Dq F .x/ D q Ls Dq Iq n X sn s m1 n˛ nm n˛ C D L I F .x/ D I F .0 / q s q q q .1 q/n .1 q/m mD1
D
n X s˛ s m1 ˛m C ˚.s/ D F .0 / ; q .1 q/˛ .1 q/m mD1
(9.3) t u
proving the theorem.
Theorem 9.2. If q Ls f .x/ D ˚.s/ then the q-Laplace transform of the Caputo fractional q-derivative is given by ! n1 r X s˛ r C .1 q/ c ˛ Dq f .0 / rC1 ˚.s/ : q Ls Dq f .x/ D .1 q/˛ s rD0 Proof. Since c Dq˛ f .x/ D .1 q/Dqn f .x/ n˛1 .x/, then by (1.88) and (1.93) we obtain q
.1 q/n˛ n q Ls .Dq f .x// s n˛ ! n n X .1 q/n˛ s m1 s nm C D ˚.s/ Dq f .0 / s n˛ 1q .1 q/m mD1 ! n1 X s˛ s rC˛1 r C : D ˚.s/ Dq f .0 / .1 q/˛ .1 q/rC˛ rD0
Ls c Dq˛ f .x/ D
t u Lemma 9.3. If q Ls f .x/ D ˚.s/ then the q-Laplace transform of the Riemann– Liouville sequential q-derivative of order m˛, 0 < ˛ < 1, is given by
q
Ls Dqm˛ f .x/ D p m˛ .s/
m1 1 X ˛.m1r/ 1˛ ˛r s ; p Iq Dq f .0C /; p D 1 q rD0 1q
where the operator Dqk˛ is defined in (8.74).
9.1 q-Laplace Transform of Fractional q-Integrals and q-Derivatives
273
Proof. The proof of this lemma follows by induction on m and by using (9.2).
t u
Similarly, we have the following lemma: Lemma 9.4. If q Ls f .x/ D ˚.s/ then the q-Laplace transform of the Caputo sequential q-derivative of order m˛, 0 < ˛ < 1, is given by m˛ c m˛
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