Analytical and Computational Study of Non-Linear Partial Differential Equations in Fluid Dynamics Applications

This study provides an analytical and computational investigation of selected nonlinear partial differential equations arising in fluid dynamics, with emphasis on the Burgers equation, the Navier-Stokes equations in simplified geometries, and the Korteweg-de Vries equation, applying both perturbation methods and numerical discretisation techniques to characterise solution behaviour in regimes of practical importance. Nonlinear partial differential equations underlie the mathematical description of nearly all fluid flow phenomena of engineering and environmental significance, yet the gap between rigorous mathematical analysis and computational implementation remains a recurrent challenge in Nigerian postgraduate mathematics education. The Hopf-Cole transformation is derived and applied to obtain exact solutions of the viscous Burgers equation, providing a benchmark for validating numerical discretisations including finite difference, finite element, and spectral methods. The structure and properties of travelling wave solutions of the Korteweg-de Vries equation are analysed using phase plane methods, and soliton solution families are characterised. For the incompressible Navier-Stokes equations in a two-dimensional lid-driven cavity geometry, the vorticity-stream function formulation is solved numerically using a compact finite difference scheme, and bifurcations in flow topology as the Reynolds number is increased from laminar to transitional regimes are characterised. Computational implementations are developed in Python. Keywords: nonlinear PDEs, Burgers equation, Navier-Stokes, Korteweg-de Vries, perturbation methods