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Abstract: We give a general result about the relation between approximate symmetry and approximate homotopy symmetry for perturbed PDEs. We show that the two couple equations derived by approximate symmetry method and approximate homotopy symmetry method are connected by a transformation. Consequently, acting the transformation on the known solutions constructed by approximate symmetry method, we directly obtain approximate homotopy series solutions whose accuracy can be controlled by adjusting the convergence-control parameter. Applications to the Cahn–Hilliard equation illustrate the effectiveness of the transformation. [Copyright &y& Elsevier]

Abstract: In this paper, we successfully construct the new exact traveling wave solutions of the coupled Schrödinger–Boussinesq equation by using the extended simplest equation method. The exact traveling wave solutions with double arbitrary parameters, are expressed by the hyperbolic functions, the trigonometric functions and the rational functions respectively. When the parameters are taken as special values, some solitary wave solutions are derived from the hyperbolic function solutions. [Copyright &y& Elsevier]

Abstract: In this paper we present the generalized Adomian–Rach theorem and the generalized Rach–Adomian–Meyers modified decomposition method for solving multi-order nonlinear fractional ordinary differential equations. We consider different classes of initial value problems for nonlinear fractional ordinary differential equations, including the case of real-valued orders and another case of rational-valued orders, which are solved by the present method. This method can treat any analytic nonlinearity. The coefficients of the solution in the form of a generalized power series are determined by a convenient recurrence scheme, which does not involve integration operations compared with the classic Adomian decomposition method. [Copyright &y& Elsevier]

Abstract: In this paper, an extended simplest equation method is proposed to seek exact travelling wave solutions of nonlinear evolution equations. As applications, many new exact travelling wave solutions for several forms of the fifth-order KdV equation are obtained by using our method. The forms include the Lax, Sawada–Kotera, Sawada–Kotera–Parker–Dye, Caudrey–Dodd–Gibbon, Kaup–Kupershmidt, Kaup–Kupershmidt–Parker–Dye, and the Ito forms. [Copyright &y& Elsevier]