Mathematical Methods and Fluid Mechanics: Block 3

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Block 3 includes units 9-11, focusing on a class of differential equations exemplified by the wave equation, diffusion equation, and Laplace's equation, commonly encountered in fluid mechanics and applied mathematics. Unit 9 addresses second-order partial differential equations, detailing their classification into three standard types and demonstrating how to reduce them to standard form. It also explores general solutions, including d'Alembert's solution for the wave equation. Unit 10 delves into Fourier series, an essential method for function approximation. It begins with trigonometric Fourier series and illustrates their application, alongside separation of variables, in solving initial-boundary value problems related to the diffusion and wave equations. The unit later expands to Fourier series from Sturm-Liouville problems, including Legendre series. Unit 11 focuses on Laplace's equation, a specific second-order partial differential equation that models the flow of irrotational, inviscid fluids past rigid boundaries. It presents solutions to Laplace's equation and their interpretation in fluid flow scenarios, such as the flow around cylinders and spheres.