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Abstract
the interest has been developed in the numerical treatment of partial differential equations (PDEs) describing non-linear wave phenomena so much in recent years. these equations are used to model nonlinear dispresive waves in a wide range of applications areassuch as water wave modles, laser opticss ad plasma physics, the Buossinesq equation in the simplest class of mathematical model that contains physical effects as wave refreaction , diffraction and dispresive non-linear wave interactions,such as the generation of garmonicwave a combination of the spliting method and quintic B--spline Galerik finite element of the spiliting methods are used to solve the boussinesq equations in this thesis . this numerical methods is then applied to the dood and bad Bossinesq equations, and the improved Bossinesq equations. time integration of the resulting systems is effected using leap -frog approximination in time. simulations including propagation of solitary waves and interaction of two solitary wave are studied and represent .
results reqrded demonstrate high degree of conservation through tyhe calculation of the invariants and error norms . Comparisons with earlier worh are presented whenever possible to determine the consrevation properities of the Algorithm .