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.