Homotopy Perturbation Method with an Auxiliary Term

and Applied Analysis 3 Accordingly, we can construct a homotopy equation in the form ü ω2u p ( u3 −ω2u ) 0. 2.6 When p 0, we have ü ω2u 0, u 0 A, u′ 0 0, 2.7 which describes the basic solution property of the original nonlinear equation, 2.3 . When p 1, 2.6 becomes the original one. So the solution procedure is to deform from the initial solution, 2.4 , to the real one. Due to one unknown parameter in the initial solution, only one iteration is enough. For detailed solution procedure, refer to 5 . If a higher-order approximate solution is searched for, we can construct a homotopy equation in the form ü 0 · u pu3 0. 2.8 We expand the solution and the coefficient, zero, of the linear term into a series in p: u u0 pu1 pu2 · · · , 2.9 0 ω2 pa1 pa2 · · · , 2.10 where the unknown constant, ai, is determined in the i 1 th iteration. The solution procedure is given in 5 . 3. Homotopy Equation with an Auxiliary Term In this paper, we suggest an alternative approach to construction of homotopy equation, which is ̃ Lu p ( Lu − ̃ Lu Nu ) αp ( 1 − p)u 0, 3.1 where α is an auxiliary parameter. When α 0, 3.1 turns out to be that of the classical one, expressed in 2.2 . The auxiliary term, αp 1 − p u, vanishes completely when p 0 or p 1; so the auxiliary term will affect neither the initial solution p 0 nor the real solution p 1 . The homotopy perturbation method with an auxiliary term was first considered by Noor 15 . To illustrate the solution procedure, we consider a nonlinear oscillator in the form d2u dt2 bu cu3 0, u 0 A, u′ 0 0, 3.2 where b and c are positive constants. 4 Abstract and Applied Analysis Equation 3.2 admits a periodic solution, and the linearized equation of 3.2 is u′′ ω2u 0, u 0 A, u′ 0 0, 3.3 where ω is the frequency of 3.2 . We construct the following homotopy equation with an auxiliary term: u′′ ω2u p [( b −ω2 ) u cu3 ] αp ( 1 − p)u 0. 3.4 Assume that the solution can be expressed in a power series in p as shown in 2.9 . Substituting 2.9 into 3.4 , and processing as the standard perturbation method, we have u′′ 0 ω u0 0, u0 0 A, u0 0 0, 3.5 u′′ 1 ω u1 ( b −ω2 ) u0 cu0 αu0 0, u1 0 0, u ′ 1 0 0, 3.6 u′′ 2 ω u2 ( b −ω2 ) u1 3cu0u1 α u1 − u0 0, 3.7 with initial conditions