Fractional Variational Iteration Method for Fractional Fornberg-Whitham Equation and Comparison with the Undetermined Coefficient Method
Bao Siyuan *
Department of Engineering Mechanics, School of Civil Engineering, Suzhou University of Science and Technology, Suzhou, Jiangsu 215011, P.R. China.
Deng Zi-chen
Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710072, P.R. China and State Key Laboratory of Structural Analysis of Industrial Equipment, Dalian University of Technology, Dalian, Liaoning 116023, P.R. China.
*Author to whom correspondence should be addressed.
Abstract
The paper presents two methods for solving the fractional Fornberg-Whitham (FFW) equation. Based on the peaked solutions of FW equation, suppose the solution’s variable-separated form, and the FFW equation is transformed into a constant fractional differential equation (FDE). To solve the transformed equation, first, the fractional variational iteration method (FVIM) is used. Secondly, the undetermined coefficient method is used to expand the solution of the constant FDE. The expansion is based on the Generalized Taylor formula. Also the solutions are yielded for FFW. It should be pointed out that two cases of the order of fractional derivative between 1 and 2 and that between 0 and 1 are discussed respectively for the transformed FDE. Last, we give two numerical examples by using the two presented methods. The results show that the results agree well by both two proposed methods, and the two methods are high efficient in solving FFW.
Keywords: Differential transform method, Generalized Taylor formula, Caputo fractional derivative, Fractional differential equations, fractional Fornberg-Whitham equation.