Titlebook: q-Fractional Calculus and Equations; Mahmoud H. Annaby,Zeinab S. Mansour Book 2012 Springer-Verlag Berlin Heidelberg 2012 33D15, 26A33, 30

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,Riemann–Liouville ,-Fractional Calculi,ed by Al-Salam (Proc. Am. Math. Soc. 17, 616–621, 1966; Proc. Edinb. Math. Soc. 2(15), 135–140, 1966/1967) and by Agarwal (Proc. Camb. Phil. Soc. 66, 365–370, 1969). We give rigorous proofs of existence of the fractional .-integral and .-derivative. Therefore we establish a .-analogue of Abel’s integral equation and its solutions.

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Mahmoud H. Annaby,Zeinab S. MansourFirst detailed rigorous study of q-calculi.First detailed rigorous study of q-difference equations.First detailed rigorous study of q-fractional calculi and equations.Proofs of many classical unproved

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0075-8434 onal calculi and equations.Proofs of many classical unproved.This nine-chapter monograph introduces a rigorous investigation?of .q-.difference operators in standard and fractional settings. It starts with elementary calculus of .q-.differences and integration of Jackson’s type before turning to .q-.

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Fractional ,-Leibniz Rule and Applications,ly when α is an integer. In this respect, We extend Purohit’s result for any .. We end the chapter with deriving some .-series and formulae by applying the fractional Leibniz formula mentioned and derived earlier in the chapter.

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,-Integral Transforms for Solving Fractional ,-Difference Equations,010; Wyss, J. Math. Phys. 27(11), 2782–2785, 1986). This chapter is devoted to the use of the .-Laplace, .-Mellin, and ..-Fourier transforms to find explicit solutions of certain linear .-difference equations, linear fractional .-difference equations, and certain fractional .-diffusion equations.

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,-Difference Equations,point ., .. Then, as applications of the main results, we study linear .-difference equations as well as the .-type Wronskian. These results are mainly based on (Mansour, .-Difference Equations, Master’s thesis, Faculty of Science, Cairo University, Giza, Egypt, 2001). This chapter also includes a s

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