Indian Journal of Industrial and Applied Mathematics
  • Year: 2013
  • Volume: 4
  • Issue: 1a

Symmetry Reductions and Exact Solutions of Modified B-Family Equation

  • Author:
  • K. Singh1, R. K. Gupta2, Sachin Kumar3,, Anupma4
  • Total Page Count: 9
  • Published Online: Jun 1, 2013
  • Page Number: 52 to 60

1Department of Mathematics, Jaypee University of Information Technology, Waknaghat, P.O. Dumehar Bani, District Solan, 173215, Himachal Pradesh, India

2School of Mathematics and Computer Applications, Thapar University, Patiala-147004, Punjab, India

3School of Mathematics and Computer Applications, Thapar University, Patiala-147004, Punjab, India

4Department of Applied Sciences, Ferozepur College of Engineering and Technology, Ferozshah, Punjab, India

*Corresponding author Email id: sachin1jan@yahoo.com

Abstract

The b-family equation ut - uxxt + (b + 1)uux = buxuxx + uuxxx was introduced by D.D. Holm and M.F. Struley, which describes the balance between the convection and the stretching for small viscosity in the dynamics of 1-dimensional non-linear waves in fl uids. In this paper, we first performed the direct method introduced by Clarkson and Kruksal (1989) for symmetries of the modified b-family equation ut - uxxt + (b + 1)u2ux = buxuxx + uuxxx. Second, using symmetries of the equation, similarity reductions are obtained and we show some exact solutions.

Keywords

Modified b-family equation, Direct method, Symmetry reductions, Exact solutions