New Exact Solutions of KdV Equation in an Elastic Tube Filled with a Variable Viscosity Fluid

In this paper, treating the arteries as a prestressed thin walled elastic tube with a stenosis and the blood as a Newtonian fluid with variable viscosity. Using the perturbation method, the KdV equation with variable coefficients are obtained.With the aid of the coordinate transformation, the reduced equation admits a progressive wave solution with variable wave speed. An extended tanh function method is used with a computerized symbolic computation for constructing the new exact travelling wave solutions of the reduced equation. The main idea of this method is to take full advantage of the Riccati equation which has more new solutions. The variations of radial displacement, fluid pressure with the distance parameter and speed wave are obtained. The method is straightforward and concise and its applications is promising.

[1]  M. A. Abdou,et al.  The extended homogeneous balance method and its applications for a class of nonlinear evolution equations , 2007 .

[2]  M. A. Abdou,et al.  New solitons and periodic wave solutions for nonlinear evolution equations , 2006 .

[3]  H. Demiray.  Solitary waves in prestressed elastic tubes , 1996 .

[4]  G. Rudinger.  Shock Waves in Mathematical Models of the Aorta , 1970 .

[5]  C. Dai,et al.  Quantum Soliton Solutions of Quantum Zakharov Equations for Plasmas , 2005 .

[6]  M. A. Abdou,et al.  New explicit and exact traveling wave solutions for two nonlinear evolution equations , 2008 .

[7]  W. Malfliet.  Solitary wave solutions of nonlinear wave equations , 1992 .

[8]  T. J. Pedley,et al.  The Fluid Mechanics of Large Blood Vessels: Preface , 1980 .

[9]  M. A. Abdou.  A generalized auxiliary equation method and its applications , 2008 .

[10]  L. Tian,et al.  Nonsymmetrical Kink Solution of Generalized KdV Equation with Variable Coefficients , 2008 .

[11]  Abdul-Majid Wazwaz,et al.  The tanh method for traveling wave solutions of nonlinear equations , 2004, Appl. Math. Comput..

[12]  Y. Hashizume.  Nonlinear Pressure Waves in a Fluid-Filled Elastic Tube , 1985 .

[13]  Chuntao Yan.  A simple transformation for nonlinear waves , 1996 .

[14]  M. A. Abdou.  Further improved F-expansion and new exact solutions for nonlinear evolution equations , 2008 .

[15]  Hilmi Demiray,et al.  Wave propagation through a viscous fluid contained in a prestressed thin elastic tube , 1992 .

[16]  M. A. Abdou.  The extended tanh method and its applications for solving nonlinear physical models , 2007, Appl. Math. Comput..

[17]  M. A. Abdou,et al.  New periodic wave solutions via extended mapping method , 2009 .

[18]  Ji-Huan He.  Variational iteration method – a kind of non-linear analytical technique: some examples , 1999 .

[19]  A. Rachev.  Effects of transmural pressure and muscular activity on pulse waves in arteries. , 1980, Journal of biomechanical engineering.

[20]  H. Demiray,et al.  Weakly nonlinear waves in a prestressed thin elastic tube containing a viscous fluid , 1999 .

[21]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[22]  Shigeo Yomosa,et al.  Solitary Waves in Large Blood Vessels , 1987 .

[23]  S. A. El-Wakil,et al.  Modified extended tanh-function method for solving nonlinear partial differential equations , 2002 .

[24]  M. A. Abdou.  On the variational iteration method , 2007 .

[25]  T. Moodie,et al.  Waves in nonlinear fluid filled tubes , 1984 .

[26]  M. A. Abdou,et al.  New exact travelling wave solutions using modified extended tanh-function method , 2007 .

[27]  Tay Kim Gaik,et al.  Forced Korteweg-de Vries–Burgers equation in an elastic tube filled with a variable viscosity fluid , 2008 .

[28]  M. A. Abdou,et al.  New Applications of the Homotopy Analysis Method , 2008 .

[29]  M. A. Abdou,et al.  Nonlinear fluid flows in pipe-like domain problem using variational-iteration method , 2007 .

[30]  M. Anliker,et al.  Nonlinear analysis of flow pulses and shock waves in arteries , 1971 .

[31]  Abdul-Majid Wazwaz,et al.  The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations , 2005, Appl. Math. Comput..

[32]  M. A. Abdou,et al.  Construction of periodic and solitary wave solutions by the extended Jacobi elliptic function expansion method , 2007 .

[33]  Solitary waves in prestressed elastic tubes , 1996 .

[34]  Ji-Huan He.  Variational iteration method—Some recent results and new interpretations , 2007 .

[35]  R. S. Johnson.  A non-linear equation incorporating damping and dispersion , 1970, Journal of Fluid Mechanics.

[36]  M. A. Abdou,et al.  NEW PERIODIC SOLUTIONS FOR NONLINEAR EVOLUTION EQUATIONS USING EXP-FUNCTION METHOD , 2007 .