CMST 19(4) 235-243 (2013) DOI:10.12921/cmst.2013.19.04.235-243 Formulation and Solution of Space-Time Fractional KdV-Burgers Equation Essam M. Abulwafa, Ahmed M. Elgarayhi, Abeer A. Mahmoud, Ashraf M. Tawfik Theoretical Physics Research Group, Physics Department Faculty of Science, Mansoura University, Mansoura 35516, Egypt E-mail: abulwafa@mans.edu.eg Received: 6 August 2013; revised: 18 November 2013; accepted: 6 December 2013; published online: 17 December 2013 Abstract: The space-time fractional KdV-Burgers equation has been derived using the semi-inverse method and Agrawal’s variational method. The modified Riemann-Liouville definition is used for the fractional differential operators. The derived fractional equation is solved using the fractional sub-equation method. Key words: Fractional Euler-Lagrange equation, space-time fractional KdV-Burgers equation, modified Riemann-Liouville fractional definition, fractional sub-equation method I. INTRODUCTION All forces in nature are nearly non-conservative: dissipa- tive and/or dispersive forces. Classical mechanics, using inte- ger differential equations, treated conservative forces while the non-integer differential equations can be used to describe the non-conservative forces. Fractional calculus is a field of mathematics that grows out of the traditional definitions of calculus. Fractional calculus has gained importance during the last decades mainly due to its applications in various fields of science and engineering. Some of the areas of present day applications of fractional calculus include fluid flow, rheol- ogy, dynamical process in self-similar and porous structures, diffusive transport akin to diffusion, electrical networks, prob- ability and statistics, control theory of dynamical systems, vis- coelasticity, electro-chemistry of corrosion, chemical physics, optics, and signal processing, and so on [1-7]. There are different kinds of fractional integration and dif- ferentiation operators. The most famous one is the Riemann- Liouville definition [8-11], which has been used in various fields of science and engineering successfully, but this def- inition leads to the result that constant function differenti- ation is not zero. Caputo put definitions which give zero value for fractional differentiation of constant function, but these definitions require that the function should be smooth and differentiable [8-11]. Recently, Jumarie derived defini- tions for the fractional integral and derivative called modified Riemann-Liouville [12-15], which are suitable for continuous and nondifferentiable functions and give differentiation of a constant function equal to zero. The modified Riemann- Liouville fractional definitions are used effectively in many different problems [16-20]. It was shown that non-integer derivatives in the La- grangian describe non-conservative forces. Riewe [21, 22] derived a method using a fractional Lagrangian that leads to a fractional Euler-Lagrange equation that is, in some sense, equivalent to the desired equation of motion. Hamil- ton’s equations are derived from the Lagrangian and are equiv- alent to the Euler-Lagrange equation. Further study of the frac- tional Euler-Lagrange can be found in the works of Agrawal [23-25]. He presented generalized Euler-Lagrange equations for unconstrained and constrained fractional variational prob- lems. Baleanu and coworkers [26, 27] used the fractional Euler-Lagrange equation to model fractional Lagrangian and Hamiltonian formulations. El-Wakil et al derived the time fractional forms of some mathematical-physics equations [28] using Agrawal’s variational method [23-25] and used them to describe the electrostatic potential in some plasma systems [29].