Abstract:
Travelling waves appears as solutions of mathematical physics models in a wide range of scientific applications that range from chemical reactions to water surface gravity waves. The first appearance of the sine-Gordon equation, is in the study of differential geometry of surfaces with Gaussian curvature K = – 1 gained its significances, because of the collisional behaviors of solutions that arise from these equations. The equation also appears in the propagation of fluxions in Josephson between two superconductors, then in many scientific fields such as the motion of a rigid pendola attached to a stretched wire, solid state physics, nonlinear optics and stability of fluid motion. The term sin u is the Josephson current across an insulator between two superconductors. The double sine – Gordon equation appears in solution theory of Deoxyribo Nucleic Acid molecular. The sinh – Gordon appears integrable quantum field theory, kink dynamics, and fluid dynamic. The double sinh – Gordon equation has lot of scientific applications as well. The kdv equation and the modified kdv equation are the leading equations that paved the way for the development of the solitary wave theory. In this thesis we aim to compute travelling wave solutions to two models problem, namely the combined sine - cosine – Gordon equation and the double combined sine – cosine – Gordon equation, to investigate the Boussinesq – double sine – Gordon equation, the Boussinesq – double sinh – Gordon equation, and the Boussinesq liouville Eq. (BL – I) and (BL – II), with objectives of two folds. The first goal is to conduct an analysis on these equations to deriver more exact solitary wave solutions. Secondly, we aim to emphasize the power of the variable separated ODE method that will be employed here to investigate the modified kdv – sine – Gordon equation and the modified kdv – sinh–Gordon equation with two objectives the first goal is to derive more exact solitary wave solution in addition to that obtained previously. Secondly, we aim to investigate the modified kdv – sinh–Gordon equation to formally solve this equation. We adapted for this work a strategy that depends on a variable separated ODE. The method was developed by Sirendaoreji et al. and used by Fu et al. and by Wawaz, the variable separated ODE method has been fully described. The method has established scientific value and reliability, and the previous work emphasized its power for equations that involved sine, cosine, hyperbolic sine, and hyperbolic cosine functions. The method convert the problem from nonlinear partial differential equations to a separable ordinary differential equations. We recommend to apply this method for solving system of non-linear differential equations.
Description:
A Dissertation Submitted to the University of Gezira in fulfillment of the Requirements for The Award of The Degree of Master of Science InMathematics department of Mathematics ,Faculty of Mathematical and Computer Science February, 2016