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B?cklund Transformation of Fractional Riccati Equation and Infinite Sequence Solutions of Nonlinear Fractional PDEs

DOI: 10.1155/2014/572052

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Abstract:

The B?cklund transformation of fractional Riccati equation with nonlinear superposition principle of solutions is employed to establish the infinite sequence solutions of nonlinear fractional partial differential equations in the sense of modified Riemann-Liouville derivative. To illustrate the reliability of the method, some examples are provided. 1. Introduction Recently, nonlinear fractional differential equations increasingly are used to describe nonlinear phenomena in fluid mechanics, biology, engineering, physics, and other areas of science [1–3]. Much efforts have been spent in recent years to develop various techniques to deal with fractional differential equations. However, for the nonlinear differential equations including fractional calculus, the analytical or numerical results are usually difficult to be obtained. It is therefore needed to find a proper method to solve the problem of nonlinear differential equations containing fractional calculus. In the past, several methods have been formulated, such as Adomian decomposition method [4, 5], variational iteration method [6, 7], homotopy perturbation method [8, 9], differential transform method [10, 11], and fractional subequation method [12–14]. S. Zhang and H.-Q. Zhang [12] first proposed a new direct method called fractional subequation method in solving nonlinear time fractional biological population model and ( )-dimensional space-time fractional Fokas equation, based on the homogeneous balance principle and Jumarie’s modified Riemann-Liouville derivative. In this paper, based on the B?cklund transformation technique and the known seed solutions, we will devise effective way for solving fractional partial differential equations. It will be shown that the use of the B?cklund transformation allows us to obtain new exact solutions from the known seed solutions. 2. B?cklund Transformation of the Fractional Riccati Equation and Nonlinear Superposition Principle Firstly, we give some definitions and properties of the modified Riemann-Liouville derivative [15] which are used in this paper. Assume that , denote a continuous (but not necessarily differentiable) function, and let denote a constant discretization span. Jumarie defined the fractional derivative in the limit form where This definition is close to the standard definition of the derivative (calculus for beginners), and as a direct result, the th derivative of a constant, , is zero. An alternative, which is strictly equivalent to (1) is as follows: Some properties of the fractional modified Riemann-Liouville derivative that were

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