Some exact solutions of KdV equation with variable coefficients M.S. Abdel Latif * Department of Applied Mathematics, Astrakhan State University, 20a Tatishchev St., Astrakhan 414056, Russia article info Article history: Received 13 July 2010 Accepted 24 July 2010 Available online 3 August 2010 Keywords: Variable-coefficient KdV equation Extended mapping transformation method Soliton solutions Exact solutions abstract In this paper, the extended mapping transformation method is used to obtain some new exact solutions of a variable-coefficient KdV equation arising in arterial mechanics. The obtained solutions include soliton solutions, periodic solutions and rational solutions. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction This paper is devoted to study the variable-coefficient KdV equation u t þ f ðtÞuu y þ gðtÞu yyy þ kðtÞu y þ lðtÞu ¼ hðtÞ; ð1Þ where f(t), g(t), l(t), k(t) and h(t) are arbitrary functions of the variable t. In arterial mechanics, Eq. (1) was considered as follows: 1. By treating the arteries as prestressed, tapered, thin-walled, long and circularly conical elastic tubes and considering the blood as an incompressible inviscid fluid, the propagation of weakly nonlinear waves in such a fluid-filled elastic tube obeys the following equation [1–3], u s þ l 1 uu n þ l 2 u nnn þ Asl 3 u n ¼ 0: ð2Þ The values of l 1 , l 2 and l 3 depend on the initial deformation of the tube material. Eq. (2) is a special case of Eq. (1). 2. Eq. (1) models the pulse wave propagation through fluid-filled tubes with elastic walls, which takes into account the elasticity of the wall as well as the tapering effects [4]. 2. The extended mapping transformation method [5] Consider a given variable-coefficients nonlinear evolution equation (NEE) with independent variables x and t and depen- dent variable u Hðu; u t ; u x ; u xt ; u xx ; ...Þ¼ 0: ð3Þ It is assumed that Eq. (3) has solutions of the form 1007-5704/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.07.023 * Permanent address: Engineering Mathematics and Physics Dept., Faculty of Engineering, Mansoura University, Mansoura, Egypt. Tel.: +7 9170836794. E-mail address: m_gazia@hotmail.com Commun Nonlinear Sci Numer Simulat 16 (2011) 1783–1786 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns