International Scholarly Research Network ISRN Applied Mathematics Volume 2011, Article ID 682381, 9 pages doi:10.5402/2011/682381 Research Article Analytical Solution for the Differential Equation Containing Generalized Fractional Derivative Operators and Mittag-Leffler-Type Function V. B. L. Chaurasia and Ravi Shanker Dubey Department of Mathematics, University of Rajasthan, Jaipur 302004, India Correspondence should be addressed to Ravi Shanker Dubey, ravishankerdubey@indiatimes.com Received 26 March 2011; Accepted 10 May 2011 Academic Editor: M. F. El-Sayed Copyright q 2011 V. B. L. Chaurasia and R. S. Dubey. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We discuss and derive the analytical solution for the fractional partial differential equation with generalized Riemann-Liouville fractional operator D α,β 0,t of order α and β. Here, we derive the solution of the given differential equation with the help of Laplace and Hankel transform in terms of Fox’s H-function as well as in terms of Fox-Wright function p ψ q . 1. Introduction, Definition, and Preliminaries Applications of fractional calculus require fractional derivatives of different kinds 1– 9. Differentiation and integration of fractional order are traditionally defined by the right-sided Riemann-Liouville fractional integral operator I P a and the left-sided Riemann- Liouville fractional integral operator I P a- , and the corresponding Riemann-Liouville fractional derivative operators D P a and D P a- , as follows 10, 11:  I μ a f  x 1 Γ ( μ )  x a f t x - t 1-μ dt ( x>a; R ( μ ) > 0 ) , 1.1  I μ a- f  x 1 Γ ( μ )  a x f t t - x 1-μ dt ( x<a; R ( μ ) > 0 ) , 1.2  D μ a± f  x  ± d dx  n I n-μ a± f  x ( R ( μ ) ≥ 0; n   R ( μ )  1 ) , 1.3