The Proceedings of The 4 th Annual International Conference Syiah Kuala University (AIC Unsyiah) 2014 In conjunction with The 9 th Annual International Workshop and Expo on Sumatran Tsunami Disaster and Recovery – AIWEST-DR 2014 October 22-24, 2014, Banda Aceh, Indonesia 86 A SEMI ANALITYCAL SOLUTION OF BOUSSINESQ EQUATION USING ASYMPTOTIC METHOD CUT DINI SYAHRANI 1 , MARWAN RAMLI 1,2 *, SAID MUNZIR 1,2 1 Department of Mathematics 2 Dynamic Application and Optimization Group Syiah Kuala University, Darussalam-Banda Aeh, Indonesia, 23111 *Email: marwan.math@unsyiah.ac.id or ramlimarwan@gmail.com Abstact. This study aims to determine solution of Boussinesq equation which is approximated by using asymptotic expansion method. Nonlinearity of Boussinesq equation causes the solution is not easily determined, so the solution is approached through its linearity. This method is in the form of power series expansion up to third- order, where each term of the series is linear. Furthermore, the finding solution is compared with the solution that was found by Mohyud in previous study. The result of this comparison showed that there were similarities of the two solutions. However, differences occur in phase as modeled in the solution of each method Keywords: Boussinesq equation, asymptotic expansion method, resonance term Introduction The theory of PDEs has long been one of the most important fields in mathematics. It is essentially due to the frequent occurrence equation and the wide range of applications of PDEs in many branches of physics, engineering and other sciences (Myint-u and Debnath, 2007). One of nonlinear PDEs is Boussinesq equation. It is nonlinear PDEs which describes the propagation of long waves in shallow water under gravity propagating in both direction (Wazwaz, 2009).The Boussinesq equation was introduced by Joseph Valentin Boussinesq (1842- 1929) in 1872. (1) The Boussinesq equation used in this study is equation (1). It is called the good Boussinesq equation or well- posed because it only has a single solution (Wazwaz, 2009). Mohyud (2008) in his paper titled “Exp-Fuction Method for Generalized Traveling Solutions of Good Boussinesq Equation” found a solution of this equation by using Exp-function method. Different from previous studies, in this study, solution of Boussinesq equation will be determined by using the asymptotic expansion method. This method is an expansion in form of the power series against amplitude elevation. Here the asymptotic expansion is limited only to third-order. Mathematical Model The ’good’ Boussinesq equation which describes the propagation of a long wave in shallow water in both directions has the form (2) with is elevation of the wave. We approximate the solution of equation (2) by using an expansion up to third order in the power series of the wave elevation following form (3) with and describe type of first order, second order and third order while is a positive small number representing the order of magnitude of the wave amplitude.