Eur. Phys. J. Plus (2018) 133: 201
https://doi.org/10.1140/epjp/i2018-12030-2

Regular Article

Numerical treatment of the Benjamin-Bona-Mahony equation using Alpert multiwavelets

¹ Department of Mathematics, Faculty of Science, University of Maragheh, P.O. Box 55181-83111, Maragheh, Iran
² Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), 45137-66731, Zanjan, Iran
³ Research Center for Basic Sciences & Modern Technologies (RBST), Institute for Advanced Studies in Basic Sciences (IASBS), 45137-66731, Zanjan, Iran
⁴ Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
⁵ Young Researchers and Elite Club, Ilkhchi Branch, Islamic Azad University, Ilkhchi, Iran

^* e-mail: j_manafianheris@tabrizu.ac.ir

Received: 24 January 2018
Accepted: 25 April 2018
Published online: 29 May 2018

Abstract

This paper presents a numerical technique for solving the nonlinear Benjamin-Bona-Mahony equation. As a first step, we discretize the time by approximating the first-order time derivative via -weighted scheme. A system of ordinary differential equations is obtained and we solve this system using the wavelet Galerkin method by use of the Alpert multiwavelets. To this aim, the multiresolution analysis is used to construct the Alpert multiwavelets system and we introduce the wavelet transform matrix to decrease computational time. We use the energy method to prove that the time discrete scheme is unconditionally stable and convergent in time variable. An algorithm is proposed to achieve the desired error. Illustrative examples exhibit the efficiency of our method. The method produces accurate results and is easy to implement.