Numerical study on nonlinear burgers and modified burgers equations using b splines

Abstract

Nonlinear partial differential equations emerge very frequently while modelling physical processes and are newlinehighly useful in simulating and understanding a vast majority of phenomena in science and engineering. They newlineplay a prominent role in the modelling of almost every physical, biological or technical process ranging from newline newlineatomic interactions to celestial motion. However, they can exhibit very chaotic behaviours, and hence under- newlinestanding their physical characteristics by solving them analytically tend to be very complicated. Studies on newline newlinedifferential equations indicate that solving a partial differential equation by explicit formulas may not always newlinebe possible. In many cases, it is challenging to justify even the elementary properties of existence, uniqueness newline newlineand well-posedness of solutions of nonlinear partial differential equations. Extensive research has been car- newlineried out in developing several numerical techniques for studying the nature and solutions of partial differential newline newlineequations. However, many of these methods fail when the equations involved are highly nonlinear or when newlinethe nature of equations is extremely chaotic. Hence, there prevails a need for more studies and investigations newlinein this area and also the construction of more efficient and accurate numerical methods. newline newlineOne of the most significant nonlinear partial differential equations, which is at the heart of fluid flow mod- newlineelling, is the Navier-Stokes equation. The general flow of a Newtonian incompressible fluid may be repre- newlinesented by Navier-Stokes equation in three dimensions as: and#8706;U newlineand#8706;t newline+Uand#57344;and#57350;U = and#8722; 1 newlineand#57358; newlineand#57350;p+ and#956; newlineand#57358; newlineand#57350;2U, newline newlinewhere, U = U(x, y,z,t) and#8712; R3 is the velocity vector, and#57358; represents the density of fluid, p denotes the pressure newlinevector, and and#956; is the viscosity. A closed form solution of the Navier-Stokes equation is unknown to date. newlineJohannes Martinus Burger, a Dutch scientist, devised a simplified form of the Navier-Stokes equation by newlinedropping the pressure term. newline