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Partial Differential Equations & waves

Partial Differential Equations & waves Professor Sir Michael Brady FRS FREng Michaelmas 2005. Analysing physical systems Formulate the most appropriate mathematical model for the system of interest this is very often a PDE. Diffusion of charge, flow of heat, absorption of a drug Propagation of waves across water, electrical networks, with/without loss of energy steady state no further change in stress analysis, heat or fluid flow, . Figure out the appropriate boundary conditions, apply them We will recall from ODEs: a single equation can have lots of very different solutions, the boundary conditions determine which Solve the PDE. In this course, solutions will be analytic = algebra & calculus Real life is not like that!! Numerical solutions include finite difference and finite element techniques This is what a large part of Engineering science & practice is about. but why Partial Differential Equations A physical system is characterised by its state at any point in space and time u ( x, y, z , t ), temperature in here, now u State varies over time: t 2u State also varies over space: things like x y Surely, we need to relate these variations to each other u u 2.

…but why partial differential equations A physical system is characterised by its state at any point in space and time u(x, y,z,t), temperature in here, now t u ∂ ∂ State varies over time: x y u ∂ ∂ ∂2 State also varies over space: things like

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